EARTH PRESSURES AGAINST RIGID RETAINING WALLS IN THE MODE OF ROTATION ABOUT BASE BY APPLYING ZERO-EXTENSION LINE THEORY

Transcription

1 EARTH PRESSURES AGAINST RIGID RETAINING WALLS IN THE MODE OF ROTATION ABOUT BASE BY APPLYING ZERO-EXTENSION LINE THEORY Morshedi S.M., Ghahramani A. 2, Anvar S.A. 2 and Jahanandish M. 2 Department of Geotechnical Engineering, Shiraz Univ., Shiraz, Iran 2 Department of Civil Engineering, Shiraz Univ., Shiraz, Iran *Author for Correspondence ABSTRACT This paper illustrates the use of the zero-extension line theory for determining distribution of active and passive pressures from sand backfill behind a rigid retaining all rotating about its base. These data demonstrate a comparatively linear distribution in the active case. Hoever, because of higher remaining pressure on the loer all (due to the lack of formation of the active state), the point of application of the resultant force stands loer than H/3 (here H is the height of the all). In the passive case, the pressure distribution is approximately parabola-shaped. It also lacks a passive state formation on the loer all, hich causes the location of the resultant force to increase in height. A comparison is made among the obtained results, solutions resulting from Coulomb theory, and recent analytical and numerical research. These comparisons sho that the magnitude of the active force is nearly 2 percent greater than the Coulomb solutions in the passive case. The resultant force is about 75 percent smaller than the Coulomb solution, but agrees ell ith results obtained from Lancellotta and Shiau. Applying the Coulomb theory is not reliable in either case. Keyords: Zero-Extension Lines, Earth Pressure, Retaining Wall, Rotational Mode, Finite Difference INTRODUCTION Knoledge of the magnitude and distribution of earth pressures against retaining alls is important in the design of many civil engineering structures. Common methods for stability analysis in geotechnical problems (including retaining alls) focus more attention on the properties and behavior of soil in the ultimate state. A number of these methods assume a failure surface and compute the necessary force for moving the edge above the assumed surface. In these methods, it is supposed that the soil fails in accordance ith the Mohr-Coulomb criteria. In the Coulomb limit equilibrium, the failure line in the backfill consists of a straight line. Hoever, Rankine assumed that the failure lines are formed throughout the soil mass. Boussinesq examined the problem ith the same assumption and consideration of the all roughness. Various laboratory studies have assessed the classical theories of the lateral earth pressure as applied to the design of retaining alls (Caltabiano et al., 999; Fang and Ishibashi, 986; Matsuo et al., 978; Sherif et al., 982). These investigations demonstrated that the distribution pattern of lateral pressure depended on the mode of all movement. Therefore, the magnitude and point of application of the lateral force for various all movements (rotation about top, rotation about base and translation) are not equal, and these amounts can be different from the Coulomb solutions (Fang and Ishibashi, 986). The progressive nature of failure in soil and the necessity of knoing the load-deflection behavior at loads other than the limit loads have caused researchers to focus on the strain field. Researchers studied the directions here the linear strain is zero. These directions, called zero extension lines (ZEL), have many applications in understanding soil deformation (James and Bransby, 97; Roscoe, 97). A simple pattern of ZEL has been used for finding the strain field behind a model retaining all, ith good agreement beteen the predictions and observations (James and Bransby, 97). Success in the prediction of a strain field by the ZEL led the researchers to implement this theory in obtaining the mobilized strength at different points of the soil mass (Roscoe, 97). This information as necessary for obtaining the stress field. It as this idea that led to the development of the method of associated field, hich orks Copyright 24 Centre for Info Bio Technology (CIBTech) 2456

2 ell in the prediction of load-deflection behavior (Atkinson and Potts, 975; James et al., 972; Serrano, 972). Hoever, this method has a fe problems: ) It requires the use of an iterative process of computations to achieve convergence and compatibility beteen the and fields. 2) It requires elaborate interpolation routines because the velocities and stresses ere not computed at the same points. Attempts have been made to find a ay to calculate the stress field by the ZEL alone. To alternatives have been presented for this problem. In the first, stresses ere calculated by considering the force equilibrium of the soil elements beteen the ZEL (Habibagahi and Ghahramani, 979). In the second, equilibrium-yield equations ritten along the stress characteristics ere transferred into the ZEL directions. The second method has more applications and has been shon to lead to the same results as the first one (Anvar and Ghahramani, 997). The present study deals ith the determination of the magnitude and distribution of active and passive pressures behind a rigid retaining all rotating about its base. For this purpose, equilibrium-yield equations along the ZEL ere transferred into incremental form, then used in riting a computer program in the MATLAB softare environment for draing ZEL mesh and calculating active and passive pressures behind a all. To assess the ZEL method, the obtained results from this method have been compared ith Coulomb-theory solutions. Zero-Extension Line Theory If the soil displacements in the x and z directions are represented by U and W respectively, strains are represented by,,, and compressive strain is considered positive, then: x z xz U W U W x ; z ; xz () x z z x The Mohr circle of strain is shon in Figure. If is the angle of dilation of soil, then sin is related to the volumetric strain, v, and the maximum shear strain, max, by sin v max (2) Figure : Mohr circle and principle directions of strains It is assumed that the direction of and coincide. If the origin of lines is P on the Mohr strain circle, then linear strain is zero along the PA and PB directions. These are the zero extension lines. It is Copyright 24 Centre for Info Bio Technology (CIBTech) 2457

3 ith the direction. We can use the Mohr strain circle and algebraic manipulations to determine the to strain characteristic directions. For the positive direction, dz tan (3) dx For the minus direction, dz tan (4) dx Accordingly, the field of zero extension lines and strain characteristics coincide. Through the aid of the matrix calculations, the folloing relation beteen U and W along both characteristic directions is obtained: clear that both of these lines make the angle 4 2 du. dx dw. dz (5) From Equations (3), (4) and (5), there exists the possibility of calculating U and W at point C if these parameters and are knon at points A and B (Figure 2). Therefore, if the field of ZEL and the displacements at the boundaries are knon, the displacements at the interior points of the field can be calculated. Figure 2: Strain characteristics Equilibrium Equations along Zel Consider a soil element under stress state,,, as shon in Figure 3. It is assumed that the soil is at x z the limit state of equilibrium at the same time as is the angle that direction makes ith x axis. xz Figure 3: Stress field and principle directions Figure 4: Mohr circle of stress Copyright 24 Centre for Info Bio Technology (CIBTech) 2458

4 The resulting Mohr circle is the same as shon in Figure 4, in hich the origin of planes is P. Thus, there are to directions for yield: PA and PB, both of hich connect point P to the tangency points of yield lines ith the Mohr circle. These directions that make angle 4 2 ith axis are yield lines, hich coincide ith stress characteristics. The equilibrium equations of the soil element shon in Figure 3 hen subjected to body forces X and Z are as follos: x xz X ; xz z Z (6) x z x z Through the Mohr stress circle and algebraic manipulation, the folloing relationships beteen u and along the stress characteristic lines are gained: Along the plus characteristicsdz dx tan : du 2 u tan c d X tandz dx Z tandx dz c c u c tan dx dz dx dz z x z x Along the minus characteristicsdz dx tan : du 2 u tan c d X tandz dx Z tandx dz Assume that axis and that and and (hich coincide ith axis) (Figure 5). c c (8) u c tan dx dz dx dz z x z x denote the plus and minus stress characteristics that make angle ith denote the plus and minus zero extension lines that make angle ith axis (7) Figure 5: ZEL and stress characteristics lines For any function f of the variable x, z then takes its directional derivatives ith respect to the plus and minus stress characteristics and ZEL. Thus, the folloing equations are obtained: f f cos sin x. f cos sin f z (9) Copyright 24 Centre for Info Bio Technology (CIBTech) 2459

5 f f cos sin x. () f cos sin f z From Eqs. (9) And (), yield-equilibrium equations along stress characteristics and extensive algebraic and trigonometric manipulations, the folloing equations containing the differential of u and along and are be obtained: For plus direction, du 2u tan c d d X tan. dz. dx Z tan. dx dz u c tan tan. d d cos c tan. dc d cos For minus direction, du 2u tan c d d X tan. dz dx Z tan. dx dz u c tan tan. d d cos c tan. dc d cos here sin.sin sin sin cos ; ; (3) cos.cos cos.cos cos These equations are the stress equilibrium equations along the ZEL. It is orth noting that Eqs. () And (2) on ZEL reduce exactly to Eqs. (7) and (8) on stress characteristics if, hich holds for the associative flo of soils. This creates an analytical tool ith the knon values of the stress state u and, displacements U and W and coordinates x and z at the points A and B along minus and plus ZEL directions. We can find these parameters at the point C (Figure 2). Practical Examples and Discussions Equations (3), (4), (5), () and (2) ere transferred into incremental form, then used in riting a computer program in the MATLAB softare environment for draing ZEL mesh and calculating of the active and passive pressures behind a all. For these purposes, e need a dilation angle and an initial friction angle for the backfill material. A dense sand, hen exposed to the shear, shos a peak and critical shear strength. As shon by Cole (967), the angle of dilation remains constant during a large portion of a shear test; hence, the results presented in Figure 6 ere considered as the input for the ZEL program. This gives a 5 and an initial 25 at a void ratio of.534. The passive and active cases simulate the loading and unloading conditions on an element of soil. Because of this, the results of shear tests simulating the loading condition cannot be used to predict active behavior (Ghahramani and Clemence, 98). The results of several tests indicate that, although the peak of sin for the active and passive cases is nearly equal, the shear strain for the passive peak is 3 to 8 Copyright 24 Centre for Info Bio Technology (CIBTech) 246 () (2)

6 times larger than the shear strain for the active peak of sin. Therefore, it is recommended to use shear strain results such as in Figure 6 for the passive case and to use the same curve but a reduced on the order of /3 for the active case. sin j e-2-6.e-2-4.e-2-2.e-2.e Shear Strain () Shear Strain () (a) (b) Figure 6: a) sin ; b) curves for a dense sand (Cole, 967) Volumetric Strain (n) A general calculation process is as follos: a rotation angle,, is applied to the all. From the displacements (U & W) at the points along the all, e can obtain the displacements throughout the ZEL mesh. After the shear strain is calculated at the points of mesh from Eq. (), the angle of internal friction developed at these points can be obtained from a simple shear test. Finally, from the developed friction angle and Eqs. () and (2), the magnitudes of u and at each point of the mesh and therefore the pressures behind the all can be determined. To assess and compare the results obtained from the ZEL program ith the classical theory of earth pressure, to examples have been examined for the active and passive cases. In these examples, the height of the all is 2 m, the unit eight of the backfill material is8.5 2 KN m 3 KN m, the intensity of surcharge is 5 and the number of mesh divisions considered is2. Active Case: The active earth pressure distribution behind a all rotating about its base is generally believed to be hydrostatic, and experimental evidence has proven this for most of the all depth (Sherif et al., 984). Hoever, the stress condition near the bottom of the all demonstrates a more complicated situation. Figure 7a shos typical change of active pressure distribution as a function of all rotation angle. As is shon in Figure 7b, the active earth pressure decreases rapidly soon after the all starts to rotate up to.5. It then increases at a much sloer rate as the all rotation angle increases. When the upper parts of the all enter the active state at a rotation angle of about.3 in Figure 7a, the remaining higher pressures at the loer parts of the all lead the point of application of the resultant force to a loer position. In these stages, the top of the all translation is about.5 H. This implies that the bottom point of the all ill never be able to completely enter the active stage because it requires a great translation, hich opposes the adequate serviceability of the retaining all. This problem as also pointed out by Sheriff et al., (984). The obtained magnitude of the total active thrust from the ZEL kn KN analysis for this all is.4, hich differs from the Coulomb solution of 8.45 by about 2 %. Figure 7c shos the changing point of application of the resultant force as a function of all rotation angle. It is necessary to mention that the Coulomb theory does not directly determine the lateral pressure distribution, but the triangular shape of the linear backfill can be shon. From this interpretation of the m Coulomb theory, the point of application of the resultant force is.74 from the base. Because of higher Copyright 24 Centre for Info Bio Technology (CIBTech) 246

7 m pressure at the loer part of the all, the point of application falls to a loer location.72 from the base of the all. According to the Coulomb theory, the resultant force is inclined at an angle from perpendicular to the back face all, hile the results given from ZEL analysis sho that this angle is a little smaller than. If, in the active state, the shear strength is equal through the hole height of the all (i.e., max ), then the results obtained from the ZEL analysis ill precisely match the Coulomb solution (see Figure 7a). Such an occurrence, hoever, is practically impossible. In addition, a continuous mass of dense sand ill degenerate into quasi-rigid blocks separated by rupture surfaces soon after the shear strain in the mass exceeds that required to cause the sand to reach its peak stress ratio condition. m Figure 8a shos the initial and deformed ZEL net for the rotation angle.5 of a 2 all. In the active case, shon in Figure 8b, the ZEL nets for a smooth all and a rough all are the approximately the same. This implies that the roughness of the all has very little effect on the active pressures. To demonstrate this aspect, results are given in Table for ZEL analysis at different magnitudes of, along ith solutions from Coulomb theory and Lancellotta s equation (Lancellotta, 22). Table : Active pressure Coefficient k a Coulomb Solution Lancellotta ZEL Analysis / / / Passive Case: Figures 9a and 9b sho the changes of passive pressure distribution and passive resultant force, respectively, as functions of all rotation angle. It is evident from Figure 9b that the passive pressure first increases due to shearing and then decreases, similar to the results of the simple shear test. In this case, the formation of a fully passive state at the all base is very difficult, if not impossible. At a rotation angle of about., the upper part of the all reaches the passive state, hile the mobilized friction angle in the loer parts is much less than max. The strain is high near the top of the all but becomes much smaller at greater depth. As a consequence, high stresses ill be generated at the top of the all and cause the resultant force to rise to higher location. At later stages, strain in the sand near the top of the all is large and falls belo its peak value. Thus the stresses at the top of the all decrease belo their early peak value. No rupture surfaces are predicted near the toe of the all, even after 5 of all rotation (. ). As seen in Figure 9a, the passive pressure distribution is not simply triangular. This change of earth pressure distribution causes the resultant force to rise to a higher location. Figure 9c shos the change of the point of application of resultant force as a function of all rotation angle. If according to Figure 9b, the pressure distribution for 2.4 is chosen as the passive case, and the necessary displacement for the formation of passive state is about.4h, hich equals cm 8. for a m 2 all. Copyright 24 Centre for Info Bio Technology (CIBTech) 2462

8 Height of the Wall (m) Coulomb Solution ZEL Analysis ith max ZEL Analysis for ZEL Analysis for ZEL Analysis for ZEL Analysis for ZEL Analysis for Active Pressure (Kpa) (a) 25 Total Active Thrust (KN) ZEL Analysis Coulomb Solution Rotation Angle (Deg.) (b) Point of Application from the Base (m) Rotation Angle (Deg.) (c) ZEL Analysis Coulomb Solution Figure 7: a) Distribution of Active Pressure at Different Wall Rotation Angle; b) Active Resultant Force versus Wall Rotation Angle; c) Point of Application of Resultant Force versus Wall Rotation Angle Copyright 24 Centre for Info Bio Technology (CIBTech) 2463

9 Height of the Wall (m) Mixed Zone Goursat Zone Rankine Zone Height of the Wall (m) Distance from the Back of the Wall (m) Distance from the Back of the Wall (m) (a) (b) Figure 8: a) Initial and Deformed ZEL Net at Active Condition ith 2; b) ZEL Net ith KN The resultant force from this analysis is hich, in comparison ith the Coulomb theory equal KN to 77.24, demonstrates the overestimation and lack of safety of the Coulomb solutions. The Coulomb theory estimated greater passive force of about 75 %, hich as confirmed in previous investigations (Fang and Ishibashi, 986; Ghahramani and Clemence, 98; Habibagahi and Ghahramani, 979; Harr, 966; Jahanandish, 988; Jahanandish et al., 989; James and Bransby, 97; James and Bransby, 97; James et al., 972; Lancellotta, 22; Matsuo et al., 978; Roscoe, 97; Serrano, 972; Sherif et al., 984; Sherif et al., 982). m The point of application of the resultant force as calculated at.73 from the base, hereas the m Coulomb theory as calculated.74 from the base. m Figure a shos the initial and deformed ZEL net for the rotation angle.2 for the 2 all. In the passive case, shon in Figure b, the ZEL net for a smooth and a rough all are different. This implies that the roughness of the all has a strong effect on the passive pressures. This condition is for the case 2, but as shon in Figure c, the error of the Coulomb method increases ith the increase, and hen becomes greater than 2, the error amount of this method goes beyond %. Furthermore, the results obtained from ZEL analysis for different magnitudes of ith solutions resulted from numerical and analytical solutions of Lancellotta and Shiau (Shiau et al., 28) are given in Table 2. Table 2: Passive pressure Coefficient k p Coulomb Solution Lancellotta Shiau / / ZEL Analysis 2/ Copyright 24 Centre for Info Bio Technology (CIBTech) 2464 LB UB

10 Height of the Wall (m) Coulomb Solution ZEL Analysis ith max ZEL Analysis for ZEL Analysis for ZEL analysis for ZEL Analysis for ZEL Analysis for Passive Pressure (Kpa) (a) 8 7 Total Passive Thrust (KN) ZEL Analysis Coulomb Solution Rotation Angle (Deg.) (b) Point of Application from the Base (m) ZEL Analysis Coulomb Solution Rotation Angle (Deg.) Copyright 24 Centre for Info Bio Technology (CIBTech) 2465 (c) Figure 9: a) Distribution of Passive Pressure at Different Wall Rotation Angle; b) Passive Resultant Force versus Wall Rotation Angle; c) Point of Application of Resultant Force versus Wall Rotation Angle

11 Height of the Wall (m) Mixed Zone Goursat Zone Rankine Zone Height of the Wall (m) Distance from the Back of the Wall (m) (a) Distance from the Back of the Wall (m) (b) Coulomb Results Calculated From ZEL Passive Thrust (Kpa) Friction Angle Beteen Wall and Soil (Deg.) (c) Figure : a) Initial and Deformed ZEL Net at Passive Condition ith 2; b) ZEL Net ith ; c) Variation of Coulomb Solutions and ZEL Results versus Friction Angle beteen Wall and Backfill It is necessary to mention that the numerical analysis of Shiau is related to the translational mode, hile previous investigations demonstrated that the pressure values in this mode are greater than the rotational one. CONCLUSIONS On the basis of the results obtained from the examined examples ith the ZEL program, one can conclude that: In the mode of rotation about the base for a dense sand backfill, the formation of the active and passive states occur at the displacement of about.4h and.5h from the top of the all, respectively, hich agrees ith the presented magnitudes in the geotechnical references. Because the formation of the active state at the all base is practically impossible, the presented values of the active force by the Coulomb theory are less than the real ones, and the results obtained from ZEL analysis are greater by about 2 %. Copyright 24 Centre for Info Bio Technology (CIBTech) 2466

12 The existence of remaining higher pressures at the loer part of the all causes the point of application of the resultant force to fall to a loer location, and the pressure distribution is approximately linear. The resultant force direction makes a 7.3 angle ith the vertical direction crossed to the back surface of the all, hich is less than the soil-all friction angle In the passive case, because of the loer levels of displacement at the loer parts of the all, the applied pressures are much less than the anticipated pressures by the Coulomb theory. The results obtained from the ZEL analysis demonstrate that the Coulomb solutions are about 75 % greater than the real quantities. The passive pressure distribution has a parabola shape, and the point of application of the resultant force is about H 3 from the base of the all. When it is assumed that the max has been mobilized through the hole height of the all, the results obtained from the ZEL analysis are equal to the Coulomb solution in the active case. In the passive case, the Coulomb solutions are greater by about 6 % than the ZEL results. REFERENCES Anvar SA and Ghahramani A (997). Equilibrium Equations on Zero Extension Lines and Their Application to Soil Engineering. Iranian Journal of Science and Technology 2() Transaction B -34. Atkinson JH and Potts DM (975). A Note on Associated Field Solutions for Boundary Value Problems in a Variable -Variable Soil. Geotechnique 25(2) Behpoor L and Ghahramani A (987). Zero Extension Line Theory of Static and Dynamic Bearing Capacity. Proceedings of 8 th Asian Regional Conference on Soil Mechanics and Foundation Engineering, Kyoto, Japan Caltabiano S, Cascone E and Maugeri M (999). Sliding Response of Rigid Retaining Walls. Proceedings of 2 nd International Conference on Earthquake Geotechnical Engineering Lisboa, Portugal. Cole ERL (967). The Behaviour of Soils in Simple Shear Apparatus. Thesis Presented to the University of Cambridge, Cambridge, England, in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy. Fang YS and Ishibashi I (986). Static Earth Pressures ith Various Wall Movement. ASCE Journal of Geotechnical Engineering 2(3) Ghahramani A and Clemence SP (98). Zero Extension Line Theory of Earth Pressure. Journal of Geotechnical Engineering Division, ASCE 6(GT6) Habibagahi K and Ghahramani A (979). Zero Extension Line Theory of Earth Pressure. Journal of Geotechnical Engineering Division, ASCE 5(GT7) Harr ME (966). Foundations of Theoretical Soil Mechanics (McGra-Hill, Ne York, USA). Jahanandish M (988). Zero Extension Line Net and Its Applications in Soil Mechanics. M.Sc Thesis, Shiraz University, Shiraz, Iran. Jahanandish M, Behpoor L and Ghahramni A (989). Load-Displacement Charactristics of Retaining Walls. Proceedings of 2 th International Conference on Soil Mechanics and Foundation Engineering, Rio de Janiro, Brazil James RG and Bransby PL (97). Experimental and Theoretical Investigation of Passive Earth Pressure Problems. Geotechnique, London, England 2() James RG and Bransby PL (97). A Velocity Field for some Passive Earth Pressure Problems. Geotechnique, London, England, 2() James RG, Smith IAA and Bransby PL (972). The Prediction of Stresses and Deformations in a Sand Mass Adjacent to a Retaining Wall. Proceedings of 5 th European Conference on Soil Mechanics and Foundation Engineering, Madrid, Spain Copyright 24 Centre for Info Bio Technology (CIBTech) 2467

13 Lancellotta R (22). Analytical Solution of Passive Earth Pressure. Geotechnique, London, England, 52(8) Matsuo M, Kenmochi S and Yagi H (978). Experimental Study on Earth Pressure of Retaining Wall by Field Tests. Soils and Foundations, Japanese Society of Soil Mechanics and Foundation Engineering 8(3) Roscoe KH (97). The Influence of Strains in Soil Mechanics. Geotechnique, London, England, 2(2) Serrano AA (972). Generalization of the Associated Field Method. Proceedings of 5 th European Conference on Soil Mechanics and Foundations Engineering, Madrid, Spain Sherif MA, Fang YS and Sherif RI (984). K A and K Behined Rotating and Non-Yielding Walls. ASCE Journal of Geotechnical Engineering Division () Sherif MA, Ishibashi I and Lee CD (982). Earth Pressures against Rigid Retaining Walls. ASCE Journal of Geotechnical Engineering 8(GT5) Shiau JS, Augarde CE, Lyamin AV and Sloan SW (28). Finite Element Limit Analysis of Passive Earth Resistance in Cohessionless Soils. Soils and Foundations, Japanese Society of Soil Mechanics and Foundation Engineering 48(6) Copyright 24 Centre for Info Bio Technology (CIBTech) 2468