Rheology and Micromechanical Analysis of Granular Media Composed of Platy
Particles: A Step Toward the DEM Simulation of Clayey Soils
Mauricio BOTON a, Nicolas ESTRADA a, Emilien AZÉMA b, Farhang RADJAÏ b, and
Arcesio LIZCANO c
a Universidad de los Andes, Bogota, Colombia
b Université de Montpellier II, Montpellier, France
c SRK Consoulting, Vancouver, Canada
Abstract. This work was carried out in the framework of a larger project devoted to the micro-mechanical modeling of clayey soils by means of discrete element simulations. Here, we study the rheology and microstructure of granular media composed of platy particles. The particles are three-dimensional square plates, approximated as spheropolyhedra. Several samples composed of particles of different levels of platyness (related to the ratio of length to thickness) were numerically prepared and sheared up to large deformations. Specifically, we revisit and analyze in detail an ordering phenomenon described briefly in a previous paper [1] and investigate its consequences in terms of the alignment of particles and cluster formation. We find that particle alignment is strongly enhanced by the degree of platyness and leads to the formation of face-connected clusters. Due to dynamics, this spontaneous clustering is a robust characteristic of granular systems composed of platy particles even in the absence of attraction forces.
Keywords. Discrete element methods, platy particles, alignment, clustering
1. Introduction
Simulations using discrete element methods (DEM) constitute a powerful analysis tool in various domains of science and engineering. One of these domains is soil mechanics [2], for which DEM simulations are useful because they allow for exploring—based on the scale of the grains and their interactions—various phenomena that have been investigated experimentally for decades but still lack a clear and physically based explanation. Examples of such phenomena are internal friction, strain localization, strain-rate dependent behavior, creep, and stress relaxation [3].
Most investigations using DEM carried out during the past two decades have been restricted to coarse soils, while clayey soils have traditionally been left out of the scope of this analysis tool. This asymmetry can be explained by two essential differences between coarse and clayey soils. The first of these differences is particle shape. Coarse soils typically have bulky shaped particles that can be approximated by simple shapes such as disks or spheres, which are easy to deal with in numerical simulations. Clayey soils typically have platy shaped particles, which are difficult to manage efficiently in
numerical simulations. The second difference is the interaction between particles. The particle-particle interactions in coarse soils can be reduced to contact forces and prescribed as contact laws such as the linear spring-dashpot model and the Coulomb law. The particle-particle interactions in clayey soils include, besides contact forces, electrostatic repulsion and van der Waals’ attraction, which require an integration procedure over the particles’ surfaces that can be extremely time consuming in the computer. In conclusion, due to a complex particle shape and a complex interaction between particles, DEM simulations of clayey soils are difficult and, thus, rare [4–7].
This work was carried out in the framework of a larger project devoted to the mechanical behavior of clayey soils by means of DEM simulations. In [1], we presented the effect of the platy shape of particles on the mechanical behavior and microstructure of a granular material. We found that both the mechanical behavior, and the microstructure depend strongly on the platyness of the particles. In particular, we showed that the principal phenomenon underlying this dependency is the alignment of the particles along a particular direction and that this ordering phenomenon emerges even for shapes that deviate only slightly from that of a sphere.
The aim of the present work is, first, to revisit and study in detail the ordering phenomenon described in the previous paragraph and, second, to explore the consequences of this phenomenon in terms of particle-particle interactions and the emergence of local structures within the system. As in our first paper, our strategy is to single out the effects of particle shape, as compared to those of the interactions between particles. For this reason, we consider only contact forces between the particles and do not take into account neither the electrostatic repulsion nor the van der Waals’ attractive forces. In this sense, our findings are rather general and can be extrapolated to other granular materials composed of flat particles (some examples of such materials are granular soils derived from shale, edible seeds such as lentils, and industrial goods such as PET flakes).
The simulated particles are square plates, which we approximate as spheropolyhedra [8–10] simulated by means of the soft-sphere Molecular Dynamics method. The degree of platyness is described by a parameter , related to the ratio of length to thickness, and is varied systematically from , which corresponds to spherical particles, to , which corresponds to particles 17 times longer than thick. Several monodisperse assemblies were built numerically, one for each value of η, and then sheared in the quasistatic limit up to the “critical state” (as termed in the soil mechanics literature), where the memory of the initial state is fully erased and a steady shear state is reached. In this state, we analyze the particles’ orientations, the particle-particle interactions, and the emergence of local structures characterized by face-face interactions.
In the following, we introduce the numerical model in Sec. 2. In Sec. 3, we present our results, and, finally, in Sec. 4, we end with a summary of the most salient results followed by a brief discussion.
2. Numerical Model
2.1. Platy Shaped Particles
The particles are square plates with rounded edges, built as spheropolyhedra resulting from sweeping a sphere around a polyhedron. Mathematically, this
ηη = 0
η = 0.94
M. Boton et al. / Rheology and Micromechanical Analysis of Granular Media1358
corresponds to a Minkowsky addition of a sphere with a polyhedron (for some examples of non-spherical particles built as spheropolyhedra, see [11] and [8] for cylinders; [12] for particles with shapes as complex as that of a cow; [9] for cylinders, tetrahedra, and intersecting cylinders; and [13] for irregular polyhedra). Specifically, our particles are spheroplates resulting from a Minlowsky addition of a square plate and a sphere. Each particle has three kinds of constitutive entities: (1) four vertices, (2) four edges, and (3) one plane, as shown in Fig. 1(a).
Figure 1. (a) Scheme of a spheroplate and its constitutive elements. (b) Definition of the maximum and
minimum radii, and respectively.
The platyness of these spheroplates is defined as , where and are, respectively, the maximum and minimum radii of the spheroplate as defined in Fig. 1(b) ( is also called the spheroradius of the spheroplate). Note that varies from 0 for a sphere to 1 for an infinitely thin plate. The platyness is related to the more common aspect ratio through the simple expression .
2.2. Simulation Method
We employed the Molecular Dynamics method, adapted first by Cundall and Strack for simulating granular materials [14]. In applying Molecular Dynamics to spheroplates, we should distinguish the contacts between different elements (vertices, edges, and faces) of two interacting particles. Each interaction represents single or multiple contacts, each contact occurring between two elements belonging to either of the spheroplates. All possible contacts are resolved by considering two cases: a contact between two edges and a contact between a vertex and a face [15]; see Fig. 2.1.
The contact forces are calculated using the linear spring-dashpot model and the Coulomb law. Then, the interaction forces are calculated by adding the contact forces exerted at each contact point. For a detailed explanation of the contact laws employed, please see [1].
R r
R r
r
λ
η η = (R− r)/R
η = 1− 1/λ
ηη
M. Boton et al. / Rheology and Micromechanical Analysis of Granular Media 1359
(2.1) (2.2)
Figure 2.1 (a) Interactions comprising only edge-edge contacts (stars). (b) Interactions comprising only vertex-face contacts (diamonds). (c) Interactions comprising both edge-edge and vertex-fact contacts.
Figure 2.2 Sample construction and isotropic compression.
2.3. Sample Construction and Shear Test
Twelve monodisperse samples made up of 8000 spheroplates are built. The difference between these samples is their platyness, which varies from to 0.94. The spheroradius is set to and the density is set to for all samples. The normal and tangential stiffnesses are set to . Note that the particle size, density, and stiffness are typical of fine grained soils [3–7]. As recommended in [16], the damping coefficients are calibrated in order for the restitution coefficient in binary collisions to be around 0.6. However, it has been shown that this parameter has practically no effect in the rheology if the system is sheared quasistatically and if the particles are frictional; see [17, 18]. The friction coefficient is set to , which is close to that of kaolinite clay [3–7].
Initially, the particles are placed at the nodes of a cubic grid of side and each of them is randomly oriented. Then the samples are isotropically compressed under a stress ; see Fig. 2.2. Once the system attains static equilibrium, the lateral walls are removed and replaced by periodic boundaries. The samples are then sheared by imposing a constant velocity and a constant confining stress to the upper wall (see Fig. 3.1), allowing for the volume of the sample to vary during the test. The particles in contact with the walls are “glued” to them in order to avoid strain localization at the boundaries. In all simulations presented in this paper the gravity is set to zero in order to get homogeneous stress fields inside the packings.
r = 30 nmη = 0
2700 kg/m3
1.5× 10−3 N/μm
0.58 � tan(30o)2√2R
σ0
vw σw
M. Boton et al. / Rheology and Micromechanical Analysis of Granular Media1360
(3.1) (3.2)
Figure 3.1 Shear test. Figure 3.2 Shear stress ratio as a function of the shear strain for all values of .
As mentioned in Introduction, the focus of this work is on the mechanical behavior in the steady state. Therefore, the samples are sheared up to a large cumulative shear strain , where is the horizontal displacement of the upper wall and its vertical position is . Figure 3.2 shows the shear stress ratio , where the stresses and are, respectively, the shear and normal stresses at the moving wall, as a function of , for all values of the platyness . We see that all packings reach the steady state, since, at the end of the shear test, fluctuates around a mean value. As shown in [1], the shear stresses and velocities are such that the test is carried out in the quasistatic limit and the strain is homogeneously distributed in the sample.
3. Results
In this section, we present the results of our simulations and analyses. To ensure that these measures are representative of the steady state, all presented quantities correspond to the average over the last 40% of cumulative shear strain, which corresponds to shear strains from to 2.5.
3.1. Particles’ Alignment
The particle orientation is defined as the orientation of the vector , normal to the particle face. The distribution of particle orientations can be represented by the probability density function of particles whose vector is along the solid angle . Figure 4 shows for all samples at a shear strain . Firstly, we can see that, as the particles platyness increases, the distributions become more anisotropic, indicating that the number of particles aligning their faces along a particular direction increases. Secondly, we can see that, as the particles platyness increases, the principal direction of the distributions gradually changes from that of the major principal stress to the vertical direction. Thirdly, it is interesting to note that this type of ordering appears even in the samples composed of particles with very low values of platyness (e.g., for ), whose shape deviates only slightly from that of a sphere.
γ
γ
mmm
mmmΩ
τw/σw η
γ = xw/yw � 2.5 xw
τw/σw
τw
τw/σw
γ = 1.5
Pm(Ω)Pm(Ω) γ = 2
η = 0.14
yw
ησw
M. Boton et al. / Rheology and Micromechanical Analysis of Granular Media 1361
Figure 4. Probability density functions of particle orientations for all values of at a shear strain
.
3.2. Face-face Interactions Between Particles
In the previous subsection, we showed that, as the particles platyness increases, more and more particles tend to align their faces along a preferential direction. This implies that, as the particles platyness increases, the probability of forming face-face interactions also increases. Figure 5 shows the proportion of face-face interactions as a function of the platyness . We can see that rapidly increases with , from 0 to approximately 0.15, confirming that as platyness increases the proportion of face-face interactions also increases.
Figure 5. Proportion of face-face interactions as a function of the platyness . Error bars indicate the
standard deviation.
3.3. Formation of Clusters of Particles
In the previous subsection, we showed that, as the particles platyness increases, the proportion of face-face interactions also increases. This suggests the formation of local structures characterized by face-face interactions, such as clusters of particles with touching faces. Figure 6 shows various snapshots of the clusters composed by two, three, four, and five particles, in three samples with platyness = 0.80, 0.50, and 0.25.
γ = 2
ζζ
ζ
Pm(Ω)
η η
η
η
η
M. Boton et al. / Rheology and Micromechanical Analysis of Granular Media1362
We can see that, the number of clusters increases with , as also does the number of particles composing these clusters.
Figure 6. Snapshots of clusters composed by two, three and four or more particles in three samples with
platyness = 0.25, 0.50 and 0.80.
4. Summary and Discussion
In summary, by means of discrete element simulations, we revisited and studied in detail the ordering phenomenon described in [1] for monodisperse packings of 8000 particles of platy shape. The platyness was varied from 0, corresponding to a sphere, to 0.94 for particles that are 17 times longer than thick. The samples were sheared up to a large shear deformation and analyzed in the steady state.
Our results indicate that particle platyness enhances the spontaneous alignment of the particle faces. Due to dynamics in continuous shearing, this ordering phenomenon emerges even in systems composed of particles with very low platyness with a shape differing only slightly from spherical shape. This kind of ordering had previously been observed in experiments [23] and numerical simulations with two-dimensional elongated particles [24–26]. Regarding the privileged direction of alignment, we found that as platyness increases this direction gradually changes from that of the major principal stress to a direction that close to perpendicular to the shear direction. In other
η
η
M. Boton et al. / Rheology and Micromechanical Analysis of Granular Media 1363
words, the larger axis of the particles tends to align with the average velocity field). This evolution of the privileged direction of alignment is consistent with the observations reported previously [23, 24, 27].
The occurrence of face-face interactions may be attributed to this alignment of the particles due to continuous shearing. The increase of these interactions has also been observed in experiments [28] as well as in numerical simulations with elongated particles in 2D [25].
Finally, we showed that the random occurrence of such interactions leads to the formation of face-connected clusters. But even at high platyness, the proportion of face-face interactions is too low to allow for percolating clusters throughout the sys- tem. Cluster formation in this system is interesting because, among other reasons, this kind of structures are frequently observed in clayey soils such as kaolinite clays [4]. In addition, it is also surprising that these clusters emerge here uniquely as a consequence of particle shape even in the absence of attraction forces between the particles. But it is obvious that they should be more stable and reinforced in the presence of attraction forces as is the case in kaolinite clays. This is the goal of our ongoing investigation in view of enriching our model of clays. We acknowledge financial support by the Ecos-Nord program (Grant No. C12PU01).
References
[1] M. Boton, E. Azéma, N. Estrada, F. Radjaï, and A. Lizcano, Phys. Rev. E 87, 032206 (2013). [2] M. D. Bolton, in Proc. International Workshop on Soil Crushability, edited by N. Y. Hyodo M.
(Yamaguchi University, Japan, 1999), pp. 1–24. [3] J. K. Mitchell and K. Soga, Fundamentals of Soil Behavior, third edition (Wiley, 2005). [4] A. Anandarajah, Engineering Geology 47, 313 (1997), ISSN 0013-7952.A.N. [5] A. Anandarajah, Powder Technology 106, 132 (1999), ISSN 0032-5910. [6] A. Anandarajah, Computers and Geotechnics 27, 1 (2000), ISSN 0266-352X. [7] M. Yao and A. Anandarajah, Journal of Engineering Mechanics 129, 585 (2003). [8] L. Pournin, M. Weber, M. Tsukahara, J.-A. Ferrez, M. Ramaioli, and T. M. Liebling, Granular Matter 7,
119 (2005), ISSN 1434-5021, 10.1007/s10035-004-0188-4. [9] S. A. Galindo-Torres, F. Alonso-Marroquín, Y. C. Wang, D. Pedroso, and J. D. Muñoz Castaño, Phys.
Rev. E 79, 060301 (2009). [10] F. Alonso-Marroquín, EPL (Europhysics Letters) 83, 14001 (2008). [11] P. A. Langston, M. A. Al-Awamleh, F. Y. Fraige, and B. N. Asmar, Chemical Engineering Science 59,
425 (2004), ISSN 0009-2509. [12] F. Alonso-Marroquín and Y. Wang, Granular Matter 11, 317 (2009), ISSN 1434-5021. [13] S. Galindo-Torres, D. Pedroso, D. Williams, and L. Li, Computer Physics Communications 183, 266
(2012), ISSN 0010-4655. [14] P. A. Cundall and O. D. L. Strack, Géotechnique 29, 47 (1979). [15] J. Ghaboussi and R. Barbosa, International Journal for Numerical and Analytical Methods in
Geomechanics 14, 451 (1990), ISSN 1096-9853. [16] S. Luding, Granular Matter 10, 235 (2008), ISSN 1434-5021. [17] G. MiDi, The European Physical Journal E 14, 341 (2004). [18] F. Radjai, F. Dubois, and F. Dubois, Discrete-Element Modeling of Granular Materials (John Wiley &
Sons, 2011), ISBN 9781848212602.
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Behavior of Buried Pipes: Numerical Simulation vs Experimental Test
Stephane do N. SANTOSa,1 , Denise M. S. GERSCOVICH a and Bernadete R. DANZIGER
a a
Universidade do Estado do Rio de Janeiro
Abstract. Analytical and experimental methods have some theoretical and practical limitations. On the other hand, numerical methods are capable to simulate staged constructions and to analyze together materials with different constitutive models, supplying practical necessities of geotechnical projects and complementing analytical and experimental models. In this article comparisons between experimental and numerical results have been made. The numerical modeling was performed on PLAXIS 3D software, based on Finite Elements Method. The experimental test was made by Costa [1] at São Carlos Engineering School/USP and involved buried pipes undergoing loss of support or elevation in a localized region along its length. Tests have been performed with physical models comprising dry and pure dry sand and a tube resting on a rigid trapdoor base located at the center of its length. The models were equipped with devices for measuring deflections and strains in the pipe, and total stresses in soil mass and in the lower boundary of the model.
With the improvement of computational tools some limitations have been overcome, allowing the analysis of geotechnical problems under the three dimensional point of view. The present work aims to simulate numerically, under 3D optics, the arching effect in a granular soil mass induced by a localized action (loss of support or elevation). In addition, the influence of these localized actions on a flexible buried pipe was analyzed, verifying the deflections on its walls and the stress variation in its surroundings. Results of the numerical simulation were compared to ones of a campaign of instrumented tests made by Costa [1], with and without pipe, using reduced model, at São Carlos Engineering School/USP.
2. Experimental Tests
The experimental program was prepared using a metal chamber with a trap door at its bottom. The chamber had 560mm of internal width and high and 1400mm of length, as
showed in Figure 1. The trap door could be vertically displaced upward or downward, simulating localized process of elevation and loss of support.
The trap door’s vertical movement was made in little increments of displacement, and at the end of each displacement the instrumentation measurements were registered.
(a) General view. (b) Lateral view.
Figure 1. Box test from EESC/USP, Costa [1]. General and lateral view.
The test campaign was organized in two phases. In the first (serie C), the tests were realized without pipe, fulfilling the chamber only with soil, analyzing different relative densities (Dr), subjected to different values of overload and different trap door geometries. In the second (series D and E), tests were realized with the pipe, analyzing different relative densities, submitted to different values of surcharges, with the geometry of the trap door constant. The chamber was fulfilled with sand by pluviation. The surcharge was applied using a PVC inflatable bag, reinforced by polyester fiber, made by Sansuy S.A.
In serie C tests, the instrumentation system was comprised of total stress cells placed in the lower boundary of the model (interface cells I), as showed in Figure 2. In series D and E tests a displacement transducer made with strain-gages was used, allowing simultaneous measurements of radial displacements in eight distinct points for each 45° direction around the cross section. In order to measure stress on soil mass around the pipe, total stress cells (inclusion cells M) were used around two cross sections (S1 and S2), as showed in Figure 3.
Figure 2. Interface cells disposition at the base of the test box, Costa [1].
Trap door’s handle
S. do N. Santos et al. / Behavior of Buried Pipes: Numerical Simulation vs Experimental Test1366
Figure 3. Inclusion cells disposition around the two cross sections analyzed, Costa [1].
A pure sand, called “Areia Itaporã”, was used. Following the rules of ABNT NBR-12004/90 [2] and NBR-12051/91 [3], maximum (emax) and minimum void ratios (emin) were specified as 0,87 and 0,50, respectively. Maximum (γd,max) and minimum dry specific weights (γd,min) were 17,7 kN/m³ and 14,2 kN/m³. The shear strength parameters, obtained from conventional triaxial compression tests, are presented in Table 1. Table 1. Soil parameters from Itaporã sand, Costa [1].
Soil type Dr (%) �3 (kPa) �'p (°) �cr (°) E50 (MPa) �
Soft 50 50 38 ... 27,2 0,37
100 36,7 33,4 35,2 0,34 200 34 31 38,5 0,38
Dense 100 50 39,9 34,6 35,9 0,44
100 39,2 33,1 40,6 0,41 200 38,2 31,8 49,3 0,42
PVC commercial pipes with 75mm diameter and 2mm thickness were used in the
models. The mechanical properties of pipes were obtained by parallel plates test (ASTM D 2412-02 [4]), giving a elasticity modulus of the material equal to 1,91 GPa (Costa [1]).
3. 3D Numerical Modeling
The computational program PLAXIS/3D, based on Finite Elements Method (FEM), was used as numerical tool.
The geometry of the mesh tried to faithfully reproduce the dimensions of the chamber described before. The boundary conditions were established so that lateral horizontal displacement was restricted at the walls. A surface element was created on the bottom of the model to which were assigned predicted gradual displacements, simulating the stages of movement of the trap door. Figure 4 shows the geometry used with the trap door at the bottom of the chamber and the pipe placed near to it.
From the tests campaign without pipe C2 and C5 tests were selected and from the tests campaign realized with the pipe, tests D7 and E6 were selected. Table 2 presents the characteristics of each test.
Stress and displacements of the pipe were measured in the locations indicated in Figure 5. In numerical stage, displacements were registered in the positions 1, 3 and 5, corresponding respectively to the bottom, the water line and the top of the pipe.
S. do N. Santos et al. / Behavior of Buried Pipes: Numerical Simulation vs Experimental Test 1367
Figure 4. Test geometry.
(a) Stress. (b) Displacements.
Figure 5. Measurements points of PVC pipe deformation.
Table 2. Characteristics of the analyzed tests.
Materials Test Dr (%) q (kPa) Trapdoor’s length, Lv
Trapdoor’s direction
Soil C2 100 100 3B Down C5 100 100 3B Up
Soil and pipe
D7 50 100 3B Down E6 50 100 3B Up
4. Main Results
Results obtained in numerical predictions were compared to the obtained in physical model. In order to facilitate the analysis, vertical tensions are normalized in relation to vertical initial tension (�v/��vi) and the displacement of the trap door is normalized in relation to its width ( /B). Continuous lines represent experimental results and dashed lines numerical ones.
4.1. Arching Effect on Soil
Results showed below presents the arching effect by vertical stress variation in the soil localized inside the trap door’s limits and in its surroundings. The active and passive arching was analyzed, represented respectively by the trap door’s movement downward and upward.
Trap door
S1 – Section of the symmetry axis
560 mm
S. do N. Santos et al. / Behavior of Buried Pipes: Numerical Simulation vs Experimental Test1368
4.1.1. Loss of Support – Test C2
Numerical results reproduced quite well the experimental behavior. Figure 6 presents results of the evolution of vertical stress with the trap door’s translation in two places (I1 and I3), in longitudinal direction of the trap door. One observes a relief of almost 70% of initial stress, inside the trap door, after a translation of 0,15B. Under loss of support, the vertical stress inside the trap door suffers abrupt reduction, these efforts being transferred to adjacent regions. With the progression of the movement, this mechanism is being expanded to more distant regions.
Figure 7 shows the vertical stress variation in the region outside the trap door’s limits (positions I4, I5, I6, I7, I8 e I9). It can be seen that both models, physical and numerical, presented an increase of stress followed by a decrease for little
ments.
(a) Position I1 (b) Position I3
Figure 6. Normalized vertical stress vs displacement relative to the trap door - Test C2: Dr = 100% q= 100 kPa, trap-door Lv/B = 3.
(a) Transversal elements (b) Longitudinal elements
Figure 7. Normalized vertical stresses vs displacement relative to the trap door - Active condition – external to trap door.
4.1.2. Localized Elevation – Test C5
Figure 8 shows the increase in vertical stress inside of trap door, when it is moved upward. In the edge of the trap door (position I3), the vertical stress exceeds more than five times the initial value. This could be assigned to the proximity of the shear zone. Numerical results indicated an increase of stress that is lower than that verified experimentally.
Outside the trap door (Figure 9 (a)), both experimental and numerical results indicate a significant decrease in vertical stress in position I4 and a less marked relief in position I5. In this region, there is a good agreement between the prediction and the
S. do N. Santos et al. / Behavior of Buried Pipes: Numerical Simulation vs Experimental Test 1369
experimental behavior. The influence of trap door’s displacement is been reduced with the increase of the
distance between the analyzed point and the trap door. In Figure 9(b), for example, the positions I8 and I9 are not even mobilized.
Figure 8. Vertical stress variation in the interior of the trap-door – Test C5 Dr = 100% and q = 100 kPa with
trap-door Lv/B = 3.
(a) Transversal direction (b) Longitudinal direction
Figure 9. Stress variation at the exterior of the trap-door – Test C5 Dr = 100% and q = 100 kPa with trap-door Lv/B = 3.
4.2. Effect of Loss of Support and Elevation on Pipe
Deflections along the pipe’s wall and stress variation in soil around the pipe, deriving from trap door’s movement, were analyzed in the positions indicated in Figure 5. The deflections correspond to the measured displacement divided by the pipe’s average diameter. A positive displacement means that the measured point moved to the center of the pipe.
0,0
1,0
2,0
3,0
4,0
5,0
6,0
-0,7-0,6-0,5-0,4-0,3-0,2-0,10,0
� v/�
vi
/B (%)
I1 NUM I1 EXPI2 NUM I2 EXPI3 NUM I3 EXP
-0,2
0,0
0,2
0,4
0,6
0,8
1,0
1,2
-0,7-0,6-0,5-0,4-0,3-0,2-0,10,0
� v/�
vi
/B (%)
I4 NUM I4 EXPI5 NUM I5 EXP
0,0
0,2
0,4
0,6
0,8
1,0
1,2
-0,7-0,6-0,5-0,4-0,3-0,2-0,10,0
� v/�
vi
/B (%)
I6 NUMI6 EXPI7 NUMI7 EXPI8 NUMI8 EXPI9 NUMI9 EXP
S. do N. Santos et al. / Behavior of Buried Pipes: Numerical Simulation vs Experimental Test1370
4.2.1. Loss of Support – Test D7
Figure 10 compares numerical and experimental results, corresponding to positions M1 to M3, in test D7. It is good to remember that positions M1 and M3 correspond to the left and the right side of the pipe, respectively, and the position M2 is placed on top of the pipe.
No matter where the measure instrument is placed, all curves showed a sharp reduction of the stress in the beginning of the process of displacement of the trap door. This reduction is, as expected, less sharp above the pipe. The greater stiffness of the pipe compared to the stiffness of the soil decreases the transference of deformations in the region above the pipe. On the other hand, the positions M1 and M3 are less affected for the presence of the pipe. In all analyzed positions a good proximity between numerical and experimental results was verified.
(a) Position M1 e M3. (b) Position M2.
Figure 10. Stresses in soil mass close to section S1; numerical x experimental.
Figure 12 shows the pipe’s deflection, increased by a factor of 5, for a relative displacement of 1%, along its longitudinal axis for three distinct positions. This relative displacement represents the initial condition of the behavior of the system. Numerical results were compatibles with experimental ones.
Figure 11. Loss of support - Pipe deflections for /B = 1%.
4.2.2. Localized Elevation – Test E6
Figure 12 presents the stress variation with the trap door’s movement at positions M2 and M3, on test E6. On both models one observes a lower increase in stresses on top of the pipe and higher on the water line. On top of the pipe (position M2) this increase was less than twice. On the water line (position M3) the increase was more expressive, being more than three times the initial stress in the end of the translation.
Figure 13 shows the profile of pipe’s deflection increased by a factor of 7 along its axis for three distinct positions. The deflections are shown as related to a relative displacement of 2% that correspond to the initial condition of the behavior of the system.
-0,2
0,0
0,2
0,4
0,6
0,8
1,0
0 10 20 30 40 50
��h/��
hi
/B (%)
M1 - NUM
M1/M3 - EXP
M3 - NUM
-0,2
0,0
0,2
0,4
0,6
0,8
1,0
0 10 20 30 40 50
�v/�
vi
/B (%)
M2 - NUM
M2 - EXP
-4
-2
0
2
4
6
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4
d* (%
)
Position (m)
5 - NUM 3 - NUM 1 - NUM5 - EXP 3 - EXP 1 - EXP
S. do N. Santos et al. / Behavior of Buried Pipes: Numerical Simulation vs Experimental Test 1371
(a) Position M2. (b) Position M3.
Figure 12. Stress variation in soil close to S1; numerical x experimental.
The numerical simulations of the trap-door displacement tests without pipe, named C2 and C5, satisfactorily reproduced the experimental results. When the pipe was present, tests named D7 and E6, the stress redistribution around the pipe with the trap-door displacement pointed out to the same stress variation on experimental and numerical predictions. The pipe displacements presented the same tendency and experimental and numerical results of the same order. It has been observed that the computational model reproduced adequately the arching effect mobilized at the actual test box model, conceived by EESC/USP. The numerical simulation with PLAXIS 3D has shown to be capable to predict the behavior of embedded pipes. However, the quality of results will depend on the adequate choice of the constitutive model and, mainly, the soil parameters.
References
[1] Y.D.J. Costa, Physical modeling of buried pipes subjected to localized loss of support or elevation. Ph.D. thesis, School of Engineering of São Carlos, University of São Paulo, Brazil, 321 p, 2005. In Portuguese.
[2] ABNT NBR-12004. Soil – Determination of maximum void ratio of non cohesive soils. Rio de Janeiro, 1990. In Portuguese
[3] ABNT NBR-12051.Soil – Determination of minimum void ratio of non cohesive soils. Rio de Janeiro, 1991. In Portuguese.
[4] ASTM D 2412-2. Standard Test Method for Determination of External Loading Characteristics of Plastic Pipe by Parallel-Plate Loading, 2002.
0,5
1,5
2,5
3,5
4,5
0 1 2 3 4
� v/�
vi
/B (%)
M2 - NUM
M2 - EXP
0,5
1,5
2,5
3,5
4,5
0 1 2 3 4
� h/�
hi
/B (%)
M3 - NUM
M3 - EXP
-4
-2
0
2
4
6
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4
d* (%
)
Position (m)
5 - NUM 5 - EXP 3 - NUM3 - EXP 1 - NUM 1 - EXP
S. do N. Santos et al. / Behavior of Buried Pipes: Numerical Simulation vs Experimental Test1372
