Geosynthetics and Geosynthetic-Engineered Soil Structures, Symposium sponsored by the ASCE Engineering Mechanics Division, honoring Prof. R.M. Koerner, McMat 2005, Baton Rouge, Louisiana June 2, 2005.

Limit analysis of reinforced soils and limit loads on reinforced soil slabs

Radoslaw L. Michalowski1, Fellow ASCE, and Chardphoom Viratjandr2

ABSTRACT Soil reinforcement is a practical solution to construction over weak soils. Application of the kinematic approach of limit analysis to reinforced soil is reviewed, both for traditional reinforcement with strips or sheets of geosynthetic material and for fiber-reinforced soils. The development of the failure criteria for isotropic and anisotropic fiber composites is presented. The kinematic approach is applied to evaluating limit loads on reinforced foundation soils. Patterns of reinforced soil failure determined in earlier laboratory tests are utilized. A method for practical calculations of the bearing capacity is presented, and the depth of placement of the geosynthetic layers is recommended. The method is extended to two-layer foundation soils with the upper layer reinforced either with geogrids or fibers, or with both. Preliminary results indicate that fiber reinforcement can be as effective as geogrids. Application of this concept is envisioned in construction of unpaved roads, temporary aircraft parking facilities, or when placing footings over weak soils. Foundation soils with combined reinforcement include layers of geogrid placed in the fiber-reinforced fill, to produce a slab capable of carrying large loads. To analyze the limit state of the slab, a model of fiber-reinforced soil was used first to determine the properties of the fiber-reinforced fill. The contribution of the geogrid to the strength of the slab was then determined by considering two possible modes of failure: slip and tensile rupture. The framework of the kinematic approach of limit analysis was used to arrive at the limit loads on reinforced slabs. INTRODUCTION Application of limit analysis to reinforced soils proved to be a robust method for determining the strength of reinforcement necessary to avoid collapse of slopes or retaining walls. The kinematic approach of limit analysis as applied to reinforced soil is reviewed in

1 Professor, Dept. Civil & Env. Engrg., University of Michigan, 2340 G.G. Brown Bldg., Ann Arbor, MI 48109-2125; tel: 734 763-2146, fax: 734 764-4292, e-mail: rlmich@umich.edu. 2 Graduate Research Assistant, Dept. Civil & Env. Eng., University of Michigan, Ann Arbor, MI 48109-2125.

the paper. Both traditional reinforcement in the form of strips or sheets of geosynthetic material and the fiber reinforcement are considered.

Stability of reinforced foundation soils was addressed recently (Michalowski 2004) using the kinematic approach of limit analysis. Of particular interest here is mitigating the effect of weak clays overlaid by a layer of granular soils, a practical problem encountered in the construction of roads and airfields, but also a problem appearing in the placement of foundation footings. Preliminary results for foundation slabs reinforced with fibers and combined fiber and geogrid reinforcement are presented. LIMIT ANALYSIS OF REINFORCED SOIL This section is not meant to be a comprehensive state-of-the-art report on application of the kinematic approach of limit analysis to reinforced soils, but it is a description of the essential elements of the method. The theorems of limit analysis (Hill 1951, Drucker et al. 1952) have been used intensely in plastic forming of metals, structural engineering, and in geotechnical engineering, but applications in the latter have not been universally accepted. This is surprising, considering that the alternative is the less rigorous ‘limit equilibrium method.’ We review here the kinematic approach as it is relatively straightforward in soil reinforcement applications, whereas the static approach cannot be easily used because of the difficulties with constructing statically admissible stress fields. Introducing kinematically admissible strain rate field k

ijε& and associated stress kijσ ,

one can prove that, in any kinematically admissible mechanism, the rate of plastic internal work is not less than the rate of work of true external loads (1) k k k

ij ij i i i iV S V

dV T v dS X v dVσ ε ≥ +∫ ∫ ∫&

The proof of this inequality requires that the material conform to a convex yield condition (such as the Mohr-Coulomb function) and the deformation be governed by the normality rule. The inequality in (1) indicates that if a kinematically admissible collapse mechanism can be found for which the rate of work of external forces exceeds the rate of internal (dissipated) work, the structure will collapse. This is essentially the kinematic theorem of limit analysis. The contribution of the reinforcement to the stability of a reinforced soil structure can be included on the left-hand-side of inequality (1) as a dissipation term due to tensile rupture of the reinforcement or due to slip of the reinforcement within the soil mass. The dissipation rate in a unit of volume of soil can be calculated as (see, e.g., Davis 1968) 1 3( ) cosd cε ε φ= −& & (2)

40

where 1ε& and 3ε& are the rates of major and minor principal strains, and c and φ are the cohesion and the internal friction angle, respectively. If the deformation localizes, the dissipation rate per unit area of the failure surface becomes [ ] cosd c v φ= (3)

with [v] being the magnitude of the velocity jump vector. If a reinforcement layer intersects a failure surface, the rate of dissipation is dependent on the mode in which the reinforcement fails. The graph in Fig. 1 illustrates a reinforcement layer intersecting a failure surface interpreted here as a shear band of thickness t. If the reinforcement of strength Tt (force per unit width) fails in tension (or ruptures), the work dissipation rate per unit width of reinforcement can be calculated as (Michalowski 1998)

/ sin

0

| | [ ] cos( )t

t l tD T dx T vη

ε η φ= =∫ & − (4)

where t is the thickness of the shear band, η is the angle of cross section (Fig. 1), and | |lε& is the rate of longitudinal strain of reinforcement (the modulus of lε& is taken to assure positive work).

φ

t

x

ηη−φ

A B

[v]GeosyntheticReinforcement

Velocity "Jump"

Shear Band

Figure 1 Reinforcement intersecting a shear band.

When multiple layers of reinforcement are used, for instance in reinforced slopes of height H, it may be more convenient to introduce an average (or distributed) strength kt defined as

tt

nTkH

= (5)

where n is the number of layers. Such strength distributed over the height of the slope makes it possible to calculate the dissipation rate in the reinforcement during failure using a relatively straightforward integration, rather than summing up the dissipation in individual layers. This is illustrated in Fig. 2. However, this convenient manner of calculating internal

41

work in inequality (1) may not be accurate enough if a small number of layers is used, or, when the reinforcement slip failure mode occurs. Design of the reinforced soil structures entails determining the strength of the reinforcement to be used and also the extent of it (length). While the calculations of the reinforcement strength yield a rigorous bound, the calculations of the length (using kinematic approach) lead typically to a reasonable but approximate result. This is because the length considerations require determining the pull out force, which can only be estimated approximately, since the stress acting on the reinforcement is not known. The pull out force in the reinforcement layer i in Fig. 2(a) can be calculated, per unit width, as 2pT elμσ= (6)

where μ is the friction coefficient between the reinforcement and the soil, σ is the average normal stress on the reinforcement’s “effective” part, and le is the effective length of the reinforcement as indicated in Fig. 2(a).

φ

O

H

B

CA

D

θ

β

θ

o

hθ

r

r

rh

o

Vφ

Vo

cos φcos(θ − φ)

dθr

rdθ

.

θ

(a)

(b)

ω

Lle

i

iz

layer i

Figure 2 Reinforced slope: (a) analysis with discrete reinforcement layers, and (b) infinitesimal element for integration of the internal work using average strength.

Including the slip mode of reinforcement in the analysis is not a trivial task as the effective length le is dependent on the mechanism itself, which, in turn, has to be found during an optimization procedure. The magnitude of the average normal stress acting on the reinforcement is typically set to be equal to the overburden stress. Consequently, the rate of internal work during pull out of a single layer can be calculated similar to eq. (4), with the exception that the tensile strength is now replaced with the pull out force Tp (per unit width of reinforcement, as in eq. (6))

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[ ] cos( )pD T v η φ= − (7)

Complexities in the analysis of reinforced soils are introduced by the fact that there may be more than one realistic failure mode and that during failure not all reinforcement layers may be failing in tension or pull out mode; rather, some may be ruptured and others may slip (Michalowski 1997a). Economic design of reinforcement requires that the safety of the structure is comparable with respect to both tensile failure and slip, or any combination of thereof. Introducing fiber reinforcement into the soils requires that a model of fiber-reinforced soil be developed. For limit analysis purposes it is the yield condition that is of primary importance. The kinematic approach is then applied in a manner similar to that for soils without reinforcement. The development of the failure criterion for fiber-reinforced soils is reviewed in the next section. FIBER REINFORCEMENT OF SOILS Consider granular soil (sand) as a matrix of a fiber-reinforced composite. The length of fibers is at least one order of magnitude larger than the size of the grains, and the distribution of the fibers is uniform in space. The fibers can be distributed uniformly in all directions giving rise to a composite that is macroscopically uniform and isotropic, or a preferred bedding plane may cause the composite to be anisotropic. Isotropic fiber-reinforced sand For the purpose of deriving the failure function for the fiber-reinforced soil consider a deforming cylindrical specimen, Fig. 3(a), with the deformation governed by the normality rule and the plastic potential identical to the yield condition of the matrix. Under these assumptions no internal work is dissipated by the frictional matrix, but the non-zero term on the left-hand-side of inequality (1) will appear due to frictional slip of the fibers in the matrix.

(a)(b)

σ1

σ3

x

y

z

A

D

BC

O

dV

n

dS

θ0

αθ0

R0

σ1

σ3

r

θ

Figure 3 Fiber-reinforced soil: (a) cylindrical specimen, and (b) integration space.

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Consider triaxial compression of the specimen in Fig. 3(a). Once the plastic state is reached, the fibers slip within the matrix material (sand), and the slip work can be determined, assuming fibers to be inextensible. The process of integration of this work rate becomes tractable if the specimen is mapped onto the integration space, Fig. 3(b), where all fibers are moved (without rotation) so that they all originate at center O. The fibers are not shown in Fig. 3(b) so as to not obscure the geometry of the integration space. Most fibers in the specimen are subjected to tensile stress, except for portions of the specimen where the matrix strain in the direction of fibers is compressive. The surface separating the portions of the specimen with fibers in tension and that with fibers in compression can be determined by angle of zero radial strain. This surface is conical in shape and its location depends on the dilatancy angle of the matrix material (surface BCO in Fig. 3(b)). Finally, the yield condition of the fiber-reinforced composite can be found from the balance of the work rate of the “external” load and internal dissipation

1 ( )ij ij ijV

D dV

σ ε ε= ∫& & V (8)

where the integral on the right-hand-side denotes the dissipation rate in the entire specimen and V is the specimen volume. The details of this procedure can be found in Michalowski and Zhao (1996) for a plane strain process, and in Michalowski and Čermák (2003) for an axi-symmetric process. As the result of using the “macrohomogeneity” rule in eq. (8), the following yield condition was derived for the fiber-reinforced sand (Michalowski and Čermák 2003)

1 1 3

tan 60

6 tanw p

w

M Kf

Mρη φ

σρη φ

σ+

= − =−

(9)

where 0sinpM K θ= (10)

and

10

1 sin and tan1 sin 2

pp

KK φ θ

φ−−

= =+

(11)

Other parameters in eq. (9) are: ρ and η are the volumetric content of fibers and their aspect ratio, respectively, and φ and φw are the internal friction angle and the fiber-matrix friction angle, respectively. For a less practical case, when the fibers fail in tension, the yield condition becomes

2

2 2 0 02 1 1 3 3 1 3(1 ) ( ) 0

3 6 tanp pw

f K K Mρσ ρσσ σ σ σ σ ση ϕ

= + − − − + + =M (12)

44

where σ0 is the yield point of the fiber material. Since it was assumed that the fibers are distributed uniformly in all directions, the resulting yield condition is isotropic, Fig. 4. The stress space in Fig. 4 has co-ordinates 1 3( ) /p 2σ σ= + , 1 3( ) / 2q σ σ= − , and xyτ , all normalized by 0ρσ . The practical case of application of fiber reinforcement of soils involves fiber slip. As function (9) is linear in 1 3,σ σ , this function can be described with an equation similar to the Mohr-Coulomb function with a “macroscopic” internal friction angle φ incurred from eq. (9) as

1 tan 62 tan

6 tanw p

w

M KM

ρη φ2πφ

ρη φ− +

= −−

(13)

Application of fiber reinforcement in soils will then involve a simple calculation of the macroscopic friction angle φ , and the use of well-established methods of analysis for the soil governed by the Mohr-Coulomb yield condition.

,

0.4

0.4

1.0

pρσo

qρσo

composite

matrix(linear)

0.6

rzτρσo

Figure 4 Yield surface for isotropic fiber-reinforced sand. Development of the yield condition for an anisotropic fiber-reinforced soil will be outlined in the next subsection. Anisotropic fiber-reinforced sand The model of fiber-reinforced sand presented above included uniform and isotropic distribution of fibers. We now consider a distribution that is macroscopically uniform but

45

anisotropic. This is often a practical case because the fibers assume preferred orientation due to the method of their deposition and compaction (e.g., rolling). A realistic distribution

-0.5

x

y

z

x

y

θρ(θ)

(a)

(b)

Figure 5 Anisotropic distribution of fibers with horizontal preferred plane (Michalowski and Čermák 2002).

of fibers is presented in Fig. 5. This distribution has a horizontal (x,z) preferred plane, and it includes no vertical fibers. The mathematical form of this function can be written as ( ) cosnBρ θ ρ= θ (14)

where B and n are the parameters of the distribution, and ρ is the volumetric content of fibers defined as the ratio of the fiber volume Vf to the volume of the entire specimen V

fVV

ρ = (15)

The composite becomes “more anisotropic” with an increase in n. The volumetric content of fibers can be written as

/ 2

/ 2

( )o

o

d

d

π

π

ρ θ θρ

θ=

∫

∫ (16)

The distribution of fibers can change in the process of deformation; however the volumetric content must remain ρ . Description of such a process will result in varying B and n in eq. (14). However, the two parameters are not independent, since, at any time in the

46

process, they must satisfy eq. (16). Notice that the composite becomes isotropic when n = 0, and, for this special case, B = 1. Development of the failure surface for anisotropic distribution of fibers is now more elaborate (Michalowski 1997b, Michalowski and Čermák 2002), and we present only the final result here as the surface in the stress space, Fig. 6.

0.20.2

pρσo

qρσo

composite

matrix(linear)

xyτρσo

(anisotropic)

0.2

Figure 6 Yield surface for anisotropic fiber-reinforced sand.

Both isotropic and anisotropic fiber-reinforced sand will be considered when analyzing bearing capacity of fiber-reinforced soils. Calculations of the bearing capacity of foundation soils reinforced with “traditional” reinforcement and the fiber reinforcement is discussed next. GEOSYNTHETIC-REINFORCED FOUNDATION SOILS Limit loads on reinforced foundation soils have been investigated for 30 years now, yet no method has been generally accepted in design. The strength of foundation soils reinforced with metal strips was investigated by Binquet and Lee (1975) using model strip footings (3-inch in width) over a granular soil. They indicated that both the bearing capacity and settlement can be improved by a factor of 2 to 4 compared to unreinforced sand. Huang and Tatsuoka (1990) distinguished two effects dependent on the reinforcement length: a “deep footing effect” associated primarily with short reinforcement, and a “wide slab effect.” The benefit of geosynthetic reinforcement for soil improvement was confirmed in field-scale experiments on square footings by Adams and Collin (1997).

47

(a)

(b)

(c)

(d)

Figure 7 Failure pattern of a foundation soil reinforced with one layer of geosynthetic strips: (a) vertical displacement increments, (b) horizontal increments, (c) shear strain increments,

and (d) volumetric increments (Michalowski and Shi 2003).

Predictions of limit loads on reinforced foundation soils have been attempted primarily in the context of unpaved roads (Giroud and Noiray 1981, Burd 1995), and mitigating an adverse influence of a soft soil underlying a stronger fill (Love et al. 1987). While reinforcement affects the mechanism of failure to some degree, the characteristic features of the collapse process resemble those of unreinforced soil, with clear regions of nearly rigid displacement and narrow bands of localized strain (Michalowski and Shi 2003). A pattern of displacements within sand reinforced with one layer of geosynthetic strips (laboratory-scale experiments) is shown in Fig. 7; the system is loaded with a near-the-peak force.

48

(a)

(b)

(c)

(d)

Figure 8 Advanced phase of failure pattern: (a) vertical displacement increments, (b) horizontal increments, (c) shear strain increments, and (d) volumetric increments

(Michalowski and Shi 2003).

The color (or gray) scale in Fig. 7 indicates the intensity of displacements (a,b) or strain increments (c,d), and it is meant to be interpreted here qualitatively. The vertical displacements in Fig. 7(a) clearly indicate the regions of the kinematic pattern associated with the classical Prandtl (1920) static solution. Horizontal displacements in Fig. 7(b) confirm this characteristic pattern, but there is a clear influence of the reinforcement that prevents the very top layer of soil (that above the geosynthetics strips) from moving sideways. The presence of reinforcement gives rise to a horizontal shear band along the geosynthetic layer. A pair of symmetric shear bands propagates from the corners of the footing; these shear bands trace a contour of the region of distributed shear clearly seen in Fig. 7(c). As the shear in granular materials is associated with dilatancy, the contours of the region where volume increases (Fig. 7(d)) coincide with those for shear strain.

49

The advanced phase of the reinforced foundation soil failure process is shown in Fig. 8. The pattern is no longer symmetric, and a substantial region of sand on the right-hand-side of the footing is moving up and sideways. The reinforcement then slips on the left-hand-side (pull out), leaving a distinct pattern of shear banding (Fig. 8(c)).

Reaching the peak force in laboratory experiments is often associated with a

symmetric deformation, as in Fig. 7, followed by a “one-sided” mechanism in a post-peak regime, Fig. 8. These patterns resemble those for unreinforced soils. Hence, it seems reasonable to explore the modeling of reinforced foundation soils using mechanisms qualitatively similar to those for soils without reinforcement (Michalowski 1997c), but let the specific geometry be modified in a process of optimization to account for the presence of reinforcement. The kinematic approach of limit analysis was proposed recently for calculations of bearing capacity of reinforced foundation soils, and the solutions are described in Michalowski (2004) and Michalowski and Viratjandr (2005). The highlights of those solutions are presented here.

B

V0

b

dφ

Figure 9 Failure mechanism of a foundation soil with one layer of reinforcement.

When analyzing the foundation soil collapse, two qualitatively different mechanisms need to be considered: one associated with the slip of the reinforcement in the soil, and the second one where the reinforcement ruptures. While the second one yields a strict upper bound to loads that cause failure of the foundation soil, the slip mechanism gives a reasonable estimate to limit loads, but it cannot be considered as a rigorous upper bound. This is because the pull-out force is dependent on the normal stress distribution on the reinforcement-soil interface, which is not known, and it has to be assumed a priori.

50

Figure 10 The effect of reinforcement depth (Michalowski 2004).

The collapse mechanism in Fig. 9 was considered in calculations with one slipping reinforcement layer. The calculations revealed that the contribution of reinforcement is very much dependent on the depth of reinforcement and the internal friction of the soil φ, as illustrated in Fig. 10. For each soil characterized by φ there is an optimum depth past which the mechanism is forced entirely to the soil layer above the reinforcement, and the increase in the contribution of reinforcement to the bearing capacity decreases quickly with a further increase in the reinforcement depth. The recommended depth of a single layer of reinforcement is suggested in Fig. 11(a).

The procedure for deriving an expression for the bearing capacity of reinforced

foundation soils was described in a recent paper (Michalowski 2004), and it is given here in the following form

(17)

( ) ( )1

1

1 121

ni

c c qnii

pi

dp c N nf M q N n M B N Md BMB

γμ γ μμ =

=

⎡ ⎤⎛ ⎞= + + + + +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦−∑

∑ where Nc, Nq, and Nγ are the bearing capacity factors for a strip footing over an unreinforced soil (given later in eqs. (24)-(26)), di = the depth of reinforcement layer i, n = number of layers, μ = soil-reinforcement friction coefficient, and fc = cohesion bond factor. Factors M and Mp were found through numerical simulations using limit analysis, and they are provided below. The magnitudes of these factors depend on the internal friction angle of the soil φ, and the number of reinforcement layers; for one layer of reinforcement

1.31.6 (1 8.5 tan )M φ= + (18)

51

Figure 11 Recommended depth/spacing of reinforcement for 1, 2, and 3 layers.

and

21.5 1.25 10pM φ−= − ⋅ (19)

here φ is in degrees. It is emphasized that the expressions in Eqs. (18) and (19) are wapplicable under conditions consistent with assumptions used in numerical calculations: the reinforcement length b is 4 times the footing width, and its depth is in the recommended range indicated in Fig. 11. For 2 layers of the reinforcement coefficients M and Mp can be approximated with the following expressions

1.31.1 (1 10.6 tan )M φ= + (20) And

30.75 6.25 10pM φ−= − ⋅ (21) and for 3 layers

1.30.9 (1 11.9 tan )M φ= + (22) and

52

30.50 6.25 10pM φ−= − ⋅ (23) φ in degrees). The expressions in Eqs. (20) through (23) are applicable for the

Coefficients Nc , Nq , and Nγ vary dependent not only on φ, but also on c/γB and q/γB

(reinforcement length b/B = 4 and its spacing as recommended in Fig. 11(b,c). (as they do for footings without reinforcement, Michalowski 1997c). However, to preserve tractability of the design recommendations, it is recommended that these coefficients are taken as functions of φ only (conservative assumption). Following Prandtl (1920) and Reissner (1924)

( )1 cotc qN N φ= − (24)

2 ttan4 2qN e anπ φπ φ⎛ ⎞= +⎜ ⎟

⎝ ⎠ (25)

c in Eq. (24) is a convenient form represented as a function of Nq; historically, however, N

Prandtl determined the contribution of cohesion to the bearing capacity first (neither Prandtl nor Reissner used the notion of bearing capacity factors). Of known proposals for Nγ , a recent solution (Michalowski 1997c) is suggested here

0.66 5.11tan tanN e φγ φ+= (26)

he formula in Eq. (26) is a closed-form approximation of the numerical results based on

The expression in eq. (17) takes into account the slip of the reinforcement as the

Tlimit analysis, and it is consistent with the method proposed here for evaluating the reinforcement effect in foundation soils. mode of failure, and will be referred to as a case of strong reinforcement. A case of weak reinforcement involves tensile failure of geosynthetics. To simplify the calculations, replace the true strength of the reinforcement layers with a distributed strength defined as

tt

Tks

=

where Tt is the strength of a single layer and s is the reinforcement spacing. To make this approximation realistic, a minimum of 3 layers needs to be used with spacing of about 0.3B for sand or gravel, and 0.2B for cohesive soils. The incipient mechanism of failure considered is similar to that in Fig. 9, and the expression for the bearing capacity can be written in the following form

12c q t rp cN qN BN k Mγγ ′= + + + (27)

where

53

tan

2(1 sin )rM eπ φ φ

φ⎛ ⎞+⎜ ⎟⎝ ⎠′ = + (28)

and the remaining coefficients are given in eqs. (24-26). Once the bearing capacity is evaluated according to eq. (17) and eq. (27), the lower of the two must be considered in design. A practical approach includes finding the appropriate strength of reinforcement and the number of layers in order to achieve a required safety level. FIBER –REINFORCED FOUNDATION SOILS Isotropic distribution of fibers Increase in the bearing capacity due to the use of the isotropic fiber reinforcement can be addressed using the model of the fiber-reinforced soil described earlier in this paper (see Fig. 4 and eq. (13)). The use of such reinforcement may be useful in granular soils overlaying a weak clay. A schematic of a failure mechanism is illustrated in Fig. 12, and it is similar to that considered earlier in Michalowski and Shi (1995), but with the upper granular layer now reinforced with fibers.

B

tFiber-reinforced sand

Weak clayD1

D2

D3

E

FA

φφ

V4

V0

V1V2

V3[V]0 [V]1

[V]2

G1 G2 G3

[V]3A'

Figure 12 Failure mechanism for a two-layer foundation soil.

54

This mechanism is constructed in such a manner so that the blocks that extend through both layers can move with a uniform velocity, thanks to the piece-wise linear geometry of the interfaces between the blocks. We first consider the isotropic fiber reinforcement with the composite being characterized with the macroscopic internal friction angle φ , eq. (13). The bearing capacity of the foundation soil in Fig. 12 is bound by bearing capacities of the homogeneous fiber-reinforced sand, and the bearing capacity of the uniform foundation

200

160

120

80

40

00 1 2 3 4 5

t/B =

5.0

t/B = 4.0

t/B = 3.0

t/B = 2.0

t/B = 1.0

t/B = 0

cu

γB

γBp

prsand

p clay

Figure 13 Bearing capacity of foundation soil with the upper fiber-reinforced

layer (φ = 45°) of thickness t over weak clay. clay. This is illustrated in Fig 13 for fiber-reinforced soil with φ = 45° over a layer of weak clay characterized by c/γB (c – cohesion): the horizontal line describes the bearing stress (p/γB) of a uniform fiber-reinforced sand, whereas the straight line (marked t/B = 0) represents the bearing stress on the clay alone (the slope of this line is 5.14). The bearing capacity for the specific combination of the two soils must be in the known range between the two extreme cases. Calculations performed repeatedly for a variety of combinations of the clay depth and material properties revealed that it is possible to interpolate between the two extreme cases, but the interpolation is not linear. The following interpolation rule was found to work with good accuracy

tanA

clay clayrsand

crit

tp ppp B

tB B B BB

φ

γ γ γ γ

⎡ ⎤⎢ ⎥⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎢ ⎥= + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(29)

55

where pclay and prsand are the extreme values of the bearing stress on uniform clay and fiber-reinforced sand, respectively, t is the thickness of the reinforced sand layer, and A is an interpolation parameter found from parametric analysis. Parameter A is dependent on overburden stress (q/γB) and φ . At the time of writing this paper we had limited computational results with φ more typical of unreinforced sand (for instance A = 3.25 when φ = 30° and q/γB = 0). It is expected that the interpolation rule as in eq. (29) will allow avoiding a large number of charts that otherwise would be needed to present results for a wide range of parameters. Anisotropic fiber reinforcement Calculations for anisotropic fiber-reinforced soils are similar, in principle, to those in the previous Section, except that now the internal friction angle at a failure surface depends on the inclination of this surface. Consequently, the optimization process in the search for the least upper bound to the limit load becomes more elaborate. COMBINED FIBER AND GEOSYNTHETIC-REINFORCED FOUNDATION SOILS The method used to calculate the bearing capacity of the foundation soil was extended to include combined fiber and geogrid reinforcement. The specific problem solved is that in Fig. 12 where the upper layer of granular soil is reinforced with both the fibers and 3 layers of geogrid. The soil underneath is a weak clay, described by c/γB (c = cohesion), and is assumed to behave in an undrained manner. Preliminary results are illustrated in Fig. 14. The granular fill is characterized with the internal friction angle φ = 30°, and for fibers:

tan wρη φ = 0.5 (φw = fiber-matrix interface friction angle). The three solid lines in Fig. 14 show the bearing stress (p/γB) for cases of no reinforcement, isotropic fiber reinforcement, and anisotropic fiber reinforcement (both with the same volumetric content of fibers). Clearly, the anisotropic fiber-reinforcement (with horizontal being the preferred bedding plane) leads to the largest increase in bearing capacity. The anisotropic distribution of fibers is described here by eq. (14) with n = 10.

56

0

5

10

15

20

25

30

35

40

45

50

0 0.5 1 1.5 2 2.5 3

c/÷ B

p/B

No reinforcement

3 layers of geogrid

Isotropic fiber reinforcement

Isotropic fiber and geogrid

Anisotropic fiber

Anisotropic fiber and geogrid

p/γ B

c/γB

Figure 14 Two-layer foundation soil: dimensionless limit contact stress p/γB as a function of c/γB (t/B = 0.9).

The bearing capacity of the foundation soil with the sand layer reinforced with 3 geosynthetic layers is marked with the dashed lines. The use of 3 geosynthetic layers appears to be equivalent to using anisotropic fibers. A combination of fibers and 3 layers of geogrid leads to a further substantial increase in limit stress. FINAL REMARKS The kinematic approach of limit analysis is an effective way to assess the contribution of geosynthetic reinforcement to the stability of geotechnical structures. When using this technique, two modes of reinforcement failure must be considered: tensile rupture and slip of the reinforcement in the matrix material. The former can be rigorously included in the kinematic approach, whereas the slip (or pull out) requires additional assumptions regarding stress distribution on the reinforcement-sand interface, leading to relaxing the rigorous nature of limit analysis. Still, the method yields reasonable results.

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Limit analysis is an effective method in analysis of stability whether a “traditional” reinforcement is used (in form of geogrid or geotextile), or the fiber reinforcement is considered. However, before stability analyses of fiber-reinforced structures can be attempted, a yield condition for fiber-reinforced soils needs to be known. Such yield criteria have been developed for both isotropic and anisotropic distributions of fibers. The analysis of limit loads on two-layer foundation soils indicates that fiber reinforcement may have an effect on the bearing capacity equivalent to using geogrid layers. Fiber reinforcement seems to be underutilized in geotechnical engineering practice. Acknowledgements The material in this paper is based upon work supported by the National Science Foundation under Grant No. CMS-0096167. The work was performed also while the first author was supported by the Army Research Office, grant No. DAAD19-03-1-0063, and the second author was supported by The Royal Thai Government Scholarship. This support is greatly appreciated. REFERENCES Adams, M. T. and Collin, J. G. (1997) “Large model spread footing load tests on

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