Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F 601) A BOUNDARY ELEMENT SOLUTION FOR SINGLE PILE SUBJECTED TO COMBINED AXIAL AND TORSIONAL LOADINGS S. Basack, Associate Professor, Bengal Engineering & Science University, Howrah, India, [email protected]S. Sen, Post Graduate Student, Bengal Engineering & Science University, Howrah, India, [email protected]ABSTRACT: Torsional loading on piles are introduced due to eccentric horizontal forces on the supporting structures. Structures such as, tall buildings, bridge piers, offshore platforms and electric transmission towers, can be subjected to significant torsional and axial loadings due to eccentric lateral forces and superstructural load respectively. Inadequate design of the piles against combined effect of axial and torsional loading may seriously affect the serviceability and safety of these structures with disastrous consequences. The research reported herein presents a boundary element solution for single pile response subjected to axial as well as torsional loadings. INTRODUCTION Foundation design philosophy provides adequate factor of safety against bearing failure (bearing capacity criteria) and acceptable displacement of the foundation base and the structure (serviceability criteria). Deep foundation (e.g. pile foundation) is adopted when subsoil adjacent to ground surface have insufficient bearing capacity and stiffness to carry the superstructure load and selection of shallow foundation in above mentioned circumstance causes collapse of the structure due to bearing failure or excessive settlement. Pile foundation is considered as the best solution to encounter such situation and globally accepted as one of the best alternative. Eccentric horizontal forces on the supporting structures induce torsional loading on pile. Apart from axial load, structures such as, tall buildings, bridge piers, offshore platforms and electric transmission towers are subjected to remarkable torsional forces due to eccentric lateral loading from ship impacts, high-speed vehicles, wind and wave actions, and other sources of loading. Improper design of piles against the combined effect of axial and torsional loadings may seriously influence the serviceability and safety of these structures with disastrous consequences. Although significant theoretical as well as experimental investigations have been carried out by some researchers on pile-response under torsional load, works on piles subjected to combined effect of axial and torsional loads are rare. Significant contributions to study the response of pile under torsional load alone are available [1, 2, 3, 4, 5 and 6]. On the contrary, the contribution on pile-response under combined effect of axial and torsional loads is rather limited [7]. This work has aimed to bridge up this gap. The work reported herein represents a boundary element analysis to study the response of single pile embedded in clay, under combined effect of axial and vertical loads. The numerical model developed has been validated and further parametric studies have been conducted. SOIL STRESS-STRAIN BEHAVIOUR Following the analysis of Duncan and Chang [8], the stress strain behaviour of soil in shear (τ-γ relationship) has been idealized as hyperbolic having a reduction factor R f till the ultimate value of the shear stress τ u is attained. The secant modulus of soil at a certain point is given by ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = u f t s R G G τ τ 1 where, G t is the initial tangent modulus of soil and R f is a hyperbolic soil parameter with a range of 0.8 – 1.0 [9]. BOUNDARY ELEMENT ANALYSIS To arrive at specific solutions for pile-soil interactive response under combined torsional and vertical loads, a boundary element analysis similar to a previously developed model [10] is carried out. As given in Fig.1, the pile is longitudinally discretized into n number of equal elements throughout its length such that the thickness of each element is δ (= L/n). Each of these i th element is assumed to be acted upon by horizontal shear stress τ t (i) due to the applied torque T t and vertical shear stress τ v (i) resulting from the applied vertical load V t , uniformly over the surface of the entire element. The primary objective of this analysis is to evaluate the values of τ t (i) and τ v (i) considering pile-soil slip and hence to determine the angle of twist at the central nodal points of all such elements. Analysis For Static Torsion Alone From moment equilibrium condition of the element under torsional load and following the relation of analysis of Randolph [2], the governing differential equation for no-slip condition has been re-written in finite difference form as: () p p t i i i G J i D 2 2 2 2 1 1 τ π δ θ θ θ = + − − + (1)
Transcript
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F 601)
A BOUNDARY ELEMENT SOLUTION FOR SINGLE PILE SUBJECTED TO COMBINED AXIAL AND TORSIONAL LOADINGS S. Basack, Associate Professor, Bengal Engineering & Science University, Howrah, India, [email protected] S. Sen, Post Graduate Student, Bengal Engineering & Science University, Howrah, India, [email protected] ABSTRACT: Torsional loading on piles are introduced due to eccentric horizontal forces on the supporting structures. Structures such as, tall buildings, bridge piers, offshore platforms and electric transmission towers, can be subjected to significant torsional and axial loadings due to eccentric lateral forces and superstructural load respectively. Inadequate design of the piles against combined effect of axial and torsional loading may seriously affect the serviceability and safety of these structures with disastrous consequences. The research reported herein presents a boundary element solution for single pile response subjected to axial as well as torsional loadings. INTRODUCTION Foundation design philosophy provides adequate factor of safety against bearing failure (bearing capacity criteria) and acceptable displacement of the foundation base and the structure (serviceability criteria). Deep foundation (e.g. pile foundation) is adopted when subsoil adjacent to ground surface have insufficient bearing capacity and stiffness to carry the superstructure load and selection of shallow foundation in above mentioned circumstance causes collapse of the structure due to bearing failure or excessive settlement. Pile foundation is considered as the best solution to encounter such situation and globally accepted as one of the best alternative. Eccentric horizontal forces on the supporting structures induce torsional loading on pile. Apart from axial load, structures such as, tall buildings, bridge piers, offshore platforms and electric transmission towers are subjected to remarkable torsional forces due to eccentric lateral loading from ship impacts, high-speed vehicles, wind and wave actions, and other sources of loading. Improper design of piles against the combined effect of axial and torsional loadings may seriously influence the serviceability and safety of these structures with disastrous consequences. Although significant theoretical as well as experimental investigations have been carried out by some researchers on pile-response under torsional load, works on piles subjected to combined effect of axial and torsional loads are rare. Significant contributions to study the response of pile under torsional load alone are available [1, 2, 3, 4, 5 and 6]. On the contrary, the contribution on pile-response under combined effect of axial and torsional loads is rather limited [7]. This work has aimed to bridge up this gap. The work reported herein represents a boundary element analysis to study the response of single pile embedded in clay, under combined effect of axial and vertical loads. The numerical model developed has been validated and further parametric studies have been conducted.
SOIL STRESS-STRAIN BEHAVIOUR
Following the analysis of Duncan and Chang [8], the stress strain behaviour of soil in shear (τ-γ relationship) has been idealized as hyperbolic having a reduction factor Rf till the ultimate value of the shear stress τu is attained. The secant modulus of soil at a certain point is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
ufts RGGττ1
where, Gt is the initial tangent modulus of soil and Rf is a hyperbolic soil parameter with a range of 0.8 – 1.0 [9]. BOUNDARY ELEMENT ANALYSIS
To arrive at specific solutions for pile-soil interactive response under combined torsional and vertical loads, a boundary element analysis similar to a previously developed model [10] is carried out. As given in Fig.1, the pile is longitudinally discretized into n number of equal elements throughout its length such that the thickness of each element is δ (= L/n). Each of these ith element is assumed to be acted upon by horizontal shear stress τt(i) due to the applied torque Tt and vertical shear stress τv(i) resulting from the applied vertical load Vt , uniformly over the surface of the entire element. The primary objective of this analysis is to evaluate the values of τt(i) and τv(i) considering pile-soil slip and hence to determine the angle of twist at the central nodal points of all such elements. Analysis For Static Torsion Alone From moment equilibrium condition of the element under torsional load and following the relation of analysis of Randolph [2], the governing differential equation for no-slip condition has been re-written in finite difference form as:
( )pp
tiii
GJiD
22 2
211 τπ
δθθθ
=+− −+ (1)
S. Basack & S. Sen
where, θ is the angle of twist of the top surface of the pile element and JpGp is the torsional rigidity of pile material.
Fig.1 Boundary element discreatization of pile.
where, θ is the angle of twist of the top surface of the pile element and JpGp is the torsional rigidity of pile material. From the above equation, correlations for all pile elements are established and compiling all of the following matrix equation is obtained: [ ]{ } { }aM =θ (2)
where, [M] is a coefficient matrix of order (n+1) x (n+1), {θ} is an unknown column matrix of the order (n+1) x 1 and {a} is an augment vector of the order (n+1) x 1. Using the correlation of Randolph [2], all the θi in the Eq. (2) have been replaced in terms of τt(i). Rearranging the terms and coupling the modified correlations thus developed, the following matrix equation has been formulated:
[ ]{ } { }bB t =τ (3) where, [B] is a coefficient matrix of order (n+1) x (n+1), {τt} is an unknown column matrix of the order (n+1) x 1 relevant to the horizontal shear stress acting on pile elements and {b} is an augment vector of the order (n+1) x 1. The values of the unknown torsional shear stress τt(i) obtained above have been compared with their relevant ultimate values τu(i). The elements, for which the magnitude of the horizontal shear stress exceeds the corresponding ultimate value, are assumed to be slipped and the relevant shear stresses are replaced in magnitude by the ultimate values. Since
correlation proposed by Randolph [2] is no more valid for the slipped elements, the matrix Eq. (3) above has been modified accordingly and the values of τt(i) for remaining unslipped elements are determined using the modified equations. The procedure is recycled which may led to progressive slippage of the pile elements.
After the elemental horizontal shear stress τt(i) are evaluated, the elemental angle of twists, θi are obtained using Eq. (2). At ground surface, the angle of twist has been obtained using parabolic extrapolation from the known values of θ1 , θ2 , and θ3.
Analysis for Static Axial Load Alone
Following the analysis of Mattes and Polous [10] and using moment equilibrium condition of the element under axial load the following matrix equation has been developed for no-slip condition:
[ ]{ } { }cC =ρ (4)
where, [C] is a coefficient matrix of order (n+1) x (n+1), {ρ} is an unknown column matrix of the order (n+1) x 1 and {c} is an augment vector of the order (n+1) x 1. Following the analysis of Randolph (2003) [9], Eq. (2) is modified in terms of τv(i) as,
[ ]{ } { }dD v =τ (5)
where, [D] is a coefficient matrix of order (n+1) x (n+1), {τv} is an unknown column matrix of the order (n+1) x 1 relevant to the vertical shear stress acting on pile elements and {d} is an augment vector of the order (n+1) x 1. To incorporate the pile-soil slippage, the procedure which is adopted is same as for torsional loading.
Combined Torsional and Axial Loads
In case of combined effect of torsional and axial load, the resultant shear stress induced on pile element is expressed as:
( )[ ] ( )[ ] ( )[ ]iii vt τττ +=22
(6)
Hence, the torsional analysis described above has been modified by altering the ultimate elemental shear stress as follows:
( ) ( )[ ] ( )[ ] 22iii vuut τττ −=
(7)
where, tτu(i) is the ultimate elementary shear stress to be used for torsional analysis.
Similarly, the analysis for vertical loading described earlier has been modified by altering the ultimate elemental shear stress as follows:
A Boundary Element Solution for Single Pile Subjected to Combine Axial and Torsional Loading
( ) ( )[ ] ( )[ ] 22iii tuuv τττ −=
(8)
where, vτu(i) is ultimate elementary shear stress to be used for analysis for vertical load. The details are available elsewhere [11].
Flowchart
The flowchart of the programme developed to compute the present analysis is presented in Fig. 2.
Fig.2 Flowchart of the computer programme..
VALIDATION To validate the model developed, a comparison was carried out with the available analytical results [7] and presented in Fig. 3. It is observed that, pile head axial load-settlement response for a constant torque (4 MNm) computed from present model have a reasonable good agreement with these existing results [7].
PARAMETRIC STUDY For parametric study the soil and pile parameters were taken from Basack [12] and presented in Fig. 4. The Fig.5. depicts the profiles for angle of twist induced in the pile at a typical value of L/D = 30 and Vt/Vu0 = 0.4. As the applied torque ratio Tu/Tu0 increases from 0.2 to 0.6, the normalized angle of twist observed to vary in the range of 0.0 - 0.37.
Fig.3. Comparison of computed load-settlement response with the analytical model of Georgiadis and Saflekou (1990).
Fig.4 Assumed soil and pile parameters for clay (after Basack, 2012)
The Profiles are essentially parabolic and the effect of Rf is in significant. The nature of change in normalized shear stress, τ(z)/αcu0 with L/D = 30 for clay with normalized depth z/L for vertical load ratio Vt/Vu0 = 0.4 and torsional load ratio Tu /Tu0 = 0.4 is shown in Fig. 6. As observed, the horizontal and resultant shear stresses increase linearly till the normalized peak values (τ(z)/αcu0) of 1.35 and 1.62 are attained respectively for a normalized depth of about 0.38,
Computed (Georgiadis and Saflekou (1990)
Rf = 0.85 Present Rf = 0.95 model
Input parameters for pile, soil and loading
Start
Analysis for torsion only
Compute τv(i)
Are current τv(i)same as
previous iteration ? Compute
tτu(i)
Compute
tτu(i)
Print output data
Want incremental
torque ?
Stop
Yes No
Yes No
Compute vτu(i)
S. Basack & S. Sen
and sharply decreases thereafter. The vertical shear stress is found to decrease with depth following a curvilinear pattern from 1.0 to 0.78. For a normalized depth beyond 0.8, the horizontal stress is significantly low due to which the values of the vertical and resultant shear stresses almost coincide. The effect of Rf is not remarkable except in the ranges of 0 ≤ z/L ≤ 0.2 and 0.8 ≤ z/L ≤ 1.0.
CONCLUSION A boundary element solution for predicting the pile-soil interactive performance under combined torsional and axial loadings has been developed considering hyperbolic stress-strain response of soil and interface slippage. The comparison of numerical results with existing analytical studies justifies the validity of the proposed model. From the parametric studies, it is observed that the profile of angle of twist decreases with increase of normalized depth. On the other hand, horizontal and resultant shear stress profiles increases linearly to a peak value followed by a sharp curvilinear decrement. In case of vertical shear stress the profile observed to follow a non-linear trend. REFERENCES 1. Poulos, H. G. (1975), Torsional Response of Piles, Jl. of
Geotech. Engrg. Div., ASCE, 101 (GT10), 1019-1035. 2. Randolph, M. F. (1981), Piles Subjected to Torsion, Jl.
of Geotech. Engrg. Div., ASCE, 107 (GT8), 1095-1111. 3. Hache, R. A. G. and Valsangkar, A. J. (1988), Torsional
resistance of single pile in layered soil, Jl. of Geotech. Engrg., ASCE, Vol. 114 (2), 216-220.
4. Kong, L. G. and Zhang, L. M. (2007), Centrifuge modeling of torsionally loaded pile groups, Jl. of Geotech. and Geoenv. Engrg., ASCE, 133 (11), 1374-1384.
5. Wang, K., Zhang, Z., Leo, C. J. and Xie, K. (2007), Dynamic Torsional response of end bearing pile in saturated poroelastic medium, Computers and Geotechnics, Elsevier, 35, 450-458.
6. Azadi, M. R. E., Nordal, S. and Sadein, M. (2008), Nonlinear behaviour of pile-soil subjected to torsion due to environmental loads on jacket type platforms, Jl. of WSEAS Transaction on Fluid Mechanics, 4 (4), 390-400.
7. M. Georgiadis, S. Saflekou, Piles under axial and torsional loads, Computers and Geotechnics, Elsevier, 9(1990) 291-305.
8. Duncan and Chang, (1970) Nonlinear analysis of stress and strain of soil, Jl. of Soil Mech. and Found. Engrg. Div., ASCE, 96 (SM5), 1629-1653.
9. Randolph, M. F. (2003), Load Transfer Mechanism of Axially Loaded Piles, Technical Manual, Centre for Offshore Foundation Systems, The University of Western Australia.
10. Mattes, N. S., and Poulos, H. G., (1969), Settlement of single compressible pile, Jl. of Soil Mech. and Found. Engrg. Div., ASCE s, 95 (SM1), 189-207.
11. Sen, S. (2012), A Mathematical Solution for Static and Cyclic Torsional Loading on Pile Embedded in Elastio-Plastic Medium, ME Thesis, Bengal Engineering and Science University, Howrah, India.
12. Basack, S. (2010), a boundary element analysis on the influence of krc and e/d on the performance of cyclically loaded single pile in clay, Latin American Jl. of Solids and Structures, 7, 265-284.
Fig. 5 Variation of normalized angle of twist with normalized depth for different torsional load ratio.
Rf = 0.85 Rf = 0.95
L/D = 30 Krt = 0.35 X 10-4 Vt / Vu0 = 0.4
Fig.6 Variation of normalized shear stress with normalized
depth.
, τ(z)/αcu0
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F 603)
STABILITY ANALYSIS OF 18 M DEEP EXCAVATION USING MICRO PILES Amit Srivastava, Asst. Professor, Dept of Civil Engg, Jaypee University of Engg & Tech, Guna, [email protected] Pawan Kumar, Project Assistant, Department of Civil Engineering, IISc, Bangalore, [email protected] G. L. Sivakumar Babu, Professor, Department of Civil Engineering, IISc, Bangalore, [email protected] ABSTRACT: An 18 m deep excavation for a 3-storey basement structure for a shopping mall in Bangalore is proposed to be stabilized using micro piles. The offset of the deep excavation from the adjoining buildings is in the range of 5m and foundation pressures of existing buildings are estimated to be 200 kPa. For the stability analysis of the given deep excavation, in situ soil properties are evaluated from the field and or laboratory test results and implemented in commercially available finite element based software tool PLAXIS 2D. Stability analysis is performed by considering in situ soil following Mohr-Coulomb constitutive behaviour and modelling micro piles as plate elements. Global factor of safety of the given deep excavation problem is evaluated using “Strength Reduction Technique” available as an inbuilt option in the numerical package and information on estimated deformation values are reported. INTRODUCTION Micropiles were first introduced in Italy by an Italian contracting company as Pali radice and later the technique was brought to North America for performing several underpinning jobs in the city of New England, Massachusetts. Since mid 1980s, micropiles have been used mainly as elements for foundation support to resist static and seismic loading conditions, and as in situ reinforcements for slope and excavation stability. FHWA [1] provided a unique and innovative classification system for micropiles based on two main criteria: (a) Philosophy of behaviour (design), and (b) Method of grouting (construction). A detailed review of literature on micropiles is provided in [2]. A micropile is a small-diameter (typically less than 300 mm), drilled and grouted replacement pile that is typically reinforced [3]. Generally, micropiles are applicable when there are problems with using conventional deep foundation systems. These problem conditions include: obstructions, adjacent structures, limited access job sites, and other shaky areas like caves, sinkholes, underground rivers. For example, micropiles are commonly the preferred foundation choice in the challenging areas that feature nearby buildings and difficult access. The unique characteristics of micropile offers advantages when other deep foundation systems are not applicable include: • Limited access situations due to size of equipment • Environmentally sensitive projects because they create
relatively little disturbance to the surrounding area • Seismic Retrofit • Arresting Structural Settlement • Resisting Uplift/Dynamic Loads • Underpinning • Reticulated Pile Wall Installation techniques vary depending on the load bearing specifications of the project. The selection of the installation technique depends largely on soil conditions and load transfer requirements.
Common Uses of Micropiles • To replace deteriorating foundation systems • To provide extra support for structures during renovation • To provide pile foundations where access, geology or
environment prevent the use of other methods • To support structures affected by adjacent excavation,
tunnelling or de-watering activities • To provide a fast, effective alternative to more traditional
underpinning methods Benefits of Micropiles • Can be installed through most ground condition,
obstruction and foundation at any incline. • Ensure minimum vibration or other damage to
foundation and subsoil. • Can be installed in as little headroom as 6' and close to
existing walls. • Depending on situation, could actually allow facility
operations to be maintained during construction. • Simple and economical connection to existing and new
structures. • Can be preloaded to working load before connecting to
particularly sensitive structures. MICROPILES INSTALLATION PROCESS FHWA has produced the Micropile Design and Construction Guidelines Implementation Manual and a Micropile Design and Construction NHI course [4]. The objective is to provide “practitioner oriented” technical guidance needed to: do micropile design, produce construction specifications, conduct construction inspection and integrity testing, develop cost estimates and select contracting methods; to facilitate and speed the implementation and cost-effective use to micropiles on transportation projects. The installation of micropiles begins with first determining the geological condition by doing a detailed subsurface investigation at the site. Based on known soil data, the design engineer will design and specify the types of micropiles to be
Amit Srivastava, Pawan Kumar, G. L. Sivakumar Babu
used. The step by step procedure followed is briefly indicated below: 1. After marking the position of the micropiles on the
ground by a competent land surveyor, the drilling rig is manoeuvred to the micropile position (Fig. 1).
2. Drilling will start and the types of drilling tools used depend on the soil and site condition. In dry ground and rocky condition is expected, the drilling tool known as Down The Hole Hammer (DTHH) is used. This is a pneumatic tool and works much like a jack hammer. Suitable for drilling in almost any soil condition and particularly best in rock.
3. Once the required drilling depth is reached, the drilled hole is flushed clean to remove any remaining debris before the drilling rods are removed from the drilled hole. Then a prefabricated reinforcement steel rod or rods with grouting hose attached is then lowered into the drilled hole.
4. After that water is first pumped through the grouting hose to make sure the hose is not blocked during the lowering process. It is also a way to give the drilled hole a final cleaning. If all is good, water will be seen being displaced and flows out smoothly from the hole.
5. Then grout made of cement and water mixture is pumped into the drilled hole. This is known as grouting. This is done using a grouting pump under pressure. During grouting water in the hole will be displaced and flows out smoothly.
6. The grouting hose is given a jerk to loosen it and slowly raised at the same time grout is continuously pumped into the hole. Grouting will stop once good grout is seen coming out from the hole. When this happens, the grouting hose is removed. Fig. 1 shows these steps for micropiling for better illustration purposes.
Fig. 1 Drilled micropiles installation process 7. PROBLEM DEFINITION An 18 m deep excavation for the proposed 3-storey basement structure for a shopping mall in Bangalore is to be stabilized using soil nailing technique. The offset from the adjoining buildings is in the range of 5m and foundation pressures of the existing buildings are estimated to be 200 kPa. As micropiles enable the full utilization of the space and also
support excavations with adequate factor of safety, it is suggested to use for stabilization of excavation work. Finite Element Method tool, PLAXIS 2D, is utilized to evaluate the global stability and provide information on deformation pattern of deep excavation. SITE CONDITIONS The soil investigation provides the borehole information as well as SPT values. There is no ground water table. The report indicates that the encountered material in general is disintegrated rock in many locations. Disintegrated rock, at times being particulate, normally cannot stand without support and beyond a certain height depending on cohesion and will cave in. It is likely the excavation of 18m is likely to collapse due to saturation of excavated areas. To prevent collapse, use of micropiling is suggested. RECOMMENDATIONS Improving the stability of excavations using micropiling is an appropriate solution for the present case. The suggested systems are designed based on the data supplied by the clients. The following are the recommendations: 1. Micro-piles in the form of steel pipes of 125mm dia with
6mm thickness, spaced at 25cm and 27m length are suggested.
2. Horizontal component of shear resistance provides resistance for induced shear forces due to excavating and loading. In addition 4 rods of 20mm tor steel rods are provided in the annular space of micropile to provide additional shear resistance and can be positioned in the central space of the pile. The space in the pile and between the steel rods can be filled up with grouting. Micropiles have a spacing of 25cm between the piles to a depth of 27m, the factor of safety is 1.446.
3. The spacing between the piles can be nailed with short driven nails of 1m length and 16m dia if loose pockets exist and shotcreted using appropriate wire mesh to prevent erosion of soil between the piles.
4. In the monsoon, migration of surface run off towards the excavation should be prevented with appropriate measures as indicated in precautions. Additional toe ditches shall be provided to properly drain of the water during construction and shall be maintained until a permanent retaining wall is constructed in front of the stabilized excavation.
5. Excavation in the water logged conditions shall be done carefully by draining out water from the excavated areas using appropriate pumping scheme.
6. If loose soil is encountered at any stage of excavation, restrict the depth of excavation to about 0.75m. If the soil has tendency to fall, spray cement on the surface so that the stability of excavation is maintained till nail is inserted.
7. Any deviation in the soil profiles from the geotechnical reports and anomalous conditions may be brought to the notice of the consultant. Consultant may be contacted in case of clarifications.
Type paper title on odd pages except1st page, sentence case, Times 8 italics, aligned right, in full or brief, in one line
STABILITY ANALYSIS USING FEM The stage construction responses, global stability and deformation pattern in any deep excavation problems are generally predicted by finite element method (FEM) using 2D or 3D numerical modelling. The easiest and fastest way is to define a 2D plane strain model using PLAXIS -2D. Finno et. al. [5] observed that when the ratio of excavated length to excavated depth of a wall is greater than six, the results of plane strain simulations yield the same displacements in the centre of that wall as those analyzed by a 3-D simulation. Stability analyses are conventionally assessed using Limit Equilibrium (LE) methods and lately the Finite Element (FE) method has been found to be suitable in performing stability calculations. Griffiths and Lane [6] highlighted that the FE method provides a more powerful alternative to traditional LE methods in assessing stability in their study of unreinforced or reinforced slopes and embankments. The stability of excavation is assessed in terms of factor of safety, which is obtained through strength reduction technique [5]. In this approach, factor of safety is taken as a factor (F) by which the soil shear strength parameters, i.e., cohesion (c) and angle of internal friction (φ), is reduced (c1, φ1) to bring the slope on the verge of failure.
11c cF
= (1)
11
1tan tanF
− ⎛ ⎞φ = φ⎜ ⎟⎝ ⎠
(2)
Further, prediction of the deformation behavior of a soil-nailed structure through FEM is required to ensure that displacement limits set by the construction requirements are not exceeded. For predicting deformation using FEM, one has several possibilities to model the constitutive behavior of in situ material; the most commonly used is “Hardening soil model” [8, 9] for deep excavation problems. However, if all the input parameters for HS-model are not available, alternatively Mohr- Coulomb material model can be used [8]. Facings and micropiles can be modeled as elastic materials using plate element. 15-node triangular elements can be used for generating finite element mesh. Briaud and Lim [11] provided information about where to place the boundaries so that their influence on the results of the numerical simulation of soil nail wall can be minimized. They suggested that bottom of the mesh is best placed at a depth where soil becomes notably harder (say at a depth D below the bottom of the excavation). Based on the studies of Briaud and Lim [9], if D is not exactly known, D can be taken as two to three times the vertical depth of excavation H. Further, for known values of D and H, width of excavation We can be taken equal to three to four times D and the horizontal distance from wall face to the end of mesh boundary Be can be chosen equal to three to four times (H + D) of the dimensions. The most important input material parameters for plate elements are the flexural rigidity (bending stiffness) EI and the axial stiffness EA. Plate structural elements are rectangular in shape with width equal to 1 m in out-of-plane direction. Since, the micropiles are circular in cross-section
and placed at designed horizontal spacing, it is necessary to determine equivalent axial and bending stiffness for the correct simulation. Fig. 2 shows the plan view of micropiles arrangements per meter length of excavation. Estimation of axial and bending stiffness
Fig. 2 plan view-micropiles arrangements per meter length of excavation Calculation of EI for equivalent plate element Moment of Inertia of each pipe section (I1) = (π/64) × (D1
4-D2
4) = (π/64) × (1254-1194) = 2140539.077 mm4 There are 4 such pipes per meter length, hence moment of inertia of 4 pipe sections about X-X = 4 × 2140539.077 = 8562156.308 mm4 Moment of inertia of each HYSD bar (I2) = (π/64) × d4 = (π/64) × 204 = 7853.981 mm4 There are 4 bars in each pipe and there for total no of bars in each meter length of wall = 20 Moment of inertia of all the bars (HYSD) about XX axis = 20 × 7853.981 mm4 =157079.63 mm4 Total moment of inertia of the assemble of 4 pipes and 20 HYSD bars = 8562156.308 mm4 + 157079.63 mm4 = 8719235.94 mm4 Elastic modulus of steel = 200 × 109 N/m2 EI value of the pipes and bars = 200 × 109 N/m2 × 8719235.94 mm4 = 1743 KN-m2/m Calculation of EA for equivalent plate element X-section area of one pipe = (π/4) × (D1
2-D22) = (π/4) ×
(1252-1192) = 1149.82 mm2 There are 4 such pipes per meter length, hence x-sectional area of 4 pipe sections = 4 × 1149.82 mm2 = 4599.29 mm2 X-sectional area of each HYSD bar (I2) = (π/4) × d2 = (π/4) × 202 = 314.16 mm2 There are 4 bars in each pipe and there for total no of bars in each meter length of wall = 20 X-sectional area of all the bars (HYSD) = 20 × 314.16 mm2 = 6283.18 mm2 Total X-sectional area of the assemble of 4 pipes and 20 HYSD bars = 4599.29 mm2 + 6283.18 mm2 = 10882.48 mm2 Elastic modulus of steel = 200 × 109 N/m2 EA of the assembly per meter length = 200 × 109 N/m2 × 10882.48 mm2 = 2176495.39 KN/m
Amit Srivastava, Pawan Kumar, G. L. Sivakumar Babu
Table 1 Properties of in situ soil mass Property Values Stiffness E′ 50.0 × 103 kN/m2 ν 0.32 Strength c′ref 20.0 kN/m2 φ′ 25° Advanced Stiffness E′inc 10.0 kN/m2/m γref 43.0 m Strength c′inc 10.0 kN/m2/m γref 43.0 m
The soil properties used in the numerical analysis is provided in Table 1. Figure 3 shows the deformation contours of 18 meter deep excavation supported with micropiles. It can be noted that the maximum deformation predicted is 23.41 mm.
Fig. 3 deformation pattern of micropiles supported deep excavation The maximum and minimum bending moments in the plate elements were obtained as 3.773 kNm/m (element 9 at node 1693) and -10.62 kNm/m (element 9 at node 1691), respectively. The maximum and minimum values of axial forces were obtained as 0.382 kN/m (element 1 at node 2590) and -562.4 kN/m (element 9 at node 1691), respectively. The maximum and minimum values of shear force were obtained as 13.59 kN/m (element 10 at node 1691) and -67.44 kN/m (element 9 at node 1691), respectively. Initially, the stability of the excavation is checked without micropiles and 200 kPa surcharge load. The factor of safety value is obtained as 0.325, which is not acceptable as it is less than 1.0. Hence, stabilization of this deep excavation using micropiles is essentially required The factor of safety value for the excavation with support system is evaluated as 1.62, which is more than the minimum acceptable limit of 1.5.
CONCLUSION The paper presents the FEM analysis of 18 m deep excavation, which is stabilized using micropiles. It is demonstrated through FEM analysis that the safety of deep excavation with micropiles is considerable improved. To model micropiles, plate element is utilized and a calculation procedure is demonstrated to evaluate the equivalent EA and EI values of plate element representing micropiles arrangement. The material behaviour is modelled as Mohr-Coulomb. It is concluded that 18 m deep excavation can be stabilized with the provision of micropiles with high quality construction, and good quality control are taken by the field engineers. REFERENCES 1. FHWA (1997), Micropile Design and Construction
2. Abdul Karim Elsalfiti (2011). Skin friction of micropiles embedded in gravelly soils. MS thesis submitted in the Department of Building, Civil and Environmental Engineering, Concordia University, Montreal, Quebec, Canada.
3. Bruce, D.A., Bruce, M.E.C., and Traylor, R.P. (1999), High Capacity Micropiles – Basic Principals and Case Histories. GeoEngineering for Underground Facilities. Proc. of the 3rd National Conference of the Geo-Institute of the ASCE. Geotechnical Special Publication No. 90, Urbana-Champaign, IL, June 13-17, pp. 188-199.
4. NHI Micropile Design and Construction course # 132078. http://www.nhi.fhwa.dot.gov
5. Finno, R.J., Atmatzidis, D.K.. and Roboski, J.F. (2007), Three-dimensional effects for supported excavations in clay, Journal of Geotechnical Engineering Division, ASCE, 115(8), 1045–1064.
6. Griffiths, D.V. and Lane, P.A. (1999), Slope stability analysis by finite elements, Geotechnique, 49(3), 387-403.
7. Matsui, T. and San, K-C. (1992), Finite element slope stability analysis by shear strength reduction technique, Soils and Foundations, 32(1), 59-70.
8. Shanz, T., Vermeer, P.A. and Bonnier, P.G. (1999), Formulation and verification of the Hardening Soil Model, In proceedings (Editor R.B.J Brinkgreve), Beyond 2000 in Computational Geotechnics, Balkema,Rotterdam, 281-290.
9. Brinkgreve, R.B.J. (2002), Plaxis finite element code for soil and rock analysis: Manual, Balkema: Rotterdam.
10. Plaxis (2010). Plaxis User Manual, Delft University of Technology & Plaxis bv The Netherlands.
11. Briaud, J-L and Lim, Y. (1997), Soil nailed wall under piled bridge abutment: simulation and guidelines, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 123(11), 1043 – 1050.
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F 604)
ANN BASED PREDICTION AND SENSITIVITY ANALYSIS OF CBR VALUES
Sina Borzooei : Graduate student, Department of Civil Engineering, J.N.T.U, Hyderabad M. R. Madhav: Professor of Civil Engineering, Department of Civil Engineering, J.N.T.U, Hyderabad V. Padmavathi: Associate Professor of Civil Engineering, Department of Civil Engineering, J.N.T.U., Hyderabad Srinivasulu S.: Associate Professor of Civil Engineering, Department of Civil Engineering, J.N.T.U., Hyderabad
ABSTRACT: In this paper, Multilayer Feed Forward Back propagation neural network was developed for the prediction of CBR values using soil index parameters to reduce the amount of CBR testing done in industry. The data set consist of more than 200 series of experimental test results consisting of sieve analysis, liquid limit and plastic limit, OMC and maximum dry unit weight from modified Proctor compaction tests, and soaked CBR values. The data set was divided in to two subsets, one for CBR less than 15 and the other for CBR greater than 15. A parametric study was also carried out to evaluate the sensitivity of CBR values due to the variation of the most influential input parameters. INTRODUCTION
CBR value is one of the important parameters for the design of flexible pavement. It is a load-deformation test performed in the laboratory and/or field and the results are used to determine the thickness of flexible pavement, base and other layers for a given loading. Because the test requires large amounts of material and is time consuming to perform and considering testing cost and disposal of tested material, several researchers have tried to find prediction models which can approximately predict the values of CBR from easily measurable physical properties of natural soil. Artificial neural networks have a remarkable quality of learning the relationship between the input and output data and can solve many complicated engineering challenges. Goh [1] developed ANNs for predicting the highly complex liquefaction potential of soil and Ghaboussi et al. [2] described the intrinsic constitutive relationships of sand using ANN. In this paper, two Multilayer Feed Forward Back propagation neural
networks were developed for the prediction of CBR values, using soil index properties: G (percentage of gravel), S-C (percentage of coarse sand), S-M (Percentage of medium sand), S-F (percentage of fine sand), Si-Cl (percentage of silt and clay), LL (liquid limit), and PL (plastic limit). GEOTECHNICAL CHARACTERISATION TESTS For the purpose of this study, more than 222 soaked CBR, modified Proctor compaction and soil classification tests were carried out based on ASTM standards for nine different soil types (CH, CI, SC-SM, SP, SP-SM, SP-SC, GC-GM, SC, SM) collected from different regions in Hyderabad, (Table 1). The ranges of various parameters are: CBR: 2-80%; MDD: 1.73 – 2.28 kN/m3; OMC: 5 – 16%; LL: 20 – 62; PL: 0 – 38; Fines content: 8 – 69%; fine sand: 7 – 46%; medium sand: 3 – 57%; coarse sand: 3 – 72%; gravel: 0 – 71%.
Table 1 Typical experimental data used in modeling
S.No Grain Size Distribution Atterberg Limits Modified Proctor
Sina Borzooei, M.R.Madhav, V.Padmavathi, Srinivasulu S.
ARTIFICIAL NEURAL NETWORKS
Artificial neural networks (ANN) are data processing systems. ANNs have the same network structure as the human brain consisting of many neurons connected to each other. Each connection (between neurons) has a weight. For modeling, the input data records are fed into the network. Through the modeling process, connections gain in each iteration, adopting themselves to the input data. MATLAB, a mathematical computing software having ANN toolbox which has inbuilt ANN architectures, learning, training functions, was used for the developing the networks in this study. DEVELOPING ANN MODEL FOR PRESENT STUDY In this study two networks are considered for predicting CBR values less than 15 (114 sets of data) and the other for CBR greater than 15 (108 sets of data) for total 222 data. As data sets have in different ranges, to make database acceptable to the neural network and increase its accuracy, the normalization process was carried out. For prevention of over fitting, early stopping, random data Division was used by authors, which in that the input data randomly is divided so that 60% of the
samples are assigned to the training set, 20% to the validation set, and 20% to the test set. Because Multi Forward Neural Network (MFNN) has been found successful in various engineering problems as a general function approximator, therefore it is selected as type of network in this study. The most prevalent and successful learning algorithm, used to train MFNNs in areas such as function approximation (nonlinear regression), pattern classification and system modeling is the Back Propagation (BP) algorithm which is used in this study. For both networks, the input layer has 9 neurons; each one is representing an input variable of the problem. The output layer has only single neuron, which is CBR value. Single hidden layer is chosen, it has been proven that MFNNs with one hidden layer can approximate any function in geotechnical engineering applications effectively (Reference), and number of neurons located in this layer is determined by trial and error approach to give the correlation coefficient(R) of training and testing samples as maximum. The results of this trial and error procedure for predicting CBR less than 15 and greater than 15, are illustrated in Figure 1.
Fig.1 Trial and error procedure for networks
RESULTS AND DISCUSSION
In the present study, correlation coefficient(R), root-mean-square error (RMSE), and mean absolute relative error (MARE) which each of them describing different aspect of the developed network, were used to compare the networks and to
evaluate the efficiency of models and for the sensitivity analysis purpose. Comparison of the predicted CBR values and the actual CBR values, for training and testing set of data respectively can be seen in Figures 2 and 3.
A NN Based Prediction and Sensitivity Analysis of CBR Values
Fig.2 Comparison the predicted CBR and its actual values for CBR less than 15 for training and testing data
Fig.3 Comparison the predicted CBR and its actual values for CBR greater than 15 for testing and training data
SENSITIVITY ANALYSIS
Sensitivity analysis is a technique used to determine how different values of an independent variable will impact a particular dependent variable under a given set of assumptions. Each of the inputs were increased and decreased by 1, 2, 3, 4, 5 units and the effect of this change were evaluated by studying and recording percentage of change in R, RMSE and MARE .By referring to this
changing percentage, the input variables can be ranked for their contribution to the output. The results are presented for
for both networks in Figures 4 and 5.
Sina Borzooei, M.R.Madhav, V.Padmavathi, Srinivasulu S.
Fig.4 Result of 1% change in inputs in term of correlation coefficient for network with CBR less than 15
Fig.5 Result of 1% change in inputs in term of correlation coefficient for network with CBR greater than 15
SUMMARY AND CONCLUSION A new approach was presented to predict CBR based on some physical properties of soils. Two separate Feed Forward Back propagation networks were developed to predict the compaction properties of the soil using more than 200 sets of data consist of sieve analysis results and Atterberg limits. Comparisons with the experimental results indicated that the accuracy of the developed models is satisfactory. It was observed that for predicting CBR values less than 15, the best results were obtained from a network with one hidden layer and ten Hidden Neurons with R=0.9831, RMSE=0.6242, MARE=0.1089 and for predicting the CBR value greater than 15, the network with one hidden layer and seven Hidden Neurons with R = 0.9971, RMSE = 1.2154, MARE = 0.0313 indicates good accuracy. A parametric study was also carried out to evaluate the sensitivity of CBR values due to the variation of the most
influential input parameters. Based on the analysis carried out and the model developed, it was concluded that the most significant parameter in predicting CBR values less than 15 is Liquid Limit, followed by Plastic Limit, fine sand percentage in that order and the most effective parameters in predicting CBR value greater than 15 is MDD followed by OMC and silt and clay and fine sand percentage. REFERENCES
1. Ghaboussi, J., Garrett, Jr J.H. and Wu, X. (1991). Knowledge-based modeling of material behavior with neural networks. J. Engrg. Mech., ASCE, 117(1):132-53. 2. Goh, A.T.C. (1994). Seismic liquefaction potential assessed by neural network. J. Geot. Engrg, ASCE, 120(9):1467-1480.
Proceedings of Indian Geotechnical Conference December 13-15,2012, Delhi (Paper No. F608)
ANALYSIS OF NEARBY RIGID STRIP FOOTINGS ON ELASTIC SOIL BED SUBJECTED TO INCLINED LOAD
L. S. Nainegali, Research Scholar, Department of Civil Engineering, IIT-Kanpur, India, [email protected] P. K. Basudhar, Professor, Department of Civil Engineering, IIT-Kanpur, India, [email protected] P. Ghosh, Associate Professor, Department of Civil Engineering, IIT-Kanpur, India, [email protected] ABSTRACT: Using finite element analysis, an attempt has been made to study the settlement and rotational characteristics of two closely spaced rigid strip footings subjected to inclined loading and resting on the surface of a homogeneous, isotropic soil-foundation treating it as a semi infinite linearly elastic half space. Parametric study is made by varying the clear spacing between the two footings of identical width and the angle of inclination of the respective footing load. The results are presented in terms of efficiency factors defined as the ratio of settlement/rotation of interfering footings to that of an isolated footing. It is observed interference has a significant effect on settlement and rotation of the footings in comparison to that of an isolated one. INTRODUCTION Analysis of bearing capacity and settlement of isolated shallow foundations is one of the widely studied fields of geotechnical engineering. However, very often foundations are laid in close proximity owing to structural and functional requirements. In such situations the stress isobars of individual footings may overlap and interfere with each other affecting the behaviour of the footings in the group in comparison to that of an isolated footing. Stuart, 1962 [1] in his pioneering work studied the effect of interference on the ultimate bearing capacity (UBC) of strip footings resting on sand using limit equilibrium method. Numerical methods like method of stress characteristic, upper bound limit analysis, finite difference program, and finite element analysis have been used as well by many researchers [2-11] to study the interference phenomenon; moreover experimental studies were conducted [12-20]. Most of the studies reported in literature [1-20] on the interference of footings are for vertical loads only. However, footings are generally acted upon by both horizontal and vertical loads, the resultant being inclined. Therefore in the present study of interference effect on settlement and rotational characteristics of rigid strip footings is taken up as a part of an ongoing investigation. PROBLEM DEFINATION Two rigid strip footings of identical width (B) designated as left (BL) and right footing (BR) and closely placed at a clear spacing (S) rest on the surface of soil-foundation considering it to be homogeneous, isotropic and semi infinite linearly elastic half space. The two footings (left and right) are subjected to inclined load, P at an inclination of θL and θR respectively. The footings with the loads are shown in Fig. 1. The vertical settlement and the rotation of the footings under such circumstance are to be predicted. The influence of different parameters such as spacing between the footings, inclination of the loads on the response of the footings are to be estimated considering the interference effect. Therefore, parametric study is made by varying these parameters presenting the results in terms of efficiency factors for
rotation and settlement as given in Eq. 1 and Eq. 2 respectively wherein a, b stands for left and right footing.
Fig. 1 Problem definition sketch
LRotation of left footing owing to interference
Rotation of isolated footing of width, load inclination as of
left footing
I =
(1a)
RRotation of right footing owing to interference
Rotation of isolated footing of width, load inclination as of
right footing
I =
(1b)
LMaximum settlement of left footing owing to interference
Maximum settlement of isolated footing of width,
load inclination as of left footing
ζ =
(2a)
RMaximum settlement of right footing owing to interference
Maximum settlement of isolated footing of width,
load inclination as of right footing
ζ =
(2b) ANALYSIS As the length of the strip footing is very large with respect to its width, the problem is one of plane strain. As such, 2-D finite element analysis is carried out developing an object oriented computer program (in Matlab2008a) to predict the response of the footings under inclined loads. Selection of Finite Element Mesh and Domain Size Plane strain finite element formulation of elasticity problems can be found in any standard finite element book [21] and
L. S. Nainegali, P. K. Basudhar, P. Ghosh
therefore not detailed here for reasons of space and brevity. The footings considered are geometrically symmetrical but the combinations of loads and inclinations (excluding case-2 and case-3) are unsymmetrical. As such, a domain that includes both the footings as shown in Fig. 2 is considered for the analysis and discretized by a number of 4-noded isoperimetric elements. With reference to [22] suitable boundary conditions are assigned at the far ends of the foundation soil. Along the vertical boundaries AB and CD (being very far off from the applied loads), the horizontal displacement is considered to be zero (u = 0), but allowance has been provided for possible vertical displacement (v) however small it may be. At the bottom of the soil domain (along BD being at a great depth) displacements along both the horizontal and vertical directions are restricted to zero (u = v = 0). Finite element analysis results are greatly influenced by the element size and the soil domain. To study this aspect sensitivity analysis is conducted varying the element size and the distance of the far ends of the soil domain. It has been found that the far ends in X and Z direction if considered at a distance of 10 times the width of the footing from outer edge of both footings respectively, the influence of far ends on the computed results are insignificant. Similarly if the depth of soil domain is taken to be very large with respect to the width of the footings (15 time the width), the results are not affected and for all practical purpose it may be considered to be at infinite depth. Rectangular elements of uniform size of 0.25 m are taken up in soil domain EFHG and in rest of the domain element size is increased by an aspect ratio of 1.2. RESULTS AND DISCUSSION The variation of load inclination (which is never greater than 1800) leads to combinations as follows. Rotation is defined as the ratio of differential settlement between the respective footing edges to that of its width. Rotation is clockwise when the load inclination is less than 900and is anticlockwise for inclination greater than 900.
1. θ = θL = θR (ex. θ = 300) 2. θL < 900 and θR > 900 (ex. θL = 300 and θR = 1500) 3. θL > 900 and θR < 900 (ex. θL = 1500 and θR = 300) 4. θL = 900 and θR < 900 (ex. θL = 900 and θR = 300) 5. θL = 900 and θR > 900 (ex. θL = 900 and θR = 1500)
Results were obtained with Young’s modulus, E = 30 MPa, Poisons ratio, µ = 0.3 and width of footing, B = 1 m. The correctness of the developed computer program has been validated with the standard problem from [23] and the results are found to be within the acceptable range of 8-10% error. This difference is possibly due to the approximations involved in considering the size of the domain, its discretization and difference in the computer model. The details of the study are not presented here. The variations of efficiency factors for rotation and settlement characteristic with clear spacing ratio (S/B) for case-1 are presented in Fig. 3a and Fig. 3b respectively. It is observed from Fig. 3a that with increase in S/B ratio efficiency factor, IL decreases where as IR increases; this signifies that in the zone of interference, rotation of left footing is higher compared to that of the right footing; as both the footings undergo clockwise rotation, the rotation of left footing is magnified by the presence of the right footing and the percentage difference between left and right footing at S/B = 0.5 is about 40% compared with that of left footing. Fig. 3b shows that the efficiency factors of settlement, ζL and ζR are roughly identical and decreases with increase in S/B ratio. The settlement of interfering footings decreases with increase in spacing as has been observed by [11]. At higher spacing efficiency factors attains a value equal to one wherein the interfering footings act as an isolated footing. The percentage difference in maximum settlement of left footing between S/B ratio equals to 0.5 and 14 compared with S/B = 0.5 is about 25%. For case-2 and 3, the footing geometry and loading condition are mirror image representing the symmetry in the two footings. However for the analysis whole footing and soil domain is considered and it is observed that the rotation magnitude and settlement of left and right footing are identical. Therefore the rotation and settlement efficiency factors of the two footings will also be identical. For the same Fig. 4a,b shows the variation of efficiency factors of rotation and maximum settlement respectively with respect to S/B ratio. It is noted that for case-2 both the efficiency factors (IL = IR and ζL = ζR) decreases with increase in clear spacing ratio.
Fig. 2 Footings, foundation soil domain with boundary conditions
Analysis of nearby rigid strip footings on elastic soil bed subjected to inclined load
However for case-3 with increase in S/B ratio, the rotational efficiency factor (IL = IR) increases and settlement efficiency factor (ζL = ζR) decreases and attains a value equal to one. For case-2, maximum rotation and settlement of the footings occurs compared with those for case-3. This may be due to the reason that, in case-2 the inclination of load on left and right footings are such that their effect on the rotations and settlement at any point especially along the central axis are additive. But in case-3 their individual effects try to nullify each other depending on the magnitude of the force and its inclination. Therefore S/B required for the two footing to behave as isolated footings is higher for case-2 than for case-3. At S/B = 0.5, the percentage difference between case-2 and 3 compared with case-2 for rotational and settlement efficiency factors are 35% and 16% respectively.
Fig. 3a Variation of efficiency factors, IL and IR with S/B for case-1
Fig. 3b Variation of efficiency factors, ζL and ζR with S/B for case-1
Fig. 4a Variation of efficiency factors, IL and IR with S/B for case-2 and case-3
Fig. 4b Variation of efficiency factors, ζL and ζR with S/B for case-2 and case-3 Figure 5a show the variation of rotational efficiency factor of right footing with that of clear spacing ratio for case-4 and case-5. Though the left footing is vertically loaded due to interference from the right footing some rotation is observed however for isolated footing loaded vertically, the settlement is uniform (zero rotation). As such, rotational efficiency factor of left footing from Eq. 1 cannot be calculated and not presented here. It is seen from the Fig. 5a that with increase in S/B ratio, IR increases for case-4 and decreases for case-5 and both footings act as isolated at certain spacing at which IR = 1. In case-4, right footing will be having clockwise rotation (θR < 900) which drives the soil towards the left footing however this is resisted by the left footing. Henceforth less rotation is observed in zone of interference for case-4 and vice versa in case-5 and the percentage decrease in rotation of right footing for case-4 compared with case-5 is about 64% at S/B = 0.5.
Fig. 5a Variation of efficiency factor, IR with S/B for case-4 and case-5
Fig. 5b Variation of efficiency factors, ζL and ζR with S/B for case-4 and case-5
L. S. Nainegali, P. K. Basudhar, P. Ghosh
Similarly the variation of settlement efficiency factors of left and right footings with clear spacing for case-4 and case-5 are presented in Fig. 5b and it can be observed that both ζL and ζR decreases with increase in S/B ratio and attains a constant value equal to one at higher spacing at which the footings can be considered isolated. In the zone of interference at a particular S/B, the settlement efficiency factors of case-5 are higher than that of case-4. In case-5 the right footing will be having anticlockwise rotation (θR > 900), this derives the foundation soil outward of right footing wherein no resistance is offered and load inclination on right footing is such that it adds effect of interference on settlement. However in case-4 it is vice versa, right footing will be having clockwise rotation which derives the foundation soil towards left footing and the presence of left footing resists the same and load inclination is such that it nullifies the settlement due to interference. Henceforth efficiency factors in case-5 are higher signifying higher rotation and settlement compared with case-4. In general on the whole the clear spacing ratio required for the two footings to be isolated is between 10 and 12 for all the cases considered in the analysis. CONCLUSIONS Based on the results and discussions as presented above the following conclusions are drawn. Settlement of the two interfering footings is higher compared with that of an isolated footing. For all possible inclinations of the load as considered, settlement efficiency factors decrease with increase in the clear spacing between the footings. The rotational efficiency factors (for left or right footing), increases with increase in clear spacing between footings if the rotation of footing is such that it is resisted by the presence of other footing and vice versa if rotation is not resisted. The two footings act as isolated footings when the clear spacing ratio between them is greater than 10 to 12. REFERENCES 1. Stuart, J.G. (1962), Interference between foundations
with special reference to surface footings in sand, Geotechnique, 12(1), 15-23.
2. Graham, J., Raymond, G.P., and Suppiah, A. (1984), Bearing capacity of three closely-spaced footings on sand, Geotechnique, 34(2), 173-182.
3. Kumar, J., and Ghosh, P. (2007), Ultimate bearing capacity of two interfering rough strip footings, Int. J. Geomech., 7(1), 53-61.
4. Kumar, J., and Ghosh, P. (2007), Upper bound limit analysis for finding interference effect of two nearby strip footings on sand, Geotech. Geol. Eng., 25, 499-507.
5. Kumar, J., and Kouzer, K.M. (2007), Bearing capacity of two interfering footings, Int. J. Num. Analyt. Methods in Geomech., 32, 251-264.
6. Kouzer, K.M., and Kumar, J. (2010), Ultimate bearing capacity of a footing considering the interference of an existing footing on sand, Geotech. Geol. Eng., 28(4), 457-470.
7. Kumar, J., and Bhattacharya, P. (2010), Bearing capacity of interfering multiple strip footings by using lower bound finite elements limit analysis, Computers and Geotechnics, 37, 731-736.
8. Lee, J., Eun, J., Prezzi, M., and Salgado, R. (2008), Strain influence diagrams for settlement estimation of both isolated and multiple footings in sand, J. Geotech. Geoenviron. Eng., 134(4), 417-427.
9. Mabrouki, A., Benmeddour, D., Frank, R., and Mellas, M. (2010), Numerical study of the bearing capacity for two interfering strip footings on sands, Computers and Geotechnics, 37(4), 431-439.
10. Nainegali, L.S., and Basudhar, P.K. (2011), Interference of two closely spaced footings: A finite element modeling, ASCE Geotechnical Special Publiactions, Geo-forntiers: Advances in Geotechnical Engineering, Dallas, Tx.
11. Ghosh, P., and Sharma, A., (2010), Interference effect of two nearby strip footings on layered soil: theory of elasticity approach, Acta geotechnica, 5, 189-198.
12. Das, B.M., and Larbi-Cherif, S. (1983), Bearing capacity of two closely-spaced shallow foundations on sand, Soils and Foundations, 23(1), 1-7.
13. Das, B.M., Puri, V.K., and Neo, B.K. (1993), Interference effects between two surface footings on layered soil, Transportation Research Record, 1406, pp. 34-40.
14. Deshmukh, A.M., (1979), Interaction of different types of footings on sand, Indian Geotech. J., 8, 193-204.
15. Khing, K.H., et al. (1992), Interference effect of two closely spaced shallow strip foundations on geogrid reinforced sand, Geotech. Geol. Eng., 10, 257-271.
16. Kumar, J., and Bhoi, M.K. (2008), Interference of multiple strip footings on sand using small scale model tests, Geotech. Geol. Eng., 26, 469-477.
17. Singh, A., Punmia, B.C., and Ohri, M.L. (1973), Interference between adjacent square footings on cohesionless soil, Indian Geotech. J., 3(4), 275-284.
18. Selvadurai, A.P.S., and Rabbaa, S.A.A. (1983), Some experimental studies concerning the contact stresses beneath interfering rigid foundations resting on a granular stratum, Can. Geotech. J., 20, 406-415.
19. Saran, S., and Agarwal V.C. (1974), Interference of surface footings in sand, Indian Geotech. J., 4(2), 129-139.
20. West, J.M., and Stuart, J.G. (1965), Oblique loading resulting from interference between surface footings on sand, Proc. 6th Int. Conf. Soil Mechanics, Montreal, 2, 214-217.
21. Bathe, K.J. (1996), Finite element procedures, Prentice-Hall, New-Jersey.
22. Potts, D.M., and Zdravkovic, L. (1999), Finite element analysis in geotechnical engineering, Thomas Telford Publications, London.
23. Desai, C.S., and Abel, J.H. (1972), Introduction to the finite element method: A numerical method for engineering analysis, Van Nostrand Reinhold Co., New York.
Proceedings of Indian Geotechnical Conference December 13-15, 2012,Delhi (Paper No. F-609)
LOAD BEARING CAPACITY OF A FOOTING RESTING ON THE REINFORCED FLY ASH SLOPE
K.S. Gill, Associate Professor, Deptt. of Civil Engg., GNDEC Ludhiana, India, [email protected] A.K. Choudhary, Associate Professor, Deptt. of Civil Engg., NIT Jamshedpur, India, [email protected] J.N. Jha, Professor and Head, Deptt. of Civil Engg., GNDEC Ludhiana, India, [email protected] S.K. Shukla, Associate Professor and Program Leader, School of Engineering, ECU, Australia, [email protected]
ABSTRACT: In several parts of the world, the disposal of waste materials like fly ash is a great problem. The application of fly ash as structural fills in foundations is one of the best solutions to disposal problems, because wastes can be used in large volumes. There may be difficulty due to poor load-bearing capacity of fly ash, especially when footings are rested on the top of the fly ash fill slope; but inclusion of polymeric reinforcements as horizontal sheets within the fill may be one of the most viable solutions for improving the load-bearing capacity of reinforced fly ash slope. The aim of present investigation is to find out the efficacy of a single layer of reinforcement in improving the load-bearing capacity when incorporated within the fly ash embankment slope. Increase in load-bearing capacity of reinforced slope was found in the laboratory with varying embedment depth and edge distance of footing from slope crest. The experimental results were compared with numerical results obtained by using commercial software PLAXIS version 9.0. INTRODUCTION Use of geosynthetic reinforcements for improving load-bearing capacity of foundation has been extensively reported in literature. In reality, there are many situations when foundations need to be located either on the top of a slope or on the slope itself: For example, foundations of bridge abutments and the foundations constructed on hill slopes. When a footing is constructed on a sloping ground, the bearing capacity of the footing may be significantly reduced depending upon the location of the footing with respect to slope. One of the possible solutions for improving the bearing capacity would be to reinforce the sloped fill with geogrid layers. Another problem, which civil engineers are facing, is the decreasing availability of good construction sites and it has led to the increased use of low-lying areas filled up with borrow soil. In several parts of the world, the disposal of waste materials like fly ash is a great problem and requires a large land area. Acquiring open lands for disposal in developing countries like India is difficult due to small land-to- population ratio. In the areas of thermal power plants as well as in near-by areas, the fly ash fill can be used to elevate the foundation level of footings in low-lying areas. Fly ash when used as structural fills or as embankments offers several advantages over borrow soils. It is light in weight, and exerts a low pressure on subgrade as a fill material: a well compacted embankment made of fly ash would exert only 50% of the pressure on a soft subgrade as a fill of equivalent height using coarse granular material. Additionally, construction with fly ash is less sensitive to compaction-moisture content than that of the fine grained soils commonly used in structural fill. Fly ash being non-plastic will also solve the problem of dimensional instability as exhibited by plastic soils. Further properties of fly ash from a given source are likely to be more consistent as compared to the soil from natural borrow areas [1]. Despite having several advantages,
there may be difficulty in constructing a stable structure on fly ash fill due to poor load-bearing capacity of ash, especially when footings are rested on the top of the fly ash fill slope. An inclusion of polymeric reinforcements as horizontal sheets within the fill may be one of the most viable solutions for improving the load-bearing capacity of reinforced fly ash slope. Some experimental studies on load carrying capacity behavior of footing resting on a reinforced fly ash slopes are available in the literature [1-5]. In recent past, numerical analyses such as finite difference and finite element method (FEM) have become popular in design practices. Some preliminary studies based on FEM have been attempted and reported in literature [6-10]. However despite many attempts, no obvious method for determination of ultimate bearing capacity of strip footing resting on reinforced slope is available and therefore much investigation still remains to be carried out. In view of limited information available on this aspect in the literature, the aim of present investigation is to find out the efficacy of single layer reinforcement in improving the load-bearing capacity when incorporated in a fly ash embankment slope. For this purpose, laboratory model tests were carried out by varying the embedment depth from the top and edge distance from slope crest. A numerical analysis was also conducted using the PLAXIS software (version 9.0) to verify the model test results. Table 1 gives the variables of the investigation. Table 1. Variables of the investigation
Variables (Laboratory test and numerical analysis) Type of test Constant
LABORATORY MODEL TEST Fly ash procured from the Tata Iron and Steel Company Limited (TISCO), Jamshedpur, was used in the investigation. The properties of fly ash were: 68% silt, 28% sand, maximum dry unit weight 9.34 kN/m3, optimum moisture content (OMC) 48%, apparent cohesion (c) 20 kPa and angle of internal friction (φ) 14º. Commercially available polypropylene model geogrids (0.27 mm thick and 300 mm wide) having an average tensile strength (EA) of 4.0 kN/m and tie-soil friction angle (φμ) equal to 35º were used as reinforcing elements. A series of plain strain model tests were conducted on unreinforced and reinforced fly ash model slopes. The experimental set up and test procedure is available in detail in literature [1]. The geometry of the test configuration has been shown in Fig. 1. NUMERICAL APPROACH A series of two dimensional finite element analysis (FEA) using the PLAXIS software (version 9.0) were performed. Mohr-Coulomb model theory was applied for numerical analysis on reinforced fly ash slope in order to verify the laboratory model test results. The PLAXIS
750m
m
Bed of testtankSlope angle
compacted fly ash
Soil slope
ReinforcementFooting
Side wall of test tank
Load
Fig. 1 Schematic view of the test configuration software (version 9.0) allows automatic generation of 15 node triangle plane strain elements for the soil. The parameters used for numerical analysis is well defined in PLAXIS manual and depicted in Table 2. Figure 2 show a typical deformed mesh for a reinforced case at an optimum condition indicating slope geometry and boundary condition. Plastic points for reinforced cases are shown through Figure 3-5 when the edge distance (De) of the footing is equal to footing width (B), slope angle (β) equal to 45˚and embedment depths (z) are 0.25B, 0.75B and 3.0B respectively. RESULTS AND DISCUSSION The comparison of experimental and numerical results has been shown through Figures 6-9. A typical variation of pressure and settlement ratio with and without soil reinforcement at different embedment depth (z/B) is presented in Figure 6. In this series, all the tests were performed on a 100 mm wide footing placed at an edge distance; De = 1.0B from the slope crest at slope angle β = 45˚. It can be seen in Figure 6 that the ultimate embedment depth (z) up to certain value and thereafter any further increase in z does not enhance the ultimate bearing
bearing pressure of the footing increases with an increase in Table 2. Parameters used in numerical analysis
Parameters Wooden footing EA [kN/m] 88200.00 EI [kNm2/m 36.01 Mp [kNm/m] 1E15 Np [kNm/m] 1E15 Parameters Geogrid EA [kN/m] 4.0
Fig. 2 Slope geometry, generated mesh and boundary condition
Fig. 3 Plastic points for reinforced case (z/B=0.25) capacity of the footing. The experimental results are also in good agreement with findings of numerical analysis. In Figure 7, it is clear that maximum improvement in bearing capacity ratio (BCR) occurs when embedment ratio (z/B) is equal to 1.0 for both experimental and numerical studies. Any further increase in z/B ratio results in a decrease in BCR.
Load bearing capacity of a footing resting on the reinforced flyash slope
Fig. 4 Plastic points for reinforced case (z/B=0.75)
Fig. 5 Plastic points for reinforced case (z/B=3.0)
Fig. 6 Comparison of experimental and numerical results Bearing capacity ratio (BCR = qR/qo) is defined as the ratio of the footing ultimate pressure for reinforced slope (qR) to the footing ultimate pressure for the corresponding unreinforced slope (qo). The results from finite element analysis also confirm this and the mechanism can be explained through Figures 3-5. When the reinforcement layer is placed at a very shallow depth (z/B = 0.25), the overburden pressure on the geogrid layer is inadequate in offering the necessary anchorage resistance to the geogrid against pullout force
[Figure 3]. But when the reinforcement is placed at a higher depth (z/B = 3.0), the plastic failure zones do not extend
Fig. 7 Bearing capacity ratio vs embedment ratio
Fig. 8 Experimental vs numerical bearing capacity ratio
Fig. 9 Variation of ultimate bearing capacity with EDR much below the reinforcement layer. The shear failure of soil takes place above the reinforcement layer, thus rendering it ineffective [Figure 5]. However when reinforcement lies
Gill, Choudhary, Jha and Shukla
between 0.5 to 1.0 (z/B = 0.75), the reinforcement enables much better load distribution over a larger area below the reinforced zone, and a more adequate anchorage resistance can be mobilized under the higher overburden pressure [Figure 4]. It would appear that the plane of reinforcement acts as a plane of weakness. This load-transfer mechanism seems to reach the optimum when embedment ratio (z/B) is approximately 1.0. From Figure 7, it can also be inferred that the location of the geogrid layer at a depth greater than 2 to 2.5 times the footing width does not lead to any significant improvement in the load-carrying capacity. Figure 8 shows a comparison of experimental and numerical bearing capacity ratios. From the figure it can be observed that the experimental BCR coincides closely with the numerical BCR. This consistency provides some confidence in the reliability of the results obtained from model test. The trend line is also shown in Figure 8. Similar results were obtained for other edge distance too. Figure 9 shows a variation of the ultimate bearing capacity (UBC) with respect to the edge distance ratio (EDR) when both N and z/B are equal to 1.0. In this figure experimental and analytical results have also been plotted for a comparison. The trends of variation shown by analytical and experimental results are similar. In general, the ultimate bearing capacity of footing increases with an increase in edge distance from the slope crest for all the cases of slopes considered in the present investigation. This change in bearing capacity of footing with its location relative to the slope crest may be due to the passive resistance of soil offered by the reinforcement from the slope side. It can be noticed through Figures 7-9 that the experimental value obtained from model tests coincides closely with the numerical value obtained from the PLAXIS, thus confirming the validity of numerical analysis. CONCLUSIONS The following conclusions may be drawn from the present study and are applicable to the situations considered here only. 1. Fly ash can be used effectively as an embankment fill. 2. The insertion of a geogrid reinforcement layer at a
suitable location within the slope fill considerably improves the load carrying capacity of footing located on such slopes. The bearing capacity ratio initially increases with an increase in embedment ratio up to z/B =1.0 and thereafter it decreases irrespective of the edge distance of the footing from the slope crest.
3. The location of the single geogrid layer at a depth greater than 2.5 times the footing width does not improve the load-carrying capacity significantly.
4. The edge distance has a significant effect on the load- carrying capacity of unreinforced and reinforced fly ash slopes. In the present study the improvement was significant up to 3B.
5. Experimental observations are found to be in good agreement with numerical results.
REFERENCES 1. Choudhary, A.K., Jha, J.N. and Gill, K.S. (2010),
Laboratory investigation of bearing capacity behavior of strip footing on reinforced fly ash slope, Geotextiles and Geomembranes, 28(4), 393-402.
2. Choudhary, A.K. and Verma, B.P. (1999), Stability of loaded footings on reinforced fly ash slopes, Proceeding, Indian Geotechnical Conference,145-147
3. Choudhary, A.K. and Verma, B.P. (2000), Footings on reinforced sloped fills, Proceeding, Indian Geotechnical Conference, 331-332
4. Choudhary, A.K. and Verma, B.P. (2001), Analysis of footings behaviour on reinforced sloped fills, Proceeding, Indian Geotechnical Conference, 227-230
5. Choudhary, A.K., Verma, B.P. (2001), Behavior of footing on reinforced sloped fill, Proceedings, International Conference on Landmarks in Earth Reinforcement, Japan, 535-539.
6. Jha, J.N., Choudhary, A.K. and Gill, K.S. (2010), Stability of strip footing on reinforced fly ash slope, Proceeding, 6th International Congress on Environmental Geotechnics, 2, 1160-1165.
7. Gill, K.S., Choudhary, A.K., Jha, J.N. and Shukla, S.K. (2011), Load bearing capacity of the footing resting on a reinforced flyash slope, Proceedings, International Conference on Advances in Geotechnical Engineering (ICAGE), Perth, Australia, 531-536.
8. Gill, K.S., Choudhary, A.K., Jha, J.N. and Shukla, S.K. (2011), Load bearing capacity of the footing resting on a multilayer reinforced flyash slope, Proceeding, Indian Geotechnical Conference, Vol. II, 819-822 9. Gill, K.S., Kaur, A., Choudhary, A.K., and Jha J.N. (2011), Numerical study of footing on single layer reinforced slope, Proceeding, Indian Geotechnical Conference, Vol. II, 839-842 10. Gill, K.S., Choudhary, A.K., Jha, J.N. and Shukla, S.K.
(2012) Load bearing capacity of the footing resting on the flyash slope with multilayer reinforcements, Proceedings of GeoCongress, Oakland, USA, 4262- 4271
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F610)
ESTIMATION OF FIELD COMPACTION PARAMETERS S.K. Shukla, Associate Professor & Program Leader, School of Engineering, ECU, Perth, WA, [email protected] J.N. Jha, Professor & Head, Deptt. of Civil Engg., GNDEC Ludhiana, India, [email protected] K.S. Gill, Associate Professor, Deptt. of Civil Engg., GNDEC Ludhiana, India, [email protected] A.K. Choudhary, Associate Professor, Deptt. of Civil Engg., NIT Jamshedpur, India, [email protected] ABSTRACT: This paper describes the current Indian and Australian practices of the estimation of field compaction parameters (maximum dry unit weight and optimum moisture content) based on the laboratory compaction tests, which do not consider large-size particles of the field soil samples. The study indicates that in the absence of realistic estimation procedure, some pavements have failed due to the excessive settlement. A detailed derivation of improved expressions for determining the field compaction parameters is presented. The improved expressions would be useful for the pavements and earthworks and for developing the standards on the compaction tests for the field applications. INTRODUCTION In the laboratory, the compaction test is generally performed to obtain the values of compaction test parameters, namely the optimum moisture content and the maximum dry unit weight, which are required for achieving maximum densification of the soil in field with a given compaction energy per unit volume of the soil. In most compaction test procedures, depending on the size of the compaction mould, a fraction of the soil sample having particle size larger than a specific value, say d0, is discarded. For example, in the standard Proctor compaction test, the soil particles coarser than 19 mm are discarded before compacting soil in the standard laboratory compaction mould [1-4]. If the fraction removed is significant, the laboratory optimum moisture content and the maximum dry unit weight determined for the remaining soil are not directly comparable with the field values. This paper describes the current Indian and Australian practices of the estimation of field compaction parameters based on the laboratory compaction tests. Additionally a detailed derivation of improved expressions for determining the field compaction parameters is presented for the field applications. CURRENT PRACTICES IN INDIA AND AUSTRALIA The pavement subbase and base materials consist of natural sand, moorum, gravel, crushed stone, or a combination thereof depending upon the grading required as per the field requirements. Materials like crushed slag, crushed concrete, brick and kankar are also used as subbase and base materials, especially in rural roads. The Ministry of Road Transport and Highways of the Government of India recommends three gradings of subbase materials with soil particle size varying from less than 75 μm to 75 mm [5]. The compaction of subbase/base materials is recommended to be done by rollers; the rolling should be continued till the dry unit weight achieved is at least 98% of the maximum dry unit weight for the material determined as per IS2720 (Part – 8) [2]. It is important to note that IS2720 (Part – 8) [2] does not allow particles larger than 19 mm. It is stated that the removal of small amounts of particles (up to 5%) retained on the 19 mm
sieve will affect the density only by amounts comparable with the experimental error involved in measuring the maximum dry unit weight. However, the exclusion of a large proportion of particles coarser than 19 mm may have a major effect on the unit weight and the optimum moisture content obtained compared with that obtainable with field soil as a whole. There is at present no generally accepted method of test calculation for dealing with this difficulty in comparing laboratory compaction test results with those obtained in field. For soils containing larger proportions of particles larger than 19 mm, but up to 37.5 mm, the use of a bigger mould (2250 ml) may avoid major errors. According the Australian Practice [3-4], the laboratory compaction is conducted over a range of moisture content to establish the maximum mass of dry soil per unit volume achievable for a standard compactive effort (596/2703 kJ/m3) and its corresponding moisture content. The compaction procedure is applicable to that portion of a soil that passes the 37.5 mm sieve. Soil that passes the 19 mm sieve is compacted in a 105 mm diameter compaction mould. Soil that contains more than 20% of material retained on the 19 mm sieve is compacted in a 152 mm diameter mould. Corrections for oversize material (not more than 20% of material, on a wet basis, retained on the 37.5 mm sieve) are made in accordance with AS1289.5.4.1-2007 [6]. The field maximum dry unit weight and field moisture content are calculated from the following equations [6]:
( )wcdLdF G
ppγγγ
+−
=11
(1)
and
LF wpw )1( −= (2)
Shukla, Jha, Gill and Choudhary
where, dFγ is the field value of maximum dry unit weight;
dLγ is the laboratory value of maximum dry unit weight; p is the percentage of coarser fraction (larger than d0) discarded from the soil; cG is the specific gravity of discarded coarser soil particles; wγ is the unit weight of water;
Fw is the field value of optimum moisture content; and Lw is the laboratory value of optimum moisture content. Eqs. (1) and (2) were presented by Hausmann [7] assuming the coarse fraction (larger than d0) to be dry and no change in the volume of pore air after removal of the coarse fraction. These assumptions cannot always be appropriate for the field applications of Eqs. (1) and (2). Hausmann has stated that assuming zero moisture in the coarse fraction may lead to overestimating the field dry unit weight, which may not be desirable. The details presented here clearly show that there is currently no realistic procedure for calculating the field values of compaction test parameters, especially when the oversize materials consists of a significant part of the soil to be compacted in field. The inaccurate estimation of field compaction parameters has probably been one of the major causes of pavement settlement failures in some roads. Fig. 1 shows a typical failure of a very long section of the newly constructed bituminous pavement of the National Highway (NH) No. 2 in Varanasi during 2007 – 2008.
Fig. 1 A typical pavement settlement failure of the NH-2, Varanasi PROPOSED EXPRESSIONS Figure 2 shows the phase diagrams for the field and the laboratory compacted soil samples. In Fig. 2, in addition to the weights and volumes of the three phases, unit weights are also shown beneath the phase labels. When the coarser fraction, larger than size d0 (e.g. 19 mm), is removed, it also takes away some water associated with its water content. In addition, there is also possibility of some change in the air void volume when the soil is compacted without this coarse fraction. All these are reflected in Fig. 2.
(a)
(b) Fig. 2 Phase diagrams: (a) the field compacted sample and (b) the laboratory compacted sample [8] In the context of Fig. 2, in addition to the notations defined in the previous section, notations are defined as follows: Gf is the specific gravity of the fine soil particles (smaller than d0) in the field/laboratory soil sample; Va is the volume of the air in voids of the field soil sample; VF is the total volume of field soil sample; VL is the total volume of the laboratory soil sample; wc is the water content of the coarse soil particles in the field soil sample, Ws is the weight of the soil particles in the field sample; Wwc is the weight of the water with coarse soil particles in the field soil sample; Wwf is the weight of the water with fine soil particles in the field/laboratory soil sample; α is the ratio of volume of the air in voids of the laboratory sample to that in the field soil sample, (
wcG γ ) is the unit weight of the coarser fraction of soil particles in the field
Estimation of field compaction parameters
soil sample; and (wfG γ ) is the unit weight of the finer
fraction of soil particles in the field/laboratory soil sample
From Fig. 1(b), the laboratory dry unit weight and the water content can be obtained as
L
sdL V
Wp)1( −=γ (3)
and
s
wfL Wp
Ww
)1( −= (4)
The corresponding maximum field dry unit weight can be obtained as
F
sdF V
W=γ (5)
where
wc
s
w
wcaLF G
pWWVVVγγ
α ++−+= )1( (6)
with
dL
sL
WpVγ
)1( −= (7)
By substituting Eq. (6) with Eq. (7) into Eq. (5), the maximum field dry unit weight is obtained as
wcw
c
s
a
dL
dF
Gppw
WVp
γγα
γ
γ++
−+
−= )1(1
1 (8)
where
s
wcc pW
Ww = (9)
From Fig. 1(b), we get
wfsw
wf
s
L
s
a
Gp
WW
WV
WV
γγα −
−−=1
(10)
Substitution of values from Eqs. (3) and (4) into Eq. (10) provides
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−
−−
−=
wfw
L
dLs
a
Gpwpp
WV
γγγα1)1(11
(11)
Substitution of Eq. (11) into Eq. (8) gives
⎟⎟⎠
⎞⎜⎜⎝
⎛ −⎟⎠⎞
⎜⎝⎛ −
−−⎟
⎠⎞
⎜⎝⎛ −
−++
−=
wfw
Lc
wcdLdF Gp
wppw
Gpp
γαα
γαα
γαγγ11
)1(111 (12)
Assuming βαα=
−1, Eq. (12) can be expressed as
( )( ) ( )
wfw
Lc
wcdLdF Gpwppw
Gpp
γβ
γβ
γγβ
γ−
−−−
+++−
=1)1(111 (13)
From Fig. 1(a), the field optimum moisture content, wF, can be expressed as
s
wc
s
wf
s
wcwfF W
WWW
WWW
w +=+
= (14)
Using Eq. (4) and (9), Eq. (14) can be expressed as
cLF pwwpw +−= )1( (15) Eqs. (13) and (15) provide improved expressions for calculating the maximum dry unit weight and the optimum moisture content, respectively, of the field sample based on the test values obtained from the laboratory compaction test on the laboratory sample which does not contain soil particles larger than the maximum size limit of the compaction mould.
If the removal of the coarse fraction from the field sample does not alter the volume of the air present in voids of the remaining soil for the laboratory test, then α = 1. For this case, Eq. (13) reduces to
( )w
c
wcdLdF
pwG
ppγγγγ
++−
=11
(16)
and Eq (15) remains unaltered. If the removal of the coarse fraction from the field sample does not alter the volume of the air present in voids and the removed coarse particles are dry, then α = 1 and 0=cw . For this case, Eq. (13) and (15) reduce to Eqs. (1) and (2), respectively, as presented by Hausmann (1990).
CONCLUSIONS There is currently no realistic procedure to estimate the field compaction test parameters based on the laboratory compaction tests which have limitations of the particle size. This causes inaccurate estimation of the maximum dry unit weight and the optimum moisture content of the field soils, especially for soils used in subbase and base materials. In the authors’ experience, this has probably been one of the major
Shukla, Jha, Gill and Choudhary
causes of the excessive pavement settlement failure of roads. The expressions [Eqs. (13) and (15)] proposed by Shukla et al. [8] for the field values of maximum dry unit weight and the optimum moisture content as presented here in detail are quite suitable for field applications. The proposed expressions require the values of the parameters α and wc in addition to the laboratory values of compaction parameters ( dLγ and wL) for calculating the field values of the maximum
dry unit weight ( dFγ ) and the maximum moisture content (wF). The water content (wc) of the coarse fraction, removed from the field soil sample for the laboratory test, can be determined in the laboratory as a routine test, but the appropriate value of α should be considered with caution. REFERENCES 1. IS: 2720 (Part 7) (1980), Determination of Water
Content – Dry Density Relation Using Light Compaction, Bureau of Indian Standards, New Delhi, India.
2. IS: 2720 (Part 8) (1983), Determination of Water Content – Dry Density Relation Using Heavy Compaction, Bureau of Indian Standards, New Delhi, India.
3. AS 1289.5.1.1 (2003), Determination of the Dry Density/Moisture Content Relation of a Soil Using Standard Compactive Effort, Standards Australia, Sydney, NSW, Australia.
4. AS 1289.5.2.1 (2003), Determination of the Dry Density/Moisture Content Relation of a Soil Using Modified Compactive Effort, Standards Australia, Sydney, NSW, Australia.
5. Ministry of Road Transport & Highways, Government of India (2001), Specifications for Road and Bridge Works, 4th ed., Indian Roads Congress, New Delhi, India.
6. AS 1289.5.4.1 (2007), Compaction Control Test – Dry Density Ratio, Moisture Variation and Moisture Ratio, Standards Australia, Sydney, NSW, Australia.
7. Hausmann, M.R. (1990), Engineering Principles of Ground Modification, McGraw-Hill, New York.
8. Shukla, S.K., Sivakugan, N., Gandhi, M. And Ahmad, M.K. (2009), Improved expressions for field values of compaction test parameters, Geotechnique, 59(10), 851-853.
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F611.)
Determination of limit state function in SPT-based liquefaction analysis using genetic programming for reliability analysis
Pradyut Kumar Muduli, Research Scholar, Civil Engineering Department, NIT, Rourkela, [email protected] Sarat Kumar Das, Associate Professor, Civil Engineering Department, NIT, Rourkela, [email protected] ABSTRACT: The present study discusses about the evaluation of liquefaction potential of soil based on standard penetration test (SPT) data obtained after 1999 Chi-Chi, Taiwan, earthquake using evolutionary artificial intelligence technique, genetic programming (GP). A comparative study of the developed GP model with available ANN and SVM models for prediction of liquefied and non-liquefied cases in terms of percentage success rate with respect to the field observations is discussed. The developed GP model can be used to evaluate the cyclic resistance ratio (CRR) of a soil and thus, the factor of safety against the liquefaction occurrence in the future seismic event using the available SPT data by the geotechnical practicing engineers.The developed SPT based limit state function also forms the basis for the developemnt of reliability based method for evaluation of liquefaction potential of soil. INTRODUCTION Soil liquefaction phenomena have been observed in many historical earthquakes after first large scale observations of damage caused by liquefaction in the 1964 Niigata, Japan and 1964 Alaska, USA, earthquakes. Since 1964 a lot of work has been done to explain and evaluate the liquefaction hazard. Though different approached like cyclic strain based, energy based and cyclic stress based [1] are in use, the stress based approach is the most widely used method for evaluation of liquefaction potential of soil. Seed and Idriss [2] first developed a simplified empirical model using laboratory and field observations in earthquakes which presents a limit state and separates liquefaction cases from the non-liquefaction cases. Due to difficulty in obtaining high quality undisturbed samples and cost involved therein further development of this simplified method was made using standard penetration test (SPT) based field test data [3]. Cetin [4] developed probabilistic model for evaluation of liquefaction potential using SPT data. The 1998 National Center for Earthquake Engineering Research (NCEER) workshop published reviews of SPT based methods [5]. Artificial intelligence techniques such as artificial neural network (ANN) [6, 7] and support vector machine (SVM) [8] have been used to develop liquefaction prediction models, based on SPT and cone penetration test (CPT) database and are found to be more efficient compared to statistical methods. However, the ANN has poor generalization, attributed to attainment of local minima during training and needs iterative learning steps to obtain better learning performances. The SVM has better generalization compared to ANN, but the parameters ‘C’ and insensitive loss function (ε) needs to be fine tuned by the user. Moreover these techniques will not produce an explicit relationship between the variables and thus the model obtained provides very little insight into the basic mechanism of the problem. In the recent past genetic programming (GP) and its variants, based on the Darwinian theory of natural selection are being used as alternate artificial intelligence (AI) techniques.
GP models have been applied to some difficult geotechnical engineering problem [9, 10] with success. However, its use in liquefaction assessment is very limited [11]. In the present study an attempt has been made to develop a limit state function for assessing cyclic resistance ratio (CRR) of soil and to evaluate the liquefaction potential of soil in terms of factor of safety (Fs) against liquefaction occurrence based on a database consisting of post liquefaction SPT measurements and field manifestations [12] using GP . A comparative study is also made with available ANN and SVM models. METHODOLOGY SPT-Based Method for Prediction of Liquefaction Potential The common deterministic methods are based on determination of factor of safety (Fs) against the liquefaction occurrence and is defined as Fs=CRR/CSR7.5, where, CSR7.5 = cyclic stress ratio adjusted to the benchmark earthquake (moment magnitude, Mw=7.5) as presented by Youd et al. [5]. In deterministic approach liquefaction and non-liquefaction cases are predicted on the basis of corresponding Fs ≤ 1 and Fs > 1 respectively [7]. Genetic Programming Genetic Programming is a pattern recognition technique where the model is developed on the basis of adaptive learning of provided data, developed by Koza [13]. It mimics biological evolution of living organisms and makes use of principle of genetic algorithm (GA). In traditional regression analysis the user has to specify the structure of the model whereas in GP both structure and the parameters of the mathematical model are evolved automatically. It provides a solution in the form of tree structure or in the form of compact equation based on the provided data set. A brief description about GP is presented for the completeness, but the details can be found in Koza [13]. GP model is composed of nodes, which resembles to a tree structure and thus, it is well known as GP tree. Nodes are the elements either from a functional set or terminal set. A
Pradyut Kumar Muduli, Sarat Kumar Das
functional set may include arithmetic operators (+, ×, ÷, or -), mathematical functions (sin(.), cos(.), tanh or ln(.)), Boolean operators (AND, OR, NOT etc), logical expressions (IF, or THEN) or any other suitable functions defined by the users. Whereas the terminal set include variables (like x1, x2, x3, etc) or constants (like 3, 5, 6, 9 etc) or both. The functions and terminals are randomly chosen to form a GP tree with a root node and the branches extending from each function nodes to end in terminal nodes as shown in Figure 1. Initially a set of GP trees, as per user defined population size, are randomly generated using various functions and terminals assigned by the user. The fitness criteria are calculated by the objective function i.e. quality of the each individual in the population competing with rest. At each generation a new population is created by implementing various evolutionary mechanisms like reproduction, crossover and mutation to the functions and terminals of the selected GP trees. The new population then replaces the existing population. This process is iterated until the termination criterion; a threshold fitness value or maximum number of generations; is satisfied. The best GP model, based on its fitness value that appeared in any generation, is selected as the result of genetic programming.
Fig. 1 Typical GP tree representing function (5X1+X2)2
The general form of GP model can be presented as:
( )[ ] 0∑1
,, bn
iibXfXFpLI +
== (1)
where LIp = predicted value of liquefaction index (LI)[7], F= the function created by the GP process referred herein as liquefaction index function, X = vector of input variables = {(N1)60, CSR7.5} , (N1)60= corrected blow count as presented by Youd et al.[5], bi is constant, f is a function defined by the user and n is the number of terms of target expression. In the present study GP model is developed using Matlab [14]. RESULTS AND DISCUSSION Development of the GP model for evaluation of liquefaction potential of soil In the present study post earthquake field observations and the SPT data collected, from various areas of Taiwan as per
Hwang and Yang [12] are used. The database consists of total 288 cases, 164 out of them are liquefied cases and other 124 are non liquefied cases. Out of the above data 202 cases are randomly selected for training and remaining 86 data are used for testing the developed model. Here in the GP approach normalization or scaling of the data is not required which is an advantage over ANN /SVM approach. GP Model for Liquefaction Index In the present study a GP based model is developed to evaluate LI using SPT based liquefaction field performance dataset; LI = 1 for liquefaction and LI = 0 for non-liquefaction[7] . In the GP procedure a number of potential models are evolved at random and each model is trained and tested using the training and testing cases respectively. The fitness of each model is determined by minimizing the root mean square error (RMSE) between the predicted and actual value of the output variable (LI) as the objective function,
( )n
n
ipLILI
f∑=
−
= 12
(2)
where n = number of cases in the fitness group. If the errors calculated by using Eq. 2 for all the models in the existing population do not satisfy the termination criteria, the evolution of new generation of population continues till the best model is developed as discussed earlier. The best LIp model was obtained with population size of 3000 individuals at 150 generations with reproduction probability = 0.05, crossover probability = 0.85, mutation probability = 0.1 and with tournament selection (tournament size=7). In the GP model development it is important to make a tradeoff between accuracy and complexity in terms of number of gene and depth of GP tree. In this study optimum result was obtained with maximum number of genes as 3 and maximum depth of GP tree as 4. The developed GP model can be described as Eq. (3) and shown below.
( ) ( )( ) ( )
( ) 964.05.7exp601089.02
5.76015105.1
6015.7152.55.72.8824.2
−−⎟⎟⎠
⎞⎜⎜⎝
⎛−×+
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
CSRN
CSRN
NCSR
tanhCSRtanhpLI (3)
In the present study when LIp value is greater than or equal to 0.5 the case is considered to be liquefied otherwise non-liquefied. It is evident from the results presented in Table 1 that the proposed GP based LI models are able to learn the complex relationship between the liquefaction index and its main contributing factors with a very high accuracy. It can be noted that the performances of LI model for training and testing data are comparable and the successful prediction rates are 94.55% for training and 94.19% for testing data. The classification accuracy of the ANN model [15] is 94.55% and
Determination of limit state function in SPT-based liquefaction analysis using genetic programming for reliability analysis
88.37% for training and testing data respectively. Similarly the liquefaction classification accuracies for SVM model [15] are 96.04% and 94.19% for training and testing dataset respectively. As it is important that the efficiency of different models should be compared in terms of testing data than that with training data [16] and thus, in this study the comparison of efficacy among the developed GP, ANN [15] and SVM[15] models, are done for the testing data only and it is found that GP based prediction model (Eq.3) is better than that of ANN model on the basis of rate of successful prediction of liquefaction and non liquefaction cases and is also at par with SVM model. This GP model (Eq. 3) is further used for the development of proposed CRR model. Searching for artificial points on the limit state boundary curves The developed GP model for LIp as given by the Eq (3) is used to search for points on the unknown boundary curve separating liquefied cases from the non–liquefied ones following the search technique developed by Juang et. al [7]. In the present study 71 generated data points are obtained using optimization technique. These artificial data points are used for the development of limit state or the boundary curve. GP Model for CRR Similarly as mentioned earlier here a multigene GP is adopted for development of CRR model using artificially generated 71 data points, out of which 52 data points are selected randomly for training and rest 20 numbers for testing. The several CRR models were obtained with population varying from 1000 to 3000 individuals at 100 to 300 generations with reproduction probability = 0.05, crossover probability = 0.85, mutation probability = 0.1, maximum number of genes = 2 to 4, maximum depth of GP tree = 3 to 4, and tournament selection (size=7). Then developed models were analyzed with respect to engineering understanding of CRR of soil and after careful consideration of various alternatives the following expression was found to be most suitable for the prediction of CRR.
( ) ( )[ ] ( )[ ]( )[ ] ( ) ( ) 84.52
601001102.0
601
00968.06011701.0exp096.6
6011615.0018.4601601114.0
−−−−+
++=
NN
N
NtanhNtanhNCRR (4)
Based on the statistical performances(correlation coefficient (R), Nash-Sutcliff coefficient of efficiency (E), root mean square error (RMSE), average absolute error (AAE) and maximum absolute error (MAE) ) as presented in Table 2, it can be noted that the performances of GP based CRR model for training and testing data are comparable showing good generalization of the developed model. Fig. 2 shows the developed GP limit state curve, separating the liquefied and non-liquefied cases of the data base. Thus, CRR can be calculated by this model using only one input parameter, (N1)60. The performance of the developed GP based CRR model is evaluated in deterministic approach by calculating the Fs for each case of field performance of the database as discussed earlier and a prediction is considered to be successful if it agrees with the field manifestation. The
success rate in predicting liquefied case is 98% and that for non-liquefied case is 88% and the overall success rate in the deterministic approach is found to be 94%, as shown in Table 3, which is also in agreement with the results of GP based classification model (Eq.3) as presented in Table 1. Due to both parameter and model uncertainty there is some probability of occurrence of liquefaction in a particular case even if Fs >1. Keeping this in view proposed GP based limit state model for CRR and the CSR equation as presented by Youd et al.[5] may further be used to evaluate the probability of occurrence liquefaction for each Fs and can be done by reliability based analysis.
0 10 20 30 40 500.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Limit state curveNon-liquefied cases Liquefied cases
Cyc
lic S
tress
Rat
io (C
SR7.
5)
Corrected blow count, (N1)60
Fig. 2 Showing developed GP limit state curve separating liquefied cases from non-liquefied cases CONCLUSION Case histories of soil liquefaction due to 1999 Chi-Chi earthquake is analyzed using, the genetic programming to predict the liquefaction potential of soil. The efficacy of the developed GP based classification model for LI is compared with the available ANN and SVM models and it is found that GP model is better than the ANN model on the basis of rate of successful prediction of liquefaction and non liquefaction cases and is also at par with SVM model. Close rate of successful prediction for training and testing data for the developed CRR model shows good generalization capabilities of GP approach. This compact CRR model can be used in a spreadsheet by the geotechnical professionals for evaluation of liquefaction potential of soil in terms of Fs in future seismic event in deterministic approach. The CRR model can further be used for evaluation of liquefaction potential in probabilistic frame work by reliability based analysis. However, it needs more study with new data sets of different liquefaction case histories to confirm or disprove the present findings.
Pradyut Kumar Muduli, Sarat Kumar Das
Table 1 Comparison of results of developed GP based LI model with ANN and SVM models (Samui and Sitharam, 2011) Table 2 Statistical performances of developed GP based CRR model Table 3 Performance of the developed GP based CRR model REFERENCES 1. Krammer, S. L. (1996), Geotechnical earthquake
2. Seed, H. B. and Idriss, I. M. (1971), Simplified procedure for evaluating soil liquefaction potential, Jl. of the Soil Mechanics and Foundations Division, ASCE, 97(9), 1249-1273.
3. Idriss, I. M. and Boulanger , R. W. (2010), SPT-based liquefaction triggering procedures, Report No. UCD/CGM-10/02,Department of Civil and Environmental Engineering, College of Engineering, University of California at Davis
4. Cetin, K. O. (2000), Reliability based assessment of seismic soil liquefaction initiation hazard, Ph.D dissertation, University of California, Berkerly, California.
5. Youd, T.L., Idriss, I.M., Andrus, R.D., Arango, I., Castro, G., Christina, J.T., Dobry, R., Liam Finn, W.D., Hrder Jr., L. F., Hynes, M.E., Ishihara, K., Koester, J.P., Liao, S.S.C., Marcuson III W. F., Martin G. R., Mitchell J. K., Moriwaki Y., Power, M.S., Robertson, P.K., Seed, R.B., and Stoke II, K. H. (2001), Liquefaction resistance of soils: summary report from the 1996 NCEER and 1998 NCEER/NSF workshops on evaluation of liquefaction resistance of soils, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 127(10), 817-833.
6. Goh, A. T. C. (1994), Seismic liquefaction potential assessed by neural networks, Jl. of Geotechnical Engineering, 120(9), 1467-1480.
7. Juang, C. H., Yuan, H., Lee, D. H. and Lin, P.S. (2003), Simplified Cone Penetration Test- based method for evaluating liquefaction resistance of soils, Jl. of Geotech. and Geoenv. Engineering, 129(1), 66-80.
8. Pal, M. (2006), Support vector machines-based
modeling of seismic liquefaction potential, Jl. for Numerical and Analytical Methods in Geomechanics, 30, 983-996.
9. Javadi, A. A., Rezania, M. and Nezhad, M. M. (2006), Evaluation of liquefaction induced lateral displacements using genetic programming, Jl. of Computers and Geotechnics, 33, 222-233.
10. Rezania, M. and Javadi, A. A. (2007), A new genetic programming model for predicting settlement of shallow foundations, Canadian Geotech. Jl., 44, 1462-1473.
11. Das, S. K. and Muduli, P. K. (2011), Evaluation of liquefaction potential of soil using genetic programming, In: Proc. of the Golden Jubilee Indian Geotechnical Conference, Kochi, India 2: 827-830.
12. Hwang, J. H. and Yang, C. W. (2001), Verification of critical cyclic strength curve by Taiwan Chi-Chi earthquake data, Soil Dynamics and Earthquake Engineering, 21, 237-257.
13. Koza, J. R. (1992), Genetic programming: on the programming of computers by natural selection, The MIT Press, Cambridge, Mass.
14. MathWork Inc. (2005), Matlab User’s Manual, Version 6.5, TheMathWorks, Inc, Natick.
15. Samui, P. and Sitharam, T. G. (2011), Machine learning modelling for predicting soil liquefaction, Nat. Hazards Earth Syst. Sci., 11, 1–9.
16. Das, S. K. and Basudhar, P. K. 2008. Prediction of residual friction angle of clays using artifical neural network, Engineering Geology, 100( 3-4), 142- 145.
Model Input variables
Performance in terms of successful prediction (%)
GP ANN SVM GP ANN SVM Training data Testing data
LI (N1)60, CSR7.5 94.55 94.55 96.04 94.19 88.37 94.19
Data R E AAE MAE RMSE Training 0.999 0.998 0.010 0.030 0.012 Testing 0.998 0.996 0.013 0.044 0.016
Performance in terms of successful prediction (%) Liquefied cases(164) Non-liquefied cases(124) Over all (288)
98 88 94
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No .F 612)
INELASTIC RESPONSE OF SHALLOW FOUNDATIONS SUBJECTED TO ECCENTRIC LOADING WITH NON-LINEAR WINKLER MODEL
M. Padmavathi , Assistant Professor, Department of Civil Engineering, JNTU Hyderabad, e-mail: [email protected] V. Padmavathi, Associate Professor, Department of Civil Engineering, JNTU Hyderabad, e-mail: [email protected] M. R. Madhav, Professor Emeritus, Department of Civil Engineering, JNTU Hyderabad, e-mail: [email protected] ABSTRACT: The response of footings subjected to eccentric loads has not been adequately studied. Published analyses are primarily based on the assumption of linear elastic behavior but the actual load - displacement relationship for foundations on or in the ground is nonlinear. This paper investigates the response of rigid surface foundations on nonlinear (hyperbolic) soil model. The study was motivated by the need to develop macroscopic foundation models that can realistically capture the nonlinear behavior of shallow foundations subjected to vertical eccentric loading. A parametric study quantifies the affect of eccentricity on vertical displacements, rotation and contact stress response of the foundation soil. The results compared well with published data. INTRODUCTION Tall structures are very often subjected to moments due to wind or dynamic forces which can cause the structures to tilt or rotate as a whole. The structure resting on or in the soft ground having low deformation modulus is more prone to instability. In most of the cases, the foundations are rigid and stability depends on moment-rotation relationship of the soil-foundation system. All the analyses so far available assume that the ground is elastic, i.e., the modulus of deformation under compression is the same as that for stress reduction or unloading. However it is well known that these two moduli are not the same. During compression, the soil undergoes both plastic and elastic deformations, while only the elastic part of the total deformation is recovered during unloading. Consequently, the modulus for unloading is much larger than that for compression. Numerous experimental results from plate and pile load tests have established this type of behavior of soil. According to many building code guidelines, the seismic rehabilitation of reinforced concrete buildings located in moderate and high seismic risk zones requires consideration of the interaction between the structure and the supporting soil. The estimation of the settlement and the rotation response in the nonlinear regime of the soil may be a key component in the assessment of the seismic performance of a building. REVIEW OF LITERATURE Very limited data exist in the literature regarding the settlements and/or rotations of foundations beyond their linear response (Georgiadis and Butterfield 1988; Nova and Montrasio 1991, 1997). As a first approximation, soil-structure interaction has conventionally been taken into account by assuming a linear response of the soil (Veletos and Verbic 1974). Most of the relationships for determining rotation of rigid footing due to moment are obtained on the basis of Winkler (one parameter) or the elastic continuum models based on Boussineq’s or Mindlin’s expressions. Weismann (1972) derives expressions using Winkler’s
model, for tilt of rigid foundations of rectangular and circular shapes. Rotation due to moment loading on smooth, rigid circular footing is obtained by Borowicka (1943) for a finite layer for semi-infinite soil. Moment loading on rigid, rectangular footing on elastic half space is considered by Lee (1962). Numerous studies have been performed on the bearing capacity of a shallow foundation under inclined eccentric loading. Guided by experimental results, Butterfield and Gottardi (1994) have proposed a solution for a shallow foundation on a sand layer. STATEMENT OF THE PROBLEM A rigid footing of length, L, resting on the surface of ground subjected to an eccentric load, P, at an eccentricity, e, from the left end of the footing is considered (Figure 1(a)). The foundation - soil response is represented by a series of independent springs as in Winkler Model (Figure 1(a)). The footing settles by δl at the left end due to eccentric load, P, and rotates through an angle θ due to the moment, M (Figure 1(c)). The ground below one part of the footing undergoes further compression the loading side while on the other side experiences stress reduction Figure 1(c). A modular ratio, μ= kc L/qmax, is defined where kc is moduli of subgrade reaction in compression, L is the length of the footing and qult is the ultimate stress below the footing respectively. ANALYSIS The eccentric load, P is acting at a distance ‘e’ from the left end of the footing (Fig.1 (a)). The footing undergoes a deformation along with the rigid body rotation (Fig. 1(c)). By integrating the contact pressure distribution (Fig. 1b)) one can get the load P applied on the footing. Considering the non-linear behaviour of the soil subgrade based on Winkler model the following equations are obtained.
M.Padmavthi, V. Padmavathi & M.R.Madhav
dx
qxlk1
BxlkPL
0
max
c
c∫ ++
+= )(
)(θδ
θδ (1)
dxxl1
xlBlk
P L
0xc∫= ++
+=
)()(
2 θδμθδ
(2)
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
+++
−=l
lPδμ
θδμμθθμ 1
)(1ln1* 2 (3)
Similarly the Moment, M, is given by the following equation
dx
qxlk1
xBxlkML
0
max
c
c∫ ++
+= )(
)(θδ
θδ (4)
dx
qlx
lllk
1
xBlx
lllk
ML
0
max
c
c∫
++
+=
)(
)(
θδ
θδ (5)
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛
+++
+
+−
=
lll
M
δμθδμδμ
μθμθ
θμ1
)(1ln)1(2
2)(
21*
2
23 (6)
)()(*)(
)(
max xl1xlq
qxllk1
xllkq
q
max
c
c
θδμθδμ
θδθδ
+++
=⇒+
+
+= (7)
(8)
M*= P*(e) M*- P*(e) = 0 (9) P* = P**/μ (10) θ can be estimated from Eq. (9), by substituting for P* and M* from Eqs. (3) and (6) respectively. where µ = dimensionless parameter = kcL/qmax , kc = modulus of subgrade reaction in compression, L = length of the footing, qult = ultimate stress in the subgrade soil, δl= deformation of the footing at the left end due to eccentric loading, θ = rotation of the footing due to moment, B= width of the footing, x= distance from the left end of the footing,
lδ =Normalized displacement at the left end of the footing due to eccentric loading = δl/L, x = normalized distance=x/L.P* =Normalized load=P/kc Bl2, M*= normalized Moment= M/kc Bl3,e = eccentricity, P**= Ratio of applied load to ultimate load =P/Pmax, e' = normalised eccentricity = e/L and e′′= normalised eccentricity from centre of the footing.The range of values used for the analysis are P** is from 0 to 0.9, eccentricity e′ is from 0.5 to 0.7 from the left end of the footing and µ is taken as 10, 30, 100, 500 etc. RESULTS AND DISCUSSION Fig. 2 illustrates the variation of load with displacement for different normalised eccentricities, e′′. The ultimate load decreases with increase in eccentricity, e′′. This reduction in ultimate load may be due to the reason that increase in e′′ is causing the footing to tilt towards the loading side. This causes the nonuniform contact presuure distribution below the footing as shown in Fig. 1(b). For an applied eccentricity, e'′ = 0.02 the maximum displacement is 0.07 where as it is 0.015 when the e′' is 0.1. The load, at which the footing experiences maximum displacement on the loading side, is defined as the critical load, Pcr. If the applied load on the footing exceeds the critical load Pcr, part of the footing on the side opposite to the loading gets unloaded causing a reduction in the contact area or the efective size of the footing at an applied normalised eccenricity e′′. The reduction in P** is about 30% for e′′ is 0.1 with respect to concentric (e′′= 0.0) load.
)*( 2
))1(/()1(μθθμ
μμθδ+−=
+−−+=Perwhere
rrl
Fig. 1(a) Footing subjected to eccentric loading
Footing P
Winkler’s Spring, kc
M e
Fig. 1(b) Contact Pressure Distribution below Footing
L
θ
Xδl
Fig.1(c) Displacement under the Footing
Inelastic response of shallow foundations subjeted to eccentric loading based on non linear Winkler model
Fig. 2 Effect of eccentricity on load-displacement for μ =100 Fig. 3 shows the effect of µ on load - displacement response for e′'=0.05. Increase in µ value corresponds to increase in the subgrade modulus of the foundation soil. This is clearly shown in Figure 3. The magnitude of failure or the ultimate load increases with increase in µ. The displacement of the footing is higher for lower values of µ. Fig. 4 shows the variation of normalised stress q* with normalised distance from the left end of the footing for µ of 100 and e'′ of 0.05. The stress is a nonlinear function of the applied load P** (= P/Pult) and the normalised distance X. The contact stress increases with increase in the load applied on the footing. The normalised stresses are about 0.35 and 0.8 on the left and on the loading sides respectively.
Fig. 4 Variation of normalized stress q* with normalized distance (µ=100, e′′ = 0.05) The displacement increases with the normalized load P**. Fig. 5 shows the variartion of normalised displacement with normalised distance X for µ =100 and e′′ = 0.05. The normalised displacement is more on the loading side of the footing and increases with the increase in normalised load P** applied on the footing. The plot is given upto a maximum value of Pcr of 0.65. At the centre of the footing (X = 0.5), the normalised displacement increases from 0.0025 to 0.0215 for the applied normalised load P** of 0.2 to 0.65 respectively.
Fig. 5 Variation of normalized displacement with normalized distance (µ=100, e′′ = 0.05)
µ= 10
30
µ=1000
0.35
0.7
0 0 . 5 1P**
0. 4 0. .5 0. 6
Pcr = 0 .65
P** = 0. 2
0
0.45
0 .9
0 0. 5 1
Normalised distance from left end, X
Nor
mal
ised
stre
ss, q
*
0 .02
0 .05
e′′ = 0.1
e′′ = 0.0
0
0.05
0.1
0 0 . 5 1P**
Nor
mal
ised
dis
plac
emen
t und
er lo
ad
P** = 0.2
Pcr = 0 . 65
0.5
0.6
0
0.025
0.05
0 0 . 5 1Normalised distance from left end, X
Nor
mal
ised
dis
plac
emen
t
Fig. 3 Effect of μ on load- displacement for e′'= 0.05
M.Padmavthi, V. Padmavathi & M.R.Madhav
Fig. 6 Comparison of Predicted Load - Displacement Response with Experimental Values (µ= 99, e′′= 0.0416) Fig. 6 shows the comparison of predicted load - displacement response with experimental values of Georgiadis & Butterfield (1987) for µ= 99 and e′′= 0.0416).The plot shows very good agreement between the predicted and experimental values. The proposed model captures the inherent nonlinear response of the soil and the coupled nature of displacements and rotations of rigid surface foundations subjected to eccentric loading. CONCLUSIONS 1. The variation of normalised stress, q* with displacement to
eccentric vertical load is nonlinear and depends on subgrade modulus, ultimate resistance, the rotation of the footing, θ, and displacement, δ.
2. The load-dispalcement responses are similar for all e′′ but the ultimate loads decrease with increase in normalised eccentricity e′′ . The ultimate load decreases by 30% for e′′ = 0.1 corresponding to the value for concenric load.
3. The ultimate load ratio P** increases with increase in µ. REFERENCES
1. Borowicka, H. (1943). Uber ausmittig belaste starre
platen aufelastic-isotropem Unter grund. Ingenier-Archiv, Berlin, Vol. 1, 1-8.
2. Butterfield, R. and Gottardi, G. (1994). A complete three dimensional failure envelope for shallow footings on sand. Géotechnique 44(1), 181–184.
3. Georgiadis, M. and Butterfield, R. (1987). Displacements of footings on sand under eccentric and inclined loads, Canadian Geotechnical Journal, 25(2), 199-212.
4. Nova, R. and Montrasio, L. (1991). Settlements of shallow foundations on sand, Géotechnique, 41(2), 243 – 256.
5. Nova, R. and Montrasio, L. (1997). Settlements of shallow foundations on sand: geometrical effects, Géotechnique, 47(1), 46 – 60.
6. Veletsos, A.S. and Verbic, B. (1974). “Basic response functions for elastic foundations”, Journal of Engineering Mechanics, ASCE, 100(EM2):189-202
8. Borowicka, H. (1943). Uber ausmittig belaste starre platen aufelastic-isotropem Unter grund. Ingenier-Archiv, Berlin, Vol. 1, 1-8.
9. Butterfield, R. and Gottardi, G. (1994). A complete three dimensional failure envelope for shallow footings on sand. Géotechnique 44(1), 181–184.
10. Georgiadis, M. and Butterfield, R. (1987). Displacements of footings on sand under eccentric and inclined loads, Canadian Geotechnical Journal, 25(2), 199-212.
11. Nova, R. and Montrasio, L. (1991). Settlements of shallow foundations on sand, Géotechnique, 41(2), 243 – 256.
12. Nova, R. and Montrasio, L. (1997). Settlements of shallow foundations on sand: geometrical effects, Géotechnique, 47(1), 46 – 60.
13. Veletsos, A.S. and Verbic, B. (1974). “Basic response functions for elastic foundations”, Journal of Engineering Mechanics, ASCE, 100(EM2):189-202
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (F613)
PARTICLE SIZE BASED ASSESSMENT OF SOIL USING NEURAL NETWORK MODELING TECHNIQUE
Yeetendra Kumar, Research Scholar, MNNIT, Allahabad, [email protected] K. Venkatesh, Assistant Professor, MNNIT, Allahabad, [email protected] Vijay Kumar, Research Scholar, MNNIT, Allahabad, [email protected] ABSTRACT: Acquaintance of particle sizes and its percentage provides basis for indexing the soil. Attempts are made to use percentage finer than particle sizes ranging 2 mm, 0.075 mm, 0.002 mm in combination with other properties like consistency characteristics to define specific soil class. The selection of sizes is based on their part in defining boundary conditions between sand, silt and clay. The study is limited to fine grained soils only because bore log chart referred for input and target parameters is pertaining to sites, in general, situated in plane area abundant in said soil. The main goal of ongoing study is to develop a sense of adopting neural networks for supplementing time-consuming laboratory methods. Bore log chart for three different sites were collected and further data were divided into potential input and target vectors. Using comprehensive arrangement of input vectors or applying different permutation and combinations ANN models were developed. Multilayer feed forward neural network trained with back propagation algorithm used for ANN modeling. Obtained results were in satisfactory agreement with Indian standard soil classification criterion. Key Words: Input/target vectors, artificial neural network, back propagation learning algorithm. INTRODUCTION The prime objective of soil classification is to obtain information of engineering behavior of soil such that it may be referred for relevant engineering purposes or otherwise degree of modification may be determined for specific use. Almost all countries either adopts ASTM E 11-1961in its original form as standard soil classification system or develop own classification system with slight amendments in it. The sole reason of correction is to give place locally available soil in classification system. Classification of soil based on plasticity characteristics requires conducting a series of laboratory experiments {sieve analysis (IS: 460-1962), hydrometer analysis, liquid limit test (IS: 9259-1979) and plastic limit test} subsequently placement of results in respective category to determine specific class. Present work is an effort to examine the learning ability of ANN in defining specific soil class such that conventional method may supplemented with computational method. Some pioneer work on soil classification employing ANN technique [1,2] uses single and multi-dimensional output system, driven from them the prime objective of this paper is to compare both system, however to enhance the learning ability of ANN and use of bore-log information in classification, input space is increased many folds. Other than soil classification ANN found its way in reliability analysis of structures [3]; swell pressure and soil suction behavior [4]; site characterization [5]; soil structure interaction [6]; slope stability estimation [7]; stress strain behavior [8], stress history of clayey soil [9] etc. BACK PROPAGATION NEURAL NETWORKS ANNs are parallel processors that work on the principle of biological neurons in human brain. The advantage of this
method is its multidimensional nonlinear mapping capability of the any target parameter [10]. In its simplest form ANN consist three layers; first is Input layer; second is a hidden layer consisting neurons for processing and third is a output layer. These layers connect to each other by connection weights, which are adjustable in nature. The characteristics of a neural network come from the activation function and connection weights [11]. Out of many available neural networks feed forward (which is used in this study) network trained with back propagation learning algorithm is described below.
Fig. 1 Back propagation neural network The output from Jth node from hidden layer in fig 1;
(1) Where, i & j presents input and hidden nodes respectively
Yeetendra Kumar, K. Venkatesh & Vijay kumar
oj is o/p from the jth hidden node xi is i/p introduced to node i wij is the synaptic weight on the link between ith input and jth o/p node bj is the bias applied at the jth hidden node
The activation function for the jth hidden node may be determined using the sigmoid (or any other) function.
(2)
The o/p from the kth node may obtained by
(3) Where, wjk = is the synaptic weight on the link between jth hidden node and kth o/p node Bk is the bias applied at the kth o/p node The activation function vk for the o/p node k is (4)
The error at the kth o/p node is obtained by
(5) Correction to the weight on link between jth hidden node and kth o/p node during lth iteration is
(6) Where, η is learning rate which determines the size of weight adjustment. α is the momentum factor and used to change the weight by speeding up the convergence W (l-1) is weight during (l-1) iteration. This iteration continues until Mean Square Error reaches its minimum value. DATA SELECTION Three different sites rich in clay and silt content whereas less to negligible amount of sand portion used for model development. For the same reason, study is confined to frictional - cohesive (c-φ) soils only. For the sake of simplicity, these sites designated with name A, B & C. According to IS 1498: 1970, Three types of soil namely CL- ML (inorganic clays with less amount of inorganic silt & very fine sand), CL (inorganic clays with low compressibility) and CI (inorganic clays with medium compressibility) were available from these sites. In particular, CL & CI soils were abundant in all three sites whereas CL-ML was less to absent in site B and C respectively. In respect of these, three-soil classes some soil parameters that were used as input vectors are physical properties, SPT- N value,
percentage finer and consistency characteristics. It is customary that all properties of soil are determined on particular depth hence it is also included in input parameter. Standard notations considered for these geotechnical parameters were d for depth, N for SPT-N value; w for moisture content, ρb for bulk density; Dx for percentage of particle finer than 2 mm; Dy for 0.075mm; Dz for 0.002 mm: wL liquid limit and wp plastic limit. Table 1 shows range of input parameters for all three sites collectively. Table1-Range of input parameters
Input Parameters Minimum Maximum Depth(m) 01.50 09.00 SPT-N value(No.) 02.00 50.00 Natural water content (%) 03.18 34.60 Bulk density (gm/cm3) 01.40 02.32 % finer than 2 mm(%) 59.00 100.0 % finer than 0.075 mm(%) 46.60 98.78 % finer than 0.002 mm(%) 02.70 26.51 Liquid Limit (%) 25.00 46.00 Plastic Limit (%) 09.00 26.00
DEMARKATION OF SOIL CLASS One- Dimensional Output System For coding the soil with respect to their soil class designation, two policies were developed. Table 2 gives the details of first coding policy. Each soil class allotted certain numbers in increasing order of clay content. Demarcation values fell within limits of activation function and output ranges were predefined such that decoded value from validation output may easily be classified. Table 2 Coding of soil class Soil Class Coded Value Output Range CL-ML 0.1 0.00-0.2CL 0.3 0.21-0.4 CI 0.5 04.1-0.6
Three- Dimensional Output System Table 3 shows second coding policy i.e. a three-dimensional Table 3 Learning paradigm for 3-D output system
Parameters Training Data Pairs d 1.5 6 6 N 11 39 22 w 5.3 14 16.3 ρb 1.8 1.92 1.91 Dx 100 99.9 96.9 Dy 46.6 93 84.9 Dz 6 22 19 wL 26 33 36 wP 21 19 21 Soil Class CL-ML CL CI Three-Dimensional Output
0.9 0.1 0.1 0.1 0.9 0.1 0.1 0.1 0.9
Particle size based assessment of soil using artificial neural network modeling technique
output system. These three dimensional target matrixes were used for training the network. These matrixes developed separately for all three sites. To separate the output signal from each other, the target matrix of all three-soil class defined keeping much difference within values almost like in binary system. Instead of using 0 or 1 here used values 0.1 and 0.9 so that output may range between 0 to 1. SELECTION OF NETWORK ARCHITECTURE Feed forward network with single hidden layer of varying numbers of neurons (4 to 14) employed in the analysis. Figure 2 describes the way network treated from given set of input and target parameters.
Fig. 2 Neural network with 9 x n x 1/3 architecture The first layer presents 9 inputs bring in to the network, second layer namely hidden layer shows neurons in process. Connecting links between first - second layer and second- third layer are adaptable synaptic weights and last layer presents output. Networks trained for varying number of iterations until the minimum value of MSE at maximum regression for training, testing and validation reached. NETWORK ATTRIBUTES As discussed in previous articles back-propagation neural network (BPNN) employed for all kind of operations, in which training carried out through the minimization of the defined error function using the gradient descent approach [12]. It is advisable to use differentiable activation function because the weight update is dependent variable and relies on the gradient of error [13]. There exists many ways to improve the rate of convergence one of them is normalization, therefore datasets were normalized using following equation [14, 15, 16].
(7)
The ANN toolbox in MATLAB 7.10 (R2010a) computer added software utilized to perform the necessary computation in which learning rate (LR) and momentum term kept constant whereas connection weights kept adjustable for all the models. Range of training parameters were set at Epochs 1000;Time - Infinite; Goal - Zero; Validation Checks - 0 to1000; Gradient - 1 to 1e-10 and Mu - 0.001 to 1e+10. RESULTS AND DISCUSSION ANNs have feature of automatically dividing data into training testing and validation sets. In present case 60% of total data was reserved for training and remaining 40% data was equally divided for testing and validation by ANN itself. In addition, certain datasets reserved for all three sites to validate the developed models externally. Table 4 shows predicted soil class for some of above said reserved data. It was observed that network 9-10-1 (10 neurons in hidden layer) gave better results for all three sites. Table 2 may refer to decode soil class in its original form for one-dimensional output. Table 4 Predicted soil class for One-Dimensional system Actual Soil Class
The predicted value of soil class in the form of 3-dimensional array shown in Table 5 includes some of the validation results from all three sites. The exact procedure of decoding is comparison of predicted array with similar standard array and fall of soil group in same array. as an example for any output array 0.027, 0.893, & 0.240, the maximum value is set at two that is 0.893 and since this output matches with 3-dimensional output pattern of inorganic clay with low compressibility (CL) that is 0.1, 0.9 & 0.1 so the same class of soil will be preferred.
Yeetendra Kumar, K. Venkatesh & Vijay kumar
Table 5 Predicted soil class for Three-Dimensional system
Site C did not contained CL-ML soil hence it contained 2- dimensional array only likewise due to lack of CL-ML soil in site B said soil is not included in validation data. Table 6 gives the statistical parameters for trained and validated results. Table 6 Statistical Parameters
Site Networks MSE Overall Regression
Avg. Absolute
Error
A 9-10-1 1.924e-03 0.9335 0.025984 9-7-3 3.842e-14 0.9798 7.171e-05
B 9-10-1 5.918e-05 0.9967 0.009671 9-5-3 1.182e-11 0.9776 0.003613
C 9-10-1 1.816e-08 0.9769 0.001309 9-10-2 5.578e-13 1.0000 0.000103
CONCLUSION Table 4, 5 & 6 shows that ANN classified soil finely for each validation data and mean square error (MSE) decreased exponentially. Though both systems predicted soil class with considerable precision even based on statistical analysis it is concluded that multi-dimensional output system is better than one-dimensional system since MSE value and average absolute error in each site through multi-dimensional system is less than one-dimensional system. Network 9-10-1 that is 10 neurons in hidden layers is sufficient for developing optimal network in one-dimensional system where as 5 to 10 neurons may give optimal solution in multi-dimensional system. Overall regression coefficient (combined training, testing and validation phases) reaching unity in site C is also an indication of reliable data source. Though study is confined to fine grained soil only even it may be extended to coarse grained soil in near future depending on convenience of advanced version of ANN tool with desired computer configuration such that multi- dimensional system may be increased from 6 to 10 outputs. REFERENCES 1. Cal, Y. (1995), Soil classification by neural network,
Advances in Engineering Software. 22, 95-97.
2. Goktepe, F., Arman, H. & Pala, M. (2010), A new approach for classification of clayey soil: a case study for Adapazari region, Turkey, Scientific Research and Essay,. 5(15), 2037-2043.
3. Goh, A.T.C. & Kulhawy, F.H. (2003), Neural network approach to model the limit state surface for reliability analysis, Can. Geotech. J., 4, 1235-1244.
4. Erzin, Y. (2007), Artificial neural network approach for swell pressure versus soil suction behavior, Can. Geotech. J., 44, 1215-1223.
5. Juang, C.C., Jiang, T. and Christopher, R.A. (2001), Three-dimensional site characterization: neural network approach, Geotechnique, 51(9), 799-809.
6. Wan, S. & Yen, J.Y. (2006), The study of SSI problems in an industrial area with modified neural network approaches, Int. J. Numer. Anal. Meth. Geomech., 32(9), 1087-1106.
7. Shangguan, Z., Li, S. & Luan, M. (2009), Intelligent forecasting method for slope stability estimation by using probabilistic neural network, The Electronic Journal of Geotechnical Engineering, 13, 1-10.
8. Banimahd, M., Yasrobi, S.S. and Woodward, P.K. (2005), Artificial neural network for stress strain behavior of sandy soils: knowledge based verification, Computer and Geotechniques, 32, 377-386.
9. Kurup, P.U. and Dudani N.K. (2002), Neural network for profiling stress history of clays from PCPT data, Jl. of Geotech. and Geoenv. Engineering, ASCE, 128(7), 569-579.
10. Sezer, A. (2011), Prediction of shear development in clean sands by use of particle shape information and artificial neural networks, Expert Systems with Applications, Elsevier, 38, 5603-5613.
11. Kim, C.Y., Bae, G.J., Hong, S.W., Park, C.H., Moon, H.K. & Shin, H.S. (2001), Neural network based prediction of ground surface settlements due to tunneling, Comp. and Geot., 28, 517:547.
12. Chua, C.G. and Goh. A.T.C. (2003), A hybrid Bayesian back-propagation neural network approach to multivariate modeling, Int. Jl. Numer. Anal.Meth.Geomech., John Wiley & sons, 27, 651-667.
13. Rajshekhran, S. and Pai, G.A.V. (2010), Neural networks, fuzzy logic and genetic algorithms, PHI learning private limited, New Delhi, India.
14. Rafiq, M.Y., Bugmann, G. and Easterbrook, D.J. (2001), Neural network design for engineering applications, Comput.Struct., 79, 1541-1552.
15. Kayadelen, C. (2008), Estimation of effective stress parameter of unsaturated soils by using artificial neural networks, Int. J. Numer. Anal.Meth.Geomech., 32(9), 1087-1106.
16. Gunaydım,O. (2009), Estimation of soil compaction parameters by using statistical analyses and artificial neural networks, Environmental Geology, 57, 203-215.
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F 614)
BEHAVIOUR OF RIGID FACED REINFORCED WALLS WITH STRIP REINFORCEMENT USING 3D MODELS
Arup Bhattacharjee, Asst. Prof. Dept. of Civil Engg., Jorhat Engg. College, Jorhat, Email – [email protected] A. Murali Krishna, Asst. Prof. Dept. of Civil Engg., IIT Guwahati, Email- [email protected] ABSTRACT: In this paper, the behavior of rigid faced reinforced soil retaining walls with strip reinforcement and sheet reinforcement are simulated and studied using FLAC 3D. In modeling of rigid faced soil retaining walls, soil is modeled as elasto-plastic with Mohr-Coulomb failure criterion. The reinforcement members are modeled by using shell structural elements. Various interfaces are considered between dissimilar materials for proper interaction. The performance of rigid faced wall with strip and sheet reinforcements subjected to monotonic and dynamic loading are being studied. Behavioural aspects of the model walls are discussed in terms of displacements, horizontal pressure and octahedral shear strains under monotonic and dynamic loading. The results of model walls with strip reinforcement and sheet reinforcement are compared. INTRODUCTION Over past few decade uses of reinforced soil technologies is enormously increasing and are found to be effective even for several critical conditions compared to conventional soil structures. Reinforced soil retaining walls offer competitive solutions to earth retaining problems associated with less space and more loads posed by tremendous growth in infrastructure in recent times. They also offer improved performance in addition to the advantages in ease and less cost of construction compared to conventional retaining wall systems. The studies conducted for observing the behavior of reinforced soil retaining walls subjected to seismic shaking can be classified into three categories: experimental studies mainly based on shaking table tests and centrifuge tests, analytical studies based on pseudo-static and pseudo-dynamic approach and numerical studies. The numerical studies are conducted by using different software based on finite element and finite difference methods by many researchers [6,7,9]. Reinforced soil walls are constructed using different reinforcing elements and facing systems. Wall facing system may be: Warp facing, full height rigid facing, segmental block facing and modular block facing. Reinforcing elements may be metal strips or polymer product like geotextile, geogrid, geomembrane etc. A study was conducted static response of reinforced soil wall with strip reinforcement using FLAC [1]. In this paper, the rigid faced soil walls are modeled using three dimensional explicit finite difference software FLAC3D. Two different types of reinforcement, sheet reinforcement and strip reinforcement are considered for simulation. The patterns of sheet and strip reinforcement are explained in Fig.1. The dynamic response of rigid faced wall with sheet and strip reinforcement are examined. GENERATION OF NUMERICAL MODELS FOR RIGID FACED WALLS Rigid faced reinforced soil wall models with mat reinforcement described by [8] are considered as the reference case for the generation of numerical models. The detail of the model wall is shown in Fig. 2.
Fig.1 Rigid faced reinforced wall with (a) mat reinforcement (b) strip reinforcement. The experimental procedure of model development has been followed in development of numerical model using FLAC3D. FLAC3D is an explicit finite difference programme used for engineering mechanics problems. A rigid foundation is first generated to represent the shaking table. The rigid wall is simulated and fixed at the bottom to lateral sliding. The backfill is filled in layers of equal lifts and reinforcements are placed after each. The formulation of model is described in detail in the following subsections.
Fig.2 Test arrangement of rigid face reinforced retaining wall [8] Numerical grid A rigid zone of size 800 mm long and 50 mm thick considered at the base of the wall to represent the shaking table. A grid of size 600 mm high, 25 mm thick and 500 mm wide rigid wall is generated to represent the rigid faced
(a) (b)
Arup Bhattacharjee & A. Murali Krishna
retaining wall. A grid of size 600 mm high, 750 mm long and 500 mm wide is generated to represent the backfill of rigid faced retaining wall. The whole grid is divided into number of zones of size 25 mm each. The size of the grids is selected in such a way that, the mesh size of the model must be smaller than approximately one-tenth to one-eighth of the highest frequency component of the input wave for accurate transmission of wave through a model [5] during dynamic shaking. The construction sequence followed in generation of numerical grids is same as that of physical model. Before placing the first layer, the foundation zone is generated and brought to static equilibrium. The wall is placed over the foundation and brought to static equilibrium. Initially wall is fixed in x direction to represent the fixed support during the construction. The backfill of the model is generated at an equal lifts. The reinforcement is placed after each lift. The four layer mat reinforcements are laid along whole length of wall. In case of strip reinforcement, four layers with the horizontal spacing between 25 mm and 50mm wide strip reinforcement is 150 mm. So the vertical distance between the two reinforcing layer is 150 mm. The reinforcement is fixed with the wall to form a rigid connection between wall and reinforcement. The model is solved for static equilibrium after generation of grids of each lift. The surcharge of 0.5 kPa is applied at the top of the backfill and model is brought to equilibrium. The supports of the wall are removed after that and model is brought to static equilibrium. Figure 3 shows the numerical grid considered to simulate the rigid faced wall with mat and strip reinforcement.
Fig.3 Numerical grid of rigid faced reinforced retaining wall with (a) mat reinforcement and (b) strip reinforcement Material properties Wall The rigid wall is simulated as elastic material. The properties required for the elastic material model are mass density, shear modulus and bulk modulus. Backfill material The backfill soil is modeled as elasto-plastic material with Mohr Coulomb failure criterion. The properties required for Mohr-Coulomb material model are mass density, bulk and shear modulus, friction and dilation angle. A small cohesion value is applied to prevent premature yielding [2]. The local damping ratio of 5% is adopted for soil and wall element during dynamic analysis. Reinforcement Material (Geotextile) The geotextile layers are modeled using the geogrid structural element in FLAC3D. The geogrid elements are three nodded
shell elements that resist as membrane but do not resist bending loading. The geogrid element behaves as isotropic linear elastic material with no failure limit. The required input parameters for geogrid element in FLAC3D are: (1) elastic modulus (2) Poisson’s ratio (3) thickness of geogrid. Interface properties The interface between the dissimilar materials is modeled as linear spring-slider system with interface shear strength defined by the Mohr-Coulomb failure criterion. Two types of interfaces are used in this model: interface between the soil and rigid wall and interface between soil and reinforcement. The interface between backfill soil and rigid wall controls the relative movement between them. The relative interface movement is controlled by interface normal stiffness (kn) and shear stiffness (ks). A recommended thumb rule is that ks and kn be set to ten times the equivalent stiffness of the stiffest neighboring zone [3]. The maximum stiffness value is given by [3] as
( ) ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
Δ
+×==
min
34
max10z
Gkkk sn
(1)
Where the parameters (Δz)min , K and G are the smallest dimensions in normal direction, bulk modulus and shear modulus continuum zone adjacent to the interface respectively. This approach gives the preliminary values of the interface stiffness components, and these can be adjusted to avoid intrusion to adjacent zone and to prevent excessive computation time. The model material properties are tabulated in Table 1.
Table 1 Material properties used in numerical simulation Wall properties Mass density, kg/m3 2500 Elastic modulus, kPa 2×107
Soil properties for Mohr model Mass density, kg/m3 1630 Elastic modulus, kPa 1×104 Poisson’s ratio 0.3 Friction angle, Degrees 43 Dilation angle, Degrees 15 Cohesion, kPa 0.1 Reinforcement (Geotextile) properties Mass density, kg/m3 0.23 Thickness, m 0.001 Reinforcement stiffness, kN/m 5.2 Reinforcement (Geotextile) interface properties Coupling spring cohesion, kPa 0.1 Coupling spring friction, Degrees 29 Coupling spring stiffness, kPa 1×106
RESULT AND DISCUSSION The dynamic model studies are conducted for sinusoidal dynamic motion at 0.2g base input acceleration. The model is subjected to dynamic shaking of 20 cycles at frequency 3 Hz. Figure 4 shows the comparison of the variation of horizontal displacements, RMSA amplification factors and horizontal
(a) (b)
Behaviour of rigid faced reinforced walls with strip reinforcement using 3D models
pressure increments at different elevations obtained from of physical and numerical models. The RMSA amplification factor is the ratio of RMS acceleration values at different elevation to that of base RMS acceleration value. The RMS acceleration value can be calculated from following equation [4].
( )2
1
0
21⎥⎥⎦
⎤
⎢⎢⎣
⎡= ∫
dt
d
dttat
RMS (2)
where a(t) is acceleration time history, td is the duration of the acceleration record and dt is time interval of the acceleration record. The results obtained from numerical models shows reasonable agreement with experimental results reported by researcher [8]. However, the incremental pressure obtained from numerical model shows some significant difference with experimental results. Using the validated numerical model, rigid faced reinforced retaining wall with strip reinforcement is generated. The static and dynamic responses of rigid faced retaining wall with mat and strip reinforcement are compared and discussed.
Fig. 4 Variation of horizontal displacement, RMSA amplification factor and incremental pressure at different elevation after 20 cycles 0.2g at 3Hz dynamic motion Figure 5 shows the comparison of horizontal displacement and horizontal pressure at different elevation for unreinforced wall and reinforced wall with mat and strip reinforcement at end of construction. The maximum horizontal displacement at an elevation of 55 cm is 13.19 mm for unreinforced wall while that of reinforced wall is 0.55 mm, 0.95 mm and 0.73 mm with mat reinforcement, 25 mm and 50 mm wide strip reinforcement respectively. The horizontal displacement decreases considerably for reinforced wall. But little increase of displacement in wall with strip reinforcement compared to mat reinforcement. The horizontal pressure at an elevation of 10 cm is about 1.0 kPa for unreinforced wall as well as reinforced wall. The horizontal pressures do not show appreciable variation in unreinforced and reinforced wall.
0
20
40
60
0 5 10 15 0.0 0.2 0 .4 0.6 0.8 1 .0 1.2
Horizontal displacement in mm
Elev
atio
n in
cm
Unreinforced M at reinforcement 25mm wi de st rip reinforcem ent 50mm wi de st rip reinforcem ent
Horizontal pressure in kPa
Fig. 5 Comparison of horizontal displacement and horizontal pressure at different elevation after support removal Figure 6 shows the variation of octahedral shear strain on soil element along the length of backfill after support removal. The maximum strain of 0.14 is produced on soil element at elevation 52.5 cm adjacent to the wall for unreinforced retaining wall. The maximum strain is 0.06 and 0.04 for reinforced wall with 25 mm and 50 mm wide strip reinforcement and 0.02 for wall with mat reinforcement. This is due to movement of wall away from backfill. So a small failure zone in form of settlement of soil is formed near the wall, but is confined only on higher elevations. This is more significant for unreinforced wall and less for wall with mat reinforcement. The strain on soil elements at deeper backfill is less than 0.03 for unreinforced wall and is much lesser for reinforced wall. So no failure zone will form at deeper backfill soil.
0 .0 0
0 .0 6
0 .1 2
0 .0 0
0 .0 6
0 .1 2
0 .0 0
0 .0 6
0 .1 2
0.0 0 .2 0 .4 0 .6 0. 80 .0 0
0 .0 6
0 .1 2
A t el evatio n 52.5cm
Oct
ahed
ral s
hear
str
ain
A t elevat ion 37.5 cm
A t elevat ion 22.5 cm
Length of b ackf ill in m etre
un re info rced m at 25 mm st rip 50mm st rip
A t elevat ion 7.5cm
Fig. 6 Variation of octahedral shear strain along length of backfill at different elevation after support removal Figure 7 shows the comparison of horizontal displacement and horizontal pressure at different elevation for unreinforced wall and reinforced wall with mat and strip reinforcement after dynamic excitation. The maximum horizontal displacement at an elevation of 52.5 cm is 18.70 mm for
Arup Bhattacharjee & A. Murali Krishna
unreinforced wall while that of reinforced wall is 7.39 mm, 11.00 mm and 8.80 mm with mat reinforcement, 25 mm and 50 mm wide strip reinforcement respectively. The horizontal displacement decreases considerably for wall with mat reinforcement but increases for wall with strip reinforcement. The horizontal pressure at an elevation of 10cm is 1.60 kPa for unreinforced wall. The horizontal pressure at same elevation is 2.30 kPa for reinforced wall with mat reinforcement and 2.05 kPa and 1.83 kPa for wall with 25 mm and 50 mm wide strip reinforcement.
0
2 0
4 0
6 0
0 4 8 12 1 6 2 0 2 4 0. 5 1 .0 1. 5 2 .0 2. 5
Horizontal disp lacement in mm
Elev
ation
in cm
Incremental pressure in kPa
Unreinforced Mat re inforcement 25mm strip rein fo rcement 50mm strip rein fo rcement
Fig. 7 Comparison of horizontal displacement and horizontal pressure at different elevation after dynamic excitation
0.00
0.06
0.12
0.00
0.06
0.12
0.00
0.06
0.12
0.0 0.2 0.4 0.6 0.80.00
0.06
0.12
unreinforced mat reinforcement 25mm s tr ip 50mm strip
At elevation 52.5cm
Oct
ahed
ral s
hear
stra
in
At elevation 37.5cm
At elevation 22.5c m
Length of backfil l in metre
At elevation 7.5cm
Fig. 8 Variation of octahedral shear strain along length of backfill at different elevation after dynamic excitation Figure 8 shows the variation of octahedral shear strain on soil element along the length of backfill after dynamic excitation. The maximum strain of 0.14 is produced on soil element at elevation 52cm adjacent to the wall for unreinforced retaining wall. The maximum strain is 0.09 and 0.06 for reinforced wall with 25 mm and 50 mm wide strip reinforcement and 0.04 for wall with mat reinforcement. This is due to
movement of wall away from backfill. So a small failure zone in form of settlement of soil is formed near the wall, but is confined only on higher elevations. The vertical settlement is more for unreinforced wall than reinforced wall with strip and mat reinforcement. The backfill strain is 0.03 for unreinforced wall and less than 0.02 for reinforced wall. So some horizontal and vertical displacement will occur at higher elevations of backfill soil. CONCLUSIONS The numerical models are developed to study the behaviour of reinforced wall with mat and strip reinforcement. The following conclusions are made from present study:
1. The displacements of reinforced wall with strip reinforcement are more than the wall with mat reinforcement for both monotonic and dynamic simulation.
2. The failure surface in form of settlement of backfill will form near the wall but confined to the upper layers of backfill. The settlement of soil is more for wall with strip reinforcement than that of mat reinforcement. REFERENCES 1. Abdelouhab, A., Dias, D., and Freitag, N. (2011).
Numerical analysis of the behaviour of mechanically stabilized earth walls with different types of strips, Geotextiles and Geomembranes, Vol. 23, 116-129.
2. Hatami, K. and Bathurst, R. J. (2005). Development and verification of a numerical model for the analysis of geosynthetic-reinforced soil segmental walls under working stress conditions. Canadian Geotechnical Journal, Vol. 42, No. 4,1066-1085.
3. Itasca (2008). Fast Lagrangian Analysis of Continua3D Version 3.1 Itasca Consulting Group Inc., Minneapolis.
5. Kuhlemeyer, R.L and Lysmer, J. (1973). Finite element method accuracy for wave propagation problems, Journal Soil Mechanics and Foundations Div. ASCE, Vol. 99, No. SM5, 421-427.
6. Ling, H.I., Yang, S., Leshchinsky, D., Liu, H. and Burke, C. (2010). Finite-element simulations of full scale modular-block reinforced soil retaining walls under earthquake loading. Journal of Engineering Mechanics, ASCE, Vol.135, No. 5, 653-661.
7. Liu, H. and Ling, H.I.(2011). Seismic response of reinforced soil retaining walls and strain softening of backfill soil. International Journal of Geomechanics, ASCE, doi:10.1061/(ASCE)GM.1943-5622.0000051.
8. Murali Krishna, A. and Madhavi Latha, G. (2009). Seismic behaviour of rigid-faced reinforced soil retaining wall models: reinforcement effect. Geosynthetics International, 16, No.5, 364-373.
9. Murali Krishna A. and Madhavi Latha G. (2012). Modeling of dynamic response of wrap faced reinforced soil retaining wall. International Journal of Geomechanics, ASCE, Vol.12, No.4, doi: 10.1061/(ASCE)GM.1943-5622.0000128.
Proceedings of Indian Geotechnical Conference December 13-15,2012, Delhi (Paper No.F616)
ANALYSIS OF GEOTEXTILE TUBE FOR COASTAL ENVIRONMENT Dr Ambarish Ghosh, Professor, Bengal Engg & Science University, Shibpur [email protected] Sudhanwa Pal, Project Engineer, Development Consultant (P) Limited Kolkata, [email protected] ABSTRACT: Geotextile tubes are used in flood protection and erosion control in coastal areas. They can be used to containment of dredged materials and dewatering of slurry. This paper describes the analysis and design of geotextile tube for various engineering applications. Analysis of geotextile tube has been done using the program MATLAB based on the Plaut and Suherman (1998) method [1]. Two cases have been considered where the unit weight of slurry relative to water is taken as 1.2 and the circumference of the tubes are chosen as L=9m and 10m. The various important parameters like height, base width and tensile force in Geotextile tube have been computed. The effect of pumping pressure versus height of the tube and tensile forces have been illustrated in graphs. The major design considerations which are related to the integrity of the units during release and impact, the accuracy of placement, and the stability under current and wave attack are discussed. The various design aspects such as geotechnical design, hydrodynamic design and geotextile characteristics have been discussed. INTRODUCTION Geotextiles have recently become a new engineering material with numerous applications. One of these applications is the use of geosynthetic tubes filled with a slurry-mix, including sand, concrete, or mortar. These tubes have proven to be an economical alternative for the construction of breakwaters, groins, and temporary levees. They have also been used for slope protection along with many other engineering projects. Geotextiles are permeable fabrics which are able to hold back materials while water flows through. Geosynthetic tubes are large tubes consisting of a woven geotextile material filled with a slurry-mix. The mix usually consists of dredged material from the nearby area but can also be a mortar or concrete mix. The tubes can be used solely, or stacked to add greater height and usability. SYSTEM DESCRIPTION There are inlets at the upper part of the tube where the pumping hose is inserted. The number and interval of inlets are dependent upon the type of the soil being used [2]. Typical lengths and widths of geotextile tubes are 150-180m and 4-5m, respectively, with the effective height of 1.5-2.0m.
Fig. 1 Filling procedure in a geotextile tube
ANALYSIS OF GEOTEXTILE TUBE Analysis of geotextile tube can be done numerically or analytically. In this paper the method of analysis as done by Plaut & Suherman (1998)[1] is adopted to calculate the various design parameters for the geotextile tube. For calculation purposes MATLAB (R14) program has been used. Two cases are considered where the unit weight of slurry relative to water was taken as 1.2 and the circumference of the tube was chosen as L=9m and 10m.
Analysis and result The cross section of a tube that rests on a rigid, foundation is shown in Fig.2. The horizontal coordinate is X, the vertical coordinate is Y, the arc length from the origin at the right lift-off point is S, the contact length is B, and the cross section has total perimeter L, height H, width W, and area A.
Fig. 2 Cross section of tube on rigid foundation
Dr Ambarish Ghosh, Sudhanwa Pal
It is convenient to introduce the following nondimensional quantities:
,LBb = ,
LWw =
LHh =
,int L
Pp bot
bot γ= 2
int LTt
γ=
Where Pbot = Pressure at the bottom of the tube γint= Specific weight of the slurry T=Circumferential tension The parameter k is defined by the following expression. A plot of parameter k versus bottom pressure was given by Plaut & Suherman (1998) [1].
botptk 2
=
The result of analysis is presented in a graphical manner. Fig.3, Fig.4, Fig.5, Fig.6 describes the effect of pumping pressure on the tensile force and height of the geotextile tube.
Analysis of geotextile tube for coastal environment
Fig. 5 Pumping pressure versus tensile force for L=10m.
Fig. 6 Pumping pressure versus height for L=10m.
DISCUSSION Two cases have been considered where the unit weight of slurry relative to water is taken as 1.2 and the circumference of the tubes are chosen as L=9 m and 10 m. The various important parameters like height, base width and tensile force in Geotextile tube have been computed. It is clear from the graphs (for both cases) that with the increase of the pumping pressure the height of the tube and the tensile force are increasing and the base width is decreasing. DESIGN CONSIDERATION OF GEOTEXTILE TUBE The major design considerations are related to the integrity of the units during release and impact, the accuracy of placement, and the stability under current and wave attack. The following design aspects should be considered. Geotechnical design aspect The physical characteristics of the filling material are important factors of geotextile tube design and construction. Types of soil and degree of saturation influence the final geotextile tube shape. Field experience
has demonstrated that it is possible to fill in the geotextile tubes to 70% or 80% of the theoretical maximum circular diameter. The dredged material filled in the geotextile can be any material capable of being transported hydraulically. Naturally occurring beach or river sand is the perfect choice for structural fill. Hydrodynamic Design Hydrodynamic stability is a very important factor for coastal and near shore geotextile tube construction. Loading sources include waves, tides and winds. Geotextile structure may collapse due to overturning and sliding forces associated with waves including breaking wave, non breaking wave etc. Geotextiles characteristics The retention of fill and the structural integrity of a dredged material-filled tube are provided by geotextile envelope. Functionally, geotextile selection is based on the geotextile’s opening characteristics, which must match the fill particle size and permeability, and must have sufficient strength to resist the filling pressures. A composite fabric shell is sometimes used, since it incorporates both an inner non woven fabric for filtration and an outer woven fabric for strength. Formulation of a geotextile tube, filled with pressurized slurry or fluid, is based on the equilibrium of the encapsulating flexible shell. ADVANTAGE AND DISADVANTAGE OF USING GEOTEXTILE TUBE There are several advantages using geotextile tube which include lower cost, successful beneficial uses of dredged material, ability to use the tube in soft foundation and flexibility in working in difficult access area. The major disadvantages of using geotextile tube are lack of permanency, tendency when used incorrectly to roll or move, vulnerable to vandalism, only useful as longer term breakwaters when filled with sand material, fine-grained materials use primarily limited to contaminants storage and isolation, only appropriate in low to moderate wave energy conditions, and hard to successfully stack, especially in high tidal ranges. CONCLUSIONS Geotextile tubes may be considered for alternative structure designs in many applications. They are being considered for sills, low-crested breakwaters, the cores of dunes or rubble mound structures, containment dikes, groins, and compartmentalization structures that limit movement of sand along a beach. The successful application of geotextile tube warrants the consideration of possible loading and various geohydrological conditions in design. To prevent the geotextile tube from various adverse condition suitable armour design shall be considered.
Dr Ambarish Ghosh, Sudhanwa Pal
REFERENCES 1. Plaut, R.H., and Suherman, S. (1998). Two
dimensional analysis of geosynthetic tubes. Acta Mechanica 129, 207-218
2. Pilarczyk, Krystian W.(2003). Alternative systems for coastal protection –An overview. International Conference on Estuaries and Coasts November 9-11,2003, Hang-Zhou, China.
3. Cantre,S.(2002).Geotextile-tubes-analytical design aspects. Geotextile and geomembranes 20 (305-319).
4. Leshchinsky, D.,Leshchinsky, O., Ling, Hoe I., Gilbert, Paul A.(1996).Geosynthetic tubes for confining pressurized slurry: some design aspects. Journal of geotechnical engineering, 122(8),682-690.
5. Liu, G.S. (1981). Design criteria of sand sausages for beach defense. Proceedings, 19th Congress of the International Association for Hydraulic Research, Vol. 3, new Delhi, India, 123-131
6. Plaut, R.H., Klusman, C.R.(1999) Two-dimensional analysis of stacked geosynthetic tubes on deformable foundations.Thin walled structure 34 (179-194).
8. Shin, E.C. and Oh, Y.I., (2006), Using submerged geotextile tubes in the protection of E. Korean shore, Coastal engineering, Vol. 53, pp. 879-895.
9. Shin, E.C. and Oh, Y.I., (2007), Coastal erosion prevention by geotextile tube technology, Geotextiles and Geomembranes, Vol. 25, pp. 264-277.
10. Silverster, R. (1986). Use of grout-filled sausages in coastal structures, Journal of Waterway, Port, Coastal, and ocean Engineering, Vol.112, No-1,pp. 95-114.
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F 617)
BEARING CAPACITY OF CIRCULAR FOOTINGS ON REINFORCED FOUNDATION BEDS OVER SOFT COMPRESSIBLE GROUND
K. Rajyalakshmi Lecturer, Dept. of Technical Education (A.P.), email: [email protected] Madhira R. Madhav Professor Emeritus, JNTU Hyderabad and Visiting Professor, IITH, email: [email protected] K. Ramu Professor, JNTU Kakinada, email: [email protected]
ABSTRACT: The paper presents a method for the estimation of bearing capacity of a circular footing on the surface of a reinforced foundation bed over soft compressible clay. The proposed method modifies the Meyerhof’s theory for estimation of bearing capacity of a two layer system of dense fill over soft ground which considers punching mode of failure of footing, for upper granular beds of thickness smaller than the width of the footing, by incorporating the Vesic’s Cavity expansion theory for soft soils and also the axial resistance to pull of reinforcement. A parametric study quantifies the contributions of various parameters. INTRODUCTION Reclamation of tidal or low-lying lands typically involves laying of granular bed over the soft ground for possible constructional utilization. The estimation of bearing capacity of foundations is one of the basic and challenging problems of geotechnical engineering and is largely based on rigid-plasticity analysis. An alternative to the plasticity analysis is the Vesic’s [1, 2] cavity expansion theory that considers the compressibility/stiffness of the ground together with its shear strength. The bearing capacity of circular footings on finite or semi-infinite homogeneous soft soil has been an area of interest to many researchers. Menard [3] obtained bearing capacity solutions considering the shear modulus as well as the shear strength of soil. Vesic’s solution [1, 2] for bearing capacity, qb, of a footing based on expansion of cylindrical cavity in undrained clay under the conditions of zero average volumetric strain is
(1) where Nc
* = (lnIr +1) and Ir = G/su - the relative rigidity index. The overburden pressure, for footings on the surface of the ground is zero. Hence, Eq. 1 reduces to
(2) Madhav & Padmavathi [4] established that ground/soil being a much more complex material than metals from which the original theories have been developed, requires the consideration of stiffness as well as the strength parameters for the estimation of ultimate loads.
PROBLEM DEFINITION AND FORMULATION A circular footing (Fig. 1) of diameter, B rests on the surface of a granular layer of thickness, H, with a single layer of geosynthetic reinforcement placed in the granular bed, overlying a soft compressible clay deposit. The unit weight and the angle of shearing resistance of the granular stratum are γ and ϕ respectively while su is the undrained shear strength, G, the shear modulus of soft ground and ϕr, the interface/bond resistance between geosynthetic layer and the granular fill.
Fig.1 Circular footing resting on a Reinforced foundation bed
Method of Analysis Substituting the shape factor for circular footing, the ultimate unit bearing capacity, qbL, of a circular footing on the surface of a compressible deposit given in Eq. 2 becomes
0)1(ln2.1 uu
u ssGq += (3)
K. Rajyalakshmi, Madhira R. Madhav & K. Ramu
The shape factor for circular footings on compressible ground is assumed to be equal to 1.2, as in the case of circular footings on homogeneous ground. Meyerhof’s Method for footings on sand overlying clay Figure 2 illustrates the punching mode of failure of footings on two layered soil of sand overlying clay by Meyerhof [5]. As per the Meyerhof’s [5] theory, the bearing capacity of a circular footing on the two layered soil is
(4)
and is limited by the ultimate bearing capacity of the granular layer of infinite extent as (5)
where c = undrained cohesion, φ is the of shearing resistance of sand, γ is the unit weight of sand, B is the width of the footing, D is the depth of embedment of the footing, Ks is the coefficient of punching shearing resistance, s is the shape factor governing the passive earth pressure on a cylindrical wall, Nc, Nq and Nγ are the bearing capacity factors. For circular footings on the surface of the ground (D = 0) and assuming value of ‘s’ as equal to 1, Eq. 4 gets reduced to
(6) Bearing capacity of unreinforced and reinforced foundation beds on soft compressible ground
The bearing capacity, qug, of an unreinforced two layered system of granular fill overlying soft compressible ground, supporting a circular footing on the surface of the granular layer is obtained by incorporating the Vesic’s [1, 2] theory in the Meyerhof’s [5] equation for estimation of bearing capacity of a two layered system (with c=su) as
(7) The bearing capacity, qur*, of a reinforced two layered system of granular fill overlying soft compressible ground, supporting a circular footing, on the surface of the granular layer is obtained by summing the bearing capacity of the soft ground, granular fill and the axial tension in the reinforcement as
(8) Eqs. (7) and (8) are normalised by the undrained shear strength, ‘su’ to get Eqs. (9) and (10) respectively which are
the normalised bearing capacity factors, Ncg and Ncr* for a two-layered unreinforced and reinforced systems.
Fig. 2 Bearing capacity analysis for sand over clay, Meyerhof [5] (9)
The bearing capacity of a footing resting on reinforced granular bed overlying a soft compressible clay layer, depends on φ and H/B related to the granular layer, γB/su, related to unit weight of granular fill, width of the footing and undrained strength of the clay layer, G/su related to the clay layer and φ
r/φ & Lr/B related to the reinforcement. A parametric study is carried out to quantify the effects of various parameters on the bearing capacity of the unreinforced and reinforced two layered systems for G/su equal to 63(for Nc = 5.14), 250 and 1000.
Fig. 3 Ncg versus H/B- Effect of Relative rigidity index, G/su
Bearing capacity of Circular footings on Reinforced foundation beds over soft compressible ground
Fig. 3 presents the variation of normalised bearing capacity, Ncg of an unreinforced two layered system with H/B, for a granular fill with φ of 350, for γB/su equal to 15 and 25. Ncg values equal 6.2, 7.8 and 9.5, for G/su equal to 63, 250 and 1000 respectively, at H/B=0, for both γB/su equal to 15 and 25. Ncg values equal 11.4, 13.2 and 15.1 at H/B = 0.4 and 70.8, 74.1 and 77.8, at H/B = 1.4 for G/su equal to 63, 250 and 1000 respectively, for γB/su equal to 15. Ncg values increase to 14.6, 16.4 and 18.2, at H/B = 0.4 and 109.8, 112.8 and 116.5, at H/B = 1.4 for G/su equal to 63, 250 and 1000 respectively, for γB/su equal to 25. The normalized bearing capacity, Ncg increases with H/B. The effect of G/su, relative stiffness index on the bearing capacity of the unreinforced two layer system is marginal while that of γB/su on the bearing capacity of unreinforced two layer system is significant. Relatively softer clays or relatively wider footings with higher values of γB/su show marked improvement in bearing capacity, which increases with H/B value (Fig. 3). The variation of normalised bearing capacity, Ncr* of a reinforced two layer system with H/B, for a granular fill with φ of 350, φ
r/φ of 0.75 and Lr/B of 5, for γB/su equal to15 and 25 is presented in Figure 4. The normalized bearing capacity, Ncr* increases with H/B. Ncr* values equal 23.3, 25.1 and 26.9, at H/B = 0.4 and 112.2, 115.5 and 119.3, at H/B = 1.4, for G/su equal to 63, 250 and 1000 respectively, for γB/su equal to 15. Ncr* values increase to 34.4, 36.1 and 38.0, at H/B = 0.4 and 178.8, 181.9 and 185.6, at H/B = 1.4, for G/su equal to 63, 250 and 1000 respectively, for γB/su equal to 25. Similar results as obtained in figure 3 are obtained. While the effect of G/su, the relative stiffness index on the bearing capacity of the reinforced two layer system is negligible, that of γB/su on the bearing capacity of reinforced two layer system is significant and improves with the value of H/B (Fig.4).
Fig. 4 Ncr* versus H/B- Effect of G/su
Fig. 5 presents the variation of normalised bearing capacity, Ncg with φ, for a granular fill with H/B of 0.6 and γB/su equal to 15, for G/su equal to 63, 250 and 1000. Ncg values equal 14.4, 16.6 and 18.8, at φ = 30 degrees, 18.0, 20.0 and 22.0, at φ = 35 degrees and 25.6, 27.4 and 29.2, at φ = 40 degrees, for G/su equal to 63, 250 and 1000 respectively, for γB/su equal to 15. The normalized bearing capacity, Ncg increases with φ. The increase is gradual for 300 < φ < 350 and sharp thereafter due to the increase in denseness of the granular fill. Stiffness of the underlying ground adds to the improvement in bearing capacity of the two layered system of granular fill over soft ground. Denser granular fills on relatively stiffer grounds show improved bearing capacity.
Fig. 5 Ncg versus φ- Effect of Relative rigidity index, G/su
Fig. 6 Ncr* versus φ- Effect of Relative rigidity index, G/su
K. Rajyalakshmi, Madhira R. Madhav & K. Ramu
The variation of normalised bearing capacity, Ncr*, of a reinforced two layer system with φ, for a granular fill with H/B of 0.6, γB/su of 15, φ
r/φ of 0.75 and Lr/B of 5, for G/su equal to 63, 250 and 1000 is presented in figure 6. The normalized bearing capacity, Ncr* increases with φ. Similar results as obtained in fig. 6 are obtained. Ncr* values equal 29.3, 31.5 and 33.8, at φ = 30 degrees, 35.8, 37.8 and 39.8, at φ = 35 degrees and 46.3, 48.2 and 50.0, at φ = 40 degrees for G/su = 63, 250 and 1000 respectively. The effect of stiffness of the underlying soft ground on the bearing capacity of the reinforced two layer system is marginal.
Fig. 7 Ncr* versus Lr/B- Effect of G/su
The variation of normalised bearing capacity, Ncr*, of a reinforced two layer system with Lr/B, for a granular fill with φ of 350, φ
r/φ of 0.75, H/B of 0.6, γB/su of 15, for G/su equal to 63. 250 and 1000 is presented in figure 7. Ncr* values equal 18.0, 20.0 and 22.0 at Lr/B equal to 1, 26.9, 28.9 and 30.9 at Lr/B equal to 3 and 35.8, 37.8 and 39.8 at Lr/B equal to 5, for G/su equal to 63, 250 and 1000 respectively. The normalized bearing capacity, Ncr* increases with Lr/B. For Lr/B equal to 1, the effect of reinforcement is zero, as the reinforcement does not extend beyond the footing width. The increase in length of reinforcement beyond the width of the footing results in an increase in bearing capacity. The effect of G/su, relative stiffness index on the bearing capacity of the reinforced two layer system is significant due to the increase in bearing capacity of the ground with increase in stiffness of the ground indicated by the value of G/su (Fig. 7).
SUMMARY AND CONCLUSIONS An analysis of bearing capacity of a circular footing on the surface of a geosynthetic reinforced foundation bed over soft compressible clay is presented. Punching mode of failure proposed by Meyerhof for dense granular fill overlying clay is considered and the results for bearing capacity of a footing on the surface of a clay deposit given by Vesic incorporated in the Meyerhof’s approach, for estimating the bearing capacity of a footing on reinforced two layer system. Denser granular fills on relatively stiffer grounds and also reinforced granular fills overlying soft ground show improved bearing capacity. The increase in length of reinforcement results in an increase in bearing capacity. Relatively softer clays or relatively wider footings with higher values of γB/su show marked improvement in bearing capacity. REFERENCES 1. Vesic, A.S. (1972), Expansion of cavities in infinite soil
mass, J. Soil Mech. and Found. Div., ASCE, 98(3), 265-290.
2. Vesic, A.S. (1973), Analysis of ultimate loads of shallow foundations, J. of Soil Mech. & Found. Div., ASCE, 99(1), 45-73.
3. Menard, L. (1957), Mesures in situ des propriétés physiques des sols, Annales des Ponts et Chaussées, Paris, 14, 357- 377.
4. Madhav, M.R. and Padmavathi, V. (2008), Effect of Stiffness of Ground on Ultimate Capacity of Foundations, IGC 2008, Bangalore.
5. Meyerhof, G.G. (1974), Ultimate bearing capacity of footings on sand layer overlying clay, Canadian Geotechnical Journal, Vol. 11, 223-229.
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F618)
A STUDY ON RESPONSE OF LATERALLY LOADED PILES EMBEDDED IN LAYERED COHESIONLESS SOIL
S.K. Biswas, Assistant Professor of Civil Engineering, Jadavpur University, Kolkata-32, [email protected] S.P. Mukherjee, Professor and Head of Civil Engineering, Jadavpur University, Kolkata-32, [email protected] Moyukh De, PG Scholar of Civil Engineering, Jadavpur University, Kolkata-32, [email protected] ABSTRACT: In this paper, an attempt has been made to study some aspects of behavior of laterally loaded piles in cohesionless soil by experiments and consequently with the help of numerical study by PLAXIS 3D software. The experimental study was done on cast iron pipe model piles, varying different parameters. Consequently numerical analysis has been done for single piles with the variation of parameters within the same ranges. It was observed that the results obtained by PLAXIS 3D software agree very well with the experimental results. So, there is a good possibility to go for in depth analysis for lateral response of pile in cohesionless medium by the software to develop design charts. INTRODUCTION In civil engineering practice piles are subjected to a wide variety of loading conditions due to earthquake, wind, sea wave and the like. Thus it is understood that response of pile under lateral loading has a great importance in analysis and design of piles. Many approaches have been made by various researchers so far [1, 3]. A non-dimensional relative stiffness factor was suggested to predict the behavior of piles [3]. Design charts were developed for prediction of lateral response of piles with the help of theoretical and experimental studies [1]. With this in view an attempt has been made to examine the behavior of laterally loaded piles in layered soil in the present study. EXPERIMENTAL STUDY Material Properties Here mainly two types of materials were used – i) Sand as soil medium and ii) Cast iron pipe as model piles. A. Cast Iron Tubular Piles (Hollow) Density of Cast Iron 9.23 X 10-8 KN/mm3 Young’s Modulus Ep 66307.55 MPa. Shear Modulus τp 27130.424 MPa. Poisson’s Ratio (μs) 0.222 B. Cohesionless Soil Maximum dry Density(γdmax)
31.70gm/cc
Minimum dry Density(γdmin) 1.37gm/cc Specific Gravity (Gs) 2.696 Void ratio (loosest) emax 0.992 Void ratio (densest) emin 0.605 Shear Strength Poisson’s Ratio (μs) 0.2 The particle size distribution of the sand used and its variation of relative density with density have been plotted in Fig. 1, Fig 2 respectively. Elastic and shear parameters of
sand (i.e. elastic modulus (E), increment of Young’s modulus (Eincrement) with depth and angle of internal friction (φ),) have been investigated thoroughly for parametric studies. The variation of these parameters is presented through graphs shown in Fig. 3, Fig. 4, and Fig. 5. The φ value and E value were obtained from Vacuum Triaxial Tests done with different confining pressure of 0, 1 & 1.5 Kg/cm2. In case of sand the young’s modulus increases with depth and this is represented by Eincrement expressed in units of kN/m2/m.
Fig. 1 Particle Size Distribution Curve
Fig. 2 Calibrated Graph for relative density Vs Density
Sumit Kumar Biswas, Sibapriya Mukherjee, Moyukh De
Fig. 3 Figure showing elastic modulus of sand vs. density with respect to different confining pressure
Fig. 4 Eincrement vs. density
Fig. 5 Angle of internal friction vs. relative density The density of the sand medium for the test was controlled by the rainfall technique. Experiments have been carried out to establish the variation of density of sand with height of fall of sand for a constant flow rate (Fig 6).
Fig. 6 Height of fall vs. density
Test Programme Total 12 tests, out of which, six tests were executed for single or uniform layered and six tests were performed for double layered soil. The tests were done with variation in slenderness ratio and relative density. For double layer, variation was made also in respect of the top layer thickness expressed in terms of percentage of pile length. Test Set up and Equipment The Test set up consists of the following components (Fig.7) 1. Test tank.
Fig.7 Schematic Diagram of the Test Set Up A square aluminum tank of 100 cm X 100cm X 120cm deep for model test was fabricated to facilitate the observation of lateral response inside foundation medium. A scale was set inside the tank to check the depth of sand during filling. 2. Dial gauges 3. Hopper 4. Model piles. Test Procedure For conducting the model test sand was filled into test tank by rainfall technique with hopper. Sand pouring technique plays an important role in the process of achieving the desired density of sand bed. The reliability of results would depend upon the uniformity of density. So maintaining the respective height of fall correctly is very essential. At first the model pile was installed at the centre of the tank and then sand filling was done. Sand was filled up to the desired height as was required. Model pile test was then done by applying lateral load. The load was applied by dead loads in increments and the test was conducted till failure.
A Study on Response of Laterally Loaded Piles Embedded in Layered Cohesionless Soil
NUMERICAL STUDY BY FEM Finite Element Modeling To model the non-linear behavior of soil surrounding the pile, Mohr-Coulomb criterion has been considered. This elasto-plastic model depends on the basic geotechnical parameters. Lateral static loading was applied stepwise up to failure load (P). The load was applied along X axis. Here, the 3D finite element modeling was done with the help of PLAXIS 3D Foundation Software. A 3D mesh was created by the software by using the 15 nodded wedge elements. The main input parameters were: E (Young's modulus), ν (Poisson's ratio), φ (Friction angle), C (Cohesion), Eincrement (Increase of stiffness). The soil mass affecting the pile response diminishes when the width is greater than 40D [2]. So, in the present analysis, the width of soil mass was taken as greater than 40D i.e. (40×24) mm, where, D was the pile diameter i.e. 24 mm. Here the depth of soil is taken as 1.7 times the length of pile to consider the effect of lateral load on soil immediately below the pile. FAILURE LOAD The variation for ultimate horizontal failure load with respect to different parameters like density, slenderness ratio of piles and thickness of top layer for layered soil system in case of single pile has been studied. Earlier an empirical equation was proposed to calculate horizontal pile head deflection corresponding to the ultimate soil resistance of the pile [4], the equation is given as:- Yu =3b/80 (1) Where, b=width of the pile (in inch) and Yu =ultimate deflection (in inch). So, Yu=3× (24/25.4)/80 =0.0354 inch =0.9 mm (This accounts to about 3.75% of the initial diameter) Above method was employed to estimate the failure load in the analysis of results of both PLAXIS and experimental results. PARAMETRIC STUDY In brief variations of the following testing parameters were done for experimental and numerical studies: 1. Relative Density (from 40% to 90%) 2. Slenderness ratio (15, 20 and 25), 3. Top layer thickness in terms of percentage of pile length (from 10% to 100%) RESULTS AND DISCUSSION Load Displacement Graph The following figures (Fig.8 to Fig. 11) were obtained at the outset of the total study. The load displacement curves from the experiments are compared with the PLAXIS generated curves as shown in Fig. 8 where it is shown that the curves obtained from experiments and PLAXIS 3D are in close proximity. Also the ultimate load obtained from both the cases varies within 10%, in which the PLAXIS slightly overestimates.
Fig. 8 Typical graph of uniform soil comparing the numerical and experimental approach As the results from numerical analysis and experimental analysis are in very close proximity when compared, some more cases were studied by PLAXIS i.e. numerical analysis and the consequent results were thoroughly studied along-with the results obtained earlier to see the variation of ultimate horizontal load with different parameters. Failure load vs. Relative Density
Fig. 9 Ultimate Horizontal failure load vs. Relative density with varied slenderness ratio for pile embedded in uniform medium Fig. 9 shows the variation of failure load Vs. relative density with varied slenderness ratio. As the relative density increases, the failure load on the pile increases due to increase in soil stiffness. As the slenderness ratio increases the failure load increases with increase of embedded length. Failure load vs. top layer thickness in terms of percentage of pile length Fig. 10 shows the variation of failure load vs. top layer thickness variation with respect to pile length for type 2 soil condition.
Sumit Kumar Biswas, Sibapriya Mukherjee, Moyukh De
As the thickness of top layer increases, the failure load of the pile decreases. This decrease can be divided into two distinct zones. This is the due to the fact that in zone 1 the thickness of top layer is such that the fixity depth of the pile is substantially inside the stronger layer.
Fig. 10 Ultimate Horizontal failure load vs. Top layer thickness variation with respect to pile length Comparison between failure load vs. top layer thickness in terms of percentage of pile length
Fig. 11 Typical graph showing comparison between Ultimate Horizontal failure loads vs. Top layer thickness variation with respect to pile length Fig. 11 shows the variation of failure load versus the percentage thickness of top layer. It can be seen that as the relative density of layer 2 increases the commencement of zone 2 precedes. This is due to increase in relative density decreases the fixity depth. CONCLUSIONS The following conclusions may be drawn from the present study: 1. It was seen that the result obtained by PLAXIS3D
FOUNDATION software agrees very well with the experimental results.
2. As the relative density and slenderness ratio increases the failure load increases.
3 There is a decrease in failure load with increase in top weak layer depth.
4. With increase in density of soil of bottom layer the depth of fixity depth decreases.
REFERENCES 1. Broms, B.B. (1965), Design of laterally loaded piles,
Journal of Soil Mechanics and Foundations Division, Vol. 91, 79-99
2. Chik, Z.H., Abbas, J.M., Taha, M.R. and Shafiqu, Q.S.M. (2009). Lateral behavior of single pile in cohesionless soil subjected to both vertical and horizontal loads. European Journal of Scientific Research, Vol.29 No.2, 194-205.
3. Reese, L.C. and Matlock, H. (1956). Non-dimensional solutions for laterally-loaded piles with soil modulus assumed proportional to depth. Proceedings of the 8th Texas Conference on Soil Mechanics and Foundation Engineering, Austin, Texas, 1-41.
4. Reese, L.C. Cox, W. R., and Koop, F. D. (1974) Analysis of laterally loaded piles in sand, Proc. 6th Annual Offshore Technology Conference, Houston, paper OTC 2080, 473–483.
Proceedings of Indian Geotechnical Conference December 13-15,2012, Delhi (Paper No. F619)
A CYCLIC NON-LINEAR MODEL FOR COHESIVE SOILS P. Subramaniam, Research Scholar, Indian Institute of Technology, Madras, [email protected] Subhadeep Banerjee, Assistant Professor, Indian Institute of Technology, Madras, [email protected] ABSTRACT: The shear modulus and damping ratio are the two general parameters for clayey soils in dynamic soil behaviour characterization. It is obvious that under cyclic loading, soil behaves hysteretically in the stress-strain plane. In this regard, several mathematical models were proposed to simulate the soil behaviour under the cyclic loading conditions. For most of such models, Masing rule is often used to define the unloading-reloading behaviour of stress-strain loops. However, the framework of masing rule overpredicts the damping ratio at high strain range as noted by many researchers. The present study deals with a hyperbolic-hysteretic soil model based on Masing rule. Simple correction factor for the calculation of damping ratio was introduced and the corrected damping ratio for kaoline clay was compared with the present experimental results. Good agreement was obtained between the computed and present experimental results for a wide range of strains. INTRODUCTION It is well known that the mechanical behaviour of natural soil under dynamic loading significantly differs from those under quasi-static loading. From literatures [1, 2], it is noted that soil behaviour is non-linear, even at relatively small strains. At very small strains, shear modulus (G) is almost a horizontal straight line, indicating that shear modulus, often termed as “small strain shear modulus” or “maximum shear modulus (Gmax)” is roughly constant. Furthermore, as the strain level rises above a certain range, the shear modulus decreases significantly over a range of strains. In this regime, the soil behaviour is hysteretic, indicating limited plasticity already exists, even though the yield locus is not yet reached. Within this range of strain, G can drop by as much as 2 orders of magnitude (~100 times). Finally, at a very high strain, large-scale yielding occurs and elasto-plasticity starts to dominate soil behavior. Under cyclic loading soil produces hysteresis loop in the stress-strain plane. i.e., significant amount of applied energy is dissipated in terms of material damping due to cyclic loading such as earthquake loading, machine loading etc., Hysteresis damping ratio reveals the energy dissipated in one cycle irrespective of frequency of loading. In order to measure the material damping laboratory tests are conducted such as resonant column, cyclic triaxial and cyclic shear tests. Several mathematical models [3, 4] were proposed to simulate the soil behaviour under the cyclic loading conditions. For most of such models, Masing rule [5] is often used to define the unloading and reloading branches of hysteresis loop along with the nonlinear backbone curve to represent the stress strain behaviour of the materials under cyclic loading. The loading and unloading branch of the backbone curve is twofold drag of the backbone curve and has same geometric shape. Damping characteristics of Masing model is derived from the backbone curve. So the backbone curve cannot be modified independently. However, it was also noted that the framework of Masing rule tends to overpredict the damping ratio at moderate to high strain range [6, 7]. The present study focuses on a hyperbolic-hysteretic soil model based on Masing rule. First the competence of the
model to predict the variation of shear modulus and damping ratio for a wide strain range will be examined. Secondly a simple correction for the calculation of damping ratio will be introduced. Finally the corrected damping ratio is validated with the present experimental results. Backbone Curve The stress–strain behaviour of cohesive soils can be expressed using a hyperbolic relationship in the form of Eq. 1
(1)
Where, εs is the generalised shear strain, R is a modulus ratio given as,
(2)
Where qf is the deviator stress at failure and maximum shear modulus (Gmax) for clay is taken as [8]
(3)
Modeling the Hysteretic Behaviour of Soils In this study, Masing’s rule [5] was adopted to model the hysteretic behaviour of the soil during the unloading and reloading phases of each load cycle. Accordingly, the shapes of the unloading and reloading curves are similar to that of the backbone curve, except that (i) the scale is enlarged by a factor of 2 and (ii) the shear modulus on each loading reversal assumes a value equal to the initial tangent modulus of the initial loading (backbone) curve. Accordingly the unloading and reloading phases of each loading cycle can be given as Eqs. 4 and 5 respectively.
(4)
fqGR max=
( ) 653.0,max 1964 pG =
sf RR
Gqq
ε2max
+−=
( )2
1max
max1
2*1
*
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
+= −
f
sr
sulunload
qG
Gdqqεε
ε
P. Subramaniam & Subhadeep Banerjee
(5)
In the current constitutive relationship, the point of loading reversal was identified using Dasari’s approach [9] as follows:
Reversal angle, YYXX
YX .cos 1−=θ (6)
where, X = strain increments for all six strain components between (i-1)-th step and (i-2)-th step and Y = strain increment for all six strain components between i-th step and (i-1)-th step Accordingly, if the reversal angle θ computed from Eq. 6 is larger than 90°, stress path reversal is deemed to have occurred. EXPERIMENTAL STUDY The following sections assess the performance of the proposed constitutive model by comparing the computed results with the experimental findings. In the present study, 38mm samples of kaolin clay is used for strain controlled cyclic triaxial tests and resonant column test to obtain modulus values and damping ratios for shear strains ranging from 10-3% to 1%. Prior to cyclic shearing, a confining pressure of 200kPa was applied for all the tests and at 6 different frequencies ranging from 0.05Hz to 1.5Hz, cyclic testing was carried out. Stress-strain behaviour
-80
-60
-40
-20
0
20
40
60
80
-0.015 -0.01 -0.005 0 0.005 0.01 0.015
Dev
iato
ric st
ress
kPa
Strain (%)
Computed Experimental
Fig. 1 Comparison of computed and experimental stress-strain loops for cyclic triaxial tests (strain amplitude =1.4%)
Fig. 1 shows the measured and computed stress-strain loop for test, in which a specimen was subjected to cyclic shearing of constant strain amplitude 1.4% and loading frequency of 1 Hz. As shown on figure, despite a little over-estimate at the peak tension, the hyperbolic-hysteretic model generally predicts the peak stresses to a reasonable extent. However figure also shows that the area enclosed by the stress-strain
loop as observed in experimental result reasonably matched well with the proposed model. Modulus Reduction Curve The modulus reduction curve, derived from Eq. 1, can be represented as Eq. 7,
(7)
1E-3 0.01 0.1 1 100.0
0.2
0.4
0.6
0.8
1.0
Test results Hyperbolic hysteretic model
G/G
max
Shear Strain (%)
Fig. 2 Comparison of computed and experimental modulus reduction curves varying with shear strain
The reduction curve, shown on Eq. 7, is plotted for different strain amplitudes on Fig. 2. The continuous line in Fig. 2 represents the computed trend of the modulus reduction ratio (G/Gmax) for different strain levels associated with the initial backbone curve. The results of the laboratory cyclic triaxial and resonant column tests conducted in the present study are also plotted in the Fig. 2. Calculated modulus reduction curve from the present analyses exhibits the typical reverse S-shape trend. The computed curve does fall within the range covered by the present experimental data points. Damping ratio The damping ratio can be defined as the ratio of energy dissipated per unit volume of one cycle to the elastic strain energy stored the material. The concept of damping ratio is explained in the Fig. 3. The area of hysteresis loop can be expressed as [9]
⎥⎥⎦
⎤
⎢⎢⎣
⎡−=Δ ∫
r
WdfWε
εε0
)(8 (8)
W is the elastic strain energy stored in the loading phase:
(9)
( )2
1max
max1
2*1
*
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
+= −
f
sr
srlreload
qG
Gdqqεε
ε
)1(1
max rRGG
ε+=
)(21
rr fW εε=
A cyclic non-linear model for cohesive soils
where, )( rf ε describes the basic stress strain relationship given by Eq. 1.
Fig. 3 Typical hysteresis curve in the deviatoric stress – strain plane Hence the damping ratio is given by
WWDπ4Δ
= (10)
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−=∫
1)(
)(22 0
rr f
dfr
εε
εε
π
ε
(11)
Substituting )(εf as q from Eq. 1, D can be expressed as (12)
where, R=fq
Gmax .
The damping response given by Eq. 12 is graphically plotted on Fig. 4, together with the present experimental data. The results point out that, for almost entire strain range, the proposed model over-predicts the strain-dependent damping characteristics of clay. Such general over-prediction of damping ratio is also noted by Ishihara [10].
1E-4 1E-3 0.01 0.1 1 100
5
10
15
20
25
30
35
40
45
50
55
60
65 Test results Hyperbolic hysteretic model
Dam
ping
ratio
(%)
Shear strain (%)
Fig. 4 Comparison of computed and experimental damping ratios varying with shear strain DAMPING RATIO CORRECTION The hyperbolic-hysteretic model with Masing rule, though can predict the modulus reduction, tend to over-predict the damping ratio. Few researchers have tried to solve this. Based on plasticity index (PI) and relative consistency (Cr) two correction parameters for damping ratio was proposed [7]. The reference strain was quantified in terms of relative consistency. The model also requires the maximum damping ratio as an additional input. However the maximum damping ratio has to be find out from the experiments at the large shear strain amplitude, is not a readily available parameter. A set of equations to predict the damping ratio based on Masing rule was developed [6]. However the equations were very complex and involve several parameters. An expression for damping ratio was developed for Taipei silty clay based on Ramberg-Osgood type backbone curve [11]. Again the application of the model was limited as it was not tested for different types of soils with wide range of plasticity. Correction Parameters In the present study correction parameters are introduced as the function of plasticity index and modulus reduction. These correction parameters are related to the damping ratio based on original masing rule. Effects of loading cycles and confining pressures on damping ratio were not considered [10]. Hence the corrected damping ratio can be expressed in the form of Eq. 12,
originalcorrected DCDD *min += (12) The terms present in the eq. 12 are as follows. The constant minimum damping ratio that clay can exhibit under low strain, is termed as (Dmin). The general trend suggest that, for strain amplitudes less than 0.001%, the curves for modulus degradation and damping ratio become almost horizontal. From Fig. 4, Dmin was found to be approximately 0.9%. However the effect of minD at higher strains was found to be negligible.
( )
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−
+−= 1
1ln222
2max
2max
sin
rfr
rrf
gma
RRGq
RRGq
D
εε
εε
π
P. Subramaniam & Subhadeep Banerjee
The correction factor C depends on modulus reduction and plasticity index and can be defined as eq. 13.
B
GGAC
log
max
* ⎟⎟⎠
⎞⎜⎜⎝
⎛= (13)
Where (G/Gmax) is the modulus reduction at a specified strain level and the parameters A and B are the functions of plasticity index (PI). Vucetic & Dobry presented set of design curves for variation of modulus reduction and damping ratios with strain amplitudes for different plasticity index [1]. These well-established data sets were used to derive the expressions for parameters A and B. The expressions for A and B, as obtained from the regression analysis, are as follows.
)023.0exp(3.0056.0 PIA −+= (14) PIB 004.0log −= (15)
1E-4 1E-3 0.01 0.1 1 100
5
10
15
20
25
30 Test resullts Hyperbolic hysteretic model
with damping correction
Dam
ping
ratio
(%)
Shear strain (%)Fig. 5 Comparison of computed and experimental damping ratios varying with shear strain The detailed analysis and comparison with published results will be available [12]. CONCLUSIONS In the present study a simplified hyperbolic-hysteretic model with damping correction was proposed for cyclic loading on remoulded soft clay. The modulus reduction was computed from the model. The computed damping ratio matched well with the present experimental studies on remoulded kaolin clay. Only two additional parameters required for damping correction and both the parameters A and B depend on plasticity index & modulus reduction. REFERENCES 1. Vucetic, M. and Dobry, R. (1991), Effect of soil
plasticity on cyclic response, J. Geotech. Eng, ASCE, 117(1), 89-107.
2. Puzrin, A., Frydman, S. and Talesnick, M. (1995), Normalising degrading behaviour of soft clay under
3. Idriss, I.M., Dobry, R., Doyle, E.H. and Singh, R.D. (1978), Nonlinear behaviour of soft clays during cyclic loading conditions, J. Geotech. Eng. Division, ASCE, 104, 1427-1447.
4. Rao, S.N. and Panda, A.P. (1998), Non-linear analysis of cyclic strength of soft marine clay, Ocean Engineering, 26(3), 241-253.
5. Masing, G. (1926), Eigenspannungen und Verfestigung beim Messing, Proc., 2nd Intl. Congress on Applied Mechanics, Zurich, 332-335.
6. Darendeli, M.B. (2001), Development of a New Family of Normalized Modulus Reduction and Material Damping Curves, Ph.D. Dissertation, The University of Texas, Austin.
7. Romo, M.P. and Ovando-Shelley. E. (1996), Modelling the dynamic behaviour of Mexican clays, Proc., 11th
world conf. Earthquake eng, Mexico, 1024. 8. Viggiani, G. and Atkinson, J.H. (1995), Stiffness of fine-
grained soils at very small strains, Geotechnique, 45(2), 249-265.
9. Dasari, G.R. (1996), Modeling of the variation of soil stiffness during sequential construction, Ph.D. Dissertation, Cambridge University, United Kingdom.
10. Ishihara, K. (1996), Soil behavior in earthquake geotechnics, Clarendon Press, Oxford.
11. Lee, C.J. and Sheu, S.F. (2007), The stiffness degradation and damping ratio evolution of Taipei Silty Clay under cyclic straining, Soil Dyn. and Earthquake Eng, 27, 730-740.
12. Subramaniam, P. and Subhadeep Banerjee. (2013), A correction to damping ratio for hyperbolic-hysteretic model for clayey soil, International Journal of Geotechnical Engineering, (In Press).
Proceedings of Indian Geotechnical Conference December 13-15,2012, Delhi (Paper No. F620)
VERTICAL PULLOUT CAPACITY OF TWO INTERACTING GROUND ANCHORS IN HOMOGENEOUS COHESIONLESS SOIL DEPOSIT
G. Santhoshkumar, PG Student, IITK, [email protected] Priyanka Ghosh, Associate Professor, IITK, [email protected] ABSTRACT: The ultimate vertical pullout capacity of a group of two horizontal smooth anchors embedded at shallow depths has been found out by using method of characteristics coupled with limit equilibrium approach. Both the anchors are loaded simultaneously until the failure occurs. The effects of surcharge and density are measured in terms of Fq and Fγ, respectively, which influence the uplift resistance of anchors and are presented as functions of embedment ratio λ and the friction angle Φ. The effect of interaction phenomenon of two anchors is expressed in terms of efficiency factor ξγ and its variation with respect to different clear spacing (S) between two anchors has been computed. The results of the numerical analysis are compared with the available theoretical and experimental data reported in the literature. INTRODUCTION In many situations, anchors are generally placed in group to support structures like transmission towers, offshore mooring structures, retaining walls etc. There are number of theories available in hand for single isolated anchors. But only few studies have been carried out in case of group of anchors [1-7]. Meyerhof and Adams [1] gave a theoretical solution using limit equilibrium approach by considering a rectangular wedge of the soil, prevailing through the outer edges of the anchor. Hanna et al. [2] conducted a series of small scale model tests on circular anchors. Murray and Geddes [3] also conducted experiments on square anchors. Kumar and Kouzer [4] sought the help of upper bound method to study the interacting strip anchors using a simple rigid wedge mechanism. Kumar and Bhoi [5] studied the interference effect of group of anchors experimentally. Experiments were conducted on a single anchor by adopting the concept of plane of symmetry. Kumar and Kouzer [6] improved their previous research by incorporating finite elements and linear programming. Ghosh and Rajusha [7] worked on both static and seismic interference cases using finite element method. From different studies, it is observed that the ultimate pullout capacity of the interfering anchors reduces extensively with a decrease in the spacing between them. The present study aims to find out the vertical pullout capacity of nearby anchors placed in the cohesionless soil medium numerically. The analysis has been carried out using method of stress characteristics. The vertical equilibrium of soil is also satisfied in order to obtain the correct failure mechanism. PROBLEM DEFINITION Two closely spaced strips anchors of equal width B are considered to be embedded horizontally in a cohesionless soil medium at a depth of D from the ground surface. The clear spacing between two anchors is kept as S as shown in Fig.1. The anchor plates are considered to be perfectly smooth as the roughness of the anchor plates does not affect the pullout capacity much (Rowe and Davis [8], Merfield and Sloan [9]). Both the anchor plates are simultaneously loaded. Thus, the
present study aims to determine the uplift capacity (Pu) per unit length of the anchor plates.
Fig. 1 Definition of the problem
ASSUMPTIONS The following assumptions are considered in the analysis 1. Anchors are perfectly rigid and smooth. 2. Suction forces under the anchors are neglected. 3. The influence of anchor tie rods has not taken into account. 4. The anchor plates failed at the same instant with the same magnitude of failure load. 5. The soil obeys Mohr-Coulomb failure criterion. 6. The problem follows plain strain condition. ANALYSIS The analysis has been carried out by modifying the failure mechanism proposed by Rao and Kumar [10] for single isolated anchor. The present study considers the method of characteristics to find out the state of stress at required points and thereby obtaining the distribution of vertical stress along the surface of the anchor, applying suitable boundary conditions. By satisfying the vertical equilibrium of soil in the failure zone, the correct failure surface has been determined. The extent of soil contributing to the anchor resistance on the non interfering side (AN') of the anchor is not fixed unlike the previous investigations (Fig.2). The ultimate pullout capacity of the interacting anchor is calculated similar to Terzaghi’s bearing capacity theory and expressed in terms of uplift capacity factors.
G. Santhoshkumar, Priyanka Ghosh
(a) (b)
Fig. 2 Failure mechanism of an intervening anchor Stress Distribution Along the Anchor Considering the plane of symmetry along the centre line CL-C′L, (Fig. 1), the left portion is considered for the analysis. A logarithmic spiral failure surface is considered to commence from the outer edge of the anchor and becomes tangential to the Rankine passive zones AEL and AE′N′, of which the former zone is restricted due to the spacing S between the anchors and the latter is not restricted. Thus, the log-spiral surface becomes one of the characteristics. Following the sign conventions (Fig. 2a), the curve OE becomes the (θ+μ) characteristic. The state of stress at point E is known from the concept of Rankine passive zone. The orientation of the major principal stress (θ) at E is -π/2.
Fig. 3 Element P′P′′ on the log-spiral arc
Making use of the geometry, the ordinates and the orientation of the major principal stress along the arc are found out (Fig. 3). The state of stress along the curve (log-spiral) can be determined using the standard equations of method of characteristics [11]. Thus the state of stress at the outer edge (O) of the anchor can be established. By forming a network of slip lines and applying boundary conditions, the method of characteristics can be used to calculate the stress
distribution along the anchor OG. Similarly, the stress distribution along O′G can also be determined. Hence the ultimate pullout capacity pu along the anchor can be established. The ultimate pullout force of an interacting anchor can be calculated by,
u
0
P = p dyB
∫ (1)
Determination of Exact Failure Surface The failure surface, for which the pullout force is obtained, is checked with the help of vertical equilibrium of soil region LEOO′E′L′. The pullout force based on the vertical equilibrium can be determined by,
u P = Q W V+ + (2) Where, Q is the vertical downward forces due to surcharge (q) = gq (X )2 2
SB× + + W is self weight of the soil mass in the region LEOO′E′L′ V is the total vertical downward force along the log-spiral surfaces OE and O'E'. For different values of AF′, a number of failure surfaces are obtained and the procedure is repeated until the pullout forces obtained from the method of characteristics and vertical limit equilibrium are equal in magnitude. The corresponding failure surface gives the correct one. Non Dimensionless Factors Similar to Terzaghi’s bearing capacity equations, the ultimate pullout resistance of anchor can be expressed as,
u u qp P B 0.5 BF + qFγ= = γ (3) Where, Fq and Fγ are the ultimate uplift capacity factors corresponding to surcharge and soil unit weight, respectively. The uplift capacity factors can be determined by neglecting one of the effects i.e. Fq can be obtained by considering soil as weightless (γ = 0). Similarly, Fγ can be obtained by considering no surcharge (q = 0). In this paper, only the effect of interference due to unit weight of soil (q = 0) is reported. The interference effect can be expressed in terms of efficiency factor ξγ, which can be defined as
u
u
p of ineracting anchor of width B considering q 0p of isolated anchor of width B considering q 0γξ
==
= (4)
In the absence of surcharge, the ultimate pullout capacity of interfering anchors can be expressed as,
u up (P /B)=0.5B Fγ γ= γ ξ (5)
Vertical pullout capacity of two interacting ground anchors in homogenous cohesionless soil deposit
RESULTS The ultimate uplift capacity factors for single isolated anchors are obtained and presented in Table 1, as functions of embedment ratio (λ) and soil friction angle (ϕ). Table 1 Ultimate uplift capacity factors
λ ϕ Fq Fγ
3 30 2.94 14.12 35 3.33 15.30 40 3.70 16.41
5 30 4.26 34.82 35 5.07 39.21 40 5.91 43.37
Figs. 4-5 show the variation of efficiency factor ξγ with S/B ratio for λ = 3 and 5, respectively.
Fig. 4 Variation of ξγ with S/B for λ=3
Fig. 5 Variation of ξγ with S/B for λ=5
For any value of λ and ϕ, ξγ is observed to decrease with the decrease in S/B ratio. It is observed that the decrease is substantial when there is an increase in embedment ratio. COMPARISONS The present results are compared with the available theoretical and experimental works in Figs. 6 and 7, respectively.
Fig. 6 Comparison with analytical work The present results are observed to be little higher than those obtained from previous studies [1, 4]. Unlike the other cases, the present work has not fixed the extent of failure zone on the non interfering side of the anchor. The earlier studies were found to provide conservative results.
Fig. 7 Comparison with experimental work The present results are found to be higher than that obtained by Kumar and Bhoi [5]. The difference is found to be significantly high about 50% at minimum S/B and becomes less about 5% at maximum S/B.
S/B
G. Santhoshkumar, Priyanka Ghosh
CONCLUSIONS The interference effect of a group of two anchors is studied and expressed in terms of efficiency factor ξγ. It is observed that there is a substantial decrease in the pullout capacity of anchors with decrease in spacing between the anchors. The extent of the failure zone is found to be dependent not only on the embedment ratio (λ), but also on S/B ratio. REFERENCES 1. Meyerhof, G. G. and Adams, S. I. (1968), The ultimate
uplift capacity of foundations, Canadian Geotechnical Journal, 5(4): 225-44.
2. Hanna, T. H., Sparks, R. and Yilmaz, M. (1972), Anchor behaviour in sand, Journal of Soil Mech. Found Division, ASCE, 98(11), 1187–1207.
3. Murray, E. J. and Geddes J. D. (1987), Uplift of anchor plates in sand, Journal of Geotechnical Engineering, Vol. 113, No. 3, 202-214.
4. Kumar, J. and Kouzer, K. M. (2008), Interference effect on the vertical uplift capacity of two shallow horizontal anchors, Géotechnique, 58(10), 821-824.
5. Kumar, J. and Bhoi, M.K. (2009), Interference of two closely spaced strip footings on sand using model tests, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, USA, 134(4), 595-604.
6. Kouzer, K. M. and Kumar, J. (2009), Vertical uplift capacity of two interfering horizontal anchors in sand using an upper bound limit analysis, Computers and Geotechnics, 36: 1084-1089.
7. Ghosh, P. and Rajusha. K. (2012), Seismic interference effect of two nearby horizontal strip anchors, Natural Hazards, DOI: 10.1007/s11069-012-0187-4.
8. Rowe, R. K. and Davis, E. H. (1982), The behaviour of anchor plates in sand, Géotechnique, 32(1): 25-41.
9. Merifield, R. S. and Sloan, S. W. (2006), The ultimate pullout capacity of anchors in frictional soils, Canadian Geotechnical Journal, 43(8), 852-868.
10. Rao, K.S.S. and Kumar, J. (1994), Vertical uplift capacity of horizontal anchors, Journal of Geotechnical Engineering, ASCE, USA, 120, 1134-1147.
11. Sokolovski, V. V. (1960), Statics of soil media, Butterworths Publications, London.
NOTATION The symbols used in the figures and equations are listed below: B width of strip anchor D depth of embedment F
q ultimate uplift capacity factor due to surcharge
Fγ ultimate uplift capacity factor due to unit weight of
soil HF horizontal force acting at the plane of symmetry Hq horizontal force due to surcharge Hγ horizontal force due to soil unit weight N1 normal force component along the log spiral P
u ultimate pullout load
p vertical pressure along the anchor base p
u ultimate uplift pressure of the anchor plate
Q total vertical downward force due to surcharge q surcharge pressure r
radius of log-spiral arc at any point
ro
initial radius of log-spiral r
1 final radius of log-spiral
S clear spacing between two anchors T1 shear force component along the log spiral V total vertical downward force along the log-spiral
surfaces W weight of the soil mass LEOO′E′L′ Xg extent of failure surface on the ground α
angle formed between the radii r0 and r of the log-
spiral surface α
o angle formed between the radii r0 and r1 of the log-
spiral surface ϕ internal friction angle of soil γ unit weight of soil λ embedment ratio (D/B) μ π/4 − ϕ/2 σ
1 major principal stress
θ angle made by the major principal stress in a counterclockwise sense with the positive x-axis
ξγ
efficiency factor due to the unit weight of soil
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F 621)
ESTIMATION OF CAPACITY OF OFFSHORE PILES UNDER BOAT IMPACT
Prakasha Kuppalli, Engineering Specialist, Saudi Aramco, Saudi Arabia, [email protected]
ABSTRACT : Offshore piles are subject to boat impact, the pile capacity during which is greater than those recommended by API-RP-2A due to higher loading rate. Three approaches are generally attempted while making pile capacity estimation under boat impact: 1) Adopt a certain percentage increase in soil strength per log time cycle. 2) Estimate the shear stress required to fail a sample in one cycle of loading and use it in capacity estimation. 3) To estimate the dynamic capacity from dynamic monitoring data due to hammer impact and interpolate to find the dynamic capacity under boat impact. This paper describes the three approaches and comments on their suitability. INTRODUCTION Jacket platforms are installed offshore to provide support for various equipments and guides for drilling. These jacket platforms should transfer various loads to pile foundations below. These include environmental loads, dead and live loads and boat impact loads. Boat impact loads are the loads arising due to boats hitting the platform. These could be either boats that approach the platform for operational purposes or those which lose control and accidentally hit the platform. The time to reach the peak load under such impact is known to be very small. The resistance of soils under high rates of loading is known to be high. RGME [3] studied this issue and have recommended procedures for estimating increase in undrained strength of clays. Lunne et. al [2] present results from cyclic load tests and suggest that the capacity of Dramman clays can sustain 65% more loads if they are subject to failure in single cycle of load compared to slow monotonic loading conditions. Kraft [3] have studied the effect of rate of penetration on CPT skin and tip resistance and have concluded that both resistance increase rapidly at high rates of penetration for sands.
Problem Definition The capacity of offshore piles is generally calculated based on API-RP-2A [1] recommendations, which are based on results from unconsolidated undrained triaxial tests. These tests are run at a very slow rate and time to peak is generally about 120 secs. These recommendations are meant for calculating the pile capacity under static and environmental loads which have a period of about 10 secs. Hence the time to peak load is 2.5 secs in case of wave loading. However the peak load due to boat impact is known to occur in 0.8 secs. The hammer impact loading takes about 2 millisecs to reach the peak load. GRL [2] have presented the load – time plot during hammer impact (Refer Fig. 1). Hence relative to the triaxial test loading rate, the hammer impact rate (during pile driving) is about 60000 times faster, boat impact loading is 150 times faster and wave loading is 50 times faster. Since it is well known that the shear
strength of both clays and dense sands increase with the rate of loading. Hence the capacity of offshore pile is expected to be considerably higher for boat impact conditions. This paper evaluates the approaches and compares them and recommends the appropriate method for estimating pile capacity under boat impact.
Different Approaches As pointed out earlier three approaches can be adopted for estimating the pile capacity under impact. They are:
Based on increase in strength of soils under higher rate of loading RGME [3] have compiled various literatures available and the
Fig. 1 Load transfer during hammer impact same have been presented in Fig.2. Based on literature and series of laboratory tests on clays found in Saudi Arabia at different rates of loading, they have suggested that the strength of clays increases about 20% per every log cycle increase in rate of loading. From the loading rates mentioned above, it can be estimated that the capacity during impact would be 46% higher than that determined from an UU triaxial test. Kraft et.al, [4] based on their experiments have established that the CPT cone and friction resistances increases by 20% for a log cycle increase in rate of penetration. Hence similar
Prakasha Kuppalli increase can also be expected of dense sands which develop negative pore pressures during shearing. Hence if the pile is located in strata having clays and dense sands the capacity is expected to be about 46% higher than that estimated using API-RP-2A [1]. However, loose sands which develop positive pore pressures during shearing are not expected to show considerable increase in strength during impact loading.
Based on results from cyclic tests Lunne & Andersen [5] presents results from cyclic tests carried out under different cyclic and average stresses for Dramman clay. Their results are reproduced in Fig. 2. The diagram shows the cyclic shear strength of soils as a function of cyclic stress ratio, average stress ratio and number of cycles to failure. As impact loading can be considered as a single cycle of load, the strength under impact can be read off from the plots from Fig. 2. The plots represent data for a cyclic loading of 10 sec period (2.5 sec to peak loading). It can be seen from the plot that the cyclic strength (cyclic stress plus average stress) is at least 65% higher than for a monotonic triaxial test. This increase would be higher for higher rate of loading. This approach suggests that the increase in capacity during impact would be at least 65% higher than the static pile capacity. This is based on the
general fact that piles are subject to an average shear
Fig. 2 Effect of loading rate on strength of clays
stress of about 40% due to dead loads and environmental loads. Nf in Fig. 2 denotes the number of cycles required for failure at the stress levels indicated by the axes. They have also established similar relationships for dense sands, but have not been presented here. Though this ratio may differ for different soils, the variation is not expected to be considerable. Objections have been raised to this approach, on the grounds that API-RP-2A does not allow for increase under wave loading which has a much higher rate of loading than that occurs in a triaxial test. It may be pointed out that, the number of cycles of loading for a typical offshore situation is about 150. Going back to the plots in Figs 1 and 2, one can find that the cyclic shear strength for 150 cycles to failure would be close to static capacity based on the same average shear stress assumption mentioned earlier. This would explain the reasoning behind API not allowing higher strengths to be considered for wave loading conditions. Hence, method based on increase in shear strength due to higher rate of loading gives conservative estimate of increase in capacity.
Fig. 3 Cyclic shear strength of clays
Based on CAPWAP results during pile driving
While the above methods provide means of estimating the increase in capacity during impact before pile driving, the CAPWAP results can be conveniently used to confirm the above based on the monitoring carried out during pile
Estimation of Capacity of Offshore Piles Under Boat Impact
driving. The signal matching process of CAPWAP computes not only the static capacity but also the dynamic capacity and the total capacity. Though many damping models are available, Smith damping used in CAPWAP is expected to give conservative estimates of dynamic capacity. The data for a typical site for a restrike blow is shown in Table 1. It can be seen from the table that the dynamic resistance is usually greater than the static resistance. It has been observed from data of various locations that the skin friction dynamic resistance is always greater than 100% of static resistance while the end bearing dynamic resistance is between 30% and 100% of static end bearing resistance. This lower increase in end bearing is typical of restrike blows as full end bearing capacity might not have developed during restrike. It has been observed that even the dynamic end bearing is close to 100% static end bearing capacity during end of drive blows. Never the less the total resistance has always been found to be greater than twice the static resistance. Knowing the increase in capacity under hammer impact, one can estimate the increase in capacity due to boat impact knowing the times to peak loads mentioned earlier, assuming a logarithmic relationship. From figure 4, it can be seen that the interpolated dynamic resistance under boat impact would be 45% of static capacity. So, all the approaches indicate an increase of greater than 45% under boat impact. However a factor of 25% is taken in practice as an appropriate value, in the absence of actual data, as a conservative estimate. While the first two approaches can be adopted for design purposes, the last approach can be used for confirming the capacity based on pile monitoring tests.
Table 1. Dynamic resistance table for a typical location
Fig. 4. Interpolation for dynamic capacity under boat impact
Conclusions Following Conclusions can be drawn from the discussions in the paper: • The pile capacity during boat impact would be
considerably higher than that estimated as per API-RP-2A.
• One can adopt the higher shear strengths of soils based on laboratory tests and CPTs to account for rate effect or use the cyclic shear strength for failure in one cycle as the shear strength for estimating the boat impact pile capacity during design stage. However the former, being conservative, is used in practice.
• The boat impact capacity should be confirmed from CAPWAP analysis based on pile monitoring.
• In the absence of definitive data, an increase in capacity of 25% can be assumed as a conservative estimate.
References 1. API RP 2A (2010), Recommended guidelines for
design of offshore structures. 2. GRL associates (2012), Field report for ARBI 9,
Saudi Aramco. 3. RGME (2006), Final engineering report for Zuluf
in sands for offshore conditions, Marine Geotechnology, 61-72
5. Tom Lunne and K.H. Andersen (2007) ‘Soft clay shear strength parameters for deepwater geotechnical design, SUT conference, London 11-13 Sept.
Proceedings of Indian Geotechnical Conference December 13-15,2012, Delhi (Paper No. F622)
NUMERICAL SIMULATION OF SOIL-PILE SYSTEM SUBJECTED TO HORIZONTAL DYNAMIC LOADING
Debjit Bhowmik, Research Scholar, IIT Kharagpur, [email protected] D. K. Baidya, Professor, IIT Kharagpur, [email protected] S. P. Dasgupta, Professor, IIT Kharagpur, [email protected] ABSTRACT: The present study aims at investigating the nonlinear behavior of single hollow pile in layered soil and subjected to varying levels of horizontal dynamic load. A Finite Element Model has been developed using a commercially available FEM based software. Mohr-Coulomb plasticity is used to simulate the soil plasticity whereas the pile-material is idealized as elastic in nature in the model. Two types of motion: horizontal and rocking were estimated by this analysis. The effects of various influencing parameters like eccentric moment, and length of pile on the nonlinear dynamic response of piles are investigated. It is found that separation of pile from surrounding soil considerably affects the load carrying capacity of a pile. INTRODUCTION: Dynamic loads on pile foundation may come from different sources like seismic activity, operation of heavy machinery in factories, traffic movement in case of bridges and wave action in case of offshore structures. One of the primary objectives of pile foundation is to minimize the vibration amplitude to a permissible limit. During vibration it offers resistance through generation of stiffness and damping of pile-soil system due to the interaction between them. The study of soil-pile interaction is becoming more and more significant for more accurate and advance design of continuously evolving complex and heavy structures In the early development, the soil-pile system was idealized as a mass less equivalent cantilever and the theory of the sub-grade reaction was used for dynamic analysis of piles [1]. Subsequently lumped mass-spring-dashpot model was introduced to analyse pile foundations [2]. Later, a number of solutions have been developed for the dynamic analysis of pile foundation assuming that the behaviour of soil is linear elastic or viscoelastic in nature and the soil is perfectly bonded to the pile [3,4]. These approximate solutions are very useful in understanding the basic mechanism of dynamic pile-soil interaction. However, in reality both separation and slippage can occur due to the formation of weak bond at the contact between the soil and the pile. The finite element solutions are very powerful computationally efficient method to evaluate nonlinear dynamic soil-pile-structure system. Many researchers used a 3-D Finite Element model to obtain the pile response under dynamic loading considering the effects of material and interface nonlinearities on the dynamic behaviour of single and group piles [5,6]. The response of soil-pile system due to dynamic excitation is a complex phenomenon because of soil nonlinearity at high strain level and complex nature of pile-soil interaction involving both
slippage and separation between soil and pile. Very few researchers have predicted the length of separation between the pile and soil due to the vibration. Therefore there is a need for developing some guidelines for estimating the pile separation length under different modes of vibration with a good degree of accuracy. In the present investigation it is aimed to study the nonlinear dynamic behaviour of single piles by numerical investigation under coupled vibration. FINITE ELEMENT ANALYSIS: A finite element model using ABAQUS 6.10 has been developed to study the dynamic behaviour of a single pile driven in layered soil. The model was developed to simulate the experimental investigation carried out earlier [7]. Geometric Configuration of the model: The basic structure of the model is shown in the Figure 1 The soil mass surrounding the pile is assumed to be cylindrical in shape with radius equal to 10 times the diameter of the pile(1.0 m) and with depth 10 times the diameter of the pile below pile tip of the longest pile (3.0 m). After the main soil mass, a 0.25 m thick outer layer of infinite soil mass is used to create a boundary which will not reflect any seismic wave in the soil medium. Static load of 10 kN is used with same size and shape on top of the pile as used in experimental investigation. In the soil mass, elements are more closely spaced near the pile compared to the outer region. The pile and soil mass are discretized using 8 noded hexahedral elements. The outer periphery of the soil mass is modeled using single layer of infinite elements. Static over burden load and horizontal dynamic load have been applied to the centre of the oscillator. Material Modelling: The soil mass is idealized with elasto-plastic material property. Mohr-Coulomb plasticity model is used to model the soil plasticity. The pile is considered to be elastic. A calibration
Debjit Bhowmik, D. K. Baidya and S. P. Dasgupta
Fig. 9 Simulation of soil-pile system using ABAQUS 6.10
analysis has been performed on a single element of soil. Soil properties used in the model are closest simulation of the properties determined from different laboratory tests on soil sample collected from open pits during experimental investigation are shown in Table 1. Table 1 Soil properties determined from laboratory tests
Interaction between pile and soil: Interaction properties are defined with surface to surface interaction for top and middle layer, whereas for bottom layer node to surface interaction is considered. Tangential behavior will be governed by penalty interaction with angle of wall friction taken as 2/3 of angle of internal friction. Normal behavior is considered to be hard contact, allowing separation after contact. Cohesive behaviour with default value has been considered in case of bottom layer, as this layer predominantly consists of medium stiff clayey soil.
Model Test: Hollow close ended steel pile with 100 mm diameter and 5 mm wall thickness has been considered. Three different L/d ratios (10, 15 and 20) were chosen for this study. Sinusoidal dynamic force is applied to the centre of the oscillator. The excitation force amplitude is given as a Fourier series in Equation 1.
N
0 n 0 N 0n 1
a A [A cos n (t t ) B sin n (t t )]=
= + ω − + ω −∑ for 0t t≥
0 0a A for t t= < (1)
Where, ω = circular frequency; t = time; t0 = intial time = 0; N = 1; A0 = 0; A1 = 0 and B1 = 1. Analysis Procedure: Numerical model tests were carried out for 3 different L/d ratios of the pile with 4 exciting moments (Me) in each case. The analyses are done in three steps. First, gravity load is applied in negative Z direction only in soil mass. Then gravity load is applied to over burden mass and the pile. In the next step the pile-soil interaction has been introduced and dynamic external load is applied on a point 155 mm above the pile top to simulate the conditions of the field test. There are two modes of vibration in the coupled dynamic load analysis: a) Horizontal translation, b) Rocking motion. RESULTS AND DISCUSSION Frequency and Amplitude Analysis Typical frequency versus amplitude curves for both horizontal and rocking motion obtained from the present study are respectively shown in Figure 2 and 3 for L/d = 20 for different values of dynamic excitation intensities. Figures 2 & 3 show very similar characteristics those were found in experimental investigation [7].There are two prominent peaks in frequency amplitude response. The first peak is dominated by horizontal motion whereas the second peak is dominated by rocking motion. It is also seen from these figures that as the excitation moment (Me) increases the resonant amplitude increases but the resonant frequency decreases as was observed in the experimental results. Figures 4 & 5 show comparison of frequency amplitude response between numerical and experimental investigations. It can be seen from figures 4 and 5 that the first resonance frequencies (fn1 & Φn1) from the numerical model matched quite well with the results from experimental investigation. But the second resonance frequencies (fn2 & Φn2) are found to be larger from numerical model than that of experimental investigation. Resonance amplitudes (An & θn) matched quite well on both the cases. This may be the effect of pre-test localised separation occurred during pile driving or presence of void in soil-pile interface which in turn affects the stiffness of soil-pile system. Another possibility is that in field condition soil mass mobilized by the dynamic loading might be much greater in mass than that has
Layer Description Depth (m)
Unit weight (gm/cc)
C (Kg/cm2)
Φ (°)
I Brown medium organic sandy clay
0.0 to 0.3 1.771 0.46 23.4
II Soft yellow organic silty clay
0.3 to 1.5 1.804 0.62 20.1
III Brown medium stiff inorganic clay
Below 1.5 1.852 1.13 15.6
Numerical Simulation of Soil-Pile System Subjected to Horizontal Dynamic Loading
been idealized in numerical model and thus lesser value of second resonance frequencies have been recorded in field tests.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 5 10 15 20 25 30 35 40
Eccentric Moment = 0.125 N-mEccentric Moment = 0.248 N-mEccentric Moment = 0.366 N-mEccentric Moment = 0.477 N-m
Am
plitu
de(m
m)
Frequency (Hz) Fig. 2 Frequency vs Amplitude response for Horizontal vibration for L/d = 20
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
0 5 10 15 20 25 30 35 40
Eccentric Moment = 0.125 N-mEccentric Moment = 0.248 N-mEccentric Moment = 0.366 N-mEccentric Moment = 0.477 N-m
Frequency (Hz)
Am
plitu
de (R
ad)
Fig. 3 Frequency vs Amplitude response for Rocking Vibration for L/d = 20
0
0.05
0.1
0.15
0.2
0.25
0.3
0 10 20 30 40 50 60
Me = 0.125 N-m, Experimental
Me = 0.477 N-m, Experimental
Me = 0.125 N-m, Numerical
Me = 0.477 N-m, Numerical
Frequency (Hz)
Ampl
itude
(mm
)
Fig. 4 Comparison between experimental and numerical results for horizontal motion in case of L/d = 20 Separation Length: In numerical model horizontal displacements are recorded on points on outside surface of pile and inside surface of soil in interface region along the whole pile length. Maximum displacement on pile surface at different resonance frequencies
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0.0008
0.0009
0.001
0 10 20 30 40 50 60
Me = 0.125 N-m, Experimental
Me = 0.477 N-m, Experimental
Me = 0.125 N-m, Numerical
Me = 0.477 N-m, Numerical
Ampl
itude
(Rad
)
Frequency (Hz) Fig. 5 Comparison between experimental and numerical results for rocking motion in case of L/d = 20
Me = 0.125 at Time 6.96743Me = 0.125 at Time 6.98371Me = 0.248 at Time 6.98355Me = 0.248 at Time 6.9671Me = 0.366 at Time 6.98344Me = 0.366 at Time 6.96688Me = 0.477 at Time 6.96677Me = 0.477 at Time 6.98338
Displacement in Horizontal Direction (m)
Dep
th B
elow
Gro
und
Leve
l (m
)
Fig. 6 Maximum horizontal displacement on pile surface at Φn2 along the pile length for L/d ratio 20 for different exciting moments (Me) are recorded and plotted in Figure 6. It is evident from the figure that as exciting moment increases movement of pile increases.
The separation between pile and soil mainly takes place at the top of the pile. From the plot of horizontal movement of pile and that of soil at contact surface, separation at top region of pile is clearly visible. Figure 7 through 10 show a few such plots which indicates the variation of length of separation with dynamic load intensity at second resonance frequency for rocking mode of vibration. Non-dimensional separation length (ratio of separation length to pile diameter) versus non-dimensional amplitude (ratio of amplitude to pile diameter) are plotted in figures 11 and 12. for horizontal and rocking motion respectively. In case of horizontal motion at second resonance frequency the curve is much stiffer than that for first resonance frequency. It means that for second resonance frequency the depth of separation is much more dependent on maximum horizontal amplitude.
Debjit Bhowmik, D. K. Baidya and S. P. Dasgupta
-2.0
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0-0.8 -0.6 -0.4 -0.2 0.0 0.2
PileSoil
Horizontal Movement (mm)
Dep
thbe
low
G.L
. (m
) Length of Separation (0.78 m)
Me= 0.477 N‐m-2.0
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0-0.8 -0.6 -0.4 -0.2 0.0 0.2
PileSoil
Horizontal Movement (mm)
Dep
thbe
low
G.L
. (m
)
Length of Separation (0.93 m)
Me= 0.477 N‐m
Fig. 7 Separation at L/d 10 Fig. 8 Separation at L/d 15
-2.0
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
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-0.2
0.0-0.8 -0.6 -0.4 -0.2 0.0
Pile
Soil
Horizontal Movement (mm)
Dep
thbe
low
G.L
. (m
)
Length of Separation (0.96 m)
Me= 0.477 N‐m -2.0
-1.8
-1.6
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-1.0
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-0.4
-0.2
0.0-0.8 -0.6 -0.4 -0.2 0.0
Me = 0.248, Pile
Me = 0.248, Soil
Me = 0.366, Pile
Me = 0.366, Soil
Me = 0.477, Pile
Me = 0.477, Soil
Horizontal Movement (mm)
Dep
thbe
low
G.L
. (m
)
Fig. 9 Separation at L/d 20 Fig. 10 Separation for different moment at L/d 20
0
2
4
6
8
10
0 0.0005 0.001 0.0015 0.002 0.0025 0.003
First Resonating FrequencySecond Resonating Frequency
Maximum Horizontal Amplitude/ Pile Diameter (An/d)
Sepa
ratio
n Le
ngth
/Pi
le D
iam
eter
(S/
d)
Fig. 11 Separation Ratio vs. Non-dimensional Horizontal Amplitude for L/d 20 CONCLUSIONS: Using the results of numerical simulation the effects of different influencing parameters have been investigated. Some important conclusions that can be made from this study are summarized as follows: • Two resonant peaks are observed at two different frequencies for both horizontal and rocking component.
0
2
4
6
8
10
0 0.002 0.004 0.006 0.008 0.01
First Resonating FrequencySecond Resonating Frequency
Maximum Rocking Amplitude/ Pile Diameter (Φn/d)
Sepa
ratio
n Le
ngth
/Pi
le D
iam
eter
(S/
d)
Fig. 11 Separation Ratio vs. Non-dimensional Rocking
Amplitude for L/d 20
• For pile foundation subjected to coupled vibration, the first resonant peak is characterized by larger horizontal amplitudes and the second resonant peak by larger rocking amplitudes. • Separation between soil and pile greatly influence pile capacity. After separation pile starts to act as free head pile and subsequently looses its stiffness. • The effect of second resonant frequency on depth of separation is much greater than that of first resonant frequency. • As pile length increases, the depth of separation increases for a particular length of pile for a particular exciting moment. REFERENCES 1. Hayashi, S., Miyajima, N., and Yamashita, I. (1965),
“Lateral resistance of steel piles under static and dynamic loads.” Proceedings of the 3rd World Conference on Earthquake Engineering Vol. 2. pp. 146-167
2. Barkan, D. D. (1962), Dynamics of Bases and Foundations, McGraw-Hill Book Co. New York.
3. Novak, M. (1974), Dynamic Stiffness and Damping of Piles, Canadian Geotechnical Journal, Vol. 11, pp. 574-598.
4. El Naggar, M. H., and Novak, M. (1996), Nonlinear Analysis for Dynamics Lateral Pile Response, Soil Dynamics and Earthquake Engineering, Vol. 15, No. 4, pp. 233-244.
5. Lewis, K., and Gonzalez, L. (1985), “Finite Element Analysis of Laterally Loaded Drilled Piers in Clay”, Proc., 12th International Conference on Soil Mechanics and Foundation Engineering, Rio de Janiero, Vol. 2, pp. 1201-1204.
6. Maheshwari, B. K., Truman, K. Z., El Naggar, M. H., and Gould, P.L. (2005),”Three-Dimensional Nonlinear Seismic Analysis of Single Piles using Finite Element Model: Effects of Plasticity of Soil”, International Journal of Geomechanics, Vol. 5, No. 1, pp. 35-44.
7. Debjit Bhowmik, D. K. Baidya & S. P. Dasgupta (2011), "Coupled Motion of Soil-Pile System Under Dynamic Loading", Indian Geotechnical Conference, Kochi.
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F 623.)
PROBABILISTIC ANALYSIS OF SHEAR BEHAVIOUR OF FIBER REINFORCED RED SOIL
K.Geetha Manjari1, Research Student, Indian Institute of Science, Bangalore-500612, India, [email protected] G.L.Sivakumar Babu 2, Professor, Indian Institute of Science Bangalore-500612, India, [email protected] Sandeep Kumar Chouksey3, Research Scholar, Indian Institute of Science, Bangalore-500612, India, [email protected] ABSTRACT: Fiber reinforced soil is one of the efficient methods of improving the strength and stability of many engineering systems. This paper presents the stress-strain response of red soil reinforced with coir fiber. A series of consolidated undrained triaxial tests were performed on red soil with different percentages of randomly distributed coir fiber. Soil and fiber characteristics, their interaction are some of the major factors affecting the strength of reinforced soil. One of the important properties is the tensile resistance offered by fibers during the shearing of soil. To observe this effect, reliability analysis is carried out on the tensile resistance of fibers along the shear plane in a triaxial sample and then the increase in the shear strength of the reinforced soil as a function of tensile resistance of fibers is studied. INTRODUCTION The concept of reinforcement was developed in the late 19th century by observing the improvement in stability of soil due to the plant roots. The plant roots act as a natural source of reinforcement that takes the tensile stresses acting on soil and improve its stability. Thus soils were reinforced with different fibers and the effect of reinforcement was studied. The other reinforcing materials include natural fibers (sisal, coir, jute etc), artificial fibers (glass, steel, polypropylene etc), geosynthetics, These materials are distributed in different layers in soil to obtain the desired improvement in the strength. To study the effect of reinforcement, probabilistic analysis is carried out on the failure mechanisms in root reinforcement based on models proposed in the previous studies on root reinforcement. LITERATURE REVIEW Attempts to quantify root reinforcement of soil have been studied by the use of simple perpendicular root models developed by researchers [2, 4], which simply requires the knowledge of the tensile strength of the roots, and the cross-sectional area of fibers crossing the shear plane. So the increase in the shear strength of fiber reinforced soil due to the tensile strength offered by fibers can also be analyzed. Experimental investigations were carried out and reports showed that fiber reinforcement leads to significant improvement in strength and stiffness of soil [2, 3]. The stress strain behavior of fiber reinforced soil shows small loss of post-peak strength (i.e., greater ductility in the composite material) as compared to plain soil. The increase in the strength is a function of soil characteristics, e.g., particle size, shape, and gradation; fiber characteristics [5, 6]. A series of triaxial compression tests on soil reinforced with discrete, randomly distributed fiber influence of various properties on shear strength of reinforced soil were analyzed and presented as a mathematical model based on regression analysis of test results [7]. They reported that the strength of reinforced sand increases with increase in fiber content, aspect ratio, and soil fiber surface friction. In order to study the behavior of root reinforcement and its effect in improving stability, experimental investigations of the soil with fibers has to be
studied. From the past work on root reinforcement and experimental studies, the in-situ root reinforcement condition can be represented in the triaxial setup with randomly distributed fibers. Numerical investigations were also carried out on coir fiber reinforced sand and it reported that, presence of random reinforcing material in soils make the stress concentration more diffuse and restricts the shear band formation [8]. In the present work, a series of triaxial tests were performed on soil mixed with coir fibers. The failure mechanism of fibers along the shear plane is studied and a probabilistic analysis is carried out on the shear strength and tensile resistance of fibers in the fiber reinforced soil. PROPERTIES OF SOIL Weathered soil from the sedimentary rocks form red soil. This is the main type of soil available in Bangalore and surrounding areas. This type of soil is also available in a large region of our country. In this region, at present, a large number of structures such as embankments, highways, retaining walls etc are constructed/being constructed with this soil. Table 1: Properties of soil used in the present study
PROPERTIES OF FIBER Coir fiber is used as a reinforcing material. The main advantages of natural materials are that they are locally available, cost effective, biodegradable and hence do not create environmental problems. Hence these materials are finding increasing applications in slope stabilization and other ground improvement projects. Table2: Properties of single coir fiber used in the present study Properties Values Properties Values
Length(mm) 15 Liquid limit (%) 39
Diameter(mm) 0.25 Plastic limit (%) 26
Specific gravity 1.12 Shrinkage limit (%) 20
Tensile strength (MPa) 102 Specific gravity (G) 2.65
CONSOLIDATED UNDRAINED TRIAXIAL TEST In order to examine the performance of fiber mixed soil in undrained condition, consolidated undrained (CU) tests have been carried in triaxial apparatus. The fiber mixed soil; samples were prepared at specified percentages of fiber (0, 0.50%, 1.0% and 2.0% by dry weight of soil). The samples were then isotropically consolidated under different confining pressures of 50, 100 and 150 kPa. The samples were finally subjected to shear under undrained condition. The axial deformation was obtained with a dial gauge and pore water pressures are measured. Deviator load was applied till the specimen failed or till a strain of 15% was reached. Pore water pressures were measured during shearing of the tests. To ensure uniform pore pressures throughout the specimen, samples were sheared at a constant strain rate (approximately 0.24% per minute). Typical result in the form of stress-strain-pore water pressure obtained from the experiments for the non-reinforced and fiber reinforced soil at confining pressure of 50 kPa for coir fiber with different percentage are presented. Deviator stress vs. strain (%) response and pore pressure response for red soil reinforced with coir and glass fibers, sand mixed with plastic waste from CU tests were performed at various confining pressures. It is clear from these results that deviator stress increases as the plastic waste content increases. Further it is observed from these results that as the strain increases the deviator stress also increases. In most cases maximum deviator stress occurred at about 6 % to 8% of strain. The results show that the stress-strain behaviour is considerably improved by incorporating fibers in soil. The increase in strength is due to the confinement, which results in the increase of cohesion and friction of soil. The pore pressure curves were plotted by plotting the excess pore pressure for the various strain levels as obtained from the CU Tests. Fig1 shows the pore pressure responses for the various confining pressures.
-200
-100
0
100
200
300
400
500
0 3 6 9 12 15Strain (%)
q (k
Pa)
Plain soil0.5% fiber1.0% fiber2.0% fiber
u (k
Pa)
Fig1: Stress-strain and pore water pressure response for
various percentages coir fiber (at confining pressure 50 kPa) THEORETICAL DEVELOPMENT OF FIBER REINFORCED SOIL The behavior of fiber reinforced soil was investigated experimentally. But theoretically the behavior and mechanism of fiber in the composite is limited in the previous studies. Waldron [2] proposed a model to describe the load-deformation characteristics of soils reinforced with plant roots. He used the original Mohr-Coulomb's equation of shear strength (s = c + σ tanφ )in a modified form, for root-permeated soil as
(1) where sr = shear strength of root-permeated soil; ΔS = increase in shear strength on account of root reinforcement.
Fig2. Model of fiber reinforced sand. (a)Perpendicular fibers (b)Inclined fibers (Gray and Ohashi1983)
Probabbilistic analysis of shear behavior of fiber rweinforced red soil
The concept of root-reinforcement of soil was used by Gray and Ohashi [3] to describe the deformation and failure mechanism of fiber-reinforced soil and to find the increase in shear strength ΔS for oriented fibers crossing a shear plane. The shear strength increase from oriented fiber-reinforcement in sand was estimated by the expressions Perpendicular fibers:
(2) Inclined fibers:
(3) where �
θ - angle of shear distortion σt -tensile stress in each fiber z- width of shear zone i- initial orientation of fiber with the shear surface The model proposed by Maher and Gray [5] predicts the orientation and the quantity of fibers at any arbitrary chosen plane. The orientation of the fibers, on average was expected to be perpendicular to the plane of shear failure in triaxial compression tests. The orientation of fibers along any plane can be predicted by the statistical theory of composite [1]. The failure plane was observed to be the same as given by Mohr-Coulomb failure criteria,, at an angle of (45+φ/2) with horizontal. The average number of fibers, Nf per unit area crossing the shear plane is given by
(4) Vf-volume of fiber in the specimen/volume of soil d- diameter of fiber The tensile stress, σt, developed in fiber is given by [5]:
(5) where l- length of the fiber d- diameter of the fiber τ- skin frictional resistance τ=σconfining stress*tanδ and δ is the angle of skin friction Thus the increase in shear strength due to fiber reinforcement by force equilibrium method is given as:
(6a) for 0 < σconf < σcrit'
(6b) for σconf > σcrit'
ξ is an empirical coefficient that depends on soil parameters like the grain size, gradation etc. Based on the above equations and the experimental results an analysis is carried out on the fiber resistance in the triaxial sample. The properties of soil and fibers given in tables 1 and table 2 were substituted in equations for the tensile resistance of the composite at different confining stresses and presented in the figure below:
Fig3.Tensile strength of fibers along the shear plane In the above figure we can observe that for a fixed number of fibers per unit area along the shear plane we can observe that as the confining stress was increased, the tensile strength also increased. According to the equation (1) the increase in the shear strength due to fiber reinforcement is given by ΔS. Now from the experimental results of red soil reinforced with coir fiber, the increase in the strength is calculated. In the present study the coir fiber the dimensions specified above is considered. The number of fibers per unit area along the shear plane given by Maher and Gray [5] is calculated for the presented and tabulated below: Table3: Number of fibers along shear plane for different percentages of coir fiber
Fiber percentage(%) Number of fibers per unit area(cm2)along shear plane
0.5 9 1 18 2 36
The increase in the shear strength in equation (6a) is used to form the limit state function and it is given by the equation: R-L<0
Where R is the shear resistance offered by fibers along shear plane and S is the applied stress on the specimen. Reliability analysis was carried out for the shear strength of the composite and the fibers on an average were perpendicular to the shear plane. The applied stress was
assumed to follow lognormal distribution of mean 50kPa and standard deviation of 0.05. and the diameter to follow a lognormal distribution with mean 0.25 and standard deviation 0.125.The probability of failure for different percentages of fiber content(as a function of number of fibers) was calculated. The probability of failure for different number of fibers per unit area is shown in the Fig3.
Fig4. Probability of failure for different percntages of coir
fiber and under same loading conditions The failure mechanism of fibers in the reinforced soil is thereinforced soil is observed by finding the probability of failure of each fiber out of N number in the composite. After certain loading the fibers start to fail but the rest of the fibers take the load and distribute it among them. So if r out of N fibers that contribute to the strength then, the fibers their failure probability is calculated and shown in the Fig4.
Fig5. Probability of failure of r out of N fibers contributing to
the tensile resistance Thus from the above figure the pattern of failure along the shear plane was obtained. The behaviour of fibers along the shear plane can be observed for a stress condition. This analysis is can predict the shear behaviour of the composite
as a function of tensile resistance of soil. This analysis is helpful in predicting the contribution of each fiber along the failure plane of the fiber reinforced soil. CONCLUSIONS
1. The tensile strength of fibers along the shear plane for different confining stresses is obtained and as the confining stresses increase the tensile resistance of the fiber increased.
2. From the reliability analysis, it can be observed that as the percentage of fibers in the specimen increased, the probability of failure of specimen under the same stress condition is reduced and thus the reliability of the system increased.
3. The mechanism of failure of fibers along the shear plane is studied and as the fibers donot fail together under a given loading condition, the failure pattern is progressive. So as a fiber reaches the failure state, there is a stress redistribution among the rest of the fibers.
REFERENCES 1. Narnman, T., Moavenzadh, E, and McGarry, E (1974).
"Probabilistic analysis of fiber reinforced concrete." J. Engrg. Mech. Div., ASCE, 100(2),397-413.
2. Waldron, L. J. (1977). "Shear resistance of root permeated homogeneous and stratified soil." Soil Sci. Soc. of Am., Proc., 41, 843-49.
3. Donald H. Gray,A. M. ASCE and Harukazu Ohashi.(1983) “Mechanics of fiber reinforced sand.” J. Geotech. Engrg., ASCE.109:335-353.
4. Wu, T. H., R. M. McOmber, R. T. Erb, and P. E. Beal (1988), Study of soil-root interaction, J. Geotech. Eng., 114, 1351– 1375.
5. Maher, M. H., and Gray, D. H. (1990). "Static response of sand reinforced with randomly distributed fibers." J. Geotech. Engrg., ASCE, 116(11),1661-1677.
6. AI-Refeai, T. (1991). "Behaviour of granular soils reinforced with discrete randomly oriented inclusions." J. Geotextiles and Geomembranes.10, 319-333.
7. Gopal Ranjan, R. M. Vasan and H. D. Charan.(1996). “Probabilistic analysis of fiber reinforced soil”. J. Geotech. Engg., ASCE,122:419-426.
8. G.L. Sivakumar Babu_, A.K. Vasudevan, Sumanta Haldar (2008). “Numerical simulation of fiber reinforced sand behavior”. Geotextiles and Geomembranes 26, 181–188
Proceedings of Indian Geotechnical Conference December 13-15,2012, Delhi (Paper No. F-625)
2D FINITE ELEMENT SEISMIC ANALYSIS AN EARTHEN ROCKFILL DAM Prateek Khare, M.Tech. Student, Dept. of Earthquake Engg., IIT Roorkee, [email protected] B.K. Maheshwari, Assoc. Professor, Dept. of Earthquake Engg., IIT Roorkee, [email protected] ABSTRACT: This paper presents the effect of boundary conditions on response of an earthen rock fill dam. The rock fill dams are highly resistant to seismic loads due to their large flexibility and capacity to absorb large seismic energy therefore less vulnerable to earthquake damage. The behavior of an earthen rock fill dam under earthquake loading with horizontal rollers and dashpots along the foundation soil system was studied. The seismic analysis of the dam consists of static and dynamic 2D finite element analysis. First, the in-situ stress state conditions that exist before the earthquake occurs are established, and then its results are used for the dynamic part of analysis. Plots of the variations of the displacements at the core with dam height are shown. Performance of the dam was satisfactory in terms of the computed maximum settlements at dam crest, which were considerably smaller than the permissible values. INTRODUCTION Dams are important for a nation in terms of irrigation water they store and provide. The failure of dams can cause destruction of nearby life and property [1]. Rock fill dams have large flexibility and ability to absorb large seismic energy, which makes them resistant to seismic loads. These dams are generally confused with earthen dams, which are more vulnerable to get damaged by seismic forces [2]. Performance and safety of dams during earthquakes are of global concern, and to check the dam performance and stability, dynamic analyses of dams are required. The main aim of dynamic analysis of dam is to determine the acceleration, dynamic stresses and deformations induced in the dam by the seismic excitation. The Failure of earth dams due to earthquakes was studied by Sherard [3] and many reasons for dam failures were given by them. Significant contributions have been made by many researchers in the past towards understanding the seismic behavior of earth and rock fill dams starting by Newmark [4]. Newmark [4] and Seed [5] were first to propose methods of analysis for predicting the permanent displacements of dams subjected to earthquake shaking. With the advancements in the modern computers, finite elements and finite difference methods have been increasingly used with more advanced material models for estimating behavior of the dams [6, 7 and 8]. This paper presents the effect of two boundary conditions on response of earthen Rock fill dam. Horizontal rollers along vertical boundaries of the foundation soil and dashpot representing Lysmer-Kuhlemeyer boundary along vertical boundaries of the foundation soil are considered here to estimate the response of the dam. SECTION OF THE DAM A 36.0 m high rock fill dam is chosen for the present study. For modeling, the dam is divided into three sections, upstream & downstream shell which consists of pervious material, and impervious core and foundation soil over rigid bedrock. The width of the dam at the crest is 9.2 m. The dam
has an upstream slope of 2H: 1V and downstream slope of the dam varies from 2H: 1V to 2.5H: 1V. Berms are provided at downstream slope at 10m interval from the dam top. The central impervious core height is 33m and top width of 6.0 m having upstream and downstream slopes of 0.5H: 1V.The reservoir depth is 33m. The material properties for the shell, core and the overburden material used in the FE analysis is listed in Table 1. Table 1 Material properties used in FE analysis
Core Shell Foundation Soil
c (kPa) 90 100 50 Φ (degree) 20 42 35
Shear wave Vel. (m/s) 200 450 300
Poisson’s ratio 0.40 0.30 0.36 Mass density
(g/cc) 1.71 1.95 1.8
Young’s Modulus (MPa) 191.52 1026.67 440.64
Shear Modulus (MPa) 68.40 394.87 162.00
BOUNDARY CONDITIONS The 2D model of the dam section is shown in figure 1. The CPE4R element, a four node plane strain reduced integration element, one integration point per element, was used to reduce the computer run times. A total number of 1943 nodes and 1786 elements have been used for modeling of the dam. In the analysis, two cases have been taken. In one case horizontal roller and in other case, dashpots representing Lysmer-Kuhlemeyer boundary are used along the vertical boundaries of the foundation soil. The bottom of the model is fixed in both x and y directions. The model is analyzed for end of construction conditions and full reservoir impounded conditions. The soil in the core, shell and overburden region is assign as a linear-elastic model.
Proceedings of Indian Geotechnical Conference December 13-15,2012, Delhi (Paper No. F-625)
Fig. 1 Cross section of dam
For a 2D FE model to represents response of infinite field conditions, reflections of the seismic waves from the side boundaries have to be minimized. For this reason, the Lysmer-Kuhlemeyer boundary [9] was used along the soil boundaries to restrict the wave reflections. This boundary absorbs the vertically propagating waves in such a way that the incident wave is transmitted entirely into the soil and no waves are transmitted back. Lysmer boundary consists of simply connecting dashpots to all degrees of freedom of the boundary nodes and the other end remained fixed, as shown in Fig. 3.
(a) Horizontal Rollers (b) Dashpot
Fig. 2 Vertical Boundary Conditions Lysmer boundaries can be derived for an elastic wave propagation problem in a one dimensional semi infinite bar [10], the damping coefficient Cx can be expressed as
xC A cρ= (1)
Where A is the cross-section of the bar, ρ is the mass density and c the wave velocity. The wave velocity is taken either as shear wave velocity (cs) or compressional wave velocity (cp) depending upon the type of wave absorbed. For the two dimensional problem, this results in two damping coefficients, Cn and Ct, in normal and tangential directions, respectively.
n pC A cρ= (2)
t sC A cρ= (3)
Where
sGcρ
= (4)
( )( )( )
11 1 2p
Ec
υυ υ ρ
−=
+ − (5)
Where G is the shear modulus of the medium and is given by,
( )2 1EGυ
=+
(6)
Where E is the Young’s modulus and υ is the Poisson’s ratio. These boundary conditions are independent of frequency and are local in time and space. Use of shape functions of the neighboring finite elements instead of crude limping procedure gives rise to a narrow banded damping matrix, which is easy to implement.
ANALYSIS OF THE DAM A 2D-plane strain static and seismic finite element response analysis is carried out for the section of the dam to study the stability of dam. The 2D linear seismic analysis of the dam for Northridge (1994) acceleration time history with PGA of 0.24 g and predominant frequency 1.22 Hz has been carried out and the maximum displacement and acceleration at dam crest have been worked out.
Fig. 3 Northridge (1994) Acceleration-Time History (a) Horizontal Component (b) Vertical Component Analysis of the dam section is carried out for static loads due to self-weight, hydrostatic pressure and seismic loads to determine the deformation of the core. Stress distribution within the dam section was also analyzed. The displacement of the dam was of prime concern as excessive deformations may lead to loss of freeboard and danger of over topping of reservoir [11].
2D Finite Element Analysis of an Earthen Rockfill Dam
End of Construction Condition The dam is analyzed for static load due to self weight and seismic loads due to earthquake. Fig. 4 shows the variation of displacments at upstream and downstream face of core at end of construction stage.
Fig. 4 Variation of displacement at core at end of construction condition
The maximum horizontal displacment at upstream face was 4.95 and 4.91 cm, respectively without and with damper.The maximum vertical displacement at upstream face was 18.04 cm and 18.01 cm, respectively without and with damper. The maximum horizontal displacment at downstream face was 7.03 and 6.9 cm, respectively without and with damper and the maximum vertical displacment at upstream face was
18.18 cm and 18.12 cm, respectively without and with damper.
Reservoir Impounding The dam is analyzed for static load due to self weight, hydrostatic pressure due to reservoir impounding and seismic loads due to earthquake. Fig. 5 shows the variation of displacments at upstream and downstream face of core after the reservoir is impounded.
Fig. 5 Variation of displacement at core after reservoir impounding
The maximum horizontal displacment at upstream face was 5.66 and 5.62 cm, respectively without and with damper and the maximum vertical displacment at upstream face was 18.25 cm and 18.21 cm, respectively without and with
Prateek Khare, B.K.Maheshwari
damper. The maximum horizontal displacment at downstream face was 6.5 and 6.37 cm, respectively without and with damper and maximum vertical displacment at upstream face was 18.23 cm and 18.17 cm, respectively without and with damper.
Fig. 6 Acceleration at dam top without dampers The Fig. 6 shows the horizontal and vertical accelerations at dam top without dampers. The maximum values of the accelerations obtained were 0.205 g and 0.161 g in the horizontal and vertical directions. The responses are reduced at higher frequencies
Fig. 7 Acceleration at dam top with dampers The Fig. 7 shows the horizontal and vertical accelerations at the dam top with dampers.The maximum values of the accelerations obtained were 0.201 g and 0.159 g in the horizontal and vertical directions. CONCLUSIONS The maximum deformations occur near the top of the dam and were less than permissible values. The results obtained from the study shows that with the use of Lysmer-Kuhlemeyer boundary, the responses obtained were marginally less than that obtained for the case with horizontal rollers. The analysis predicts that if the width of foundation
soil modeled is thrice or more, then responses are not changed significantly with the use of dampers. The analysis for the dam can further be extended for non linear soil models. REFERENCES [1] Basudhar, P.K., Kameswara Rao, N.S.V., Bhookya, M.,
Dey, A. (2010), 2D FEM Analysis of Earth And Rockfill Dams Under Seismic Condition, Fifth Intl. Conf. on Recent Adv. in Geotech. Earthquake Engg. And Soil Dyn. Symposium in honor of Prof. I.M. Idriss, San Diego, California, Paper No. 4.28b 1.
[2] Paul, D.K. (2000), Seismic Safety Analysis of a High Rock-Fill Dam Subjected to Severe Earthquake Motion, 1164, 12 WCEE, Auckland, New Zealand.
[3] Sherard, J.L., Woodward, R.J., Gizienski, S.J. and Clevenger, W.A. (1963), Earth and Earth-Rock Dams, John-Wiley and Sons, New York.
[4] Newmark, N.M. (1965), Effects of earthquakes on dams and embankments, Geotechnique, 15(2), 139–159.
[5] Seed, H.B. (1966), A method for earthquake-resistant design of earth dams, J. Soil Mech. Found. Div., ASCE, 92(1), 13-41.
[6] Sengupta, A. (2010), Estimation of permanent displacements of the Tehri dam in the Himalayas due to future strong earthquakes, Sadhana Vol. 35, Part 3, June 2010, pp. 373–392. , Indian Academy of Sciences
[7] Vrymoed, J. (1981), Dynamic FEM model of Oroville dam, J. of Geotech. Eng. Div. ASCE, 107(8), 1057–1077.
[8] Zienkiewicz, O.C., Leung, K.H., Hinton, E. (1980), Earth dam analysis for earthquakes: Numerical solutions and constitutive relations for nonlinear (damage) analysis, Design of Dams to Resist Earthquake, ICE, London, 141–156.
[9] Lysmer, J. and Kuhlemeyer, R. L., (1969), Finite dynamic model for infinite media, J. of Engg. Mechanics Div., ASCE, 95 (EM4), pp. 859-877.
[10] Burman, A., Maity, D., Sreedeep, S., (2010), Iterative analysis of concrete gravity dam-nonlinear foundation interaction, International J. of Engg., Science and Tech., Vol. 86 2, No. 4, pp. 85-99.
[11] IS: 8826 – 1978, “Indian Standard Guidelines for Design of Large Earth and Rockfill Dams,” BIS, New Delhi.
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F626)
COMPUTATIONAL STUDY ON THE EFFECTS OF TANNERY WASTES ON HIGHWAY FLY ASH EMBANKMENT
K. Bandyopadhyay, Reader, Dept. of Construction Engineering, Jadavpur University, [email protected] S. Bhattacharjee, Research Scholar, Dept. of Construction Engineering, Jadavpur University, [email protected] S. Ghosh, Student, Dept. of Construction Engineering, Jadavpur University, [email protected] ABSTRACT: Flyash is generated in large quantities due to the domination of thermal power plants in power generation sector in India. Reuse of flyash is an alternative to mere disposal and construction of highway embankments is one such area where it is being efficiently used. Cement and Lime stabilization give additional strength to flyash as required for being used as a subgrade material. In many instances embankments are constructed next to waste channels and seepage of waste water containing large number of heavy metals having low pH is a common phenomenon. Numerical study on flyash embankment structures constructed on soft soil and exposed to tannery waste water is investigated in the paper. INTRODUCTION The share of thermal power plants in the power generation sector in India is around 70 percent and the largest. This generates large volumes of waste material (Flyash). Flyash is commonly used as a highway material in embankments and approaches. Embankment constructions on soft soils like clay with high groundwater level are extremely challenging and often require prior analysis. A numerical study on the construction of a flyash highway embankment on soft soil was investigated by the authors[1]. Effect of density of flyash along with cement and lime stabilization on stress and displacement characteristics were analyzed using the numerical model. Results concluded that embankment constructed with cement stabilized flyash performs better in terms of displacements and stresses generated. Waste water from tannery contains a large amount of heavy metals and has low pH. In many instances embankments are constructed next to waste channels and thus seepage of waste water is a common phenomenon. This may have an adverse effect on the overall strength of embankments constructed next to waste channels carrying tannery waste water. The changes in physical properties of flyash exposed to industrial wastewater was studied[2]. Investigation on grain size distribution revealed that clay fraction decreased from the original flyash to the exposed sample. Conversely there was a slight increase in the silt fraction in the exposed one. The percentage of fine sand in both original and exposed flyash were observed to be the same. Experiments also revealed that shear strength parameters (C, ϕ ) decreased with increase in contaminants and coefficient of permeability increased with increasing amounts of contaminants. Close microscopic examination of shape and surface characteristics of flyash and flyash exposed to industrial tannery waste water reveal that particles in unexposed samples have a well-defined boundary and appear to be generally discrete particles, the exposed samples have relatively less sharp outline and are more agglomerated in their appearance. This difference is believed to be due to deposition of chromium onto the surface of flyash particles through absorption.
The effect of tannery waste water on the behaviour of embankments made with various compacted densities of flyash with and without lime and cement stabilization on clay is investigated in the present work. Staged construction of the embankment has been effectively modelled followed by the application of overburden pressure on the structure. The parameters required for modelling of flyash has been determined in the laboratory using a prototype embankment. FINITE ELEMENT MODEL An embankment of 9 metre crest width with 2:1 side slopes has been chosen for this study. The height of the embankment is 4 metre. The ground water table is assumed at a depth of 2 metres below the ground level. An overburden pressure of 30 kN/m2 is applied on the structure. The embankment is constructed on soft clay in two lifts. Height of each lift is 2 metres and construction time for each lift is 5 days. Construction of each lift is followed by a consolidation period of 100 days during which the excess pore water pressure is assumed to dissipate. Time required for the application of overburden pressure is taken as 1 day as taken in the model. The finite element model has been created and analysed using PLAXIS 8.2 Professional geotechnical analysis software. Due to the symmetry of the problem, only one half needs to be modelled. Fifteen Noded plain strain elements have been used for discretizing both the embankment as well as the foundation material. The model discretization is shown in Fig 1.
Fig 1 Discretized model of the embankment along with the soft soil layer The deformations at the boundary of the soft soil layer is assumed to be zero. Hence the base is fixed in x and y
K. Bandyopadhyay, S. Bhattacharjee, S. Ghosh
directions. The two vertical boundaries are assumed to be fixed in x direction. The initial conditions include the existence of the phreatic level at a depth of 2 metres below the ground level. It is assumed that water can flow out from all boundaries and excess pore water pressures can dissipate in all directions. However according to the present geometry model, the left vertical boundary must be closed as it is a line of symmetry and not a true boundary. MATERIAL MODELS The embankment material consisting of different compositions of flyash has been modelled using the Mohr- Coulomb soil model. The foundation soil comprising of soft clay has been modelled using the Soft Soil Creep model. Mohr-Coulomb model Mohr-Coulomb (MC) soil model assumes perfect plasticity of material[3]. Plasticity is associated with the development of irreversible strains. The existence of plasticity can be evaluated by introducing a yield function, f, as a function of stress and strain. This yield function can be presented as a surface in the principal stress space. MC model assumes that the yield surface is fully defined by the model parameters and remain unaffected by plastic straining[3]. MC model involves five input parameters. Elastic parameters being modulus of elasticity E and Poisson’s ratio ν. Plastic parameters being cohesion c, friction angle ϕ and angle of dilatancy ψ. This model represents a ‘first order’ approximation of material behaviour. For each layer, a constant average stiffness is assumed, resulting in relatively fast computations. Beside the five model parameters, initial material conditions also play a significant role in most deformation problems. Hence proper modelling of initial conditions need to be carried out [4]. Soft Soil Creep Model The soft soil creep (SSC) model is suited for simulating soil behaviour, taking into secondary effects such as creep. Most soft soils like soft clays exhibit some amount of secondary compression. Thus SSC model is ideal for analysis of settlement problems of foundations and embankments[4]. Some basic characteristics of this model are:
• Distinction between primary loading and unloading-reloading
• Secondary (time dependent) compression • Failure behaviour according to Mohr-Coulomb
criterion SSC model involves six input parameters. Cohesion c, Friction angle ϕ, Dilatancy angle ψ as Mohr-Coulomb parameters and modified swelling index k*, modified compression index λ* modified creep index μ* as stiffness parameters; which are related to e=void ratio and Cc, Cr, Cα are compression index, swelling index and secondary compression index respectively as:
(1)
(2) (3)
Table 1 description, identification and OMC for 12 samples of stabilized and unstabilized flyash MATERIAL PARAMETERS The embankment material used in the study consisted of compacted unstabilized flyash densities and compacted flyash+lime and flyash+cement stabilized densities all exposed to tannery waste water. The detailed description of the samples is shown in Table 1. These samples were prepared in the laboratory and compacted in a prototype
Name Identification OMC 100%compactionof 1.157g/cc +Tannery waste at insitu conc.
1-A 32.00
97% compaction of 1.157g/cc +Tannery waste at insitu conc.
1-B 32.00
100% compaction of 1.145g/cc +Tannery waste at insitu conc.
2-A 36.00
97% compaction of 1.145g/cc +Tannery waste at insitu conc.
2-B 36.00
100% compaction of 1.012 g/cc +Tannery waste at insitu conc.
3-A 42.00
97% compaction of 1.012 g/cc +Tannery waste at insitu conc.
3-B 42.00
100% compaction of 0.975 g/cc +Tannery waste at insitu conc.
4-A 45.00
97% compaction of 0.975 g/cc +Tannery waste at insitu conc.
4-B 45.00
100% compaction of 1.016 g/cc with 4%lime +Tannery waste at insitu conc.
L-A 37.50
97% compaction of 1.016 g/cc with 4%lime +Tannery waste at insitu conc.
L-B 37.50
100% compaction of 1.111g/cc with 7%Cement +Tannery waste at insitu conc.
C-A 29.10
97% compaction of 1.111g/cc with 7%Cement +Tannery waste at insitu conc.
C-B 29.10
Computational Study on the Effects of Tannery Wastes on Highway Flyash Embankment
laboratory model of the embankment. Core samples were collected for determination of cohesion, friction angle, Poisson’s ratio, modulus of elasticity and permeability values required for the study. Cohesion and friction angle were determined using direct shear tests. Permeability values were determined by Falling Head test. Split tensile test[5] on the samples were conducted for determination of Poisson’s ratio. The parameters required for modelling the foundation soil was taken from available literature and is shown in Table 2[4]. Table 2 modelling parameters for foundation material Name : clay μ* = 0.002 Model: soft soil creep γunsat=15kN/m3 C = 20kN/m2 γsat = 18kN/m3 ϕ = 0.10 Kh = 1.2x10-4(m/day) ψ = 00 Kv = 1x10-4(m/day) λ* = 0.035 k* = 0.007 RESULTS AND DISCUSSION Extreme displacements
Fig 3: Horizontal displacement gradients for sample C-A
Fig 4: Vertical displacement gradients for sample C-A Table 3 shows the extreme total displacements, extreme horizontal displacements and extreme vertical displacements at the end of the calculation phase and is compared with respect to compaction densities. Vertical displacements were more than horizontal displacements in all cases. The maximum extreme total displacement for unstabilized sample was obtained at 97% compaction with γd,max=1.145 g/cc, the minimum was obtained at 97% compaction with γd,max=0.975 g/cc. Maximum percent increase in extreme total displacement with respect to the minimum extreme total displacement was 34.44%.The minimum extreme total displacement was achieved with cement stabilized sample with 100% compaction effort. The reduction in extreme total displacement for cases of lime and cement stabilization with respect to the maximum extreme total displacement was 39.15% and 50.72% respectively. However the maximum achievable reduction in field condition (97% compaction) with respect to maximum extreme total displacement was 37.84% and 47.09%, for lime and cement stabilized samples respectively. Extreme horizontal displacements are seen to increase with the decrease in dry density for unstabilized samples. Maximum reduction of 80.24% and 83.17% were noted for lime stabilized with 100% compactive effort and cement stabilized with 97% compactive effort samples respectively. Extreme vertical displacements was seen to follow a similar trend as that of extreme total displacements. The maximum reduction for unstabilized sample was 32.24%. For lime and cement stabilization, the maximum reduction was 39.08% and 50.67% respectively. However the maximum achievable reduction in field condition was 37.77% and 47.03% for lime and cement stabilized samples respectively.
Table 4 % change in extreme displacements for 12 samples with respect to unexposed flyash
Extreme displacements for tannery waste exposed samples were compared with those for unexposed samples published in [1]and have been expressed as percentage change, shown in Table 4. In all twelve cases of extreme total displacements and extreme horizontal displacements, exposed samples undergo greater displacements than unexposed samples. It was observed that percent change in total displacements on exposure to tannery waste water for unstabilized flyash was nominal. But it was significant in lime stabilized sample and the most significant in cement stabilized sample. Thus from Table 3 it is inferred that extreme total displacements are minimum for cement stabilized flyash even after exposure to tannery wastes. From Table 4, the percent increase in extreme total displacement of exposed samples compared to unexposed samples [1] is maximum for cement stabilized flyash even though the magnitude of displacement is least. Extreme stresses Table 5 shows the variations of extreme effective stresses, extreme total stress and extreme excess pore water pressure at the end of the calculation phase. Maximum extreme effective stress for unstabilized sample was seen for 100% compaction of γd,max=1.145 g/cc. Minimum effective stress was observed for 97% compaction of γd,max=0.975 g/cc (15.46% reduction with respect to maximum). For lime and cement stabilized samples, extreme effective stress further increased. Maximum extreme total stress for unstabilized sample was obtained for 100% compaction of γd, max=1.145 g/cc. However negligible change in total stress was seen for lime and cement stabilized samples. The maximum extreme excess pore water pressure in the case of unstabilized samples was observed at 100% compaction with γd, max=1.145 g/cc. The minimum achievable extreme excess pore water pressure was found to be with lime stabilized sample (100% compaction). However the minimum achievable extreme excess pore water pressure in field condition (97% compaction) was higher.
Table 5 Extreme stresses for 12 samples Identific
ation Ext.
effective stress
(kN/m2)
Ext. total stress
(kN/m2)
Ext. excess pore water
press. (kN/m2)
x10-3 1-A -139.56 -179.16 -133 1-B -136.54 -176.14 -128.75 2-A -140.14 -179.74 -137.56 2-B -137.39 -176.99 -133.56 3-A -133.90 -173.50 -126.15 3-B -131.62 -171.22 -122.46 4-A -132.20 -171.80 -125.99 4-B -118.75 -158.35 -93.55 L-A -135.77 -175.37 -110.64 L-B -134.14 -173.74 -103.70 C-A -140.17 -179.76 -87.64 C-B -137.98 -177.58 -89.79
Extreme stresses from Table 5 are compared with those for unexposed samples from[1] and are shown in Table 6.
Percentage change in extreme effective stress for unstabilized and lime stabilized samples were observed to be less than 1% and for cement stabilized sample, greater than 1%. Nominal change in extreme total stress was observed. Changes in extreme excess pore water pressure was not very significant for unstabilized and lime stabilized samples but was as high as 31% for cement stabilized samples. Table 6 % change in extreme stresses for 12 samples with respect to unexposed flyash
Identification
% change in Ext.
effective stress
% change in Ext. total stress
%change in Ext. excess
pore water press.
1-A 0.229 0.179 -0.444 1-B 0.410 0.318 1.569 2-A 0.392 0.306 1.236 2-B 0.226 0.175 0.936 3-A 0.381 0.294 1.316 3-B 0.403 0.310 1.380 4-A 0.635 0.489 2.389 4-B 0.632 0.474 1.507 L-A 0.022 0.017 2.648 L-B 0.522 0.403 6.509 C-A 1.070 0.829 31.196C-B 1.102 0.856 24.112
CONCLUSIONS Effects of tannery wastes on Flyash highway embankment has been modelled and analysed. The effect of change in unit weight, degree of compaction, cement and lime stabilization of flyash on extreme stresses and displacements are studied and discussed in the paper. Results conclude that embankments constructed with cement stabilized flyash and exposed to tannery wastes undergo lesser total displacements than those constructed with lime stabilized or unstabilized flyash. REFERENCES 1. Bandyopadhyay, K., Bhattacharjee, S. and Ghosh, S.
(2011), Numerical Approach for Analysis of Highway Flyash Embankment, Proceedings of Indian Geotechnical Conference, Kochi
2. Bandyopadhyay, K., Gangopadhyay, A., Misra, A.K., Mukhopadhyay, S.K. and Som, N. (2002), Study on the changes in physical properties of flyash exposed to industrial wastewater, Proceedings of Indian Geotechnical Conference, Allahabad
3. Potts, DM., Zdravkovic, L.,(1999) Finite Element Analysis in Geotechnical Engineering Theory, Thomas Telford
4. Plaxis version 8,(2002) Material Models Manual 5. Gnanendran, C.T. and Piratheepan, J. (2009) “Indirect
Diametrical Tensile Testing with Internal Displacement Measurement and Stiffness Determination, Geotechnical Testing Journal ASTM, 32(1), 45-44
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F627)
MEASURES TO REDUCE THE EARTH PRESSURE ON RETAINING STRUCTURES S. Bali Reddy, Research Scholar, Indian Institute of Technology Guwahati, India. email: [email protected] A. Murali Krishna, Assistant Professor, Indian Institute of Technology Guwahati, India, email: [email protected] ABSTRACT: Earth-retaining structures play important role in various infrastructure projects and for urban development. These structures will be subjected to various types of loading including the seismic loading under earthquake conditions. Among various parameters that need to be considered in the design of retaining structures, lateral earth pressure resulting from the supported backfill is the most predominant and the same is the influencing parameter on the performance of the structure under a variety of loading conditions. With the efforts of reducing the earth pressure on the retaining structures, many novel materials came into practice that are effectively serving the purpose. These materials include: expanded polystyrene (EPS) geofoam, tire shreds and tires, fly-ash etc. This paper reviews the use of various materials in reducing the earth pressures on retaining walls with main focus on EPS geofoam and tire shreds. INTRODUCTION The national planners in India have put infrastructure development on priority. This was resulted in transport planning, widening of National Highways and new roads in the country. Thus various earth structures: retaining structures/embankments/slopes will be designed and constructed in very large numbers over different areas. Among them, retaining structures take major part, being the permanent important structures. To make these retaining structures effective in their performance, it is essential to minimise the earth pressures under normal condition and also under critical seismic conditions too. Various options can be adopted to reduce the earth pressures acting on retaining structures: Use of low density backfill materials like fly-ash; Use of expanded polystyrene (EPS) geofoam or stacked tyres near wall or as fill material; use of mixed soils with tyre shreds, plastics etc. This paper reviews the use of various materials in reducing the earth pressures on retaining walls with main focus on EPS geofoam and tire shreds. Geofoam Expanded light weight foams used in geotechnical applications are described as ‘Geofoam’ Horvath [1]. Geofoam is being used as a lightweight fill and also can be used as a compressible inclusion under concrete and earth structures. In large earth structures, geofoam can protect underlying culverts, pipelines and other buried materials against unacceptable levels of stress, while maintaining a predictable amount of resistance against the overlying structure, preventing movement or subsidence. Geofoam has excellent vibration damping and excellent thermal insulation properties. It is not biodegradable. Expanded Polystyrene (EPS) geofoam is generally reported in retaining wall research studies [2-6]. Rigid soil retaining structures required to resist grater earth pressure during a seismic event than under static conditions. Inclusion of EPS geofoam effectively reduces the earth pressures acting on the wall. Figure 1 shows the typical
application of EPS Geofoam in retaining structures as a compressible inclusion function.
Fig. 1 Typical application of EPS geofoam on earth retaining structures The thickness of the geofoam material and its density/deformation modulus are the key parameters in selecting a geofoam configuration. Several researchers conducted experimental and numerical studies to verify the efficiency of geofoam in reducing the earth pressures effectively [1-2, 5, and 7]. EPS blocks are also used in fills as lightweight fill material function as shown in Fig. 2 [8].
Fig. 2 EPS blocks as lightweight fill materials [8]
Bali Reddy S. & Murali Krishna A.
Aytekin [4] conducted numerical experiments to evaluate the effectiveness of EPS geofoam as compressible inclusions in reducing the lateral earth forced on a retaining wall due to swelling backfill soil. Different thickness of geofoam (t =H/5, H/10, H/20) are considered and reported reducing lateral forces on retaining wall (Fig. 3). The transmitted lateral pressure is less when EPS backfill with a thickness of H/5m is used instead of the same thickness of the sand. The maximum difference occurred at a depth of 3.0m is approximately 146 kPa and 565% reduction on the lateral pressure would occur when EPs geofoam backfill is used with a thickness of H/5m compare of the same thickness of sand backfill.
Fig. 3 Lateral pressure distribution with different thickness of geofoam and granular fill [4] Ertugrul and Trandafir [9] performed physical and numerical experiments (Fig. 4) and concluded that stiffness and relative thickness of the EPS inclusion have the major roles in reducing the lateral earth thrusts.
Fig. 4 Physical and numerical models considered by Ertugrul and Trandafir [9] EPS geofoam was also tested for effectively minimizing the seismic lateral earth forces on retaining structures and enhancing the seismic stability of such structures. Inglis et
al. [3] reported field installation of a rigid basement wall constructed with a compressible EPS geofoam layer for the purpose of seismic-induced earth load reduction. The design of the structure was carried out using the program FLAC. The results of numerical modeling predicted that a 1-m wide layer of EPS geofoam placed between a 10 m-high wall and granular backfill could reduce lateral loads during an earthquake event by 50% compared to the unprotected wall option. Hazarika [7] conducted numerical experiments for mitigating seismic hazard on retaining structures using geofoam and concluded that up to top one fifth of the wall there is not much difference of the resulting stress for both with geofoam buffer and without geofoam buffer. When depth increases the differences also increase as shown in Fig. 5.
Fig. 5 Normalized wall height vs. lateral seismic stress [7] Bathurst et al. [5] conducted shaking table experiments to investigate the efficiency of geofoam as seismic buffers in reducing the seismic earth pressures on retaining structures. A typical experimental setup used with geofoam behind earth retaining structure shown in Fig. 6. Numerical parametric studies on use of EPS seismic buffers were performed by Zarnani and Bathurst [6]. Figure 7 shows the higher effectiveness of EPS seismic buffers in reducing the seismic earth forces at higher acceleration levels. Based on observations of Fig. 4, the reduction in total earth forces ranged from 18 to 21 %.
Fig. 6 Typical experimental setup in shaking table tests [5]
Measures to reduce the earth pressure on retaining structures
Fig. 7 Effect of geofoam in reducing seismic wall forces [10] Scrapped tires or Tire chips Tire chips or Tire shreds and Scrapped tires are light weight materials used in geotechnical applications like behind the earth retaining structures, embankments etc. and also tire chips mixed with sand used as a backfill material on earth retaining structures. Scrap tires and their byproducts are not biodegradable, not expensive, high elastic compressibility. Tire shreds are free draining. The typical field application of scrap tire showed Figs.7a, b.
Fig. 7 a Typical application of scrap tires [11] )
Fig. 7b Typical application of whole tire on earth retaining structures [12] Different researchers [11-15] are used tire chips mixed with sand, or only scrap tires or scrap tire filed with geofoam or tire chips are investigated in Field, experimental and
numerical. Both tire chips and geofoam are cushion materials. Cecich et al. [16] conducted different laboratory tests by using shredded tires with mixing sands. Using these properties, retaining wall of various heights were also designed using shredded tires as the backfill material and also designed by considering sand for comparison purposes. Table.1 shows comparison of factor of safeties with sand, shredded tire backfill material. It was concluded that, Both sliding and overturning factor of safety for the retaining walls with shredded tires were significantly more than that for use of the sand as a backfill material. And cost estimate with different height of walls and backfill materials sand and shredded tires used. Based on observations, the total construction cost saving is 67%. Table 1 Comparison of factor of safety for retaining walls with sand vs. shredded tires as backfill materials (100 ft long walls) Height of wall (ft)
Sliding factor of safety
Overturning factor of safety
Sand Shredded tire
Sand Shredded tire
10 4.15 >20 2.10 >2020 1.68 10.37 1.84 2.1230 1.54 3.35 1.65 2.14 Lee et al. [17] studied the effects of the compressible materials on the stress variation with soil depth in the backfill of retaining walls. In the study two compressible materials (recycled tire and Geofoam) were used. In the sensitivity analysis, elastic modulus values are varying. Elastic modulus was determined based on the stiffness ratio and stiffness ratio defined as (RE=E cushion /E backfill ). From the results presented in Table.2, it was observed that, the dynamic earth pressure and total earth pressure decrease when stiffness ratio decreases Table 2 Comparison of peak horizontal earth pressures obtained from numerical and field experiments [16]
Cushion type EPS Tire Elastic modulus 9387 1400 Numerical Analysis
With cushion 87 41 Without cushion 138 143 % Reduction 37 71
Filed test With cushion 10 9 Without cushion 13 30 % Reduction 23 70
Note: In table all units are in kN/m2
Various other materials Fly ash and its derived soils; geo-materials made from plastics and plastic bottles may also be used as lightweight backfill soils, which can effectively reduce the earth pressures acting on retaining structures. Hazara and Patra [19](2007) and Lal and Mandal [20](2012) are used fly ash as
Bali Reddy S. & Murali Krishna A.
backfill material for retaining structures. Graettinger et al. [21] performed laboratory and field trail tests for recycling of plastic bottles for use as a lightweight geotechnical material in retaining walls and concluded that the material tested may be useful in fills over soft soils or backfill material for retaining walls. CONCLUSIONS Lateral earth pressures acting on retaining walls are the main concern in the design and stability aspects of retaining structures. In the efforts to minimising earth pressures on retaining walls, use of various novel materials came into existence. Some of the researchers are being involved in investigating the effectiveness and possibility of using these materials in the retaining wall applications. Some of such studies on EPS geofoam and waste tire and their derived materials were presented briefly in this paper. It appears that design guidelines for use of such new materials are not established. Further more studies are essential to derive such guidelines for using novel materials in retaining wall application of geotechnical engineering. REFERENCES [1] Horvath, J. S. (1995), Geofoam Geosynthetic, Horvath
Engineering, P.C., Scarsdale, NY, 217. [2] Aytekin, M. (1992), Finite element modelling of lateral
[3] Inglis, D., Macleod, G., Naesgaard, E., and Zergoun, M. (1996), Basement wall with seismic earth pressures and novel expanded polystyrene foam buffer layer, In Proceedings of the 10th Annual Symposium of the Vancouver Geotechnical Society, Vancouver, and B.C. The Canadian Geotechnical Society, Richmond, B.C.
[4] Aytekin Mustafa (1997), Numerical Modelling OF EPS Geofoam used with Swelling Soil, Geotextiles and Geomembranes (15), 133-146.
[5] Bathurst, R.J., Zarnani, S., and Gaskin, A. (2007), Shaking table testing of Geofoam seismic buffers, Soil Dynamics and Earthquake Engineering, 27, pp. 324-332.
[6] Zarnani. S, and Bathurst R.J., (2009), Numerical parametric study of expanded polystyrene (EPS) geofoam seismic buffers, Can.Geotech.J. (46), 318-338.
[7] Hazarika Hemanta (2001) Mitigation of Seismic Hazard on Retaining Structures – A Numerical Experiment, In Proceedings of the 11th International Offshore and Polar Engineering Conference, Stavanger,459-464.
[8] Horvath, J.S. (2010), Lateral pressure reduction on earth-retaining structures using geofoam: correcting some mis-understandings. ASCE Earth retention conference; 2010.
[9] Ertugrul, O. and Trandafir, A. (2011). Reduction of Lateral Earth Forces Acting on Rigid Nonyielding Retaining Walls by EPS Geofoam Inclusions.” J. Mater. Civ. Eng., 23(12), 1711–1718
[10] Bathurst, R.J., Zarmani, S.,(2008), Numerical Modelling of EPS Seismic Buffers , 12th Intl.Conf on International
Association for Computer Methods and Advances In Geomechanics , 1-6 October, 425-432
[11] Kazuya Yasuhara (2007), Recent Japanese experiences on scrapped tires for geotechnical applications, proceedings of the international workshop on scrap tire derived geomaterials– opportunities and challenges, Yokosuka, Japan.19-42.
[12] Shi Wei et al. (2012), Study on the role of geogrid-reinforced for fly ash retaining wall basing on the analysis of FLAC3D, Advanced Material Research (365), 599-603.
[13] Tweedie et al. (1998), Tire shreds as lightweight retaining wall backfill: active conditions, Journal of geotechnical and geoenvironmental engineering, 1061-1070.
[14] Humphrey, D.N., and Tweedie, J.J., (2002), Tire Shreds as Lightweight Fill for Retaining Walls- Results of Full Scale Field Trials, In Proceedings of the Workshop on Lightweight Geomaterials, Tokyo, Japan.
[15] Tanchaisawat et.al (2010), Interaction between geogrid reinforcement and tire chip–sand lightweight backfill, Geotextiles and Geomembranes, 28, 119-127.
[16] Youwai Sompote., Bergado .T.D., (2004), Numerical analysis of reinforced wall using rubber tire chips-sand mixtures as backfill material, Computers and Geotechnics (31), 103-114.
[17] Cecich V. et.al.,(1996), Use of Shredded Tires as Lightweight Backfill Material for Retaining Structures, Waste Management & Research (14),433-451.
[18] Lee Hyun Jong., Roh Han Sung., (2007), the use of recycled tire chips to minimize dynamic earth pressure during compaction of backfill, Construction and Building Materials (21), 1016-1026.
[19] Hazra, S. and Patra, N.R. (2008), Performance of Counterfort Walls with Reinforced Granular and Fly ash Backfills: ExperimentalInvestigation,” International journal of Geotechnical and Geological Engineering, 26, 25-267.
[20] Lal, B R R., and Mandal, J. (2012), Feasibility Study on Fly ash as Backfill Material in Cellular Reinforced Walls, Electronic Journal of Geotechnical Engineering, Vol 17, No. J.
[21] Graettinger, A. J., Johnson, P.W., Sunkari, P., Duke, M. C., and Effinger, J. (2005), Recycling of plastic bottles for use as a lightweight geotechnical material", Management of Environmental Quality: An International Journal, Vol. 16 No. 6, pp.658 – 669
Proceedings of Indian Geotechnical Conference December 13-15,2012, Delhi (Paper No. F630)
FINITE STRAIN THEORY OF CONSOLIDATION OF CLAYS: FINITE VOLUME APPROACH
Rakesh Pratap Singh, Research Scholar, Civil Eng. Dept. I I T Roorkee, [email protected] Mahendra Singh, Professor, Civil Eng. Dept. I I T Roorkee, [email protected] C S P Ojha, Professor, Civil Eng. Dept. I I T Roorkee, [email protected] ABSTRACT: The Finite Strain Theory of one-dimensional consolidation finds its application with more generality for consolidation of thick clay strata, dredged fill deposits; consolidation induced solute transport through clay liners and such other similar cases. Upwind differencing, linear upwind differencing, central differencing, QUICK and min-max QUICK schemes of FVM have been used to work out the equation with explicit formulation. The consolidation equation, in the Material Coordinate system, has been solved first and then the solutions are transformed into Lagrangian and Convective coordinate system for lucid interpretation of the results. A comparative study of FDM and FVM solutions on an example problem, shows a good match in case of consolidation of dredged fills, however in case of consolidated soils, FVM solutions give the faster rate of consolidation than that of FDM. INTRODUCTION The one-dimensional finite strain consolidation theory overcomes many limitations of Terzaghi’s theory of consolidation. It takes into account not only the large strains but also the variations of the compressibility and permeability during consolidation. The equation thus developed in terms of void ratio as independent variable, is typically nonlinear and contains geometric as well as material nonlinearity [1]. Numerical solutions to this equation in the same form or in some other equivalent form have been presented by various investigators either by finite difference method or finite element method [1-5]. Fox and Berles [6] using another concept presented a piecewise linear numerical model for one-dimensional consolidation. However, it is noteworthy that the conservation laws are the time dependent systems of partial differential equations (usually nonlinear) and the finite strain one-dimensional consolidation equation by falls into this category. The finite volume (control volume) formulations uses integration over small control volumes and the flux at the interface of control volumes is represented by the same expression, thus the material is rigorously conserved [7]. This paper presents the finite volume formulation of the finite strain one-dimensional consolidation equation and its solution using FVM schemes, upwind differencing (UD), linear upwind differencing (LUD), central differencing (CD), quadratic upstream interpolation for convective kinetics (QUICK) and min-max QUICK. Further, the solutions have been obtained in time domain directly using the explicit time marching scheme.
MODEL DESCRIPTION
Basic Assumptions The basic assumptions of the theory of one-dimensional finite strain consolidation are:
1. The soil matrix is compressible, but the pore fluid and individual soil particles are incompressible.
2. The soil is homogeneous and loading is monotonic.
3. Pore fluid flow velocities are small and governed by Darcy's law.
4. The soil permeability (k) and vertical effective stress (σ’’) have the unique relationships with void ratio.
( ) (1 )
' ' ( ) ( 2 )
k k e
eσ σ
=
=
Coordinate System Lagrangian and convective coordinate system are the measure of soil solids and pore fluid matrix whereas the material coordinates are the measure of only solid particles in the matrix. The Lagrangian coordinates of a consolidating soil matrix represents initial measurements of it i.e. at time t=0 whereas the convective coordinates are the measurements at any time after the start of the consolidation i.e. for any time t>0. Thus the values of Lagrangian coordinates and material coordinates are fixed and independent of time while the convective coordinates keep on changing with time. For the conversion of coordinates from one system to other, the following relationship may be easily deducted. Consider a differential element of soil shown below.
R. P. Singh, M. Singh, C.S. P. Ojha
d a = 1 + e ( 3 )0
d = 1 + e ( 4 )
d z = 1 ( 5 )
d z 1 ( 6 )d a 1 0d z 1 ( 7 )d 1d 1 ( 8 )d a 1 0
a d az = ( 9 )1 + e ( a , 0 )0
zξ = [ 1 + e ( z , t ) ] d z ( 1 0 )
0
e
ee
e
ξ
ξ
ξ
=+
=+
+=
+
∫
∫
Governing equation The governing equation of one-dimensional consolidation, in terms of void ratio (e), permeability k (e) and effective stress σ’ (e), may be given in the following form.
s
w
'
w
γk 1(1+e) γe (11)
t z k eγ (1+e) e z
σ
⎡ ⎤⎛ ⎞− − −⎢ ⎥⎜ ⎟
∂ ∂ ⎝ ⎠⎢ ⎥= ⎢ ⎥∂ ∂ ⎛ ⎞∂ ∂⎢ ⎥⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦
Finite volume formulation The integration of Eq. (11) over the elementary control volume dV gives,
Integrating Eq. (12) using Gauss-divergence theorem and the one-dimensional consolidation, it will take the following form for ith control volume element.
Here the superscript ‘n’ denotes time element and subscript ‘i’ denotes the space elements. Finally, Eq. (14) may be rearranged in the form given below.
1 1 1i+ i+ i+2 2 2n+1 n
i i
1 1 1i- i- i-2 2 2
eβ(e) +α(e) ( )zΔte =e - (15)
Δz eβ(e) +α(e) ( )z
⎡ ⎤⎧ ⎫∂⎢ ⎥⎨ ⎬∂⎩ ⎭⎢ ⎥⎢ ⎥⎧ ⎫∂⎢ ⎥− ⎨ ⎬⎢ ⎥∂⎩ ⎭⎣ ⎦
Where, '
s
w w
γk(e) k(e) σ ( )β(e)= -1 ; α(e)=1+e γ γ (1+e) e
e⎛ ⎞ ∂⎜ ⎟ ∂⎝ ⎠
Eq. (15) may calculate the next time step value of void ratio with suitable boundary conditions for any type of linear or nonlinear relationship of permeability and void ratio, k = k (e) and effective stress and void ratio, σ’= σ’ (e), which may be obtained from the oedometer test on a soil sample in the laboratory. For calculating the values of void ratio (e) and its gradient at the elementary cell (control volume) boundaries (i+1/2 and i-1/2) following schemes have been used.
1 i 1 i+ 1 ii+ i+2 2
1 i -1 1 i i -1i - i -2 2
i i -1 i -1 i -21 1i+ i -
i+ 1 i i i -12 2
1e = e + ψ ( r ) ( e -e ) ( 1 6 )21e = e + ψ ( r ) ( e -e ) (1 7 )2
e -e e -er = ; r = (1 8 )e -e e -e
For Upwind differencing (UD) scheme; ψ(r) = 0 For Central differencing (CD) scheme; ψ(r) = 1 For Linear upwind differencing (LUD) scheme; ψ(r) = r For Quadratic upstream interpolation of convective kinetics (QUICK) scheme; ψ(r) = (3+r)/4 For Min-Max QUICK scheme; ψ(r) = max [0, [min {2r, (3+r)/4, 2}]] The gradients may be approximated as follows.
i + 1 i
1i +2
i i - 1
1i -2
e e - e (1 9 )z Δ z
e e - e ( 2 0 )z Δ z
∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠
∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠
Finite strain theory of consolidation of clays: finite volume approach
The functions α (e) and β (e) at any value of void ratio may be evaluated from the input data values (or curves) of k = k (e) and σ’ =σ’ (e) for the subject soil. Initial and boundary conditions The initial values of void ratio may be assumed consistent with the void ratio and effective stress input data set for self load or/ and any surcharge on a compressible layer. The possible boundary conditions are as follows [4]. Impermeable boundary The boundary condition where the compressible layer meets the impervious strata, there is no flow across such boundary and the following equation may be used for the purpose.
s w'
γ - γe = 0 ( 2 1 )d σzd e
∂+
∂
Semipermeable boundary This b. c. is based on the propositions that the flow coming out of lower part is equal to the flow into the upper part at the common boundary and the equal fluid pressures exist in pore water at the common boundary and these lead to the following equations.
Free draining boundary For free draining boundary, the excess pore pressure is always zero, thus effective stress is equal to total stress and the corresponding void ratio may be interpolated on the input data of void ratio and effective stress. Solution of the governing equation The above FVM formulation of the Eq. (11) has been implemented through the computer program in FORTRAN-77 for its solution. The solution, obtained in terms of material coordinates and void ratio, may be used to get convective coordinates using Eq. (10). However, the material coordinates and void ratio has only been presented here and the comparison of various FVM solutions with FDM solution for the example problem of consolidation. The example problem Disposal of dredged material is to be done at a site, 3 ft/ year in the first year, 2ft/ year in the second and third year and 1ft/ year in the fourth year. The total amount of each year will be deposited during first few weeks of each year so that it may be assumed that dumping is instantaneous in the beginning of the year. The fill is overlain by the compressible foundation of thickness 20 ft below which a semi-permeable silt layer exists that has a void ratio as 1.8 and the permeability as 1.03×10-4 ft/day. The drainage path length for this semi-permeable boundary is taken as 4.0 ft. The dredged material is assumed to have a uniform initial void ratio of 7.0 and the specific gravity of solids as 2.75. The foundation is assumed to have the specific gravity of solids as 2.83 and to be
normally consolidated under its own weight. The other input data from the oedometer tests of fill material and compressible foundation soil may be referred from Cargill [4]. The dredged fill is divided into 6 elements and compressible foundation into 10 elements. Elementary time has been taken as 1 day. RESULTS AND DISCUSSION The FVM solutions of Eq. (11), in terms of equation variables, material coordinates and void ratio (z ̴ e), have been shown below along with the FDM solution of it as given by Cargill [4]. Table-1 presents the solution for compressible foundation, the figures in bold shows the difference between the FDM solution and the FVM solutions successively for upwind differencing (UD), central differencing (CD), linear upwind differencing (LUD), quadratic upwind interpolation for convective kinetics (QUICK) and Min-Max QUICK (MQK) schemes. The similar values are shown in Table-2 for dredged fill. The absolute values of differences in the compressible foundation are comparatively less than difference values for dredged fill, but in case of the dredged fill, the differences are negative in upper segments and positive in the lower segments. Table-2 Material coordinates (z) and void ratio (e) of various schemes for dredged fill
Fig. 1 Time verses degree of consolidation curves of various schemes for compressible foundation
Fig. 2 Time verses degree of consolidation curves of various schemes for dredged fill Further, the Fig. 1 and 2 shows the variation of degree of consolidation with time for compressible foundation and the dredged fill. In case of compressible foundation the transient views of consolidation by FDM and FVM schemes are matching qualitatively, but differ quantitatively. However, in case of dredged fill it is matching quite well. If Terzaghi’s theory is considered with the following input data for the dredged fill and the foundation soil, the degree of consolidation is shown there. Dredged fill: Void ratio (e) =7.0; Permeability (k) = 8.66×10-3 ft/day; Unit weight of water (γw) = 62.4 Pound/ ft3; Effective stress gradient with void ratio (dσ׳/de) = -6.0 Pound/ ft2 Degree of consolidation (for two way drainage) = 0.94 Degree of consolidation (for one way drainage) = 0.58
Foundation Soil: Void ratio (e) =3.0; Permeability (k) = 1.21×10-3 ft/day; Effective stress gradient with void ratio (dσ׳/de) = -84.0 Pound/ ft2 Degree of consolidation (for two way drainage) = 0.17 Degree of consolidation (for one way drainage) = 0.087 The above results show that the Terzaghi’s theory solutions, FVM and FDM solutions have considerable mismatch in predicting the consolidation of the consolidated foundation soil. One way of verification of these results may be the experimental study on consolidation of thin and relatively thick samples of compacted clays and the attempt will be taken up in future. Conclusion The explicit FVM formulation of finite strain consolidation equation gives convergent and stable results like FDM formulation. The results are almost similar in case of consolidation of loose fills, but in case of soils with lower void ratios the various schemes in FVM formulation give faster rate of consolidation.
References
1. Gibson, R.E., England G.L. and Hussey, M.J.L. (1967), The theory of one-dimensional consolidation of saturated clays, Geotechnique, 17, 261-273.
2. Olson R.E. (1977), Consolidation under time dependent loading. Journal of Geotech Eng Div, ASCE, GT1, 55–60.
3. Gibson R.E., Schiffman R.L., Cargill K.W. (1981), The theory of one-dimensional consolidation of saturated clays: II, Finite nonlinear consolidation of thick homogeneous layers, Can Geotech Journal, 1981, 18(2), 280–93.
4. Cargill K.W. (1982), Consolidation of soft layers by finite strain analysis, Final report, Geotechnical laboratory, U.S. Army engineer waterways experiment station, P.O. box 631, Vicksburg, Miss., 39/80.
5. Lee P.K.K., Xie K.H. Cheung Y.K. (1992), A study on one-dimensional consolidation of layered systems, Int. Journal of Numerical and Analytical Method Geomechanics, 16, 815–831.
6. Fox, P. J., Berles, J.D. (1997), CS2: A piecewise linear model for large strain consolidation, Int. J. Numer. Anal. Meth. Geomech., 21, 453-475.
7. Botte G. G., Ritter J. A., White, R.E. (2000), Comparison of finite difference and control volume methods for solving differential equations, Computers and Chemical Engineering, 24, 2633–2654.
8. Versteeg, H.K. and Malalasekera, W. (2007), An introduction to computational fluid dynamics: The finite volume method, second edition, Pearson Education Limited, Edinburgh Gate, Harlow Essex CM20 2JE, England.
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F631)
NONLINEAR ANALYSIS OF GEOCELL REINFORCED RIGID STRIP FOOTING ON SOFT SOILS
S. Sireesh, Assistant Professor, Indian Institute of Technology Hyderabad, [email protected] M. R. Madhav, Visiting Professor, Indian Institute of Technology Hyderabad, [email protected] P. A. Faby Mole, Master’s Student, Indian Institute of Technology Hyderabad, [email protected]
ABSTRACT: Relatively recent development in the field of geosynthetics is the application of three dimensional mattresses with interconnected cells, known as geocells, to support foundations and other infrastructures. Several laboratory investigations are available to understand the behaviour of geocell reinforcement in various foundation soils. Limited number of detailed studies is available on the numerical or theoretical approach of designing the geocell reinforced foundation beds. This is attributed to the complexities involved in modelling the coherent soil-geocell mass to a great accuracy. In this paper, attempts have been made to analyse the nonlinear response of a rigid strip footing resting on a geocell reinforced soft foundation bed to the applied load. The stiffness of the soft soil and geocell layers were varied to obtain the improved load carrying capacity of the reinforced ground. Design charts in terms of non-dimensional parameters are developed to obtain the improvement in bearing capacity for a given width ratio of foundation to that of geocell. INTRODUCTION Introduction of reinforced soil below the footing can substantially increase the bearing capacity, thus obviating the necessity of a combined footing or a raft foundation [1]. Several research studies are available on laboratory model tests to provide a clear insight of the general behavioral trend of geocell reinforced soil beds [1, 2]. Besides, large scale model tests are more reliable, yet, in large scale tests, it is observed that the general mechanisms and behavior observed in the small scale model tests are only reproduced at larger scale [3]. Other approaches to predict the behavior of reinforced soil beds such as numerical simulations also provide a useful solution. However, the complexity involved in simulating the combined soil-reinforcement coherent mass properties is yet to be understood properly. As a result, alternative methods are still required to provide more accurate bearing pressure-settlement predictions. The objective of this paper is to formulate a theoretical solution to the complex soil-geocell material’s nonlinear load-settlement behavior. BACKGROUND Recently, soil reinforcement in the form of a cellular mattress (geocell) has been showing its efficacy in the fields of highway and embankment construction. Geocell mattress is a three dimensional, polymeric, honeycomb like structure of cells interconnected at joints [1]. The cell walls keep the encapsulated soil from being pushed away from the applied load and confine the soil. Because the in-filled cells are connected together, the panel acts like a large mat that spreads the applied load over an extended area, instead of directly at the point of contact, leading to an improvement in the overall performance. Several investigations have been reported highlighting the beneficial use of geocell reinforcement in the construction of foundations [1, 2, 4].
Through a series of model tests on circular footings supported on geocell reinforced sand beds overlying soft clay conducted by Dash et al. [4] demonstrated the improvement in load-deformation behavior of the geocell reinforced soft soils with varying height and width of the geocell mattress. The definition sketch of the geometry of the problem is shown in Fig. 1. Sireesh and Madhav (2011) have analysed the geocell reinforced sand layer over soft subgrades for smaller footing settlement ratios (W ≤ 1%) where the load-deflection pattern would expected to be linear [5]. Fig. 1 Definition sketch of geocell reinforced foundation bed [8] In this paper, the geocell mattress is considered as a Pasternak’s shear layer of height (H) with a shear modulus (Gg). The height and width of the shear layer is varied, as described by Dash et al. [4], to obtain the behavior of the geocell reinforced foundation system with varying geometrical properties of the mattress. This aspect has been considered in variation of Gg. The following sections briefly describe the theoretical nonlinear formulation of the geocell supported rigid strip footing on soft soil. The schematic of the problem definition is shown in Fig. 2.
Dg4
D
b
hgeocell layer
footing
u
soft clay
dense sand
HD D
Dg1 Dg2 Dg3
rigid base
Sireesh, Madhav & Fabymole
THEORETICAL FORMULATION The load settlement behavior of a rigid footing resting on an elastic half space can easily be modeled using the concept of Winkler springs, which simulates the stiffness of the foundation soil. In this case, the load will be shared by the springs supporting the load. Pasternak [6] improved the Winkler model by introducing a shear layer in between the rigid footing and the foundation soil. The shear layer is introduced to take the shear resistance of the soil into account in supporting the footing load, similar to a geocell mattress in the case of reinforced soil beds. This model is an advancement of Filonenko-Boridich model where the Winkler springs were considered to be connected through an elastic thin membrane under a constant tension. Fig. 2 Problem definition and deflected shape of the foundation system A non-linear load-settlement relationship is considered for a footing settlements (wo < 5 percent of footing width, B). For higher footing settlements, as expected, nonlinear relation between load-settlement must be assumed. Non-linear Formulation The governing equation for the load-deflection pattern of the problem considering the shear layer representing the geocell mattress, as described in Pasternak model, is presented
for 0 ≤ |x| ≤ B/2 (1)
for Bg ≥ |x|≥B/2 (2)
With X = x/B and W = w/B,
; and
Eq. 2, the governing equation reduces to
(3)
Rearrange Eq. 3 as
(4)
Using finite difference method (Crank–Nicolson method /central difference) Eq.3 has been discretized and the non-linear equation was linearized before solving using iterative Gauss-Seidel procedure. Now the load deflection equation for this formulation is
(5)
(6) (Since, X = x/B, W = δ*= w/B => dW = dw/B)
(7)
(8)
From Eq.4,
Hence,
(9)
Since the slope of the curve at Rg/2 is zero, Eq. 9 can be written as
(10)
Equation 10 depicts the complete solution for the load-settlement pattern for rigid strip footing resting on geocell reinforced sand overlying soft clay foundation. For μ = 0, the equation 10 should give the solution for linear analysis. Figure 3 shows the validation of the numerical solution. In the Fig. 3, legend with N and T represents the numerical and theoretical solutions respectively.
Shear layer (Geocell)
q(x)
H B
Bg
Rigid Footing
Rigid base
Clay (Winkler springs)
Shear layer (Geocell)
q(x)
H
w w =w0
B
Bg
Rigid Footing
Nonlinear Analysis of Geocell Reinforced Rigid Strip Footing on Soft Soils
Fig. 3 Variation of load ratio (Q*) with settlement ratio (W) for μ=0. RESULTS AND DISCUSSION Eq. 10 presents the relation between the load ratio (Q*) and the settlement ratio (W) in terms of the non-dimensional parameters α and μ. Response curves are developed for load ratio (Q*) versus footing settlement ratio (W) as functions of non-dimensional parameters (α and μ). The practical range of values for each parameter used in Eq. 10 is expressed here. In this analysis, the footing width and the thickness of the shear layer are considered to be of unit length for convenience. The modulus of subgrade reaction, ks was varied from 5,000 to 15,000 kN/m3 for soft soils [7]. The shear modulus, Gg was varied from 10 to 18 MPa based on experimental data [1]. The non-dimensional parameters (α and μ) are estimated to vary respectively between 0.5 and 2.0 with 0.5 increment for α; and 50 to 250 with 50 increment for μ. The width of the shear layer was kept constant at 11B (11 times the footing width) so that the length of shear layer from the edge of the footing is 5B. Figures 4 and 5 show the variation of W with X for different values of μ for α=2 and with different values of α for μ = 100 respectively. Both the figures depict the settlement profile of the shear layer and the underlying soft soil. A large settlement can be expected on the surface if the underlying soft layer is weaker. It can be inferred from Fig. 4 that for μ varying from 0 to 250 for a given α=2, the contribution of the length of shear layer increases. Since the value of μ introduces the nonlinearity in to the system, the influence of μ on the settlement profile of the shear layer is negligible. Similarly, for a given μ=100, the variation of α between 0.5 and 2.0 shows a significant influence of α on the settlement profile of the shear layer (see Fig. 5). It is noticed that for minimal value of α=0.5, the reinforced bed has shown excessive settlement of up to 3% out of 5% footing settlement. For all practical purposes, it can be said that the
value of α can be maintained high (>1) for stiffer foundation system.
Fig. 4 Variation of settlement ratio (W) from the edge of the footing – effect of μ for α = 2
0 1 2 3 4 50
1
2
3
4
5
Distance from Edge of FootingSe
ttle
men
t Rat
io, W
(%)
α=0.5, μ=100α=1.0, μ=100α=1.5, μ=100α=2.0, μ=100
Fig. 5 Variation of settlement ratio (W) from the edge of the footing: effect of α for μ = 100 Figures 6 and 7 show the variation of settlement ratio (W) with load ratio (Q*) for different values of μ and α respectively. From Fig. 6, it can be seen that for increase in the value of μ, the load ratio decreases for a constant value of α=2. It is also noticed that the load ratio (Q*) reduces with decrease in α. However, it is clear from Fig. 7 that for very low value of α, the reinforced bed becomes stiffer and represents a kind of elasto-plastic behavior. It can be deduced from Figs. 6 and 7 that for higher performance of the geocell reinforced sand beds over soft soils, α value should be high (> 1) and μ value should be small (< 100).
Fig. 6 Variation of settlement ratio (W) with load ratio (Q*) - effect of μ for α = 2
0 0.02 0.04 0.06 0.08 0.1 0.120
1
2
3
4
5
Load Ratio Q*
Sett
lem
ent R
atio
, W(%
)
α=0.5, μ=100α=1.0, μ=100α=1.5, μ=100α=2.0, μ=100
Fig. 7 Variation of settlement ratio (W) with load ratio (Q*) - effect of α for μ = 100
CONCLUSIONS An attempt has been made to analyse a complex system of soil-geocell mattresses supporting rigid strip footing on soft soils. The following conclusions were made from the analysis: 1. The Pasternak’s shear layer was introduced in the model
to replicate the Geocell mattress with a given shear modulus (Gg).
2. A generalized relation between load-deformation in non-dimensional form is obtained and for a given width of footing (B) and settlement ratio (W), the exact load on the footing can easily be obtained.
3. Design charts are developed for predicting the load ratio for a given settlement ratio and non-dimensional parameters μ and α.
4. The value of μ introduces the nonlinearity in to the system, the influence of μ on the settlement profile of the shear layer is negligible.
5. The value of α has huge influence on the settlement profile. Excessive settlements are observed with lower α values.
6. For higher performance of the geocell reinforced sand beds over soft soils, α value should be high (> 1) and μ value should be small (< 100).
REFERENCES 1. Sireesh, S. (2006). Behavior Geocell Reinforced
Foundation Beds, Doctoral thesis submitted to the Indian Institute of Science, Bangalore, India.
2. Dash, S.K., Krishnaswamy, N.R., Rajagopal, K., (2001), Bearing capacity of strip footings supported on geocell-reinforced sand, Geotextiles and Geomembranes, Vol. 19, pp. 235-256.
3. Milligan, G.W.E., Fannin, R.J., Farrar, D.M. (1986). Model and full-scale tests of granular layers reinforced with a geogrid. 3rd Int. Conf. on Geotextiles, Vienna, Vol, 1, 61-66.
4. Dash, S.K., Sireesh, S., and Sitharam, T.G., (2003), “Model studies on circular footing supported on geocell reinforced sand underlain by soft clay”, Geotextiles and Geomembranes, Vol. 21, pp. 197-219.
5. Sireesh Saride, and Madhav, M R (2011) ‘A Theoretical Approach for Designing Geocell Reinforced Foundations’, In the Proceeding of the Indian Geotechnical Conference, Kochi, India. Vol. 1, pp. 577-580.
6. Pasternak, P. L. (1954). On a new method of analysis on an elastic foundation by means of two parameters. (in Russian language)
7. Bowels, J. E (1997). Foundation Analysis and Design, McGraw-Hill, p. 1207.
NOMENCLATURE
Symbol Description Unit
B Footing width m Bg Width of shear layer/geocell m
Q* Load ratio Non-dim Rg Footing width ratio Non-dim w Settlement of shear layer from the
edge of the footing m
W Settlement ratio, w/B Non-dim x Distance from center of the
footing m
X Distance ratio, x/B Non-dim α2 ks.B2/Gg Non-dim
μ ks.B/qu Non-dim
Proceedings of Indian Geotechnical Conference December 13-15,2012, Delhi (Paper No. F632.)
EFFECT OF VIBRATING MASSES ON THE STEADY-STATE RESPONSE OF TWO-STORIED MACHINES
Y. Sudheer Kumar, PG Student, Indian Institute of Technology, Guwahati, [email protected] , C. M. Jibeesh, PG Student, Indian Institute of Technology, Guwahati, [email protected], B. Giridhar Rajesh, PG Student, Indian Institute of Technology, Guwahati, [email protected] , Arindam Dey, Assistant Professor, Indian Institute of Technology, Guwahati, [email protected], ABSTRACT: Gradually deteriorating spatial availability compels the use of multi-storied machines. The dynamic response of such machines are affected by the variability of the inherent parameters such as the ratio of vibrating masses, elastic stiffness, operating forces and/or operating frequencies. This paper reports the effect of the vibrating machines in terms of their masses on the analysis and determination of the dynamic response of the two-storied machines utilizing lumped parameter modeling technique. The present study does not consider the damping of the system. Based on the influence of the masses, response curves have been developed in both the time-domain and the frequency-domain. Extreme-end vibrations responses under coupled resonating condition has also been investigated and reported herein in brief. The present article provides a comprehensive insight about the dynamic behavior of two-storied machines for various mass-ratios and frequency-ratios. INTRODUCTION Industrial growth has become the most important factor for the economic and general development of the society all over the world. Almost all industrial applications have to contend with the generated vibration. Improper design of machine foundations may result in unbalanced dynamic forces that may be of significant discomfort and lead to instability. Prior to the design, it is extremely important to analyze the behaviour of such foundations. In this attempt, one of the conventional approaches has been the utilization of Lumped Parameter technique, wherein the machine-foundation system is represented by mass-spring-dashpot system. Several studies, in this regard, have been documented by Das and Ramana (2010), Rao (2006) and Saran (2006). Due to progressive space crunching of industrial sites, it may be a common picture in the nearby future to experience storey-machines, wherein a same foundation will be utilized for machines placed in the order of multiple stories. Another common example of such system is the combination of the actuator-shake table system. Giridhar Rajesh et al. (2012) have provided a detailed documentation about the dynamic response of a two-storied machine subjected to varying external dynamic loads. This paper reports the effect of varying vibrating masses on the dynamic response of the same. Such conglomerated studies aid in the development of monographs that will be serve as a guideline to the engineers related to the design of such systems. PROBLEM STATEMENT The two-storied machine has been modelled with the aid of lumped parameter system. The inherent damping of the system has been neglected in the present study. Each of the units of the coupled system is subjected to unequal operating forces, while the operating frequency is maintained identical for both the units. The underlying unit of the two-storied machine system is referred to as underlying unit (Unit 1) and is represented by mass m1 and spring stiffness k1. The unit
mounted on the top is hereby referred to as overlying unit (Unit 2) represented by mass m2 and spring stiffness k2. The operating forces effective on the units are represented as F1sin(ωt) and F2sin(ωt), where ω is the operating frequency. Under the action of these unequal operating forces, the individual units undergo a displacement of z1 and z2 respectively. Figure 1 provides the schematic diagram of the problem and the free-body-diagrams of the individual units.
ANALYSIS OF THE PROBLEM In order to maintain force equilibrium of the coupled 2DOF system, the following equations of motion are to be satisfied:
( ) ( )1 1 1 1 2 2 1 1 sinm z k z k z z F tω+ − − =&& (1)
( ) ( )2 2 2 2 1 2 sinm z k z z F tω+ − =&& (2) where, and z z& && represents the velocity and acceleration of a particular unit at any instant of time. The solution to the above equations of motion is given as:
( ) ( )1 1 2 2sin and sinz A t z A tω ω= = (3) where, 1 2and A A represents the amplitudes of displacement
Fig. 1 Schematic diagram and free-body diagrams of coupled two-storied machine systems
of the Units 1 and 2 respectively. The solution of the above systems is provided as:
( )( ) ( )
( )( ) ( )
21 2 2 2 2
1 2 2 22 2 1 2 1 2
21 2 2 1 2 2
2 2 2 22 2 1 2 1 2
,F k m F k
Ak m k k m k
F k F k k mA
k m k k m k
ω
ω ω
ω
ω ω
− +=
− + − −
+ + −=
− + − −
(4)
The amplitudes as determined above can be used to estimate the steady-state time-domain displacement, velocity and acceleration responses of the coupled two-storied machine system that are expressed as:
( ) ( )( ) ( )( ) ( )
1 1 2 2
1 1 2 22 2
1 1 2 2
sin , sin ,
cos , cos ,
sin , sin
z A t z A t
v A t v A t
a A t a A t
ω ω
ω ω ω ω
ω ω ω ω
= =
= =
= − = −
(5)
NON-DIMENSIONALITY OF THE PROBLEM In order to eliminate the parametric dimensional dependency of the obtained results, the above expressions are converted in their non-dimensional form. This procedure aids in the development of monographs and preparation of generalized analysis charts which are independent of the specific parametric values. The various non-dimensional parameters are expressed as: (a) Natural frequency of the individual Unit 1 and 2: 1 1 1 2 2 2,m mk m k mω ω= = (b) Frequency ratios: 1 1 2 2,r m r mω ω ω ω ω ω= = (c) Force ratio of the coupled system: 1 2rF F F= (d) Stiffness ratio of the coupled system: 1 2rk k k= (e) Mass ratio of the coupled system: 1 2rm m m= . Giridhar Rajesh et al. (2012) reported the utilisation of these parameters in determining the time- and frequency-domain response expressions of the system. The natural frequencies are determined by allowing the system to vibrate freely. Such a system is solved by considering the external forces on the system to be non-existent. The trivial solutions of the modified Equations 1 and 2 are expressed in a non-dimensional form as follows:
( )( )
4 21 1
4 22 1
1 . 0
1 . 0
nr r r r nr r r
nr r r r nr r r
m k k m k
m k m k m
ω ω
ω ω
− + + + =
− + + + = (6)
where, 1 1 2 2 and nr n m nr n mω ω ω ω ω ω= = are the natural frequency ratios of the coupled system. In essence, although the natural frequencies of the coupled system are invariable, the apparent difference arises due to the choice of the frequency of the individual unit used to create the non-dimensional frequency ratio. Solution of the above equations will provide the frequency ratio of the natural modes of vibration of the coupled two-storied machine system. RESULTS AND DISCUSSIONS Based on the mathematical relations, a Matlab code has been developed and an extensive investigation has been carried out
to understand the influence of the ratio of the vibrating masses m1 and m2 on the dynamic responses of the two-storied machines resting one above the other The study has been carried out to obtain results both in the time-domain and frequency-domain. Influence of mass ratio mr on natural frequencies ωnr The predetermined natural frequencies of the coupled system will depend on the stiffness-ratio (kr) and mass-ratio (mr) of the system. For example, in the present study, the dynamic responses of the coupled system have been developed for kr=5 and Fr=10. Under this condition, using Equation 6, the variation of natural frequencies of the coupled system with respect to the natural frequencies of the sole Units 1(ωnr1) & 2 (ωnr2) for various mass ratios are enumerated in Table 1. Theoretically, the displacement response of the undamped coupled system at above particular values of frequency ratios should be undefined, and the same has been depicted later in the dynamic response of the coupled system. From figure 2, it is observed that the fundamental frequency of coupled system with respect to natural frequency of sole unit 1(ωnr1 1st) & the secondary frequency of coupled system with respect to frequency of sole unit 2 (ωnr2 2nd) are converging beyond a mass ratio 5. It is also observed that the secondary frequency of coupled system with respect to natural frequency of sole unit 1(ωnr1 2nd) & the fundamental frequency of coupled system with respect to frequency of sole unit 2 (ωnr2 1st) become divergent beyond a mass ratio of 5. Hence, it is conclusive that for a given stiffness-ratio, there exists a particular mass ratio when the natural frequencies of the coupled system approach very close to each other. This mass-ratio can be termed as the optimal mass-stiffness-ratio wherein the coupled 2DOF system nearly approaches the state of a combined 1DOF system. Table 1 Natural frequency ratios corresponding to various mass ratios (mr)
Figures 3 and 4 depict the influence of mass ratio (mr) on the amplitude of the Unit 1 normalized with respect to wr1 (A’
11) and the amplitude of the Unit 2 normalized with respect to wr2 (A’
22). These figures can be suitably utilized either to decide on the parameters of the components if the displacements of particular units are pre-restricted, or also for the graphical determination of the displacements suffered by the individual components of the two-storied machines once the machineries have been pre-decided. Comparison of the two figures reveals the relatively higher effect of the mass ratio on the overlying unit as compared to that of the underlying unit. Moreover, it is also noticed that as compared
Effect of vibrating masses on the steady state response of two storied machines
to the ratio of the external operating forces [Giridhar Rajesh et al (2012)], the mass ratio has lower effect on the amplitudes of displacement of individual components. These figures also re-illustrate the shift of the natural frequencies of the coupled system with the change in the mass-ratio, as supported by the large displacements at the corresponding natural frequencies.
It can be also noted that the non-dimensional amplitude of the overlying mass is more affected by the mass ratio than the underlying mass, and this concept can be easily extended to the application of tuned vibration absorber. In contrary to the effect of force-ratio on the vibration of the coupled system [Giridhar Rajesh et al (2012)], it can be noticed herein that in most of the frequency ratios, the non-dimensional amplitude of the individual units does not deviate significantly from its minimum value. Hence, it can be conclusively stated that the optimal frequency ratio for the two-storied machines can be solely and suitably chosen based on the ratio of external forces, and investigation of its variability with the mass-ratio can be avoided. The only consideration to be kept in mind while choosing the optimal frequency ratio is to avoid the natural frequencies of the coupled system by a sufficient amount.
Influence of mr on the time-domain response of coupled system Time-domain response is an alternative representation of the frequency domain response where in the former the effect of variation of time is studied for a particular operating or a forcing frequency, whereas in the latter, the effect of variation of the forcing frequency is studied at a particular instant of time. Figures 5 and 6 represent the steady-state displacement response of the underlying unit. It is observed from Figure 5 that for a particular mass-ratio, the underlying unit undergoes large displacement at the resonating frequencies of the coupled system. For a particular frequency ratio, Figure 6 reveals the increase in the displacement of underlying unit with the increase in the mass-ratio. In both the figures, the directional reversal of the displacement of the individual units can be noted with the change in the mass-ratio and frequency ratios. However, it has to be borne in mind that such variations are definitely subjected to the choice of the combination of mass-ratio and stiffness-ratio at which the investigation is carried out
Similar observations are also made with the vibration of the overlying mass, as represented in Figures7 and 8. However, in this case, it is clearly revealed that subjected to resonance state, the overlying mass undergoes larger displacement as compared to that of the underlying mass. This forms the basic
Fig. 5 Steady-state z’11 for various ωr1 (mr =1)
Fig. 3 Influence of mr on amplitude of Unit 1 (A’11)
Fig. 2 Influence of mr on the natural frequency ratios
Fig. 4 Influence of mr on amplitude of Unit 2 (A’22)
concept of a vibration absorber, in which the vibration of the main unit is reduced or brought to rest at extremity, while the auxiliary unit vibrates with large displacement
Fig. 7 Steady-state z’
22 for various ωr2 (mr =1)
Fig. 8 Steady-state z’
22 for various mr (ωr2=1) Studies have also been carried out to investigate the effect of mass-ratio on the velocity and acceleration response on the time-domain; however, for the sake of brevity, they have not been presented herein.
CONCLUSIONS The present study investigates the effect of vibrating masses on the undamped dynamic response two-storied vibrating machines. The system forms a 2DOF mass-spring system where each individual unit is subjected to operating forces. This article reports in detail the circumstance wherein the operating forces are unequal in magnitude while maintaining identical frequency. The outcome of the present investigation has been presented in non-dimensional form utilizing ratios of contributory parameters. Based on the above discussions, the following conclusions can be drawn: • It is observed that for a stiffness ratio kr=5, an increase of
mass-ratio in the order of 104 largely affects the fundamental frequency of the underlying system (~24 times) in comparison to the secondary frequency (~4times). A reverse note is made for the overlying system.
• For a given stiffness-ratio kr=5, there exists an optimal mass ratio mr =5 where, the natural frequencies of the coupled system approach very close to each other, and the 2DOF system nearly behaves as a 1DOF system.
• In a particular frequency ratio range of 0.5-1, the normalized amplitude curves of underlying unit is less sensitive to the variation of mass-ratio as compared to the overlying unit.
• Combining the present study with the detailed investigations of the behavior of two-storied machine system for different force-ratio and stiffness-ratio, a detailed monograph can be developed which will be of two-step aid to any practicing engineer. It will help estimating the responses of the system if the absolute values of the parameters of the individual units are pre-known. On the other hand, it would also help in estimation of the parameter of an individual unit, when restrictions on the responses of the coupled unit are pre-imposed.
REFERENCES 1. B. Giridhar Rajesh, Y. Sudheer Kumar, C. M. Jibeesh and
A. Dey. Dynamic analysis of undamped two-storied machines: Influence of operating forces. Proc. of ICAMB-2012, Vellore, India. (Paper Accepted for publication)
2. B. M. Das and G. M. Ramana. Principles of Soil Dynamics. Cengage Learning, 2010.
3. N. S. V. K. Rao. Mechanical Vibration of Elastic Systems. Asian-Books, 2006.
4. S. Saran. Soil Dynamics and Machine Foundation. Galgotia Publications, 2006.
Fig. 6 Steady-state z’11 for various mr (ωr1=1)
Proceedings of Indian Geotechnical Conference December 13-15,2012, Delhi (Paper No F-635)
MECHANICS OF CEMENTED SAND BY DISCRETE ELEMENT SIMULATIONS Dinesh.S.V., Professor, Siddaganga Institution of Technology, Tumkur; [email protected] Mamatha K.H., Post Graduate Student, Siddaganga Institution of Technology, Tumkur; [email protected] Vinod J.S., Associate Professor, Department of Civil Engineering, University of Wollongong, Australia; [email protected] ABSTRACT: This paper explores the potential of Discrete Element Method (DEM) to model the behavior of cemented sand. Numerical simulations were carried out using PFC2D with assemblies consisting of about 3200 particles of radius ranging from 0.075 – 0.1mm. In the present work cementation effect is modeled by assigning bond strength value, which binds the sand particles. The effect of cementation on the mechanical behavior (stress-strain and volume change) was studied. The effect of cementation on stress dilatancy has been analyzed and reported based on numerical simulation using discrete element method. INTRODUCTION The mechanical behavior of cemented sand is an important topic and is widely studied by various researchers. Research on cemented sands has covered a wide variety of topics, including stress–strain and volumetric responses (Saxena and Lastrico (17), Clough et al. (4), Leroueil and Vaughan (12)), stiffness enhancement (Huang and Airey (7), Schnaid et al. (18)), dynamic properties (Acar and El-Tahir (1), Saxena et al. (16), Clough et al. (3), Sharma and Fahey (19)), influences of various cementing agents (Ismail (8), Ismail et al. (9), Leung (13)), critical-state features (Airey (2), Coop and Atkinson (5), Cuccovillo and Coop (6), Schnaid et al. (18), and stress–dilatancy relationships (Cuccovillo and Coop (6), Lade and Overton (4), Mántaras and Schnaid (15), Lo et al. (14)). The macroscopic behavior of stress–strain, volume change and the influence of cementing agent are well understood. But there is a need to understand the microscopic response associated with the macro-behavior and there are not many studies which brings out the mechanism of increased strength and volume change in cemented sands from particulate approach. In the present paper the results of the mechanical behavior (stress–strain, volume change) were reported based on numerical simulations using discrete element method. The simulations have been carried out using two dimensional particle flow code (PFC2D) under biaxial shear test. The cementation effect was modeled by assigning bond strength values, which binds the sand particles together. The effect of cementation on stress dilatancy was analyzed. An attempt is made to provide consistent explanations for the macro-behavior in terms of micro response. DISCRETE ELEMENT METHOD Discrete Element Method is currently used in several scientific disciplines to study the systems with inherent granularity. In geomechanics and mechanics of materials this technique was pioneered by Cundall (1971) for rock mechanics problems where continuity between elements does not exist. DEM models the granular materials as individual elements which can make and break contacts with their neighbours and are capable of analyzing interacting bodies underlying large absolute or relative motions. Its important feature is that it incorporates the coulomb’s frictional law at contacts between elements. Slippage occurs when the
tangential force at contact exceeds the critical value. The equilibrium contact forces are obtained from a series of calculations by solving Newton’s law of motion followed by force displacement law at each contact. When all forces for each contact in the assembly are updated, forces and moment sums are determined on each element, and the above process is repeated in cycles. NUMERICAL SIMULATION PROGRAM Biaxial element test with and without cement have been carried out using two dimensional discrete element software PFC2D (Itasca Consulting Group, Inc.) PFC2D simulates the mechanical behavior of a material by representing it as an assembly of circular particles. By introducing the ability to bond together adjacent particles, the procedure has been applied extended and applied to simulate complex problems in solid mechanics. Numerical simulations were carried out on assemblies having a width of 7cm and height of 14cm. The shear and normal stiffness of wall boundaries were set as 5X106N/m. About 3200 circular particles of diameter ranging from 0.075 – 0.1mm were then generated and these particles were assigned a normal and shear stiffness of 5X108N/m, density of 2650kg/m3 and inter particle friction value of 0.25. Linear contact displacement model has been employed for the numerical simulation program. In the present investigation, cementation effect was modeled by assigning bond strength values, which binds the sand particles together. Contact bond can be envisioned as a kind of glue joining to particles and can only transmit a force. The assemblies were generated by incorporating cementation effect and were assigned normal and shear bond strength of 1.5X105N/m. The assemblies were then compacted using a strain rate of 10-5/s to an initial confining pressure of 100kPa. Biaxial shear tests were then carried out on these assemblies to investigate the influence of cementation on stress dilatancy. Details of numerical simulation program are tabulated in Table 1. RESULTS AND DISCUSSIONS Figure 1 shows the mechanical responses of cemented assemblies having different bond strength under a confining pressure of 100kPa. Response of uncemented assembly under the same confinement is also presented as a reference. Figure
Dinesh .S.V, Mamatha. K.H, Vinod. J.S
1(a) shows the Deviator stress versus axial strain for uncemented and cemented assemblies. Table 1. Parameters used in DEM simulations Sl No. Parameters Selected values 1 Sand particles
1 Soil particle density 2,650 kg/m32 Initial porosity 0.20 3 Radii of particles 0.075-0.1mm 4 Inter particle friction angle 0.25 5 Normal contact stiffness 5X108 N/m 6 Shear contact stiffness 5X108 N/m 7 Normal contact stiffness
between sand and membrane particles
5X107 N/m
2 Wall 1 Normal contact stiffness 5X106 N/m
2 Shear contact stiffness
5X106 N/m
3 Cementing particles 1 Normal bond strength 1.5X105 N/m
Uncemented assembly shows normal consolidation behavior. Cemented assemblies show increased deviator stress with a pronounced peak strength followed by strain softening. The peak strength increase varies from 2 to 3 times when the bond strength was varied from 1e5 to 2e5. The peak strength is achieved at a very small strain level of 1.2% strain level. This clearly indicates higher stiffness of cemented assemblies. Beyond this, it appears that there is bond breakage and the cemented assemblies show greater strain softening. At large strain levels of the order of 35% the residual strength of cemented assemblies are slightly higher than uncemented assemblies. Figure 1(b) shows the plot of volumetric strain versus axial strain. Uncemented assembly shows lower dilation and the cemented assemblies show greater dilation. Similar observations are revealed from figure 1(c) where samples of greater bond strength show large increase in void ratio. Figure 1(d) shows the plot of stress ratio versus axial strain for uncemented and cemented assemblies. Cemented assemblies show peak stress ratio of 1.75 for assembly with greater bond strength and stress ratio at large strains are nearly uniform. This indicates that the critical state behavior of cemented and uncemented assemblies appears to be at the same state in terms of stresses.
(a)
(b)
(c)
(d)
Fig. 1 Mechanical responses of cementation under confining pressure of 100kPa STRESS DILATANCY The Dilatancy, (d) is defined as,
Mechanics of cemented sand by Discrete Element Simulation
ps
pvd
δεδε
−=
Where, svδε = Increments of plastic volumetric strain
psδε = Increments of the plastic triaxial shear strain
The sample with bond strength 1.5e5N/m at 100kPa confining pressure is selected to analyze the stress dilatancy behavior. Figures 2 (a) and (b) show the stress – strain and volumetric response of the cemented and uncemented assemblies at a confining pressure of 100kPa. The cementation effect will increase the peak strength. The deviator stress at large strain is almost same for both cemented and uncemented assemblies. The volumetric strain plot shows higher compression followed by large amount of dilation. The associated stress dilatancy relationships for these two samples are established in Figure 3. Figure 3 is presented in terms of the stress ratio (q/p) and dilatancy. The peak strength (or the peak stress ratio) and the maximum dilatancy do not occur at the same axial strain. As a matter of fact, such a delayed development of the maximum dilatancy is observed in all the cemented samples. Maximum dilatancy occurs at a strain ratio of 0.12 and the strain ratio is calculated by the axial strain at the peak strength divided by the strain where the maximum dilatancy occurs. The dilatancy at large strain level of 35% is almost zero in uncemented and cemented assemblies. (a)
(b)
Fig. 2 (a) Stress – Strain (b) Volumetric response of cemented and uncemented sand under the confining pressure of 100kPa
Fig.3 Stress dilatancy relationship of cemented and uncemented sand under the confining pressure of 100kPa Figure 4a shows the variation of specific volume with mean p of all cemented and uncemented assemblies. From the figure it can be observed that all cemented and uncemented assemblies are initially sheared from a constant specific volume during the compression phase till peak strength state and beyond the peak state samples show dilation and the specific volume increases with the increase in bond strength value. Fig 4b shows the initial and steady state points of cemented and uncemented assemblies. This figure clearly indicates the steady state points for all the assemblies and data clearly shows the magnitude of dilation and different critical state. With increase in bond strength the steady state position moves up in the specific volume and mean pˈ plot. (a)
Dinesh .S.V, Mamatha. K.H, Vinod. J.S
(b)
Fig. 4 Variation of specific volume with mean p’ for cemented and uncemented assemblies CONCLUSIONS In this study, numerical simulations are used to explore the underlying mechanisms of how cementation influences the strength and stress dilatancy behavior in cemented sand. The salient findings are as follows.
Prior to yielding the stress ratio increases rapidly from 1.4 to 1.7 and the dilatancy is hindered by the bonding network.
After yielding, the increase in stress ratio gradually becomes slower and it attains a value of 1.4 at large strain, during this phase the dilatancy speeds up.
Upto the peak strength the bond strength governs the behaviour. Thereafter the bond breakage occurs resulting in decrease in strength. But the subsequent volumetric dilation is not resulting in increase of strength.
The peak strength and the maximum dilatancy do not occur at the same strain level. The bond breakage is observed at a strain level of 0.02%/ which is the beginning point of dilation and max. dilatancy occurs at strain level of 0.039%.
The critical state position varies significantly with increase in bond strength. Higher the bond strength more dilation is observed after bond breakage.
REFERENCES
1. Acar, Y. B., and El-Tahir, A. E. (1986). “Low strain dynamic properties of artificially cemented sand.” J. Geotech. Engrg., 112(11), 1001– 1015.
2. Airey, D. W. (1993). “Triaxial testing of naturally cemented carbonate soil.” J. Geotech. Engrg., 119(9), 1379–1398.
3. Clough, G. W., Iwabuchi, J., Rad, N. S., and Kuppusamy, T. (1989). “Influence of cementation on liquefaction of sands.” J. Geotech Engrg., 115(8), 1102–1117.
4. Clough, G. W., Sitar, N., Bachus, R. C., and Rad, N. S. (1981). “Cemented sands under static loading.” J. Geotech. Engrg. Div., 104(6), 799–817.
5. Coop, M. R., and Atkinson, J. H. (1993). “The mechanics of cemented carbonate sands.” Geotechnique, 43(1), 53–67.
6. Cuccovillo, T., and Coop, M. R. (1999). “On the mechanics of structured sands.” Geotechnique, 49(6), 741–760.
7. Huang, J. T., and Airey, D. W. (1998). “Properties of artificially cemented carbonate sand.” J. Geotech. Geoenviron. Eng., 124(6), 492–499.
8. Ismail, M. A. (2000). “Strength and deformation behavior of calcite cemented calcareous soil.” Ph.D. dissertation, The Univ. of Western Australia, Perth, Australia.
9. Ismail, M. A., Joer, H. A., Sim, W. H., and Randolph, M. F. (2002). “Effect of cement type on shear behavior of cemented calcareous soil.” J. Geotech. Geoenviron. Eng., 128(6), 520–529.
10. Ladd, R. S. (1978). “Preparing test specimens using under-compaction.” Geotech. Test. J., 1, 16–23.
11. Lade, P. V., and Overton, D. D. (1989). “Cementation effects in frictional materials.” J. Geotech. Engrg., 115(10), 1373–1387.
12. Leroueil, S., and Vaughan, P. R. (1990). “The general and congruent effects of structure in natural soils and weak rocks.” Geotechnique, 40(3), 467–488.
13. Leung, S. C. (2005). “Mechanical characteristics of cemented sand—A particulate-scale study.” MPhil thesis, Dept. of Civil Engineering, Hong Kong Univ. of Science and Technology, HKSAR, China.
14. Lo, S. C. R., Lade, P. V., and Wardani, S. P. R. (2003). “An experimental study of the mechanics of two weakly cemented soils.” Geotech. Test. J., 26(3), 1–14.
15. Mántaras, F. M., and Schnaid, F. (2002). “Cylindrical cavity expansion in dilatant cohesive-frictional materials.” Geotechnique, 52(5), 337– 348.
16. Saxena, S. K., Avramidis, A. S., and Reddy, K. R. (1988). “Dynamic moduli and damping ratios for cemented sands at low strains.” Can.Geotech. J., 25(2), 353–368.
17. Saxena, S. K., and Lastrico, R. M. (1978). “Static properties of lightly cemented sand.” J. Geotech. Engrg. Div., 104(12), 1449–1465.
18. Schnaid, F., Prietto, P. D. M., and Consoli, N. C. (2001). “Characterization of cemented sand in triaxial compression.” J. Geotech. Geoenviron Eng., 127(10), 857–868.
19. Sharma, S. S., and Fahey, M. (2003). “Degradation of stiffness of cemented calcareous soil in cyclic triaxial tests.” J. Geotech. Geoenviron. Eng., 129(7), 619–629.
20. Thomas, P. A., and Bray, J. D. (1999). “Capturing nonspherical shape of granular media with disk clusters.” J. Geotech. Geoenviron. Eng., 125(3), 169–178.
Proceedings of Indian Geotechnical Conference December 13-15, 2012, New Delhi (Paper No. F636.)
EFFECT OF REINFORCEMENT STIFFNESS ON THE OBLIQUE PULLOUT BEHAVIOR OF REINFORCED SOIL
Shantanu Patra, Doctoral Student, Dept. of Civil Engng., I.I.T. Delhi. E-mail: [email protected] J. T. Shahu, Associate Professor, Dept. of Civil Engng., I.I.T. Delhi. E-mail: [email protected] ABSTRACT: With the advent of geosynthetic as extensible and quasi-extensible reinforcement, the design and construction of reinforced soil structures such as walls, embankments and slopes has gained much more momentum. Stability of these structures depends on extensibility of the reinforcement in addition to localized mobilized reinforcement force and its direction in the vicinity of the failure surface. Localized soil-reinforcement behavior again depends on kinematics of failure of these structures. However kinematics of failure is such that the failure surface intersects the reinforcement obliquely and thus causing oblique pullout of the reinforcement. Therefore obliquity of the reinforcement force should be considered for stability analysis of these structures against pullout. This paper presents an oblique pullout analysis of an extensible sheet reinforcement resting on a subgrade soil idealized by a two-parameter linear elastic Pasternak model. Effect of extensibility on deformed shape of the reinforcement and mobilized reinforcement strain are studied. Localized behaviour of reinforced soil in the vicinity of failure surface is also investigated in the present analysis. The present study removes the drawback of the earlier work by assuming more realistic soil reinforcement model characteristics.
INTRODUCTION In the last few decades, with the advent of geosynthetic product, reinforcing a soil mass with more extensible type inclusions has gained much popularity. Consequently, a large number of reinforced soil structures such as reinforced soil walls, slopes and embankments have been built as permanent structures. Stability of these structures is important and depends on extensibility of the reinforcement, localized mobilized reinforcement force and its direction in the vicinity of the failure surface. Localized soil-reinforcement behavior again depends on kinematics of failure of these structures. However kinematics of failure is such that the failure surface intersects the reinforcement obliquely (Fig. 1) and thus causing oblique pullout of the reinforcement. Therefore obliquity of the reinforcement force should be considered for stability analysis of these structures against pullout. However, most available methods do not incorporate any of these factors. As a result, these methods do not adequately describe real reinforced soil behaviour. Application of these methods requires an extra level of conservatism and sometimes results in apparent inconsistencies in interpretation of experimental data (Rowe and Ho 1993). This paper presents an oblique pullout analysis of an extensible sheet reinforcement resting on a subgrade soil idealized by a two-parameter linear elastic Pasternak model. Effect of extensibility on deformed shape of the reinforcement and mobilized reinforcement strain are studied. Localized behaviour of reinforced soil in the vicinity of failure surface is also investigated in the present analysis. The present study removes the drawback of the earlier work by assuming more realistic soil reinforcement model characteristics.
Fig. 1. Kinematics of failure of reinforced structures (Patra and
Shahu 2012b)
Y
(a) Reinforced retaining walls
(c) Enlarged View at Y
Y
(b) Reinforced slopes
α θL
Tmax
P
wL
Reinforcement (R)
Failure surface (S)
R S
R
Tangent to R
S (or Tangent to S)
AB
PROBLEM DEFINITION AND ANALYSIS An extensible sheet reinforcement of stiffness factor J* and normalized length 1.0 unit is resting on a subgrade soil (Fig. 2a) having subgrade stiffness factor DLks γμ /= and shear
stiffness factor γDLGH/*G = . The reinforcement is subjected to a normalized oblique pullout force P*(=P/THP where THP = axial pullout capacity of the reinforcement) at point B where the sliding mass intersects the reinforcement at an obliquity α (see Figs. 1a-d). The resulting horizontal extension U of the reinforcement at any point is determined from the tension-strain relationship as T*=J*dU/dX where T* is reinforcement tension at any point X on the reinforcement. For the analysis, a rigid plastic soil-reinforcement interface is assumed where angle of interface frictional resistance is φr. The soil above the reinforcement is represented by a uniform overburden stress at the top of the reinforcement.
Shantanu Patra & J.T. Shahu
Fig. 2. Schematic of the model used (Patra and Shahu 2012b)
D
(a) Reinforcement subjected to oblique force
αLΑ Βtanϕr
P
P
α
(b) Deformed modelz , w
x
θLTmax
q = γD Α
ΒwL
(c) Forces on the reinforcement
wLΒ
w0
Shear layerSprings
αp
q
Ατt
P
Reinforcement
x
τb
uL
uL
The proposed model for the analysis is shown in Fig. 2(b). The amount of tension T* in the reinforcement at end B is
*maxT with an inclination θL. Vertical displacement W at ends
A and B are W0 and WL, respectively. The displacement W and tension T* at any point along the reinforcement can be calculated by solving the following two governing coupled equations as (Patra and Shahu 2012b)
**2
2**
1 21tan
tan5.0ii
r
iiLiLi TZ
X
WWGWW
nT +
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡+⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂−=+
φ
θμ
(1)
( )
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛++
−+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+
=−+
*2*
2*
2
*,
3**
2*
11*
2*
2*
2costan4
costan2costan2
GZ
T
n
dX
dW
dX
dT
ZJ
TWWG
Z
T
W
i
iir
iixiirii
i
iir
iθφμ
θφθφ
(2)
where ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+=
*
*,* 1
J
TZ ix
i and i is the number of elements into
which the reinforcement strip is divided (i.e., nx /1=Δ ). The assumed boundary conditions are:
at X = 0, 0=dXdWi and T*= 0; and at X = 1, W = 1. (3)
Final governing equations for overall equilibrium as (Patra and Shahu 2012b)
∑
∑+
=
+
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+
∂
∂−
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+
∂
∂−+
=1
1
*,2
2*
1
1
*,2
2*
tancos2
secsintancos2
tann
iicrci
iLiL
n
iciiccirci
iLiLiL
ZX
WWGWW
ZX
WWGWWWW
φθμ
θθφθμμ
α (4)
∑+
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+
∂
∂−=
1
1
*,2
2** cos2
cos2
1 n
iicci
iLiL Z
X
WWGWW
nP θμ
α (5)
where ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+=
*
*,,*
, 1J
TZ ixc
ic , ( )[ ]iiLci WWnW −= +−
11tanθ , and
( ) 21−+= cicii θθθ subscripts c denotes elemental values at the centre of the element and subscript i is nodal number. Solution and Range of parameters Eqs. 1 and 2 are solved in conjunction with the boundary conditions (Eq. 3) and overall equilibrium equations (Eqs. 4 and 5) to obtain Wi and *
iT at all nodes. A trial and error procedure is adopted for the solution. Ranges of parameters for the analyses are taken as: μ = 500-2000 and G* = 0-1000, J* = 1-1000, φr = 20-40ο. RESULTS Mobilized strain and displacement profile Variation of reinforcement strain ε and displacement W with distance X at pullout are shown in Figs. 3 and 4, respectively, for different values of model parameters J*, G*, μ, and α. Fig. 3 shows that mobilized reinforcement strain decreases, as the reinforcement stiffness factor J* increases. The effect of J* on displacement is negligible (Fig. 4) compare to reinforcement strain. As the subgrade shear stiffness factor G* increases, reinforcement strain and displacement decreases (Figs. 3 and 4) and more uniform distribution is observed. However, for the increase in subgrade normal stiffness factor μ the reinforcement strain increases (Fig. 3) whereas displacement reduces (Fig. 4). But in both the cases, the distribution is more localized. As the obliquity α of the pullout force and angle of interface frictional resistance φr increases, the reinforcement strain and displacement also increases (Figs. 3 and 4). For higher subgrade shear stiffness factor G*, greater proportion of the applied pullout force is distributed by the subgrade soil by the interaction of neighboring soil elements (Patra and Shahu 2012a). Consequently, normal and shear stresses developed over the inner part of the subgrade reduce. As the interface shear stresses reduce pullout capacity also reduces. As a result, mobilized reinforcement strain ε and displacement W also reduce.
Effect of subgrade shear stiffness on the oblique pullout behavior of reinforced soil
Inclination factor Inclination factor IF ( αθL= , where θL is the inclination of the reinforcement at final deformed state and α is the oblique of the pullout force refer Fig. 2d), quantifies the deformation behavior of the reinforcement subjected to oblique pull. Fig. 5 shows shear stiffness factor G* versus inclination factor IF plot. As G* increase (stiffer subgrade), inclination factor IF decrease. However, as the reinforcement stiffness J* increases (stiffer reinforcement) inclination factor IF increases (Fig. 5). Inclination factor IF also increases as the obliquity α and angle of interface friction φr increases (Fig. 5). However, subgrade normal shear stiffness factor μ has negligible effect on the inclination factor (Fig. 5). A high value of inclination factor IF (=1) indicates that the orientation of the deformed reinforcement is almost the same that of the applied oblique pullout force. However, a lower value of IF (= 0) indicates almost a horizontal orientation of the reinforcement. For a high value of shear stiffness G* (> 50) inclination factor IF is very low (approaching zero value) which indicates that the reinforcement remains horizontal (IF = 0) near the failure surface (similar to the conventional method). However, as G* decreases bending of the reinforcement increases and reinforced slope becomes equal to the obliquity of pullout force at failure surface (IF = 1). But
in practice, depending on subgrade shear stiffness G* and reinforcement stiffness J*, IF lies in between these two extremes 0 and 1 (Bergado et al. 2000).
Fig. 6. Inclination factor IF versus horizontal pullout capacity *HP
Fig. 6 shows that the horizontal pullout capacity *
HP increases as the inclination factor IF increases. The horizontal component of pullout capacity *
HP also increases as the reinforcement stiffness J* decreases. As J* decreases the extension of the reinforcement increases and thus contribution of the extended part of the reinforcement also increases. Horizontal pullout capacity also increases as the obliquity of the reinforcement and angle of interface frictional resistance increases (Fig. 6). Fig. 7 shows that maximum reinforcement strain increases with the increase in inclination factor. As the reinforcement stiffness increases reinforcement strain decreases. Maximum reinforcement strain also increases with the increase in obliquity of the pullout force and with the increase in angle of interface frictional resistance. Fig. 8 shows that as the inclination factor IF increases end displacement WL increases. For lower subgrade stiffness factor μ a high value of end displacement WL is observed. Subgrade normal stiffness factor μ is found to be the single most important factor on the end displacement WL.
Fig. 8. Inclination factor IF versus End displacement WL
CONCLUSIONS 1. The horizontal component of pullout capacity *
HP increases as the reinforcement stiffness J* decreases, as contribution of the extended part of the reinforcement increases.
2. Mobilized reinforcement strain decreases, as the reinforcement stiffness factor J* increases. The effect of J* on displacement is negligible
3. Horizontal pullout capacity *HP increases as the
inclination factor IF increases. 4. The end displacement WL increases as the inclination
factor IF increases 5. Convention method of pullout analysis does not consider
extensibility of the reinforcement whereas present analysis removes this drawback and can be suitably applied for the analysis of more extensible type of reinforcement.
6. Conventional method which considers pullout capacity in the axial direction gives highly conservative value of pullout capacity where as Winkler based oblique pullout model overpredicts the pullout capacity. The present method gives a more rational and realistic value of pullout capacity.
REFERENCES 1. Bergado, D.T., Teerawattanasuk, C. and Long, P.V.
(2000). “Localized mobilization of reinforcement force and its direction in the vicinity of failure surface.” Geotextiles and Geomembranes, 18, 311-331.
2. Rowe, R. K. and Ho, S. K. (1993). “Keynote lecture: A review of the behaviour of reinforced soil walls.” Earth Reinforcement Practice,Proc., Int. Symp. on Earth Reinforcement Practice, Vol. 2, H. Ochiai, S. Hayashi, and J. Otani, eds., Balkema, Rotterdam, The Netherlands, 801–830.
3. Patra, S. and Shahu, J.T. (2012a). “Pasternak model for oblique pullout of inextensible reinforcement.” J. Geotech. Geoenviron. Eng.,http://dx.doi.org/10.1061/ (ASCE) GT.1943-5606.0000720. Posted on web ahead of print on March 2012.
4. Patra, S. and Shahu, J.T. (2012b). “Pasternak model for oblique pullout of extensible reinforcement.” Geosynthetics International, communicated.
Proceedings of Indian Geotechnical Conference December 13-15,2012, Delhi (Paper No.F637)
NUMERICAL SIMULATION OF DIRECT SHEAR TEST ON INFILLED ROCK JOINT A.K.Shrivastava, Assistant Professor, Department of Civil Engineering, DTU, Delhi, [email protected] K.S.Rao, Professor, Department of Civil Engineering, IIT, Delhi, [email protected] Ganesh W. Rathod, A. P., Department of Civil Engineering, NIT, Warangal, [email protected] ABSTRACT: Despite their frequent natural occurrence infilled discontinuities have been studied much less experimentally, perhaps because of the difficulties arising from the sampling, preparation of the sample and requirement of very good testing facility. Constitutive modelling of infilled rock joints are also difficult because joint behaviour has to consider a large number of assumption and uncertainties, which may sometimes unable to predict correctly the stress and deformation behaviour of the joints. The availability of sophisticated softwares and high speed computer has made numerical approaches of analysis popular and easy. Hence, in the present work capability of numerical study in predicting the shear behaviour of the infilled rock joints under constant normal load (CNL) and constant normal stiffness (CNS) boundary conditions is evaluated. A comprehensive numerical modelling has been performed using universal distinct element code (UDEC) based on the discrete element method of numerical analysis. The predicted shear behaviour is compared with the experimental results [1 and 2] and the detailed account of this is discussed in the present paper. INTRODUCTION The proper understanding of rock mass behaviour is important for design of underground openings in jointed rocks, stability analysis of rock slopes, risk assessment of underground waste disposal, design of foundation on rock and design of rock socketed piles. An in-situ rock mass derives its strength and deformation characteristics from physical and mechanical properties of the competent intact material and properties of discontinuities such as joints, faults, foliation surfaces or bedding planes. The physical and mechanical properties influencing shear behaviour of joints mainly are: (a) Joint roughness (b) Scale effect (size of joint) (c) Stiffness of the surrounding rock (d) Shear rate (e) Condition of the joint i.e. unfilled joint/infilled joint (f) Infill type (g) Infill thickness (h) Drainage condition of the infill material [3]. Hence rock joint models for predicting the shear behaviour has to include the influence of all these parameters. The influence of these parameters on shear strength of jointed rock has been studied by different researchers such as [1, 4, 5, 6, 7, 8, 9, 10, 11 and 12]. Based on their laboratory, analytical and numerical studies various shear strength models have been proposed in the past. It is very difficult to simulate all these factors influencing the shear behaviour of the rock joints by constitutive modelling and experimental study. A constitutive model has to consider the large number of assumptions and uncertainties or requires the input of complex parameters for analysis. The experimental studies require very good testing facilities. The availability of sophisticated software’s and high speed computer has made numerical approaches of analysis popular and easy.
Hence, in this paper an attempt has been made to model the shear behaviour of infilled joint under both constant normal load (CNL) and constant normal stiffness (CNS) conditions using UDEC. The behaviour of simulated rock joints (Plaster of Paris) with asperity angle 300-300 has been modelled in UDEC and the predicted shear strength is compared with the experimental results [1]. NUMERICAL SIMULATION Numerical method can be used to model large scale projects as well as for the physical model studies conducted on the jointed rock mass in the laboratory. In this method first geometry of the problem is established, than the estimated rock mass parameters are applied to it and finally these parameters are than be adjusted so that the output from numerical methods agrees with the observed laboratory study or behaviour of the structures as construction proceeds. The simplifications made in it to solve the systems of the differential equations either inside the continuum or at the boundaries of discretization. Hence numerical modelling leads to approximations to the correct or exact mathmetical solutions. The accuracy of these models depends upon the selection of numerical methods, geometry of the problem, loading process or history and the response of the discontinuity to the loading. Numerical methods can be broadly classified as continuum or discontinuum method [13 and 14]. There are no quantitative guidelines to determine when one method should be used instead of the other one. In continuum method the discontinuous materials are modelled by the interface elements or slide lines. However, their uses are restricted in rock mass modelling because of the limitations like, logic may break when many intersecting interfaces are used, or there may not be an automatic scheme of recognizing new contacts and this formulation may be
A. K. Shrivastava, K. S. Rao & Ganesh,W. Rathod
limited to small displacement and rotation. On the other hand discontinuum methods allow finite displacements and rotations of discrete bodies, including complete detachment and recognize new contacts automatically as the calculation progress. Excavation or construction on rock masses usually involves slip and separation along the discontinuities. Hence modelling of the discontinuities is an essential component. Influence of discontinuities on the mechanical behaviour of the unfilled and infilled rock joints were experimentally and numerically investigated by researchers like [15, 16, 17 and 18]. Therefore, in the present study, discrete element scheme, UDEC [19] is used for modelling the discontinuities. The Universal Distinct Element Code (UDEC) is a two-dimensional numerical program based on the distinct element method for discontinuum modelling. UDEC simulates the response of discontinuous media (such as a jointed rock mass) subjected to either static or dynamic loading. The discontinuous medium is represented as an assemblage of discrete blocks. The discontinuities are treated as boundary conditions between blocks; large displacements along discontinuities and rotations of blocks are allowed. Individual blocks behave as either rigid or deformable material. Deformable blocks are subdivided into a mesh of finite-difference elements, and each element responds according to a prescribed linear or nonlinear stress-strain law. The relative motion of the discontinuities is also governed by linear or nonlinear force-displacement relations for movement in both the normal and shear directions. UDEC is based on a Lagrangian’s calculation scheme that is well-suited to model the large movements and deformations of a blocky system. The detail of this is described in [19] manual. UDEC Analysis The deformability of the discontinuities between the blocks and their frictional characteristics in UDEC are modelled by spring slider system with prescribed force displacement relations enabling the normal and shear forces between the blocks to be calculated. The UDEC contains three constitutive models for the joint. First is the Coulomb slip model which provides a linear representation of joint stiffness and yield limit. It is based upon elastic stiffness, frictional, cohesive, tensile strength properties and dilation characteristics common to rock joint. Second, continuously yielding model which is more realistic than the Coulomb joint model as it accounts for nonlinear behaviour observed in physical tests. This joint model, displays a continuous accumulation of plastic displacement from the onset of shearing. The instantaneous slope (i.e. tangent to the shear stress vs. shear displacement curve) is governed not only by the stiffness k, but also by the factor F. For a given shear displacement, F depends on the distance from the actual shear stress curve (τ) and the bounding strength curve (τm) (Fig. 1). The bounding strength is given by
tanm n effPτ φ= (1)
Where, nP and effφ are the normal stress and effective
friction angle respectively. During the shearing process effφ
is continuously reduced from 0b iφ + to bφ . Where bφ and
0i are the basic friction angle and initial angle of the dilatancy respectively. In practice the factor F and the law of reduction of effective friction angle is determined empirically. The third joint model i.e. Barton-Bandis model is also a nonlinear joint model that directly utilizes index properties from laboratory tests on joints. Fig.1 Shear stress – displacement curve and bounding shear strength [19] During the numerical simulation of direct shear test on unfilled rock joints [18], it was observed that the constitutive models available for the joint in UDEC i.e. Coulomb slip or continuously yielding are not predicting well the peak shear stress as compared to experimental results. The results reflect that Coulomb slip joint model is suitable only for planar joint. It has been also observed during analysis by [2] that the asperity degradation cannot be modelled properly by continuously yielding joint model of UDEC, because of the over prediction of the dilation of the joint and hence over prediction of the corresponding normal stress. Under conventional CNL, the asperity degradation is less prominent at the same initial normal stress and at similar shear displacement; hence UDEC in present form are more appropriate for CNL than CNS conditions. Hence, exiting continuously yielding model needs to be modified to consider the effect of the CNL and CNS boundary conditions and the law of reduction of effective friction angle for infilled joint. It is modified by writing FISH function to incorporate the equation proposed by [2] based on
Shear displacement
Shea
r st
ress
(τ)
FKs
τm
Numerical Simulation of Direct Shear Test on Infilled rock Joint
the experimental results on physically simulated infilled rock joints as given by Eq. (2 , 3 and 4) for predicting the shear strength of infilled joint, normal stress corresponding to peak shear stress and decay in the effective friction angle for infilled joints.
( )'infil infilltanp n bP iτ ϕ= + (2)
n iP aP b= + (3)
'infill ln( / )n ci i x P yσ= + (4)
Where, infilpτ , nP , iP , cσ , bϕ , i and '
infilli are shear stress in MPa, normal stress in MPa, initial normal stress in MPa, uniaxial compressive strength in MPa, basic friction angle in degree, initial asperity angle in degree and effective infill asperity angle in degree respectively. The regression constants are a, b, x and y, which depend upon the initial asperity angle , cσ and normal stiffness (kn) of the rock joints. The values of these constants are given in Table 1. Table 1 Values of constants for different asperity angle, infill thickness (t) and normal stiffness of joint [17]
I t (mm)
kn (kN/mm)
a B x y R2
300
300 5 8 1 0.16 -0.26 -0.49 0.96
Modelling of Direct Shear Test on Infiled Joint Model geometry of size 297 mm X 297 mm X 125 mm is created in the UDEC software which is same as of laboratory specimen. The 300-300 asperity of rock joints having asperity height (a) as 5mm and thickness of infill material (t) as 5mm created by crack command, which makes t/a = 1. The UDEC and physical model of 5mm thick infilled joints with asperity 300-300 is as shown in Fig. 2 and 3. The properties of the material and the joint used for UDEC analysis is given in Table 2. A proper joint roughness parameter (jr) is selected, as it controls the rate at which effective friction angle decreases with plastic shear displacement. A smaller value of jr causes effective friction angle to decrease rapidly, which is resulting into smaller peak stress. Initial boundary condition is applied on the sample in such a way that the lower shear box is only allowed to move in X direction and movement in Y direction is restricted by imposing Y velocity at the bottom of the lower shear box as zero. The upper shear box is allowed to move only in the Y direction and movement in the X direction is restricted by imposing X velocity at the sides of the Upper shear box as zero. The boundary conditions are similar to the conditions used during laboratory testing [2]. Initial normal stress at the
top of the sample is applied for CNL and CNS boundary conditions as calculated from Eq. 3. The model is run for large number of cycles to reach equilibrium and give desired shear displacement to the sample. At equilibrium, the force on one side of a grid point nearly balances the opposing
Fig. 2 UDEC model geometry of infilled joint.
Fig. 3 Physical model of infilled sample [18] Table 2 Model properties used in UDEC.
Property key word in UDEC
Description Value
D Block mass density 1234 kg/m3 K Baulk modulus of
block 1.357 GPa
G Shear modulus of block
0.934 GPa
Jkn Joint normal stiffness 0.8 GPa/m Jks Joint shear stiffness 0.8 GPa/m Jen Joint normal stiffness
exponent 0
Jes Joint shear stiffness exponent
0
Jfric Joint intrinsic friction angle
38.50
Jif Joint initial friction angle
Calculated from Eq.4
Jr Joint roughness parameter
0.01mm
A. K. Shrivastava, K. S. Rao & Ganesh,W. Rathod
Force. The sample is then sheared by imposing the shear velocity on the lower sample. The average normal and shear stresses and normal and shear displacements along the joints are measured with a FISH function (av_str). RESULTS AND DISCUSSIONS The UDEC results are compared with the experimental results [1]. The average shear stress vs. shear displacement plot along the infill joints under CNL and CNS boundary conditions for 300-300 asperity joint is plotted in Fig. 3.
It is observed that the predicted peak shear stress based on modified UDEC capability is in close agreement with the experimental results, although the pre and post peak shear stress response is under estimated at higher Pi. The predicted shear stress increases with shear displacement and once the peak shear stress is reached, the shear displacement does not cause increase in the shear stress, indicating the complete shearing of the asperity at that shear displacement and after that sliding of the sample takes place. CONCLUSIONS The shear behaviour of the simulated infilled joints were analysed using UDEC. The UDEC capability is modified to accommodate the effective or realistic infilled joint friction angle and CNS conditions for proper modelling the shear behaviour of the infilled rock joints. The variations of the shear stress with shear displacement were studied for CNL and CNS boundary conditions at different initial normal stress. The UDEC predictions were compared with the experimental results for same set of joints. The results indicate that the modified UDEC is well capable of predicting the peak shear stress for infilled joints, but pre and post peak shear stress is under estimated. REFERENCES 1. Shrivastava, A.K., Rao, K.S. and Rathod, G.W. (2011),
Shear behaviour of infill joint under CNS boundary condition, IGC, Kochi, 981-984.
2. Shrivastava, A.K. (2012), Physical and numerical modelling of shear behaviour of jointed rocks under CNL
and CNS boundary conditions, Doctoral Thesis, Indian Institute of Technology, Delhi, India.
3. Shrivastava, A.K., Rao, K.S. (2009), Shear bhaviour of jointed rock: a state of art, IGC, Guntur, 245-249.
4. Patton, F.D. (1966), Multiple modes of shear failure in rock and related materials, Doctoral Thesis, University of Illinois, Urbana.
5. Ladanyi, B. and Archambault, G. (1977), Shear strength and deformability of filled indented joints, Proc. Int. Symp. on Geotechnics of Structurally Complex Formations, Italian Geotech. Assoc., Capri, Vol. l, 317-326.
6. Barton, N. (1973), Review of a new shear strength criterion for rock joints, Engineering Geology, 287–332.
7. Barton, N. and Choubey, V. (1977), The shear strength of rock joint in theory and practice, Rock Mech., 10, 1-54.
8. Saeb, S. (1989), Effect of boundary conditions on the behaviour of a dilatant rock joint, Doctoral. Thesis, University of Colorado, Boulder.
9. Saeb, S. (1990), A variance on the Ladanyi and Archambault's shear strength criterion, Proceedings of the International Symposium on Rock Joints, Loen, Norway, Barton, N. and Stephansson, O. (eds), Balkema, A.A., Rotterdam, 701-705.
10. Indraratna, B., Haque, A. and Aziz, N. (1999), Shear behaviour of idealized joints under constant normal stiffness, Geotechnique, 40(2), 189-200.
11. Welideniya, H.S. (2005), Laboratory evaluation and modeling of shear strength of infilled joints under constant normal stiffness (CNS) conditions, Doctoral. Thesis, University of Wollongong, Australia.
12. Oliveira, D.A.F and Indraratna, B (2010),.Comparison between models of rock discontinuity strength and deformation, J.of Geotech and Geoenv. Eng, 136(6), 864-874.
13. Jing, L. and Hudson, J.A. (2002), Numerical methods in rock mechanics, Int. J. Rock Mech. Min. Sci., 39, 409-427.
14. Jing, L. (2003), A review of techniques, advances, and outstanding issues in numerical modeling for rock mechanics and rock engineering, Int. J. Rock Mech. Min. Sci., 40, 283–353.
15. Cundall, P.A. (1990), Numerical modelling of jointed and faulted rock, Proc. of Int. Conf. on Mechanics of Jointed and Faulted Rock, 11.
16. Indraratna, B., Jayanathan, M. and Brown, E. T. (2008), Shear strength model for overconsolidated clay-infilled idealized rock joints, Geotechnique, 58(1), 55–65.
17. Oliveira, D.A.F. (2009), An advancement in analytical modeling of soil infilled rock joints and their practical application, Doctoral Thesis, University of Wollongong, Australia.
18. Shrivastava, A.K., Rathod G.W. and Rao K.S. (2012), Numerical simulation of direct shear test on rock sample, ASCE, GSP, 225, 2177-2186.
19. Itasca Manual (2004), User’s guide UDEC version 4.0.
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. F640)
SETTLEMENT PREDICTION OF SHALLOW FOUNDATIONS USING ARTIFICIAL NEURAL NETWORKS
M Harikumar, PG student, National Institute of Technology, Calicut, [email protected] N Sankar, Professor, Department of Civil Engineering, National Institute of Technology, Calicut, [email protected] ABSTRACT: During the past three decades, many methods have been developed to predict the settlement of shallow foundations on cohesionless soils. However, methods for making such predictions with the required degree of accuracy and consistency have not yet been developed. A realistic prediction of settlement is essential since settlement, rather than bearing capacity, generally controls the foundation system design. In this work, artificial neural networks will be used in an attempt to obtain more reliable settlement prediction. A large database of actual measured settlements is used to develop and verify the ANN model. Parameters such as footing stress, bearing capacity of soil, foundation dimensions and water table depth, which were found to have a major impact on settlement, are required to be input by the user, through interfaces designed in MS Visual Basic 2010. The settlements predicted using these parameters are then compared with the values predicted by four of the most commonly used traditional settlement prediction techniques. Comparisons were made between the ANN model and other conventional techniques of settlement prediction by means of sensitivity analyses. INTRODUCTION Settlement prediction is a major concern and is an essential criterion in the design of shallow foundations. The complexity in estimating the settlement of shallow foundations can be attributed to the uncertainty associated with the factors that affect the magnitude of this settlement, such as the distribution of applied stress, the stress–strain history of the soil, soil compressibility, and the difficulty in obtaining undisturbed soil samples. In geotechnical engineering, both theoretical and experimental methods can be found to predict the settlement of shallow foundations. But, all these methods rely upon various assumptions in geotechnical engineering and hence, the settlements predicted are often unreliable and inconsistent.ANN is a relatively a new tool in the field of prediction and forecasting and in this paper, an attempt is made to utilize ANN for settlement prediction of shallow foundations. The objectives of the paper are:
1. To develop an artificial neural network system for settlement prediction of shallow foundations based on various criteria such as the stress on the footing, bearing capacity of the soil, footing geometry and water table depth, which are commonly encountered in practical designs. 2. To validate the results by comparing the predicted settlement values with the values calculated by conventional settlement calculation techniques.
3. To conduct sensitivity and accuracy studies on the results predicted and comparison with the conventional settlement calculation techniques.
Artificial Neural Networks (ANN) ANNs are a form of artificial intelligence, which by means of their architecture, try to simulate the behaviour of the human brain and nervous system. A typical structure of ANNs consists of a number of processing elements (PEs), or nodes, that are usually arranged in layers: an input layer, an output
layer and one or more hidden layers, as shown in Fig.1. Each PE in a specific layer is fully or partially joined to many other PEs via weighted connections. The input from each PE in the previous layer (xi) is multiplied by an adjustable connection weight (wji). At each PE, the weighted input signals are summed and a threshold value or bias (θj) is added. This combined input (Ij) is then passed through a non linear transfer function (f(.)) to produce the output of the PE(yj). The output of one PE provides the input to the PEs in the next layer. The process is summed up as follows:
Ij=Σwjixi+θj ---- summation (1) yj= f (Ij) ---- transfer (2)
Fig. 1 Structure and Operation of ANN
The actual output of the network is compared with the desired output and an error is calculated. Using this error and utilizing a learning rule, the network adjusts its weights until it can find a set of weights that will produce the input/output mapping that has the smallest possible error.
ANN Model for Settlement Prediction The ANN model developed in this paper uses multilayer perceptrons (MLP) that is trained with the back-propagation training algorithm for feed forward ANNs [1]. The model has five inputs representing the footing width, B, net applied
M.Harikumar, N.Sankar
footing load, q, average blow count, N, obtained using a standard penetration test (SPT) over the depth of influence of the foundation as a measure of soil compressibility, footing geometry (length to width of footing), L/B, and footing embedment ratio (embedment depth to footing width), Df /B. The single model output is foundation settlement, S. The ANN hierarchy is shown in Fig.2.In this figure; vij represents the connection weights from the input to the hidden layer and wij, the connection weights from the hidden to the output layer.
Fig. 2 ANN hierarchy The database used for the training of the ANN model consists of 272 records, collected from literature and incorporate field measurements for settlement of shallow foundations over a wide range of footing dimensions and soil parameters. The database is summarized in Table 1. Table 1 Database collection from literature Reference No. of
cases Vargas, 1961 2 Levy and Morton, 1974 46 Burland and Burbidge, 1985 114 Picornell and del Monte, 1988 1 Papadopoulos, 1992 83 Wahls, 1997 21 Maugeri et al, 1998 5 Total 272 Data Division The ranges of the data used for the input and output variables along with statistical parameters such as the mean and standard deviation are summarised in Table 2. The available data were divided into three sets (i.e. training, testing and validation) in such a way that they are statistically consistent and thus represent the same statistical population. In total, 80% of the data were used for training and 20% were used for validation. The training data were further divided into 70% for the training set and 30% for the testing set. Before presenting the input and output variables for ANN model training, they were scaled between 0.0 and 1.0 to eliminate their dimension and to ensure that all variables receive equal attention during training.
Table 2 Data ranges used for ANN variables Model variable Min.
value Max. value
Footing net applied pressure, q (kN/m2)
33 697
Average SPT blow count, N 0 60 Footing dimensions, L/B 0.727 50.792 Footing embedment ratio, Df/B 0 3.444 Water table depth, w (m) 0 15 Measured settlement, S (mm) 0.6 254
The simple linear mapping of the variables’ practical extremes to the neural network’s practical extremes is adopted for scaling as it is the most common method for data scaling. Using this method, for each variable x with minimum and maximum values of xmin and xmax, respectively, the scaled value xn is calculated as follows:
min
max min(3)n
x xxx x
−⎛ ⎞= ⎜ ⎟−⎝ ⎠ Optimization of Weights The source code for to determine the optimized weights was developed using Turbo C compiler. Initially, random values are assigned for the weights. The output, predicted after suitable calculations using activation functions, is compared with the measured settlement. The resulting error is back propagated and weights are adjusted accordingly. Feedforward networks trained with the back-propagation algorithm have already been applied successfully to many geotechnical engineering problems [9], and are thus used in this work. Details of the back-propagation algorithm are beyond the scope of this paper and can be found in many publications. After training, the final set of optimized weights was obtained by taking the arithmetic mean of all optimized weights, and is given in Table 3. Table 3 Final set of optimized weights for the ANN Hidden layernodes
wji (weight from node i in the input layer to node j in the hidden layer)
wji (weight from node i in the input layer to node j in the hidden layer)
i=6 i=7 - - - j=8 -1.8981 -1.6224 - - - Conventional Methods of Settlement Prediction Many traditional methods for settlement prediction of shallow foundations on cohesionless soils are presented in literature. Among these, four are chosen for the purpose of assessing the relative performance of the ANN model. These include the methods proposed by Meyerhof (1974), Schultze and Sherif (1973), Bowles (1977, 1982) and Terzaghi and
Settlement prediction of Shallow foundations using Artificial Neural Networks
Peck (1948, 1967). These methods are chosen as they are commonly used and the database used in this work contains most parameters required to calculate settlement by these methods. Results and Discussions A comparison of the results obtained from the ANN model and the conventional indicates that the results given by the former are much more accurate and consistent compared to the others. The Root Mean Square Error (RMSE) and the Mean Absolute Error (MAE) of the model are also considerably less, compared to the other models, as shown in Table 4. Table 4 Comparison of ANN and Conventional methods for Settlement Prediction Error
ANN Terzaghi Schultze Bowles Meye-rhof
RMSE 32.32 267.53 58.29 41.12 83.56
MAE 0.88 4.16 1.36 1.21 2.39
Sensitivity Analysis The results of sensitivity analysis are shown in Fig.3. Plots are made between measured and predicted settlement. It is evident that for the ANN model, the measured and predicted settlement lie close to each other, whereas, considerable scattering is obtained for Terzaghi and Peck model. Similarly the analysis has been extended other models and it was concluded that the ANN model presented the best results. In order to test the robustness of the predictive ability of the ANN over a range of valid data, i.e. within the ranges of data used in the training process, the predicted settlements are examined against changes to the input variables. All input variables except one are fixed to the mean values of the data used in the database, and a set of synthetic data for the single varied input is generated by increasing the value of this input in increments equal to 5% of the total range between its minimum and maximum values. The results obtained are shown in Fig.4 and Fig.5.
The results from the analysis are compared with the common geotechnical data. It can be seen that the direction of the trends are in agreement with what one would expect based on the physical sense of settlement prediction. For example, as shown in Fig.7, there is an increase in the predicted settlement as net applied footing load, as one would expect. On the other hand, the predicted settlements decrease as the average SPT blow count, as shown in Fig.8.The analysis was also extended to the other parameters such as the footing geometry and footing embedment ratio. The results were found to agree with the geotechnical theories perfectly. This indicates that the equation is robust and can be used for predictive purposes.
0 20 40 60 80 100 120 1400
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M.Harikumar, N.Sankar
Reliability and Accuracy Analysis Accuracy is defined as the average value of calculated settlement divided by measured settlement. A value of this ratio equal to unity represents the best possible accuracy. Reliability is defined as the percentage of the cases for which the calculated settlement is greater than or equal to the measured settlement. A value approaching 100 percent represents the most desirable characteristics of reliability. The results of reliability and accuracy studies on the ANN model and other techniques of settlement calculation are shown in Fig.6.
0 1 2 3 4 5 6 7 8 9 100
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Fig. 6 Accuracy and Reliability analysis on models It can be seen that the ANN model performance lies very close to the ideal condition of Reliability 100% and accuracy 1, as compared to the other methods of settlement calculation. For all the other models, as is evident, only one of the two conditions of reliability or accuracy is satisfied. That is, as the reliability of the model increase, the accuracy decreases and vice versa. An optimum condition connecting accuracy and reliability is observed in case of the ANN model. Features of the Program interface The front end of the program was coded in Microsoft visual basic 10.The basic features of the interfaces include: a Illustrations and Bubble help feature aiding easy data entry. b. IS 8009(I)-1976, for settlement of shallow foundations subjected to symmetric static vertical loads, included for ready reference. c. ANN sensitivity and accuracy charts included. d. User action and error messages minimizing data entry error. e. Access to ANN database and the C program for ANN training. CONCLUSIONS An ANN based application for settlement prediction is attempted in this paper. The system concentrates on the settlement of shallow foundations, taking into account the significant factors affecting their settlement. The front engine is developed in MS Visual Basic 10 and the back-end coding
is performed in Turbo C compiler. The results from the model are compared with the results obtained from four conventional settlement calculation techniques that are widely in use. Sensitivity analyses performed on the model indicate consistent and accurate settlements, which are in perfect agreement with geotechnical theories. The RMSE and MAE of the model were also found to be lesser compared to the other methods. Reliability and accuracy studies confirm that the ANN model exhibits an optimum combination of reliability and accuracy. Hence the ANN model proves to be superior to the conventional methods in settlement prediction. REFERENCES 1. Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986), Learning internal representation by error propagation, Parallel Distributed Processing, D. E. Rumelhart and J. L. McClelland, eds., MIT Press, Cambridge 2. Vargas, M. (1961), Foundations of tall buildings on sand in Sao Paulo, Proc., 5th Int. Conf. On Soil Mech. & Found. Engg., Paris, 1, 841-843. 3. Levy, J. F. and Morton, K. (1974), Loading tests and settlement observations on granular soils, Conf. Settlement of Structures, Cambridge, 43-52. 4. Burland, J. B., and Burbidge, M. C. (1985), Settlement of foundations on sand and gravel, Proc. Institution of Civil Engineers, London, 78-Part 1, 1325-1381. 5. Picornell, M. and del Monte, E. (1988), Prediction of settlements of cohesive granular soils, Proc., Measured Performance of Shallow Found., Geotech. Special Publication No. 15, ASCE, Nashville, Tennessee, 55-72. 6. Papadopoulos, B. P. (1992), Settlements of shallow foundations on cohesionless soils, J. Geotech. Engrg., ASCE, 118(3), 377-393. 7. Wahls, H. E. (1997), Settlement analysis for shallow foundations on sand, Proc. 3rd Int. Geotech. Eng. Conf., Cairo, Egypt, 7-28. 8. Maugeri, M., Castelli, F., Massimino, M. R. and Verona, G. (1998), Observed and computed settlements of two shallow foundations on sand, J. Geotech. & Geoenv. Engrg., 124(7), 595-605. 9. Basheer, I. A. , Reddi, L. N. & Najjar, Y .M. (1996), Site Characterization by Neuronets - An Application to the Landfill Siting Problem, Ground Water , Vol. 34, N o. 4, pp 610-617. 10. Meyerhof, G.G.(1965), Shallow foundations, J. Soil Mech. & Found. Div., 91(SM2), 21-31. 11. Schultze, E., and Sherif, G. (1973), Prediction of settlements from evaluated settlement observations for sand, Proc.8th Int. Conf. Soil Mechanics & Found. Eng., 1, 225-230. 12. Bowles, Joseph E. (1977), Foundation Analvsis and Design, 2nd ed., McGraw-Hill, New York. 13. Terzaghi, K., and Peck, R. B, (1948), Soil mechanics in engineering practice, Wiley, New York
Proceedings of Indian Geotechnical Conference December 13-15, 2012, Delhi (Paper No. J 1012)
YİELD DESİGN APPLİED TO EARTH RETAİNİNG STRUCTURES B. Simon, Scientific director, TERRASOL, [email protected] ABSTRACT: Yield Design Theory has been worked out by Professor Salençon as a sound general framework for assessing the stability of any structure when the failure criterion of its constituent material is known. The exterior approach defined within this frame proves well suited to geotechnical structures and has been implemented in Talren4 software. Some applications are given for different kinds of earth retaining structures. They cover the evaluation of limiting active or passive pressures along retaining walls, stability of double sheet pile walls and cellular cofferdams, stability of soil nailed structures and checking of the absence of interaction between a tieback wall and its anchors. INTRODUCTION Yield Design Theory has been worked out by Salençon [1, 2, 3]. It comes as a rigorous comprehensive framework for assessing the stability of any structure when the failure criterion of its constituent material is known. It provides lower and upper bounds of the ultimate loads that can be sustained by the system, respectively by an interior approach, based on statically admissible stress fields, and by an exterior approach, based on kinematically admissible virtual velocity fields. The exterior approach proves well suited to geotechnical structures. It has been implemented as a new calculation method in the Talren4 software, which enables to check the general stability of geotechnical structures with or without reinforcement. It adds to other already implemented ones referred to as slice methods: the well known Fellenius, Bishop or perturbation methods. Results obtained by any of these methods can thus be easily compared in any situation. Application of the Yield Design Theory to earth retaining structures using this new tool appears quite fruitful as it overcomes some limits met with the other methods and also gives in some cases a better insight into failure mechanisms. THE EXTERIOR APPROACH BY THE YIELD DESIGN THEORY Its implementation in the Talren4 software In Talren4, the Yield Design exterior approach is restricted to the specific case of: • velocity fields representing movement of a rigid block
with respect to the rest of the supposedly stationary soil mass;
• Mohr Coulomb failure criterion: ϕσ+≤τ tanc (1)
Where τ =shear strength; c =cohesion; σ =normal stress; and ϕ =friction angle. J. Salençon [1] established that in this particular case, rigid motions of blocks limited by a succession of logarithmic spiral arcs: r(θ) = r0 e θtan φ sharing the same pole supplied the
best upper bound of the resisting work Prm and that it was possible to restrict analysis to these specific velocity fields. In this particular framework: • The velocity field is defined by the pole P of the spiral
arcs and the angular rotation velocity vector ω of the block; it should be noted that velocity, perpendicular to the vector radius, is not tangent to the block boundary, but inclined at an angle φ to it; this boundary can therefore not be assimilated to a slip surface;
• Along the plane tangent to the boundary, one observes that no stress state that is admissible with the Mohr Coulomb failure criterion (a point inside the "triangle" domain with green boundaries, Fig. 1) can contribute to resisting forces by more than an amount which depends on cohesion only. The maximum resisting contribution of soil in this movement is thus bounded. This contribution may be measured, in this specific case, as the maximum moment of resisting forces (Mrm) through the log spiral pole.
• A block can be in equilibrium only if the moment of all external forces applied on it (Me) is less than this upper bound of the moment of resisting forces (Mrm).
Fig. 1 Yield design exterior approach with Mohr-Coulomb failure criterion Considering the ratio F of these quantities (Mrm)/(Me): • When F is found less than one, the stability condition is
not satisfied, no equilibrium is possible, instability of the system is certain (the block fails). That is why the ratio F should rather be considered an instability factor or a “failure coefficient” [4].
B. Simon
• If on the other hand, F is found higher than or equal to one, one can only presume that equilibrium is possible: the system is potentially safe and the exterior approach must be complemented by the static interior approach under the same framework of Yield Design Theory to reduce this uncertainty.
The boundary of any block is a chain of logarithmic spiral arcs with one common pole and successive angle values φ equal to the friction angle of each layer. Any chain can be described by its intersections with the slope line (entry and exit points) and the centre angle θ between extreme radii. When the centre angle θ is nil, the pole is extended to infinity, the successive spiral arcs become straight lines and form a polyline: if the friction angle remains moreover constant across layers this becomes a segment and defines a simple wedge. When θ is positive and the friction angle in any layer is nil, then the boundary is circular. The user may choose to scan log spiral arcs with either upwards or downwards concavity.
Fig. 2 Boundary of a block as a chain of logarithmic spiral arcs sharing same pole Benefits of the yield design calculations The Yield Design exterior approach offers great advantages over the well-known “slice methods”: • No complementary assumption is required further than
the choice of the appropriate failure criterion for materials under consideration;
• It always provides upper bounds of the extreme loads: the failure load is always overestimated, which strongly characterizes this approach with respect to the Fellenius, Bishop or perturbation methods which by introducing additional constraints or weakening others do not make it possible to decide on the excess or default character of the estimated failure load;
• Its capacity to take into account situations where traditional methods are generally at fault: passive pressure equilibrium, inclined loads, overhangs, stiff reinforcing elements.
EXAMPLES OF APPLICATIONS Estimation of active and passive earth pressures Introduction of loads (especially non vertical) is a difficult matter in all stability calculations based on slice methods as the influence of loads on the distribution of stresses along failure surfaces depends on other factors than solely the failure criterion. Some arbitrarily chosen assumptions about
diffusion of loads are thus generally taken into account in these methods. Contrary to this, no other assumption needs to be made with the Yield Design method. The contribution of any point load applied to the block under study or of any surface load applied on slope boundary between its extremities is simply added to the appropriate driving or resisting cumulative moment with respect to the common pole of log spiral arcs. A straightforward application is the estimation of limiting active or passive earth pressures along the face of a cohesionless mass. Assuming this pressure is inclined at an angle δ with the normal to this face, the limiting profile is obtained by finding the maximum value σmax of a triangular distributed load in an active state equilibrium (Fig. 3a, where gravity is a driving force) or a passive state equilibrium (Fig. 3b, where gravity is a resisting force) under the condition that failure coefficient is equal to 1. Earth pressure coefficient values obtained in this way can be compared to the rigorous numerical solutions established by Kérisel and Absi [5] for any similar boundary conditions. Values obtained by the Yield Design exterior approach differ by only a few percent from these rigorous values. It is moreover observed that active earth pressure coefficient values are always underestimated while passive earth pressure coefficient values are always overestimated. This finding is consistent with the use of an exterior approach which gives upper bounds of the failure loads. With noticeable δ inclination, the most unfavourable block geometry differs from the wedge shape associated with Coulomb’s theory or Cullman’s graphical solution.
Fig. 3a and 3b Estimation of limiting active and passive pressures (φ’ = 30°, δ / φ’ = +2/3 and -2/3) This application can be readily extended to cover: • any layered soil mass; the limiting pressure profile can
be drawn layer by layer, starting from the surface. It resolves into a continuous broken line quite opposite to the discontinuous line routinely obtained by assimilating overlying layers as surcharge loads applied on top of underlying layers;
• any slope geometry as for instance when a shoulder is left against a diaphragm wall to improve earth passive pressure in an excavation;
• any situation where seepage takes place within the soil mass; seepage forces iγw are simply to be added to the
Yield design applied to earth retaining structures
gravity forces in the cumulative moment of all external forces Me with respect to the common pole;
• a pseudo-static approach of stability under seismic conditions; contribution from forces arising from the horizontal and vertical seismic coefficients ah and av is added to the cumulative moment Me of external forces; limiting pressure profiles obtained this way generalize those obtained by the Mononobe-Okabe analytical formulations.
Overall stability of double sheetpile walls and cofferdams Double sheet pile walls (or cellular cofferdams) are sometimes used to enable dry excavation in coastal or fluvial conditions. They have to resist the differential water pressure between upstream and downstream sides. With reduced distance, interaction between upstream and downstream walls can no longer be neglected and determine very complex stress conditions in the fill between them [6]. Stability is to be checked considering the whole domain. Yield Design method is well adapted for this analysis: differential water pressure which acts as a driving force on the system is to be balanced by resisting forces arising from gravity and cohesion (if any) for the work to be safe. In the example of Fig. 4, systematic search for blocks with their extremities at wall toes and upwards concavity finds the degenerated log spiral θ = 0 as the one giving the lowest failure coefficient: this corresponds to a plane slide mechanism at cofferdam basis. If this search is extended to log spiral with downwards concavity, it appears that a less favourable block is found having a centre angle θ = 60° in that particular case. This mechanism is the same as the X-convex one described by Brinch-Hansen [7].
Fig. 4 Scanning spirals with upwards/downwards concavity Stability of soil nailed walls Incapacity of slice methods to deal with overhangs does not hold for the block boundaries checked by the Yield Design method. Overhangs are obtained whenever the log spiral pole elevation is lower than the entry point. The Yield Design Theory still applies without any need for further assumption. When checking the stability of a soil nailed wall at every construction stage, one may thus find out that the critical mechanism during the second excavation stage is the one described on Fig. 5a: the log spiral pole comes close from the first row of nail direction, meaning that its contribution to the resistance moment Mrm is drastically reduced. This potentially unsafe situation is indeed what observation has unfortunately confirmed on many works on progress; this
gives credit to the prescriptions made in the French national standard for soil nailed walls [8] for that particular stage of construction. One may also find that vertical face stability is at risk at any newly excavated stage (Fig. 5b).
Fig. 5a and 5b Some unfavourable mechanisms found during construction of a soil nailed wall The French national standard [8] also calls great attention upon mechanisms extending at a very short depth under the wall toe. As an approximate way to meet that requirement, it suggests that the soil located downstream of the toe vertical plane is replaced by the limiting passive earth pressures that can develop if this plane moves downwards and then that stability of mechanisms ending in any point of this toe vertical plane is checked (Fig. 6). The same design steps can of course be followed using the Yield Design exterior approach which can provide a fairly good estimate of the limiting passive earth pressure in a first step and integrate this pressure profile in the stability calculation of the upstream part of the model in a second step.
Fig. 6 Two-block analysis of a soil nailed wall The most unfavourable mechanism is obtained by combining downstream and upstream solutions ending at the same point in the toe vertical plane. This procedure can also be applied to the 3-piece partition defined by the vertical plane originating at wall toe and another vertical plane located just behind nail ends. The stability calculation of the middle block incorporates the limiting active earth pressures on the upstream vertical boundary and the limiting passive earth pressures on the downstream vertical boundary. The different mechanisms obtained in the same example are compared on Fig. 7. It is concluded that associating the solutions in all 3 parts of the model (each obtained with a reduction factor F on c and φ around 1.4) gives a potentially more unfavourable mechanism than the 1-block mechanism commonly used (F around 1.7) or the 2-block mechanism suggested by French national standard (F around 1.5).
B. Simon
Fig. 7 Comparison of the most unfavourable block or block-combination boundaries Stability of a tieback embedded wall In order to verify the absence of interaction between an embedded wall and its anchorage (grouted anchorage, dead weight anchorage or anchor wall), Kranz [9] suggested to check the stability of the soil domain defined by a cutting made along the wall interior side (Fig. 8).
Fig. 8 Stability of a tieback embedded wall In this vertical plane, the action of the downstream soil mass consists of: • the reaction of the wall against the soil which balances
the total stresses exerted by the soil on the wall; • the forces Pi in any anchor row, which are external
actions to the investigated domain. The limiting value of the anchorage load in any anchor row is obtained as the lowest value Pdst of the force causing instability of all blocks ending in any point m of the cutting plane OB. Usually these blocks start from a specific point Ai attached to each anchorage and action of the soil mass located upstream from the vertical plane containing this point is replaced by the limiting active earth pressures on segment αi Ai. The block bottom boundary may be assumed to be a segment, a circular arc or a log spiral arc. With the segment assumption, limiting Pdst value comes straight from the force balance equation. With the circular arc assumption, one slice method is to be used provided that it can also incorporate the non vertical loads applied on either block side. The Yield Design exterior approach is perfectly suited to check the
stability of blocks [αi Ai m O] with the log spiral assumption. Its foremost benefit is to enable exploration of blocks with bottom boundaries of either positive or negative concavity. With the Yield Design approach, blocks defined by the negative concavity log spiral assumption most often lead to lower limiting destabilizing forces than with the other postulated shapes. The strong theoretical consistency of the exterior approach means that these values are true upper bounds and as a consequence it must be concluded that other commonly used approaches may sometimes err on the unsafe side. Safety factors used in conjunction with these other methods have up to now certainly compensated and hidden this potentially unsafe feature. CONCLUSIONS A new calculation method corresponding to an exterior approach under the sound theoretical framework of the Yield Design Theory by Salençon has been implemented in the software Talren4. This development has been limited to the Mohr-Coulomb failure criterion together with rigid virtual motion of one block bounded by log spiral arcs. Quite a wide range of applications to earth retaining structures has been illustrated to demonstrate the extended capacity brought by the Yield Design Theory. It covers: • earth pressures under any slope geometry, soil layering,
seepage conditions and/or set of seismic coefficients; • the stability of double sheet pile walls or cellular
cofferdams; • a sound assessment of the difficulties met during
construction of soil nailed walls as well as a better approach of the most unfavourable failure mechanism likely to develop at some short depth below the wall toe;
• a most suited frame to check the interaction between an embedded wall and its anchorages.
REFERENCES 1. Salençon J.(1983) « Calcul à la rupture et analyse limite
», Presses de l’Ecole Nationale des Ponts et Chaussées, Paris.
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