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Numerical evaluation of 3D passive earth pressure coefficients for retaining wall subjected to translation
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Computers and Geotechnics 35 (2008) 47–60 www.elsevier.com/locate/compgeo
Numerical evaluation of 3D passive earth pressure coefficients for retaining wall subjected to translation S. Benmebarek b
a,*
, T. Khelifa a, N. Benmebarek a, R. Kastner
b
a Civil Engineering Laboratory, Biskra University, BP 145 Biskra 07000, Algeria URGC-Ge´otechnique, INSA Lyon, 20 Avenue Albert Einstein, 69621 Villeurbanne Cedex, France
Received 16 May 2006; received in revised form 17 January 2007; accepted 29 January 2007 Available online 27 March 2007
Abstract The 2D passive earth pressures acting on rigid retaining walls problem has been widely treated in the literature using different approaches (limit equilibrium, limit analysis, slip line and numerical computation), however, the 3D passive earth pressures problem has received less attention. This paper is concerned with the numerical study of 3D passive earth pressures induced by the translation of a rigid rough retaining wall for associative soils. Using the explicit finite difference code FLAC3D (Fast Lagrangian Analyses of Continua), the increase of the passive earth pressures due to the decrease of the wall breadth is investigated. The results given by the present numerical analysis are compared with other investigations using limit equilibrium method, upper-bound method in limit analysis as well as experimental measures. These are presented in a form of design tables relating the geometrical parameters, soil properties, and 3D passive earth pressure coefficients Kpc(3D), Kpq(3D) and Kpc(3D) representing the effects of soil weight, surcharge loading and soil cohesion, respectively. 2007 Elsevier Ltd. All rights reserved. Keywords: Numerical modelling; 3D passive earth pressure; Soil; Retaining wall; Interaction; Failure; Behaviour
1. Introduction Numerous geotechnical structures mobilize the passive earth pressures in a three-dimensional manner such as anchor bloc, anchor plates, massif reaction, limited breadth retaining wall, retaining wall with discontinues penetration depth and laterally loaded pile caps. Safe and economical design of retaining structures requires a sound knowledge of the contact pressure exerted. The 2D passive earth pressures problem has been widely treated in the literature theoretically using different approaches (limit equilibrium, limit analysis, slip line and numerical computation), however, the 3D passive earth *
Corresponding author. Tel.: +213 33 74 20 72; fax: +213 33 74 86 87. E-mail addresses: [email protected] (S. Benmebarek), khelifat@ yahoo.fr (T. Khelifa), [email protected] (N. Benmebarek), richard. [email protected] (R. Kastner). 0266-352X/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2007.01.008
pressures problem has been received theoretically less attention apart from the work of Blum [1] using the limit equilibrium approach and Soubra and Regenass [2] using the upper bound theorem of limit analysis. This is indeed due to the difficulty to propose analytically the 3D failure mechanisms. In this paper, the 3D effect of the passive earth pressures problem is investigated using the explicit finite difference code FLAC3D [3] (Fast Lagrangian Analyses of Continua). The aim of this work is firstly to develop a numerical procedure for the analysis of a rigid rough vertical retaining wall with horizontal ground surface subjected to translation and uniform surcharge, then to investigate the influence of the wall breadth on the evaluation of the earth pressure coefficients Kpc, Kpq and Kpc representing the effect of soil weight, surcharge loading and cohesion, respectively. Numerical results and failure mechanisms, particularly the plane view, obtained from the present
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Nomenclature u w d c c K G K0 Kn Ks Pp Ppx h
angle of internal friction of the soil dilation angle of the soil angle of friction at the soil/wall interface soil cohesion unit weight of the soil bulk modulus of the soil shear modulus of the soil earth pressure coefficient at rest interface normal stiffness interface shear stiffness passive earth force horizontal passive earth force penetration depth of the wall
analysis are compared with other investigations using limit equilibrium method, upper-bound method in limit analysis as well as experimental measures. 2. Overview of previous work The 2D earth pressure theories are widely used in geotechnical engineering. They are the Coulomb Theory [4], which treats the passive earth pressure problem in terms of forces, and the Rankine Theory [5], which treats the problem in terms of stresses. Both theories are well known and are described in nearly every soil mechanics textbook. The Coulomb theory is more versatile in accommodating complex configurations of backfills and loading conditions as well as frictional effects between walls and backfills. However, both theoretical and experimental studies have shown that the Coulomb assumption of plane surface sliding is not perfectly valid when the wall is rough, especially in the passive case when interface friction is more than 1/3 of internal soil friction. The curvature of the failure surface behind the wall needs to be taken into account. Hence, Coulomb’s theory leads to largely overestimation of the passive earth pressure. The Rankine’s theory is applicable to the calculation of the earth pressure on a perfectly smooth and vertical wall, but most retaining walls are far from frictionless soil structure interface. More than 50 years ago, several researchers dealt with theoretical procedures for evaluating the earth pressure using the limit equilibrium method [6,7], the slip line method [8–11] and the upper- and lower bound theorems of limit analysis [2,12]. These different approaches generally confirm the accuracy of the Log Spiral Theory [8] for a wide range of values of the internal soil friction and the soil–structure interface friction angle. Similarly, Martin [13] and Benmebarek et al. [14] who used FLAC2D numerical analysis to evaluate passive earth pressures have found fairly close agreement with Log Spiral Theory. In spite of recent published methods, the tendency today in practice
b q 2D 3D Kp2D Kp3D Kpc(3D) Kpc(3D) Kpq(3D)
breadth of the wall surcharge on the ground surface bi-dimensional three-dimensional 2D passive earth pressure coefficients 3D passive earth pressure coefficients 3D passive earth pressure coefficients due to soil weight 3D passive earth pressure coefficients due to soil cohesion 3D passive earth pressure coefficients due surcharge loading
is to use the values given by Caquot and Ke´risel [8] and Ke´risel and Absi [9]. Most of the research effort has concentrated on a refinement of the 2D analysis with little attention given to the 3D effects, apart from the work of Blum [1] using the limitequilibrium method and recently the work of Soubra and Regenass [2] using the limit analysis theory, where threedimensional kinematically admissible failure mechanisms composed of one or several rigid blocks have been considered. However, the 3D effects have been received moderate investigation using experimentally approaches which revealed very high dispersions of the results. Ovesen [15] conducted an extensive series of passive pressure model tests to investigate the 3D effects. Ovesen’s tests were performed on compacted sand with friction angles ranging from / = 32.7 to 41.7. The maximum difference between passive and active earth pressure coefficients (Kp Ka) was 5.7 in Ovesen’s tests, and the correction factor for the 3D effects; did not exceed a value of about 2. These tests showed that passive earth pressures against short structures are higher than those predicted by conventional theory and the difference can be quite significant. Brinch Hansen [16] developed a method for correcting the results of conventional passive pressure theories for 3D effects, based on Ovesen’s test results. Duncan and Mokwa [17] have developed an Excel spreadsheet computer program based on both the Log Spiral Theory and the Ovesen–Brinch Hansen correction for 3D effects, making it applicable to both short and long breadth structures. As a conservative measure based on the result of Ovesen tests, an upper limit of 2.0 was placed by Duncan and Mokwa [17] on the value of the correction factor for the 3D effects that is used in the spreadsheet. In addition, Horn [18] has presented a synthesis of about twenty experimental research works on the 3D passive earth pressure load. The analysis of this synthesis has shown a very high dispersion of the results for the reason that the difference in the material natures, the deficiency
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
in the properties data as well as the scale effects related to the size of the experimental models. To avoid the scale effects, Belabdelouhab [19] and Meksaouine [20] have used the same test frame as well as the material ‘‘sand of Hoston’’. These tests lead thus to coherent results covering from short to long breadth wall. These experimental measures allow to avoid the scale effect by analyzing not the 3D earth pressure results but the ratio Kp(3D)/Kp(2D) obtained under the same experimental conditions. The observations of the failure surface in plan view of the tests carried out by Weissenbach [21], Belabdelouhab [19], Meksaouine [20] and Duncan and Mokwa [17] show that the progression of surface vanishes in the case of the long breadth wall and the distance of surface failure in plan view from the front of the wall does not seem to have exceeded the value of 2.5 times the wall high. In order to appreciate the accuracy of the present numerical analysis, the two theoretical approaches of Blum [1] and the upper bound solution given by Soubra and Regennas [2] for an associated flow rule Coulomb material obeying Hill’s maximal work principle as well as the experimental measures of Mekssaouine [20] and Duncan and Mokwa [17] are used for comparison. 3. Numerical modelling procedure 3.1. Case study This paper is concerned with the numerical evaluation of passive earth pressure coefficients for a rigid rough vertical retaining wall with horizontal ground surface and limited breadth subjected to translation as sketched in Fig. 1. This case was firstly treated theoretically by Blum [1] using the limit equilibrium approach. Blum’s failure mechanism is represented by an extension of the 2D Coulomb failure mechanism into the 3D case as shown in Fig. 2. According to this mechanism, the 3D passive earth force is given by: Plan view of failure surface
49 C
h/2 B
H b
A G h/2 h
D
F
E
π/4+ϕ/2
Fig. 2. Blum’s failure mechanism.
P pc
1 2 / h3 / 2 p 2 p þ þ ¼ ch b tg þ c tg 2 4 2 4 2 6
ð1Þ
From this equation it is evident that Blum solution does not take into account the soil–structure interface friction. Recently, Soubra and Regenass [2] have used the upper bound theorem of the limit analysis and have developed upper bound solutions for the 3D passive earth pressure coefficients. Three translational failure mechanisms referred to as One-Block Mechanism M1, Multiblock Mechanism Mn and Truncated Multiblock Mechanism Mnt have been considered for the calculation schemes as shown in Fig. 3. The mechanism M1 composed of one rigid block is an extension of the 2D well-known Coulomb mechanism [cf. Fig. 3a] into 3D case. The Mn requires a more elaborate failure mechanism [cf. Fig. 3b] composed of n rigid blocks, whereas the Mnt is obtained by a volume reduction of the final block in the Mn mechanism [cf. Fig. 3c]. They concluded that the Mnt mechanism which gives the least upper-bound solution is more efficient than the Mn mechanism, and for practical use, they proposed numerical results based on the Mnt mechanism for various governing parameters. 3.2. Modelling procedure
Limited breadth wall subjected to translation
Fig. 1. Case study.
The analysis of the increase of the passive pressures due to the decrease of the retaining wall breadth, as pointed out by Soubra and Regenass [2], is carried out in this paper using the commercially available three-dimensional code FLAC3D. This code uses an explicit finite difference program to study numerically the mechanical behaviour of a continuous 3D medium as it reaches equilibrium or steady plastic flow. The explicit Lagrangian calculation scheme and the mixed-discretization zoning technique [22] used in FLAC3D ensure that plastic failure and flow are modelled very accurately.
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6h
Wall
b/2 6h
6h
h
2h Z Y X
Fig. 4. Domain for FLAC3D simulation – half symmetry.
boundary corresponding to the plane at y = 0 is restricted in y-direction. The assumptions made in the numerical analysis can be summarized as follows: The soil is homogeneous and isotropic obeying Coulomb criteria plasticity. The wall of dimensions b · h (b = breadth; h = height) is vertical, and the backfill is horizontal. A translational soil–wall movement is assumed. The limit loading is reached by applying wall velocity until a steady plastic flow is obtained. The soil–structure interface friction angle is assumed to be constant.
Fig. 3. Soubra’s failure mechanisms: (a) M1, (b) Mn and (c) Mnt [2].
Accordingly, a retaining wall with limited breadth penetrated by translation in homogeneous and isotropic semi-infinite soil is considered. For this problem, half-symmetry condition is assumed in the numerical simulation. The domain used for the analysis is sketched in Fig. 4, together with its dimensions and axis. The vertical and bottom boundaries were respectively located at a distance six and three times the wall penetration in order to minimize boundary effects. For the boundary conditions applied to this domain, the displacements of the far x- and y- lateral boundaries are restricted in the horizontal x and y directions, respectively. The bottom boundary is constrained in the z-direction and the displacements of the symmetry
In order to develop an acceptable analysis scheme for later computations, preliminary simulations have been carried out by testing the sensibility of passive earth pressure forces by the mesh dimensions, the element size, the boundary conditions and the earth pressure coefficient at rest K0 as well. Fig. 5 shows an example of the mesh retained for this analysis. The grid size is fine near the wall where deformations are concentrated. The material properties, particularly stiffness assigned to an interface, depend on the way in which the interface is used. In the case of soil–structure interaction, the interface is considered stiff compared to the surrounding soil, but it can slip and may be opened in response to the loading. Joints with zero thickness are more suitable for simulating the frictional behaviour at the interface between the wall and the soil. The interface model incorporated in FLAC3D code and its components illustrated in Fig. 6 has been used to simulate the soil/wall contact. The interface constitutive model is defined by a linear Coulomb shear-strength criterion that limits the shear force acting at an interface node, normal and shear stiffnesses, tensile and shear bond strengths, and a dilation angle that causes an increase in effective normal force on the target face after the shear-strength limit is reached.
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
51
Fig. 5. Example of used mesh for domain half symmetry.
The spring in the tangential direction, the slider and the limit strength (Fig. 6) represent the Coulomb shearstrength criterion. The spring in the normal direction, the limit strength and dilation represent the normal contact. The interface has a friction angle d, a cohesion c = 0 kPa, a normal stiffness Kn = 109 Pa/m, and a shear stiffness Ks = 109 Pa/m. These values are selected to approximate the results for the case where the wall is rigidly attached to the grid. For the soil behaviour, a linear elastic-perfectly plastic associative Mohr–Coulomb model encoded in FLAC3D is adopted, requiring the specification of a shear modulus G = 22 MPa, a bulk modulus K = 60 MPa, a unit weight c = 20 kN/m3, a friction angle /, and an angle of dilation w = /. However, the dilation angle of real granular soils is generally less than the angle of friction. The potential effect of non-associative plastic flow (w p /) which may be treated numerically by FLAC3D are beyond the scope of this study. The numerical results of the 3D passive earth pressures are presented in the form of dimensionless coefficients Kpc(3D), Kpq(3D) and Kpc(3D). It should be noted that these coefficients are independent from elastic, weight soil, cohesion and surcharge values used in computation.
The proposed modelling procedure of the passive earth pressures are based on two steps: In the first one, the geostatic stresses are computed. At this stage some stepping is required to bring the model to equilibrium. The reason is that an additional stiffness from interface elements produces an imbalance that necessitates some stepping to equilibrate the model. It should be noted here that preliminary numerical tests have shown that limit passive earth pressure forces are insensitive to the earth pressure coefficient at rest K0. The influence of this coefficient is limited to the necessary displacement for reaching the limit value of passive earth pressures. In the second step, to represent rigid wall translation, a controlled horizontal velocity is applied in several steps to all the wall grid points. First, a relatively high velocity of 106 m/step is applied to the wall until a steady plastic flow state is achieved (i.e. until both conditions (i) a constant earth pressure force and (ii) small values of unbalanced forces are obtained as the number of cycles increases). As the level of error in such a FLAC3D calculation scheme depends on the applied velocity, a more-accurate earth pressure force can be obtained by reducing the wall velocity by half, and continuing to a new steady plastic-flow state. This procedure, recommended by FLAC3D manual [3], is repeated several times, particularly for soils with large values of the internal friction, until the difference between the earth passive forces calculated at two successive steady plastic-flow states becomes negligible. Using a FISH function, the horizontal passive earth pressure force Ppx acting on the wall side can be calculated as the integral of stress components for all soil zones in contact with the wall. At this stage, the horizontal earth pressure force on the wall can be recorded. Hence, the value of the passive earth pressure force Pp [2,8] can be deduced from the following relationship: Pp ¼
Fig. 6. Components of the bonded interface constitutive model.
P px cos d
ð2Þ
The generalized passive earth pressure formula [2,8] can be written as
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S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
Table 1 Comparison of present Kpc(3D) coefficients with other existing solutions b/h
/ ()
d// 0
1/3
2/3
1
Soubra
FLAC3D
Soubra
FLAC3D
Soubra
FLAC3D
Soubra
FLAC3D
0.1
20 25 30 35 40
– – – – –
6.513 10.268 16.288 25.318 40.747
– – – – –
7.199 11.96 20.146 33.875 60.25
– – – – –
7.959 13.913 24.647 44.215 87.768
– – – – –
8.763 15.804 29.483 56.015 113.548
0.25
20 25 30 35 40
5.399 7.983 11.886 20.044 50.43
4.547 6.606 9.565 14.018 20.883
6.624 10.528 17.012 29.026 70.281
5.181 7.939 12.335 19.633 32.384
8.301 14.582 27.017 54.775 140.561
5.751 9.304 15.520 26.871 49.834
10.595 20.583 42.510 95.509 240.815
6.350 10.680 19.279 35.078 65.895
0.5
20 25 30 35 40
3.726 5.229 7.443 11.984 28.594
3.391 4.666 6.502 9.080 13.101
4.538 6.849 10.604 17.354 40.954
3.896 5.680 8.372 12.798 20.281
5.629 9.379 16.634 32.487 79.700
4.378 6.756 10.626 17.642 31.961
6.994 12.776 25.085 54.064 131.753
4.814 7.913 13.108 23.645 45.457
1
20 25 30 35 40
2.887 3.850 5.221 7.954 17.676
2.770 3.639 4.875 6.584 9.199
3.487 5.001 7.391 11.518 25.317
3.180 4.356 6.275 9.140 14.208
4.279 6.760 11.418 21.308 49.269
3.583 5.192 7.910 12.561 22.057
5.139 8.798 16.273 33.202 77.015
4.010 6.159 9.796 16.647 32.698
2
20 25 30 35 40
2.466 3.159 4.111 5.939 12.218
2.459 3.096 3.952 5.109 7.015
2.956 4.069 5.777 8.600 17.499
2.805 3.735 5.073 7.606 10.957
3.593 5.435 8.787 15.683 32.329
3.164 4.493 6.543 10.109 17.006
4.171 6.746 11.764 22.607 49.371
3.515 5.212 8.082 13.464 24.903
5
20 25 30 35 40
2.211 2.743 3.444 4.730 8.942
2.262 2.797 3.489 4.459 5.80
2.633 3.505 4.801 6.849 12.808
2.568 3.314 4.470 6.13 8.80
3.133 4.542 6.965 11.519 21.145
2.876 3.957 5.731 8.350 13.28
3.561 5.456 8.948 16.035 32.361
3.194 4.536 6.812 10.885 19.602
10
20 25 30 35 40
2.125 2.604 3.222 4.327 7.851
2.103 2.512 3.218 4.030 5.095
2.524 3.315 4.473 6.266 11.244
2.480 3.181 4.275 5.773 8.105
2.954 4.178 6.209 9.87 17.226
2.793 3.821 5.610 7.913 12.331
3.348 5.004 7.958 13.730 26.424
3.102 4.412 6.751 10.675 18.108
2D passive earth pressure coefficients FLAC2D 20 25 30 35 40 Caquot
20 25 30 35 40
Soubra = after Soubra and Regenass [2]. FLAC3D = present solution. Caquot = after Caquot and Kerisel [8]. FLAC2D = after Bebmebarek et al. [14].
2.07 – 3.06 3.76 4.69
2.39 – 4.03 5.42 7.53
2.72 – 5.20 7.68 11.96
– – – – –
2.04 2.45 3.00 3.69 4.60
2.40 3.10 4.00 5.40 7.60
2.75 3.70 5.30 8.00 12.00
3.10 4.40 6.50 10.50 18.00
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
53
Table 2 Comparison of present Kpq(3D) coefficients with other existing solutions b/h
/ ()
d// 0
1/3
2/3
1
Soubra
FLAC3D
Soubra
FLAC3D
Soubra
FLAC3D
Soubra
FLAC3D
0.1
20 25 30 35 40
– – – – –
7.618 12.443 20.124 32.526 53.209
– – – – –
8.433 14.343 24.710 42.960 76.927
– – – – –
9.355 16.327 29.989 55.149 113.205
– – – – –
9.998 18.388 34.697 67.880 144.561
0.25
20 25 30 35 40
7.068 10.736 16.329 28.104 72.265
5.406 8.080 12.038 17.858 27.545
8.704 14.202 23.415 40.698 103.502
6.094 9.647 15.390 24.897 42.529
10.960 19.770 37.382 77.039 170.861
6.741 11.278 19.100 33.391 63.243
13.015 24.817 49.352 104.684 243.616
7.454 12.851 22.521 42.628 82.450
0.5
20 25 30 35 40
4.563 6.606 9.664 16.014 39.512
3.915 5.534 7.845 11.190 16.315
5.583 8.690 13.810 23.190 56.591
4.503 6.706 10.175 15.797 25.781
6.967 11.984 21.829 43.615 91.492
5.044 7.869 12.717 21.305 38.783
8.057 14.599 27.909 57.371 130.190
5.552 9.019 15.483 27.996 55.271
1
20 25 30 35 40
3.307 4.540 6.332 9.969 23.136
3.041 4.098 5.566 7.634 10.865
4.014 5.926 8.999 14.436 33.136
3.520 4.977 7.229 10.765 16.930
4.956 8.073 13.951 25.670 51.724
3.946 5.857 9.052 14.608 25.674
5.543 9.445 17.124 33.627 73.351
4.328 6.756 11.023 18.823 35.803
2
20 25 30 35 40
2.677 3.505 4.666 6.946 14.947
2.578 3.329 4.335 5.769 7.869
3.223 4.536 6.585 10.059 21.408
2.971 4.046 5.634 8.090 12.294
3.848 5.902 9.562 16.622 31.723
3.324 4.754 7.061 10.971 18.396
4.256 6.819 11.656 21.638 44.748
3.622 5.398 8.435 14.056 25.165
5
20 25 30 35 40
2.296 2.882 3.666 5.133 10.034
2.302 2.872 3.619 4.609 6.081
2.741 3.694 5.127 7.433 13.923
2.635 3.461 4.643 6.358 9.280
3.148 4.533 6.865 11.077 19.515
2.926 4.029 5.729 8.556 13.652
3.453 5.189 8.277 14.269 27.268
3.155 4.510 6.734 10.690 17.726
10
20 25 30 35 40
2.168 2.673 3.333 4.528 8.263
2.208 2.717 3.377 4.132 5.146
2.577 3.410 4.637 6.558 10.838
2.519 3.266 4.305 5.833 8.322
2.906 4.060 5.924 9.155 15.303
2.794 3.796 5.330 7.821 12.253
3.173 4.620 7.100 11.708 21.219
3.008 4.228 6.201 9.565 15.231
2D Caquot
20 25 30 35 40
2.04 2.44 3.03 3.60 4.50
2.35 3.03 4.00 5.28 7.25
2.65 3.56 5.00 7.10 10.72
2.85 4.00 5.88 8.80 14.60
Soubra = after Soubra and Regenass [2]. FLAC3D = present solution. Caquot = after Caquot and Kerisel [8].
P p ¼ K pcð3DÞ c
h2 b þ K pcð3DÞ c h b þ K pqð3DÞ q h b 2 ð3Þ
where Pp is the passive earth force; Ppx is the horizontal passive earth force; c is the unit weight of the soil; h is the penetration depth of the wall; b is the breadth of the wall; d is the soil–wall interface friction angle; c is the soil
cohesion; q is the surcharge on the ground surface; Kpc(3D), Kpq(3D) and Kpc(3D): 3D passive earth pressure coefficients that are functions of the angle of internal friction. They represent the influence of soil weight, cohesion, and surcharge loading, respectively. Accordingly, the passive earth pressure coefficients Kpc(3D), Kpq(3D) and Kpc(3D) are determined independently in three configurations outlined as follows:
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S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
Kpc(3D): The cohesive-frictional soil is replaced with a frictitious cohesionless (c = 0) soil having the same values of parametersc, / and without surcharge on the ground surface (q = 0). Following the first and the second steps of the proposed modelling procedure, the Kpc(3D)coefficient is determined from the following expression:
surcharge on the ground surface (q 6¼ 0). In this case, only the second step is necessary and the Kpq(3D) is given by the following expression:
K pcð3DÞ ¼ 2P px =c h2 b cosd;
Kpc(3D): In this case, the soil is treated as a weightless, (c = 0), cohesive medium (c 6¼ 0) with parameter / 6¼ 0 and without surcharge on the ground surface (q = 0). Also, in this case only the second step is necessary and the Kpc(3D) is determined by the following expression:
c 6¼ 0; / 6¼ 0; c ¼ 0; q ¼ 0 ð4Þ
Kpq(3D): The soil is next treated as a weightless, cohesionless medium with parameter / 6¼ 0 subjected to uniform
K pqð3DÞ ¼ P px =q h b cos d;
/ 6¼ 0; q 6¼ 0; c ¼ 0; c ¼ 0 ð5Þ
Table 3 Comparison of present computation coefficients Kpc(3D) with the theorem of corresponding states solutions b/h
/ ()
d// 0
1/3
2/3
1
TCS
FLAC3D
TCS
FLAC3D
TCS
FLAC3D
TCS
FLAC3D
0.10
20 25 30 35 40
18.183 24.54 33.124 45.024 62.220
18.014 24.129 33.068 44.913 62.138
20.403 28.591 41.041 59.893 90.457
19.874 28.076 40.405 59.210 90.163
22.879 32.775 50.101 77.203 133.584
22.791 32.455 49.797 76.136 129.063
24.546 37.067 58.099 95.196 170.732
24.183 36.574 57.306 94.084 170.014
0.25
20 25 30 35 40
12.105 15.183 19.118 24.076 31.635
11.997 15.101 19.115 24.056 31.665
13.977 18.521 24.898 34.097 49.461
13.594 18.053 24.358 33.506 48.855
15.697 21.947 31.240 46.130 74.039
15.055 21.315 30.360 45.043 72.986
17.556 25.193 37.009 59.134 96.708
16.878 24.523 36.655 58.544 96.605
0.5
20 25 30 35 40
8.009 9.723 11.856 14.552 18.252
8.003 9.701 11.858 14.582 18.023
9.606 12.214 15.865 21.101 29.501
9.188 11.756 15.305 20.380 28.978
11.035 14.636 20.184 28.870 44.888
10.587 13.994 19.426 27.760 44.010
12.330 16.975 24.818 38.238 64.316
11.909 16.316 23.976 37.433 63.607
1
20 25 30 35 40
5.608 6.644 7.909 9.474 11.757
5.584 6.591 7.824 9.467 11.720
6.905 8.506 10.763 13.915 18.952
6.655 8.030 10.245 13.330 18.395
8.018 10.322 13.836 19.306 29.265
7.754 9.903 13.261 18.464 28.292
8.967 12.122 17.093 25.138 41.114
8.626 11.710 16.410 24.766 41.045
2
20 25 30 35 40
4.336 4.995 5.777 6.811 8.186
4.312 4.963 5.772 6.798 8.166
5.397 6.509 8.000 10.095 13.427
5.157 6.259 7.738 9.678 12.987
6.309 7.956 10.387 14.112 20.591
5.710 7.294 9.896 13.461 19.780
7.028 9.210 12.610 18.330 28.436
6.710 8.776 12.130 17.823 28.692
5
20 25 30 35 40
3.577 4.015 4.536 5.154 6.056
3.55 3.990 4.524 5.174 6.031
4.473 5.255 6.283 7.622 9.835
4.283 5.023 5.991 7.315 9.438
5.216 6.402 8.080 10.663 14.937
5.018 6.116 7.699 10.167 14.467
5.744 7.306 9.664 13.523 19.570
5.513 6.986 9.364 13.234 20.204
10
20 25 30 35 40
3.319 3.682 4.117 4.473 4.941
3.305 3.674 4.109 4.266 4.721
4.155 4.836 5.698 6.872 8.693
3.985 4.757 5.425 6.575 8.324
4.853 5.902 7.389 9.614 13.270
4.654 5.666 7.084 9.223 12.806
5.341 6.701 8.741 11.916 16.597
5.091 6.396 8.531 11.831 16.478
TCS = after the theorem of corresponding states [8]. FLAC3D = present solution.
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
K pcð3DÞ ¼ P px =c h b cos d;
/ 6¼ 0; c 6¼ 0; c ¼ 0; q ¼ 0 ð6Þ
It is interesting to note that coefficients Kpc(3D) and Kpq(3D) are related by the following relationship [cf. theorem of corresponding states of Caquot (Caquot and Ke´risel [8]) in the literature: K pcð3DÞ ¼
K pqð3DÞ cos1 d tan /
ð7Þ
In order to examine the validity of Eq. (7), the present FLAC3D results of Kpc(3D) are compared to the results of this relationship (Eq. (7)) in the following section. 4. Results and discussion To investigate how the passive earth pressure coefficients are affected by the wall breadth, five values of the internal friction angle / (20, 25, 30, 35 and 40) are considered for each of the four values of soil–wall interface friction d (d// = 0, 1/3, 2/3 and 1).
55
The computed values of passive earth pressure coefficients Kpc(3D), Kpq(3D) and Kpc(3D) are listed in Tables 1– 3, respectively. The present coefficients Kpc(3D) and Kpq(3D) are compared to solutions given by Soubra and Regenass [2] reported in Tables 1 and 2 and the present coefficients Kpc(3D) are compared to the coefficients obtained by using the theorem of corresponding states (Eq. (7)). The currently used 2D passive earth pressure coefficients of Caquot and Ke´risel [8] and the FLAC2D results [14] are also reported in Tables 1 and 2 for comparison with the present results for long breadth retaining wall. These tables clearly show the sensitivity of the passive earth pressure coefficients Kpc(3D), Kpq(3D) and Kpc(3D) to the b/h ratio. These coefficients increase with the decrease of the b/h ratio. For large values of b/h, the 3D effect vanishes and the Kpc(3D) and Kpq(3D) converge to the 2D coefficients. For instance, for b/h = 10, the 3D effects expressed by the ratios Kpc(3D)/Kpc(2D) and Kpq(3D)/Kpq(2D) are less than 1.15 for the present results compared to the 2D results of Caquot and Ke´risel [8] and Benmebarek et al. [14] using numerical approach for 2D behaviour.
Fig. 7. Traces in plan view of Mnt Mechanism for / = 20, 30, 40; d// = 0 and 1; and b/h = 1 (axis unities = breadth wall) [2].
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S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
However, for Soubra and Regenass [2] solutions (Tables 1 and 2), the ratios Kpc(3D)/Kpc(2D) and Kpq(3D)/Kpq(2D) for / = 40 and d// = 0 reach 1.71 and 1.84, respectively. As observed in these Tables, the 3D effects are most pronounced for great / values for both the present computation and Soubra’s results. For b/h P 2 the 3D effects presented by the ratios Kpc(3D)/Kpc(2D) and Kpq(3D)/Kpq(2D) are less than 2 times as observed experimentally by Ovesen
and used by Duncan and Mokwa [17]. However, for b/h 6 1, the ratios Kpc(3D)/Kpc(2D) and Kpq(3D)/Kpq(2D) can be more than 2 times and increase highly with the decrease of b/h. For instance, the ratio Kpc(3D)/Kpc(2D) is equal to 1.817, 2.525, 3.661 and 6.308 and the ratio Kpq(3D)/Kpq(2D) is equal to 2.452, 3.786, 5.647 and 9.901 for b/h = 1, 0.5, 0.25 and 0.1, respectively when / = 40 and d// = 1. Soubra’s results give much higher 3D effect. For instance, in
Fig. 8. Failure mechanism for the interface friction angle d// = 0 when / = 35 and b/h = 1 visualized by: (a) displacement field vectors and (b) distribution of maximum shear strain rates.
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
the case of b/h = 0.25 when / = 40 and d// = 1, the ratios Kpc(3D)/Kpc(2D) and Kpq(3D)/Kpq(2D) are equal to 13.379 and 16.686, respectively. By comparing these results to the present ones (3.661 and 5.647), the high differences found may be explained by the use of the Mnt mechanism in Soubra’s results which seems to greatly overestimate the passive earth pressures shown by a significant change in the plan view of failure surface between / = 30 and 40 (Fig. 7). For both d// = 0 and
57
d// = 1 when / = 40 the distance of the failure surfaces from the front of the wall are 6 and 4 times the wall penetration depth, respectively. The Soubra’s Mnt mechanism for high soil friction is clearly in disagreement with experimental observations as shown by Weissenbach [21], Belabdelouhab [19], Meksaouine [20] and Duncan and Mokwa [17] where the distance of surface failure in plan view from the front of the wall does not seem to exceed the value of 2.5 times the wall depth.
Fig. 9. Failure mechanism for the interface friction angle d// = 2/3 when / = 35 and b/h = 1 visualized by: (a) displacement field vectors and (b) distribution of maximum shear strain rates.
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Figs. 8 and 9 show the present view of the failure mechanisms visualized by the displacement field and the distribution of maximum shear strain rates at the end of the run for two values of the interface friction angle (d// = 0, 2/3) when / = 35 and b/h = 1, respectively. For d// = 0 the failure surface in depth is similar to the plane surface proposed by Rankine. However for d// = 2/3 the failure mechanism is similar to the Prandtl mechanism [23] with a radial shear zone followed by a Rankine passive wedge. Fig. 10 shows, for both b/h = 1 and 5, the surface plane views of the zone mobilized by the passive earth pressures. For all present computation results, the surfaces are smooth as observed experimentally by Meksaouine [20] and they are highly different from Blum’s failure mechanism [1]. The distance of the surface failure in plan view from the front of the wall is in congruous with experimental observations. Fig. 11 shows the comparison of Kpc(3D) between the present results and those given by Blum and Soubra for / = 35; and d// = 0 and 2/3; and for different values of b/h ratio. A good agreement is observed for the case of a
smooth wall. However, for rough wall, Blum’s solutions greatly underestimate the passive earth pressure coefficients. This may be explained by the fact that the soil– structure interface friction in Blum’s mechanism is assumed smooth (d = 0). Moreover, for rough wall, the difference between the present computation results and Soubra’s results increase with the decrease of the b/h ratio. This can be explained as mentioned above by the fact that the Mnt mechanism taken by Soubra gives unrealistic failure mechanism. From 2D and 3D experimental tests conducted by Belabdelouhab [19] and Meksaouine [20], respectively, the case b/h = 0.5, / = 34.5 and d// = 0.43 shows a ratio Kp(3D)/Kp(2D) equal to 2.1. For the same case and using interpolation (Table 1), the present results give a ratio equal to 2.30, however the Soubra’s results give a ratio equal to 3.57. From this comparison, the present computation results are in accordance with experimental observations than Soubra’s ones. In Table 3, the computation results of Kpc(3D) are compared to the values of Kpc(3D) expressed by Eq. (7) obtained by using the theorem of corresponding states of Caquot.
Fig. 10. Visualization of surface plane views of the zone mobilized by the passive earth pressures for both b/h = 1 and 5.
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60 60
FLAC-3D
55
Soubra
50
Kp(3D)
59
45
FLAC-3D
40
Soubra
35 30
Blum
δ/φ = 0
25 20
δ/φ = 2/3
15 10 5 0 0
1
2
3
4
5
6
7
8
9
10
b/h
Fig. 11. Comparison with Blum and Soubra for / = 35; and d// = 0 and 2/3; and for different values of b/h ratio.
This theorem expresses the Kpc(3D) in function of Kpq(3D) for the same friction angle /. Although different calculation procedures are used, the results of both calculations are very close, with a difference less than 6%. This comparison confirms on the one hand, the validity of the theorem of corresponding states of Caquot for the case of vertical retaining wall with horizontal ground surface as pointed out by Michalowski [24] and Silvestri [25], and on the other hand the reliability of the present modeling procedure and the preliminary simulations. 5. Conclusions Numerical computations of the 3D passive earth pressure coefficients Kpc(3D), Kpq(3D) and Kpc(3D) have been performed using FLAC3D code. A number of conclusions may be drawn from this investigation: The passive earth pressures coefficients ratios Kpc(3D)/ Kpc(2D) and Kpq(3D)/Kpq(2D) increase with the decrease of the b/h ratio. The ratio may be more than 2 times for b/h 6 1. The surface plane view of the distribution of maximum shear strain rates has shown a smooth failure mechanism as observed experimentally by Meksaouine [20]. The distance of the surface failure in plan view from the front of the wall is in good agreement with experimental observations [17,20,21]. However the Soubra’s surface [2] highly overestimates this distance for great soil internal friction angle /. The comparison of the present computation results with the upper bound solution in the framework of the limit analysis theory given by Soubra and Regenass [2] has shown that these authors greatly overestimate the 3D passive earth pressure coefficients. In the case of b/h = 10, the present computation results of the Kpc(3D) and Kpq(3D) show that the maximum 3D effect is less than 15% compared to the results of Caquot and Ke´risel [8] for bi-dimensional behaviour. For b/h P 10, the 3D effect may be neglected.
The present FLAC3D results of Kpc(3D) confirm the validity of the theorem of corresponding states of Caquot for the case of vertical retaining wall with horizontal ground surface. References [1] Blum H. Wirtschaftliche Dalbenformen und deren Berechnung. Buatechnik 1932;10(5). [2] Soubra A-H, Regenass P. Three-dimensional passive earth pressures by kinematical approach. ASCE J Geot Geoenv Eng 2000:969–78. [3] FLAC3D. Fast lagrangian analysis of continua. Minneapolis: ITASCA Consulting Group, Inc.; 2000. [4] Coulomb CA. Essai sur une application des re`gles des maximas et minimas a` quelques proble`mes de statique relatifs a` l’architecture. Me´m. acad. roy. pres. divers savanta, vol. 7, 1776, Paris [in French]. [5] Rankine WJM. On the stability of loose earth. London: Philosophical Trans Royal Soc; 1857. [6] Rahardjo H, Fredlund DG. General limit equilibrium method for lateral earth forces. Can Geotech J 1984;21(1). [7] Zhu D-Y, Qian Q-H, Lee CF. Active and passive critical slip fields for cohesionless soils and calculation of lateral earth pressures. Ge´otechnique 2001;51(5). [8] Caquot A, Ke´risel J. Tables for the calculation of passive pressure, active pressure and bearing capacity of foundations. Paris: GauthierVillard; 1948. [9] Ke´risel J, Absi E. Tables de pousse´e et de bute´e des terres. 3rd ed. Presses de l’E´cole Nationale des Ponts et Chausse´es, Paris, 1990. [10] Sokolovski VV. Statics of granular media. New York: Pergamon Press; 1965. [11] Graham J. Calculation of passive pressure in sand. Can Geot J 1971;8(4). [12] Chen WF. Limit analysis and soil plasticity. Amsterdam: Elsevier; 1975. [13] Martin GR, Nad Yan L. Modelling passive earth pressure for bridge abutments. Earthquake-induced movements and seismic remediation of existing foundations and abutments. Geotech Spec Publ 1995;55:1–16. [14] Benmebarek N, Benmebarek S, Kastner R, Soubra AH. Passive and active earth pressures in the presence of groundwater flow. Ge´otechnique, The Institution of Civil Engineers, London 2006;56(3):149–58. [15] Ovesen NK. Anchor slabs, calculation methods, and model tests. Bull. No. 16, Danish Geotechnical Institute, Copenhagen, 1964. p. 5–39.
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[16] Brinch Hansen J. Resistance of a rectangular anchor slab. Bull. No. 21, Danish Geotechnical Institute, Copenhagen, 1966. p. 12–3. [17] Duncan M, Mokwa RL. Passive earth pressures: theories and tests. J Geotech Geoenviron Eng 2001;127:248–57. [18] Horn A. Re´sistance et de´placement de cule´es de ponts charge´es late´ralement. In: 5th European conf on soil mec hand found engrg, Madrid; 1972. pp. 143–5. [19] Belabdelouhab F. Etude expe´rimentale de la bute´e discontinue sur mode`le re´duit. PhD thesis, Institut National des Sciences Applique´es de Lyon, France, 1988. [20] Meksaouine M. Etude expe´rimentale et the´orique de la pousse´e passive sur pieux rigides. PhD thesis, Institut National des Sciences Applique´es de Lyon, France, 1993.
[21] Weissenbach A. Der erdwiderstand vor schmalen druckflechen. PhD thesis, Franzius Institut fur Grund und Wasserbau der Technischen Hochschule, Hannover, 1961. p. 338. [22] Cundall PA. Distinct element models of rock and soil structure. In: Brown ET, editor. Analytical and computational methods in engineering rock mechanics. London: Allen & Unwin; 1987 [chap 4]. [23] Prandtl G. Eindringungsfestigkeit and festigkeit von sneiden. Angew Math U Mech 1920;15(1). [24] Michalowski RL. The rule of equivalent states in limit-state analysis of soils. J Geotech Geoenviron Eng 2001;127:76–83. [25] Silvestri V. Limitations of the theorem of corresponding states in active pressure problems. Can Geotech J 2006;43:704–13.
Computers and Geotechnics 35 (2008) 47–60 www.elsevier.com/locate/compgeo
Numerical evaluation of 3D passive earth pressure coefficients for retaining wall subjected to translation S. Benmebarek b
a,*
, T. Khelifa a, N. Benmebarek a, R. Kastner
b
a Civil Engineering Laboratory, Biskra University, BP 145 Biskra 07000, Algeria URGC-Ge´otechnique, INSA Lyon, 20 Avenue Albert Einstein, 69621 Villeurbanne Cedex, France
Received 16 May 2006; received in revised form 17 January 2007; accepted 29 January 2007 Available online 27 March 2007
Abstract The 2D passive earth pressures acting on rigid retaining walls problem has been widely treated in the literature using different approaches (limit equilibrium, limit analysis, slip line and numerical computation), however, the 3D passive earth pressures problem has received less attention. This paper is concerned with the numerical study of 3D passive earth pressures induced by the translation of a rigid rough retaining wall for associative soils. Using the explicit finite difference code FLAC3D (Fast Lagrangian Analyses of Continua), the increase of the passive earth pressures due to the decrease of the wall breadth is investigated. The results given by the present numerical analysis are compared with other investigations using limit equilibrium method, upper-bound method in limit analysis as well as experimental measures. These are presented in a form of design tables relating the geometrical parameters, soil properties, and 3D passive earth pressure coefficients Kpc(3D), Kpq(3D) and Kpc(3D) representing the effects of soil weight, surcharge loading and soil cohesion, respectively. 2007 Elsevier Ltd. All rights reserved. Keywords: Numerical modelling; 3D passive earth pressure; Soil; Retaining wall; Interaction; Failure; Behaviour
1. Introduction Numerous geotechnical structures mobilize the passive earth pressures in a three-dimensional manner such as anchor bloc, anchor plates, massif reaction, limited breadth retaining wall, retaining wall with discontinues penetration depth and laterally loaded pile caps. Safe and economical design of retaining structures requires a sound knowledge of the contact pressure exerted. The 2D passive earth pressures problem has been widely treated in the literature theoretically using different approaches (limit equilibrium, limit analysis, slip line and numerical computation), however, the 3D passive earth *
Corresponding author. Tel.: +213 33 74 20 72; fax: +213 33 74 86 87. E-mail addresses: [email protected] (S. Benmebarek), khelifat@ yahoo.fr (T. Khelifa), [email protected] (N. Benmebarek), richard. [email protected] (R. Kastner). 0266-352X/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2007.01.008
pressures problem has been received theoretically less attention apart from the work of Blum [1] using the limit equilibrium approach and Soubra and Regenass [2] using the upper bound theorem of limit analysis. This is indeed due to the difficulty to propose analytically the 3D failure mechanisms. In this paper, the 3D effect of the passive earth pressures problem is investigated using the explicit finite difference code FLAC3D [3] (Fast Lagrangian Analyses of Continua). The aim of this work is firstly to develop a numerical procedure for the analysis of a rigid rough vertical retaining wall with horizontal ground surface subjected to translation and uniform surcharge, then to investigate the influence of the wall breadth on the evaluation of the earth pressure coefficients Kpc, Kpq and Kpc representing the effect of soil weight, surcharge loading and cohesion, respectively. Numerical results and failure mechanisms, particularly the plane view, obtained from the present
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S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
Nomenclature u w d c c K G K0 Kn Ks Pp Ppx h
angle of internal friction of the soil dilation angle of the soil angle of friction at the soil/wall interface soil cohesion unit weight of the soil bulk modulus of the soil shear modulus of the soil earth pressure coefficient at rest interface normal stiffness interface shear stiffness passive earth force horizontal passive earth force penetration depth of the wall
analysis are compared with other investigations using limit equilibrium method, upper-bound method in limit analysis as well as experimental measures. 2. Overview of previous work The 2D earth pressure theories are widely used in geotechnical engineering. They are the Coulomb Theory [4], which treats the passive earth pressure problem in terms of forces, and the Rankine Theory [5], which treats the problem in terms of stresses. Both theories are well known and are described in nearly every soil mechanics textbook. The Coulomb theory is more versatile in accommodating complex configurations of backfills and loading conditions as well as frictional effects between walls and backfills. However, both theoretical and experimental studies have shown that the Coulomb assumption of plane surface sliding is not perfectly valid when the wall is rough, especially in the passive case when interface friction is more than 1/3 of internal soil friction. The curvature of the failure surface behind the wall needs to be taken into account. Hence, Coulomb’s theory leads to largely overestimation of the passive earth pressure. The Rankine’s theory is applicable to the calculation of the earth pressure on a perfectly smooth and vertical wall, but most retaining walls are far from frictionless soil structure interface. More than 50 years ago, several researchers dealt with theoretical procedures for evaluating the earth pressure using the limit equilibrium method [6,7], the slip line method [8–11] and the upper- and lower bound theorems of limit analysis [2,12]. These different approaches generally confirm the accuracy of the Log Spiral Theory [8] for a wide range of values of the internal soil friction and the soil–structure interface friction angle. Similarly, Martin [13] and Benmebarek et al. [14] who used FLAC2D numerical analysis to evaluate passive earth pressures have found fairly close agreement with Log Spiral Theory. In spite of recent published methods, the tendency today in practice
b q 2D 3D Kp2D Kp3D Kpc(3D) Kpc(3D) Kpq(3D)
breadth of the wall surcharge on the ground surface bi-dimensional three-dimensional 2D passive earth pressure coefficients 3D passive earth pressure coefficients 3D passive earth pressure coefficients due to soil weight 3D passive earth pressure coefficients due to soil cohesion 3D passive earth pressure coefficients due surcharge loading
is to use the values given by Caquot and Ke´risel [8] and Ke´risel and Absi [9]. Most of the research effort has concentrated on a refinement of the 2D analysis with little attention given to the 3D effects, apart from the work of Blum [1] using the limitequilibrium method and recently the work of Soubra and Regenass [2] using the limit analysis theory, where threedimensional kinematically admissible failure mechanisms composed of one or several rigid blocks have been considered. However, the 3D effects have been received moderate investigation using experimentally approaches which revealed very high dispersions of the results. Ovesen [15] conducted an extensive series of passive pressure model tests to investigate the 3D effects. Ovesen’s tests were performed on compacted sand with friction angles ranging from / = 32.7 to 41.7. The maximum difference between passive and active earth pressure coefficients (Kp Ka) was 5.7 in Ovesen’s tests, and the correction factor for the 3D effects; did not exceed a value of about 2. These tests showed that passive earth pressures against short structures are higher than those predicted by conventional theory and the difference can be quite significant. Brinch Hansen [16] developed a method for correcting the results of conventional passive pressure theories for 3D effects, based on Ovesen’s test results. Duncan and Mokwa [17] have developed an Excel spreadsheet computer program based on both the Log Spiral Theory and the Ovesen–Brinch Hansen correction for 3D effects, making it applicable to both short and long breadth structures. As a conservative measure based on the result of Ovesen tests, an upper limit of 2.0 was placed by Duncan and Mokwa [17] on the value of the correction factor for the 3D effects that is used in the spreadsheet. In addition, Horn [18] has presented a synthesis of about twenty experimental research works on the 3D passive earth pressure load. The analysis of this synthesis has shown a very high dispersion of the results for the reason that the difference in the material natures, the deficiency
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
in the properties data as well as the scale effects related to the size of the experimental models. To avoid the scale effects, Belabdelouhab [19] and Meksaouine [20] have used the same test frame as well as the material ‘‘sand of Hoston’’. These tests lead thus to coherent results covering from short to long breadth wall. These experimental measures allow to avoid the scale effect by analyzing not the 3D earth pressure results but the ratio Kp(3D)/Kp(2D) obtained under the same experimental conditions. The observations of the failure surface in plan view of the tests carried out by Weissenbach [21], Belabdelouhab [19], Meksaouine [20] and Duncan and Mokwa [17] show that the progression of surface vanishes in the case of the long breadth wall and the distance of surface failure in plan view from the front of the wall does not seem to have exceeded the value of 2.5 times the wall high. In order to appreciate the accuracy of the present numerical analysis, the two theoretical approaches of Blum [1] and the upper bound solution given by Soubra and Regennas [2] for an associated flow rule Coulomb material obeying Hill’s maximal work principle as well as the experimental measures of Mekssaouine [20] and Duncan and Mokwa [17] are used for comparison. 3. Numerical modelling procedure 3.1. Case study This paper is concerned with the numerical evaluation of passive earth pressure coefficients for a rigid rough vertical retaining wall with horizontal ground surface and limited breadth subjected to translation as sketched in Fig. 1. This case was firstly treated theoretically by Blum [1] using the limit equilibrium approach. Blum’s failure mechanism is represented by an extension of the 2D Coulomb failure mechanism into the 3D case as shown in Fig. 2. According to this mechanism, the 3D passive earth force is given by: Plan view of failure surface
49 C
h/2 B
H b
A G h/2 h
D
F
E
π/4+ϕ/2
Fig. 2. Blum’s failure mechanism.
P pc
1 2 / h3 / 2 p 2 p þ þ ¼ ch b tg þ c tg 2 4 2 4 2 6
ð1Þ
From this equation it is evident that Blum solution does not take into account the soil–structure interface friction. Recently, Soubra and Regenass [2] have used the upper bound theorem of the limit analysis and have developed upper bound solutions for the 3D passive earth pressure coefficients. Three translational failure mechanisms referred to as One-Block Mechanism M1, Multiblock Mechanism Mn and Truncated Multiblock Mechanism Mnt have been considered for the calculation schemes as shown in Fig. 3. The mechanism M1 composed of one rigid block is an extension of the 2D well-known Coulomb mechanism [cf. Fig. 3a] into 3D case. The Mn requires a more elaborate failure mechanism [cf. Fig. 3b] composed of n rigid blocks, whereas the Mnt is obtained by a volume reduction of the final block in the Mn mechanism [cf. Fig. 3c]. They concluded that the Mnt mechanism which gives the least upper-bound solution is more efficient than the Mn mechanism, and for practical use, they proposed numerical results based on the Mnt mechanism for various governing parameters. 3.2. Modelling procedure
Limited breadth wall subjected to translation
Fig. 1. Case study.
The analysis of the increase of the passive pressures due to the decrease of the retaining wall breadth, as pointed out by Soubra and Regenass [2], is carried out in this paper using the commercially available three-dimensional code FLAC3D. This code uses an explicit finite difference program to study numerically the mechanical behaviour of a continuous 3D medium as it reaches equilibrium or steady plastic flow. The explicit Lagrangian calculation scheme and the mixed-discretization zoning technique [22] used in FLAC3D ensure that plastic failure and flow are modelled very accurately.
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6h
Wall
b/2 6h
6h
h
2h Z Y X
Fig. 4. Domain for FLAC3D simulation – half symmetry.
boundary corresponding to the plane at y = 0 is restricted in y-direction. The assumptions made in the numerical analysis can be summarized as follows: The soil is homogeneous and isotropic obeying Coulomb criteria plasticity. The wall of dimensions b · h (b = breadth; h = height) is vertical, and the backfill is horizontal. A translational soil–wall movement is assumed. The limit loading is reached by applying wall velocity until a steady plastic flow is obtained. The soil–structure interface friction angle is assumed to be constant.
Fig. 3. Soubra’s failure mechanisms: (a) M1, (b) Mn and (c) Mnt [2].
Accordingly, a retaining wall with limited breadth penetrated by translation in homogeneous and isotropic semi-infinite soil is considered. For this problem, half-symmetry condition is assumed in the numerical simulation. The domain used for the analysis is sketched in Fig. 4, together with its dimensions and axis. The vertical and bottom boundaries were respectively located at a distance six and three times the wall penetration in order to minimize boundary effects. For the boundary conditions applied to this domain, the displacements of the far x- and y- lateral boundaries are restricted in the horizontal x and y directions, respectively. The bottom boundary is constrained in the z-direction and the displacements of the symmetry
In order to develop an acceptable analysis scheme for later computations, preliminary simulations have been carried out by testing the sensibility of passive earth pressure forces by the mesh dimensions, the element size, the boundary conditions and the earth pressure coefficient at rest K0 as well. Fig. 5 shows an example of the mesh retained for this analysis. The grid size is fine near the wall where deformations are concentrated. The material properties, particularly stiffness assigned to an interface, depend on the way in which the interface is used. In the case of soil–structure interaction, the interface is considered stiff compared to the surrounding soil, but it can slip and may be opened in response to the loading. Joints with zero thickness are more suitable for simulating the frictional behaviour at the interface between the wall and the soil. The interface model incorporated in FLAC3D code and its components illustrated in Fig. 6 has been used to simulate the soil/wall contact. The interface constitutive model is defined by a linear Coulomb shear-strength criterion that limits the shear force acting at an interface node, normal and shear stiffnesses, tensile and shear bond strengths, and a dilation angle that causes an increase in effective normal force on the target face after the shear-strength limit is reached.
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
51
Fig. 5. Example of used mesh for domain half symmetry.
The spring in the tangential direction, the slider and the limit strength (Fig. 6) represent the Coulomb shearstrength criterion. The spring in the normal direction, the limit strength and dilation represent the normal contact. The interface has a friction angle d, a cohesion c = 0 kPa, a normal stiffness Kn = 109 Pa/m, and a shear stiffness Ks = 109 Pa/m. These values are selected to approximate the results for the case where the wall is rigidly attached to the grid. For the soil behaviour, a linear elastic-perfectly plastic associative Mohr–Coulomb model encoded in FLAC3D is adopted, requiring the specification of a shear modulus G = 22 MPa, a bulk modulus K = 60 MPa, a unit weight c = 20 kN/m3, a friction angle /, and an angle of dilation w = /. However, the dilation angle of real granular soils is generally less than the angle of friction. The potential effect of non-associative plastic flow (w p /) which may be treated numerically by FLAC3D are beyond the scope of this study. The numerical results of the 3D passive earth pressures are presented in the form of dimensionless coefficients Kpc(3D), Kpq(3D) and Kpc(3D). It should be noted that these coefficients are independent from elastic, weight soil, cohesion and surcharge values used in computation.
The proposed modelling procedure of the passive earth pressures are based on two steps: In the first one, the geostatic stresses are computed. At this stage some stepping is required to bring the model to equilibrium. The reason is that an additional stiffness from interface elements produces an imbalance that necessitates some stepping to equilibrate the model. It should be noted here that preliminary numerical tests have shown that limit passive earth pressure forces are insensitive to the earth pressure coefficient at rest K0. The influence of this coefficient is limited to the necessary displacement for reaching the limit value of passive earth pressures. In the second step, to represent rigid wall translation, a controlled horizontal velocity is applied in several steps to all the wall grid points. First, a relatively high velocity of 106 m/step is applied to the wall until a steady plastic flow state is achieved (i.e. until both conditions (i) a constant earth pressure force and (ii) small values of unbalanced forces are obtained as the number of cycles increases). As the level of error in such a FLAC3D calculation scheme depends on the applied velocity, a more-accurate earth pressure force can be obtained by reducing the wall velocity by half, and continuing to a new steady plastic-flow state. This procedure, recommended by FLAC3D manual [3], is repeated several times, particularly for soils with large values of the internal friction, until the difference between the earth passive forces calculated at two successive steady plastic-flow states becomes negligible. Using a FISH function, the horizontal passive earth pressure force Ppx acting on the wall side can be calculated as the integral of stress components for all soil zones in contact with the wall. At this stage, the horizontal earth pressure force on the wall can be recorded. Hence, the value of the passive earth pressure force Pp [2,8] can be deduced from the following relationship: Pp ¼
Fig. 6. Components of the bonded interface constitutive model.
P px cos d
ð2Þ
The generalized passive earth pressure formula [2,8] can be written as
52
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
Table 1 Comparison of present Kpc(3D) coefficients with other existing solutions b/h
/ ()
d// 0
1/3
2/3
1
Soubra
FLAC3D
Soubra
FLAC3D
Soubra
FLAC3D
Soubra
FLAC3D
0.1
20 25 30 35 40
– – – – –
6.513 10.268 16.288 25.318 40.747
– – – – –
7.199 11.96 20.146 33.875 60.25
– – – – –
7.959 13.913 24.647 44.215 87.768
– – – – –
8.763 15.804 29.483 56.015 113.548
0.25
20 25 30 35 40
5.399 7.983 11.886 20.044 50.43
4.547 6.606 9.565 14.018 20.883
6.624 10.528 17.012 29.026 70.281
5.181 7.939 12.335 19.633 32.384
8.301 14.582 27.017 54.775 140.561
5.751 9.304 15.520 26.871 49.834
10.595 20.583 42.510 95.509 240.815
6.350 10.680 19.279 35.078 65.895
0.5
20 25 30 35 40
3.726 5.229 7.443 11.984 28.594
3.391 4.666 6.502 9.080 13.101
4.538 6.849 10.604 17.354 40.954
3.896 5.680 8.372 12.798 20.281
5.629 9.379 16.634 32.487 79.700
4.378 6.756 10.626 17.642 31.961
6.994 12.776 25.085 54.064 131.753
4.814 7.913 13.108 23.645 45.457
1
20 25 30 35 40
2.887 3.850 5.221 7.954 17.676
2.770 3.639 4.875 6.584 9.199
3.487 5.001 7.391 11.518 25.317
3.180 4.356 6.275 9.140 14.208
4.279 6.760 11.418 21.308 49.269
3.583 5.192 7.910 12.561 22.057
5.139 8.798 16.273 33.202 77.015
4.010 6.159 9.796 16.647 32.698
2
20 25 30 35 40
2.466 3.159 4.111 5.939 12.218
2.459 3.096 3.952 5.109 7.015
2.956 4.069 5.777 8.600 17.499
2.805 3.735 5.073 7.606 10.957
3.593 5.435 8.787 15.683 32.329
3.164 4.493 6.543 10.109 17.006
4.171 6.746 11.764 22.607 49.371
3.515 5.212 8.082 13.464 24.903
5
20 25 30 35 40
2.211 2.743 3.444 4.730 8.942
2.262 2.797 3.489 4.459 5.80
2.633 3.505 4.801 6.849 12.808
2.568 3.314 4.470 6.13 8.80
3.133 4.542 6.965 11.519 21.145
2.876 3.957 5.731 8.350 13.28
3.561 5.456 8.948 16.035 32.361
3.194 4.536 6.812 10.885 19.602
10
20 25 30 35 40
2.125 2.604 3.222 4.327 7.851
2.103 2.512 3.218 4.030 5.095
2.524 3.315 4.473 6.266 11.244
2.480 3.181 4.275 5.773 8.105
2.954 4.178 6.209 9.87 17.226
2.793 3.821 5.610 7.913 12.331
3.348 5.004 7.958 13.730 26.424
3.102 4.412 6.751 10.675 18.108
2D passive earth pressure coefficients FLAC2D 20 25 30 35 40 Caquot
20 25 30 35 40
Soubra = after Soubra and Regenass [2]. FLAC3D = present solution. Caquot = after Caquot and Kerisel [8]. FLAC2D = after Bebmebarek et al. [14].
2.07 – 3.06 3.76 4.69
2.39 – 4.03 5.42 7.53
2.72 – 5.20 7.68 11.96
– – – – –
2.04 2.45 3.00 3.69 4.60
2.40 3.10 4.00 5.40 7.60
2.75 3.70 5.30 8.00 12.00
3.10 4.40 6.50 10.50 18.00
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
53
Table 2 Comparison of present Kpq(3D) coefficients with other existing solutions b/h
/ ()
d// 0
1/3
2/3
1
Soubra
FLAC3D
Soubra
FLAC3D
Soubra
FLAC3D
Soubra
FLAC3D
0.1
20 25 30 35 40
– – – – –
7.618 12.443 20.124 32.526 53.209
– – – – –
8.433 14.343 24.710 42.960 76.927
– – – – –
9.355 16.327 29.989 55.149 113.205
– – – – –
9.998 18.388 34.697 67.880 144.561
0.25
20 25 30 35 40
7.068 10.736 16.329 28.104 72.265
5.406 8.080 12.038 17.858 27.545
8.704 14.202 23.415 40.698 103.502
6.094 9.647 15.390 24.897 42.529
10.960 19.770 37.382 77.039 170.861
6.741 11.278 19.100 33.391 63.243
13.015 24.817 49.352 104.684 243.616
7.454 12.851 22.521 42.628 82.450
0.5
20 25 30 35 40
4.563 6.606 9.664 16.014 39.512
3.915 5.534 7.845 11.190 16.315
5.583 8.690 13.810 23.190 56.591
4.503 6.706 10.175 15.797 25.781
6.967 11.984 21.829 43.615 91.492
5.044 7.869 12.717 21.305 38.783
8.057 14.599 27.909 57.371 130.190
5.552 9.019 15.483 27.996 55.271
1
20 25 30 35 40
3.307 4.540 6.332 9.969 23.136
3.041 4.098 5.566 7.634 10.865
4.014 5.926 8.999 14.436 33.136
3.520 4.977 7.229 10.765 16.930
4.956 8.073 13.951 25.670 51.724
3.946 5.857 9.052 14.608 25.674
5.543 9.445 17.124 33.627 73.351
4.328 6.756 11.023 18.823 35.803
2
20 25 30 35 40
2.677 3.505 4.666 6.946 14.947
2.578 3.329 4.335 5.769 7.869
3.223 4.536 6.585 10.059 21.408
2.971 4.046 5.634 8.090 12.294
3.848 5.902 9.562 16.622 31.723
3.324 4.754 7.061 10.971 18.396
4.256 6.819 11.656 21.638 44.748
3.622 5.398 8.435 14.056 25.165
5
20 25 30 35 40
2.296 2.882 3.666 5.133 10.034
2.302 2.872 3.619 4.609 6.081
2.741 3.694 5.127 7.433 13.923
2.635 3.461 4.643 6.358 9.280
3.148 4.533 6.865 11.077 19.515
2.926 4.029 5.729 8.556 13.652
3.453 5.189 8.277 14.269 27.268
3.155 4.510 6.734 10.690 17.726
10
20 25 30 35 40
2.168 2.673 3.333 4.528 8.263
2.208 2.717 3.377 4.132 5.146
2.577 3.410 4.637 6.558 10.838
2.519 3.266 4.305 5.833 8.322
2.906 4.060 5.924 9.155 15.303
2.794 3.796 5.330 7.821 12.253
3.173 4.620 7.100 11.708 21.219
3.008 4.228 6.201 9.565 15.231
2D Caquot
20 25 30 35 40
2.04 2.44 3.03 3.60 4.50
2.35 3.03 4.00 5.28 7.25
2.65 3.56 5.00 7.10 10.72
2.85 4.00 5.88 8.80 14.60
Soubra = after Soubra and Regenass [2]. FLAC3D = present solution. Caquot = after Caquot and Kerisel [8].
P p ¼ K pcð3DÞ c
h2 b þ K pcð3DÞ c h b þ K pqð3DÞ q h b 2 ð3Þ
where Pp is the passive earth force; Ppx is the horizontal passive earth force; c is the unit weight of the soil; h is the penetration depth of the wall; b is the breadth of the wall; d is the soil–wall interface friction angle; c is the soil
cohesion; q is the surcharge on the ground surface; Kpc(3D), Kpq(3D) and Kpc(3D): 3D passive earth pressure coefficients that are functions of the angle of internal friction. They represent the influence of soil weight, cohesion, and surcharge loading, respectively. Accordingly, the passive earth pressure coefficients Kpc(3D), Kpq(3D) and Kpc(3D) are determined independently in three configurations outlined as follows:
54
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
Kpc(3D): The cohesive-frictional soil is replaced with a frictitious cohesionless (c = 0) soil having the same values of parametersc, / and without surcharge on the ground surface (q = 0). Following the first and the second steps of the proposed modelling procedure, the Kpc(3D)coefficient is determined from the following expression:
surcharge on the ground surface (q 6¼ 0). In this case, only the second step is necessary and the Kpq(3D) is given by the following expression:
K pcð3DÞ ¼ 2P px =c h2 b cosd;
Kpc(3D): In this case, the soil is treated as a weightless, (c = 0), cohesive medium (c 6¼ 0) with parameter / 6¼ 0 and without surcharge on the ground surface (q = 0). Also, in this case only the second step is necessary and the Kpc(3D) is determined by the following expression:
c 6¼ 0; / 6¼ 0; c ¼ 0; q ¼ 0 ð4Þ
Kpq(3D): The soil is next treated as a weightless, cohesionless medium with parameter / 6¼ 0 subjected to uniform
K pqð3DÞ ¼ P px =q h b cos d;
/ 6¼ 0; q 6¼ 0; c ¼ 0; c ¼ 0 ð5Þ
Table 3 Comparison of present computation coefficients Kpc(3D) with the theorem of corresponding states solutions b/h
/ ()
d// 0
1/3
2/3
1
TCS
FLAC3D
TCS
FLAC3D
TCS
FLAC3D
TCS
FLAC3D
0.10
20 25 30 35 40
18.183 24.54 33.124 45.024 62.220
18.014 24.129 33.068 44.913 62.138
20.403 28.591 41.041 59.893 90.457
19.874 28.076 40.405 59.210 90.163
22.879 32.775 50.101 77.203 133.584
22.791 32.455 49.797 76.136 129.063
24.546 37.067 58.099 95.196 170.732
24.183 36.574 57.306 94.084 170.014
0.25
20 25 30 35 40
12.105 15.183 19.118 24.076 31.635
11.997 15.101 19.115 24.056 31.665
13.977 18.521 24.898 34.097 49.461
13.594 18.053 24.358 33.506 48.855
15.697 21.947 31.240 46.130 74.039
15.055 21.315 30.360 45.043 72.986
17.556 25.193 37.009 59.134 96.708
16.878 24.523 36.655 58.544 96.605
0.5
20 25 30 35 40
8.009 9.723 11.856 14.552 18.252
8.003 9.701 11.858 14.582 18.023
9.606 12.214 15.865 21.101 29.501
9.188 11.756 15.305 20.380 28.978
11.035 14.636 20.184 28.870 44.888
10.587 13.994 19.426 27.760 44.010
12.330 16.975 24.818 38.238 64.316
11.909 16.316 23.976 37.433 63.607
1
20 25 30 35 40
5.608 6.644 7.909 9.474 11.757
5.584 6.591 7.824 9.467 11.720
6.905 8.506 10.763 13.915 18.952
6.655 8.030 10.245 13.330 18.395
8.018 10.322 13.836 19.306 29.265
7.754 9.903 13.261 18.464 28.292
8.967 12.122 17.093 25.138 41.114
8.626 11.710 16.410 24.766 41.045
2
20 25 30 35 40
4.336 4.995 5.777 6.811 8.186
4.312 4.963 5.772 6.798 8.166
5.397 6.509 8.000 10.095 13.427
5.157 6.259 7.738 9.678 12.987
6.309 7.956 10.387 14.112 20.591
5.710 7.294 9.896 13.461 19.780
7.028 9.210 12.610 18.330 28.436
6.710 8.776 12.130 17.823 28.692
5
20 25 30 35 40
3.577 4.015 4.536 5.154 6.056
3.55 3.990 4.524 5.174 6.031
4.473 5.255 6.283 7.622 9.835
4.283 5.023 5.991 7.315 9.438
5.216 6.402 8.080 10.663 14.937
5.018 6.116 7.699 10.167 14.467
5.744 7.306 9.664 13.523 19.570
5.513 6.986 9.364 13.234 20.204
10
20 25 30 35 40
3.319 3.682 4.117 4.473 4.941
3.305 3.674 4.109 4.266 4.721
4.155 4.836 5.698 6.872 8.693
3.985 4.757 5.425 6.575 8.324
4.853 5.902 7.389 9.614 13.270
4.654 5.666 7.084 9.223 12.806
5.341 6.701 8.741 11.916 16.597
5.091 6.396 8.531 11.831 16.478
TCS = after the theorem of corresponding states [8]. FLAC3D = present solution.
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
K pcð3DÞ ¼ P px =c h b cos d;
/ 6¼ 0; c 6¼ 0; c ¼ 0; q ¼ 0 ð6Þ
It is interesting to note that coefficients Kpc(3D) and Kpq(3D) are related by the following relationship [cf. theorem of corresponding states of Caquot (Caquot and Ke´risel [8]) in the literature: K pcð3DÞ ¼
K pqð3DÞ cos1 d tan /
ð7Þ
In order to examine the validity of Eq. (7), the present FLAC3D results of Kpc(3D) are compared to the results of this relationship (Eq. (7)) in the following section. 4. Results and discussion To investigate how the passive earth pressure coefficients are affected by the wall breadth, five values of the internal friction angle / (20, 25, 30, 35 and 40) are considered for each of the four values of soil–wall interface friction d (d// = 0, 1/3, 2/3 and 1).
55
The computed values of passive earth pressure coefficients Kpc(3D), Kpq(3D) and Kpc(3D) are listed in Tables 1– 3, respectively. The present coefficients Kpc(3D) and Kpq(3D) are compared to solutions given by Soubra and Regenass [2] reported in Tables 1 and 2 and the present coefficients Kpc(3D) are compared to the coefficients obtained by using the theorem of corresponding states (Eq. (7)). The currently used 2D passive earth pressure coefficients of Caquot and Ke´risel [8] and the FLAC2D results [14] are also reported in Tables 1 and 2 for comparison with the present results for long breadth retaining wall. These tables clearly show the sensitivity of the passive earth pressure coefficients Kpc(3D), Kpq(3D) and Kpc(3D) to the b/h ratio. These coefficients increase with the decrease of the b/h ratio. For large values of b/h, the 3D effect vanishes and the Kpc(3D) and Kpq(3D) converge to the 2D coefficients. For instance, for b/h = 10, the 3D effects expressed by the ratios Kpc(3D)/Kpc(2D) and Kpq(3D)/Kpq(2D) are less than 1.15 for the present results compared to the 2D results of Caquot and Ke´risel [8] and Benmebarek et al. [14] using numerical approach for 2D behaviour.
Fig. 7. Traces in plan view of Mnt Mechanism for / = 20, 30, 40; d// = 0 and 1; and b/h = 1 (axis unities = breadth wall) [2].
56
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
However, for Soubra and Regenass [2] solutions (Tables 1 and 2), the ratios Kpc(3D)/Kpc(2D) and Kpq(3D)/Kpq(2D) for / = 40 and d// = 0 reach 1.71 and 1.84, respectively. As observed in these Tables, the 3D effects are most pronounced for great / values for both the present computation and Soubra’s results. For b/h P 2 the 3D effects presented by the ratios Kpc(3D)/Kpc(2D) and Kpq(3D)/Kpq(2D) are less than 2 times as observed experimentally by Ovesen
and used by Duncan and Mokwa [17]. However, for b/h 6 1, the ratios Kpc(3D)/Kpc(2D) and Kpq(3D)/Kpq(2D) can be more than 2 times and increase highly with the decrease of b/h. For instance, the ratio Kpc(3D)/Kpc(2D) is equal to 1.817, 2.525, 3.661 and 6.308 and the ratio Kpq(3D)/Kpq(2D) is equal to 2.452, 3.786, 5.647 and 9.901 for b/h = 1, 0.5, 0.25 and 0.1, respectively when / = 40 and d// = 1. Soubra’s results give much higher 3D effect. For instance, in
Fig. 8. Failure mechanism for the interface friction angle d// = 0 when / = 35 and b/h = 1 visualized by: (a) displacement field vectors and (b) distribution of maximum shear strain rates.
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
the case of b/h = 0.25 when / = 40 and d// = 1, the ratios Kpc(3D)/Kpc(2D) and Kpq(3D)/Kpq(2D) are equal to 13.379 and 16.686, respectively. By comparing these results to the present ones (3.661 and 5.647), the high differences found may be explained by the use of the Mnt mechanism in Soubra’s results which seems to greatly overestimate the passive earth pressures shown by a significant change in the plan view of failure surface between / = 30 and 40 (Fig. 7). For both d// = 0 and
57
d// = 1 when / = 40 the distance of the failure surfaces from the front of the wall are 6 and 4 times the wall penetration depth, respectively. The Soubra’s Mnt mechanism for high soil friction is clearly in disagreement with experimental observations as shown by Weissenbach [21], Belabdelouhab [19], Meksaouine [20] and Duncan and Mokwa [17] where the distance of surface failure in plan view from the front of the wall does not seem to exceed the value of 2.5 times the wall depth.
Fig. 9. Failure mechanism for the interface friction angle d// = 2/3 when / = 35 and b/h = 1 visualized by: (a) displacement field vectors and (b) distribution of maximum shear strain rates.
58
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60
Figs. 8 and 9 show the present view of the failure mechanisms visualized by the displacement field and the distribution of maximum shear strain rates at the end of the run for two values of the interface friction angle (d// = 0, 2/3) when / = 35 and b/h = 1, respectively. For d// = 0 the failure surface in depth is similar to the plane surface proposed by Rankine. However for d// = 2/3 the failure mechanism is similar to the Prandtl mechanism [23] with a radial shear zone followed by a Rankine passive wedge. Fig. 10 shows, for both b/h = 1 and 5, the surface plane views of the zone mobilized by the passive earth pressures. For all present computation results, the surfaces are smooth as observed experimentally by Meksaouine [20] and they are highly different from Blum’s failure mechanism [1]. The distance of the surface failure in plan view from the front of the wall is in congruous with experimental observations. Fig. 11 shows the comparison of Kpc(3D) between the present results and those given by Blum and Soubra for / = 35; and d// = 0 and 2/3; and for different values of b/h ratio. A good agreement is observed for the case of a
smooth wall. However, for rough wall, Blum’s solutions greatly underestimate the passive earth pressure coefficients. This may be explained by the fact that the soil– structure interface friction in Blum’s mechanism is assumed smooth (d = 0). Moreover, for rough wall, the difference between the present computation results and Soubra’s results increase with the decrease of the b/h ratio. This can be explained as mentioned above by the fact that the Mnt mechanism taken by Soubra gives unrealistic failure mechanism. From 2D and 3D experimental tests conducted by Belabdelouhab [19] and Meksaouine [20], respectively, the case b/h = 0.5, / = 34.5 and d// = 0.43 shows a ratio Kp(3D)/Kp(2D) equal to 2.1. For the same case and using interpolation (Table 1), the present results give a ratio equal to 2.30, however the Soubra’s results give a ratio equal to 3.57. From this comparison, the present computation results are in accordance with experimental observations than Soubra’s ones. In Table 3, the computation results of Kpc(3D) are compared to the values of Kpc(3D) expressed by Eq. (7) obtained by using the theorem of corresponding states of Caquot.
Fig. 10. Visualization of surface plane views of the zone mobilized by the passive earth pressures for both b/h = 1 and 5.
S. Benmebarek et al. / Computers and Geotechnics 35 (2008) 47–60 60
FLAC-3D
55
Soubra
50
Kp(3D)
59
45
FLAC-3D
40
Soubra
35 30
Blum
δ/φ = 0
25 20
δ/φ = 2/3
15 10 5 0 0
1
2
3
4
5
6
7
8
9
10
b/h
Fig. 11. Comparison with Blum and Soubra for / = 35; and d// = 0 and 2/3; and for different values of b/h ratio.
This theorem expresses the Kpc(3D) in function of Kpq(3D) for the same friction angle /. Although different calculation procedures are used, the results of both calculations are very close, with a difference less than 6%. This comparison confirms on the one hand, the validity of the theorem of corresponding states of Caquot for the case of vertical retaining wall with horizontal ground surface as pointed out by Michalowski [24] and Silvestri [25], and on the other hand the reliability of the present modeling procedure and the preliminary simulations. 5. Conclusions Numerical computations of the 3D passive earth pressure coefficients Kpc(3D), Kpq(3D) and Kpc(3D) have been performed using FLAC3D code. A number of conclusions may be drawn from this investigation: The passive earth pressures coefficients ratios Kpc(3D)/ Kpc(2D) and Kpq(3D)/Kpq(2D) increase with the decrease of the b/h ratio. The ratio may be more than 2 times for b/h 6 1. The surface plane view of the distribution of maximum shear strain rates has shown a smooth failure mechanism as observed experimentally by Meksaouine [20]. The distance of the surface failure in plan view from the front of the wall is in good agreement with experimental observations [17,20,21]. However the Soubra’s surface [2] highly overestimates this distance for great soil internal friction angle /. The comparison of the present computation results with the upper bound solution in the framework of the limit analysis theory given by Soubra and Regenass [2] has shown that these authors greatly overestimate the 3D passive earth pressure coefficients. In the case of b/h = 10, the present computation results of the Kpc(3D) and Kpq(3D) show that the maximum 3D effect is less than 15% compared to the results of Caquot and Ke´risel [8] for bi-dimensional behaviour. For b/h P 10, the 3D effect may be neglected.
The present FLAC3D results of Kpc(3D) confirm the validity of the theorem of corresponding states of Caquot for the case of vertical retaining wall with horizontal ground surface. References [1] Blum H. Wirtschaftliche Dalbenformen und deren Berechnung. Buatechnik 1932;10(5). [2] Soubra A-H, Regenass P. Three-dimensional passive earth pressures by kinematical approach. ASCE J Geot Geoenv Eng 2000:969–78. [3] FLAC3D. Fast lagrangian analysis of continua. Minneapolis: ITASCA Consulting Group, Inc.; 2000. [4] Coulomb CA. Essai sur une application des re`gles des maximas et minimas a` quelques proble`mes de statique relatifs a` l’architecture. Me´m. acad. roy. pres. divers savanta, vol. 7, 1776, Paris [in French]. [5] Rankine WJM. On the stability of loose earth. London: Philosophical Trans Royal Soc; 1857. [6] Rahardjo H, Fredlund DG. General limit equilibrium method for lateral earth forces. Can Geotech J 1984;21(1). [7] Zhu D-Y, Qian Q-H, Lee CF. Active and passive critical slip fields for cohesionless soils and calculation of lateral earth pressures. Ge´otechnique 2001;51(5). [8] Caquot A, Ke´risel J. Tables for the calculation of passive pressure, active pressure and bearing capacity of foundations. Paris: GauthierVillard; 1948. [9] Ke´risel J, Absi E. Tables de pousse´e et de bute´e des terres. 3rd ed. Presses de l’E´cole Nationale des Ponts et Chausse´es, Paris, 1990. [10] Sokolovski VV. Statics of granular media. New York: Pergamon Press; 1965. [11] Graham J. Calculation of passive pressure in sand. Can Geot J 1971;8(4). [12] Chen WF. Limit analysis and soil plasticity. Amsterdam: Elsevier; 1975. [13] Martin GR, Nad Yan L. Modelling passive earth pressure for bridge abutments. Earthquake-induced movements and seismic remediation of existing foundations and abutments. Geotech Spec Publ 1995;55:1–16. [14] Benmebarek N, Benmebarek S, Kastner R, Soubra AH. Passive and active earth pressures in the presence of groundwater flow. Ge´otechnique, The Institution of Civil Engineers, London 2006;56(3):149–58. [15] Ovesen NK. Anchor slabs, calculation methods, and model tests. Bull. No. 16, Danish Geotechnical Institute, Copenhagen, 1964. p. 5–39.
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[16] Brinch Hansen J. Resistance of a rectangular anchor slab. Bull. No. 21, Danish Geotechnical Institute, Copenhagen, 1966. p. 12–3. [17] Duncan M, Mokwa RL. Passive earth pressures: theories and tests. J Geotech Geoenviron Eng 2001;127:248–57. [18] Horn A. Re´sistance et de´placement de cule´es de ponts charge´es late´ralement. In: 5th European conf on soil mec hand found engrg, Madrid; 1972. pp. 143–5. [19] Belabdelouhab F. Etude expe´rimentale de la bute´e discontinue sur mode`le re´duit. PhD thesis, Institut National des Sciences Applique´es de Lyon, France, 1988. [20] Meksaouine M. Etude expe´rimentale et the´orique de la pousse´e passive sur pieux rigides. PhD thesis, Institut National des Sciences Applique´es de Lyon, France, 1993.
[21] Weissenbach A. Der erdwiderstand vor schmalen druckflechen. PhD thesis, Franzius Institut fur Grund und Wasserbau der Technischen Hochschule, Hannover, 1961. p. 338. [22] Cundall PA. Distinct element models of rock and soil structure. In: Brown ET, editor. Analytical and computational methods in engineering rock mechanics. London: Allen & Unwin; 1987 [chap 4]. [23] Prandtl G. Eindringungsfestigkeit and festigkeit von sneiden. Angew Math U Mech 1920;15(1). [24] Michalowski RL. The rule of equivalent states in limit-state analysis of soils. J Geotech Geoenviron Eng 2001;127:76–83. [25] Silvestri V. Limitations of the theorem of corresponding states in active pressure problems. Can Geotech J 2006;43:704–13.