EVALUATION OF THE PLASTIC CRITICAL DEPTH IN SEISMIC ACTIVE LATERAL EARTH PRESSURE PROBLEMS USING THE STRESS-CHARACTERISTICS METHOD Abstract The plastic critical depth or the conventional tension crack depth has a considerable effect on the active lateral earth pressure in cohesive soils. In this paper the depth for soils has been evaluated in the seismic case using the stress-characteristics or slip-line method. The plastic critical depth was calculated on the basis of the theory of the stress-characteristics method and by considering the horizontal and vertical pseudo-static earthquake coefficients. The proposed solution considers the line of discontinuity in the stress-characteristics network. The earth slope, wall slope, cohesion and friction angle of the soil and the adhesion and the friction angle of the soil-wall interface were considered in the analysis as well. The results show that the plastic critical depths of this study are smaller than those of the other methods and are closer to the modified Mononobe-Okabe method. The effects of the wall and the backfill geometry, the mechanical properties of the soil and the pseudo-static coefficients were studied. Amin Keshavarz Persian Gulf University, School of Engineering Shahid Mahini Street, Bushehr, Iran E-mail: keshavarz@pgu.ac.ir Keywords plastic critical depth, stress characteristics, active lateral earth pressure, seismic 1 INTRODUCTION In cohesive soils, the computed active lateral earth pressure can be negative from the ground surface to some depth. The plastic critical depth is the depth where the computed soil pressure is negative from the ground surface to that depth. To calculate the active lateral earth pressure many engineers assume that the lateral earth pressure is zero from the ground surface to the plastic critical depth. Therefore, the calculation of the plastic critical depth is important in any evaluation of the active lateral earth pressure. Numerical examples showed that if the plastic critical depth is considered, the static lateral earth pressure can be more than 20 to 40 percent of the lateral earth pressure, without taking into account the plastic critical depth [1]. Peng [2] assumed that the failure surface is planar and evaluated the static active lateral earth pressure using the limit-equilibrium method. He considered the plastic critical depth and the surcharge in his study. Peng [3] modified the Mononobe-Okabe method to calculate the seismic active lateral earth pressure. He presented some equations to calculate the lateral earth pressure and the plastic critical depth in soils. Nian and Han [4] calculated the seismic active lateral earth pressure against rigid retaining walls. They used the Rankin theory and proposed an equation to estimate the plastic critical depth. In their study the retaining wall is vertical, but the ground surface can be inclined. Also, Iskander et al. [5] expanded the Rankin solution for seismic cases and proposed an equation for the plastic criti- Acta Geotechnica Slovenica, 2016/1 17. A. Keshavarz: Evaluation of the plastic critical depth in seismic active lateral earth pressure problems using the stress-characteristics method cal depth. Ma et al. [6] evaluated the lateral earth pressure and the plastic critical depth using the pseudo-dynamic method. Recently, Lin et al. [7] used the slice-analysis method to compute the nonlinear distribution of the seismic active earth pressure of cohesive-frictional soil. The stress-characteristics or slip-line method is one of the famous methods for analysing geotechnical problems. This method was presented by Sokolovski [8, 9]. Reece and Hettiaratchi [10] and Kumar and Chitikela [11] used the stress-characteristics method to compute the passive lateral earth pressure. Cheng [12] proposed a rotation of the axes in the stress-characteristics method to determine the seismic lateral earth pressure. Peng and Chen [13] applied this method to compute the active lateral earth pressure in the static case. The stress-characteristics method has also been used to evaluate the stability of reinforced soil structures [14, 15]. Table 1 summarizes the different methods for calculating the plastic critical depth. The parameters used in the table will be defined in the next sections. It is clear that most of the methods proposed closed-form solutions. Some of them need trial and error, and three of them consider all the parameters in the solution. In this paper the plastic critical depth is studied using the stress-characteristics method. Although Peng and Chen [13] used this method to compute the active lateral earth pressure, they did not consider the seismic effects Table 1. Summary of different methods to calculate the plastic critical depth. No._Proposed by Theory Parameters considered Comments E E 1 Rankin Simple Rankin e E P E - The simplest equation (Eq. (25)) kh E kv E q 0 cw 0 SW 0 2 Peng, 2012 [3] Modified e 0 P 0 - Closed form without trial and error Mononobe-Okabe kh 0 kv 0 q 0 cw 0 Sw 0 3 Ma et al., 2012 [6] Pseudo-dynamic e 0 P E -No closed form kh 0 kv 0 - Needs optimization q 0 cw 0 Sw 0 -Closed form 4 Peng and Chen, 2013 Slip line e 0 P 0 -Needs trial and error [13] kh E kv E -When P=0 solution can be found without q 0 trial and error cw E Sw E 5 Iskandet et al., 2013 [5] Expanded Rankin e 0 P 0 - Closed form without trial and error kh 0 kv 0 q E cw E Sw E 6 Nian and Han, 2013 [4] Modified Rankin e E P 0 - Closed form without trial and error kh 0 kv 0 q E cw 0 Sw 0 7 Lin et al., 2015 [7] Slice analysis method e 0 P 0 -Closed form kh 0 kv 0 -Needs trial and error q 0 cw 0 Sw 0 -Closed form 8 This study Slip line e 0 P 0 -Needs trial and error kh 0 kv 0 -When P=kh=kv=0, solution can be found q 0 without trial and error c w 18. Acta Geotechnica Slovenica, 2016/1 A. Keshavarz: Evaluation of the plastic critical depth in seismic active lateral earth pressure problems using the stress-characteristics method and the discontinuity line in the stress field. In this study, the estimation of the plastic critical depth is explained clearly, and several analyses are made in different cases. 2 THEORY_ The backfill is a c-0 soil, where c and 0 are the cohesion and the friction angle of the soil, respectively. The soil obeys the Mohr-Coulomb failure criterion. The retaining wall is rigid, and the soil-wall interface has the adhesion cw and the friction angle Sw. Figure 1 shows a soil element in the plane-strain case. There are two families of failure orientations, PA and PB, known as the negative and positive characteristics, and make a stress field. As shown, the stress-characteristics lines make the angle m = p / 4- f / 2 with the orientation of the principal stress ffj [9]. Each point in the soil media has four features, x, z, p and f, where x and z are the coordinates of the point and p is the average stress in the Mohr's circle and f is the angle between ffj and the horizontal axis (Figure 1). If the body forces are zero, the equilibrium equations along the stress characteristics can be written as [9]: 2(p tanf + c)dy + dp = 0 (1) 2(ptan f + c)dy - dp = 0 (2) 2.1 Boundary conditions Figure 2 shows the geometry of the problem. DE is the ground-surface boundary and has the surcharge q. ft is the ground-surface angle with the horizontal axis and 6 is the angle between the wall and the vertical axis. The positive signs of these angles are shown in the figure. kh and kv are the horizontal and vertical pseudo-static earthquake coefficients, respectively. The positive directions of the seismic accelerations are shown in the figure. To solve the problem, the boundary condition of the wall and ground surface must be calculated. Figure 2. The geometry of the problem. The Mohr's circles of stress along the ground surface and wall boundaries are shown in Figure 3. The normal and shear stresses on the ground surface can be written as ^ = q cosb[(1- kv) cosb - kh sinb] = Afl (3) r0 = q cosb[(1- kv )sinb + kh cosb] = A2q (4) where q is the equivalent surcharge (Eq. (18)) and (1-kv) cos b cos (d + b) A A cos d (5) (1- kv) cos b sin(d + b) cos 6 and Figure 1. The orientation of the positive (a+) and negative (a-) characteristics and the Mohr's circle of stress [16]. tand = 1- kv (6) 18. Acta Geotechnica Slovenica, 2016/1 A. Keshavarz: Evaluation of the plastic critical depth in seismic active lateral earth pressure problems using the stress-characteristics method a) b) Figure 3. Mohr's circle of stress of the a) wall and b) ground surface. The radius of the Mohr's circle on the ground surface can be written as (Figure 3b) R0 = p0sin f + Ccos f = ^/(sQ - p0) 2 + To2 (7) From Eq. (7), the average stress on the ground surface, po, is Po s0 + ccosf sinf — tJ(&0 sinf + ccosf )2 — ( cosf)2 cos2 f (8) and using the Mohr's circle, we can write h0 = p/2 + b — A s0 = P0 +(p0sinf + c cos f)cos2h0 t0 = (p0 sin f + c cos f )sin2h0 The angle f0 can be obtained from Eq. (9) as p0 sin ((5 + b) (9) y0 =-+0.5 b — 5 — sin-1 p0sinf + c cos(; (10) Referring to Figure 3a, on the retaining wall boundary hf=yf—p/2—q s f = pf —(pf sinf + c cos f)cos2hf (11) tf = cw + sf tan5w = (pf sin f + ccos f )sin2hf The angle f on the wall, f can be found from Eq. (11) as pfsin 6W + cw c°s dw yf=2+9+0.5 + sin pf sinf + ccosf (12) 2.2 Calculating the plastic critical depth Since the stresses on the left- and right-hand side of point O are different, this point is a singularity point. To obtain the depth of the plastic critical depth without computing the whole network, the singularity point must be solved. If ff>f0, the stresses are continuous everywhere, but when fff0, from Eq. (1), pf can be found as If f=0 : pf = p0 — 2c(yf — A0) else : pf =—c cotf + (p0 +c cot f)exp j—2tanf (pf — y0) (13) But if ff 0 (24) In the static case, when ^=0, the plastic critical depth can be computed without trial and error. When ff>f0, and for the static case, the results of this paper are the same as those of Peng and Chen [13]. To calculate the plastic critical depth in active lateral earth pressure problems, an equivalent surcharge can be used [13]. This uniform equivalent surcharge on the OA boundary (Figure 2) can be written as q = q+gzc (18) where zc is the plastic critical depth and y is the unit weight of the soil. The equivalent surcharge q must be computed such that the normal stress on the wall is zero. When Of = 0, from Eq. (11) and (12) [13] f Vc pf = c tan f + - ic2 - c,.2 cosf (19) If ff>f0 , the value of p0 can be obtained from Eq. (13) as if f=0 : p0 = pf + 2c(yf - y) else : p0 =— c cotf + (pf +c cot f)exp |2tanf(yf — y0 ) (20) and if ff pf + ccotf Po =-c cot f + sin2(y f-Wo ) . , -^ ^ sin2(yo - wo where from Eq. (16) Wo = 0.5 yf + y - cos Isinf coslyf - y f - y0 (21) (22) 3 RESULTS Based on the algorithm described in the previous section, a computer code was prepared. In this part of the paper different parametric analyses were made for the plastic critical depth, and the results were compared to the results of the other studies. In the static case (kh=kv=0) and for 8w=cw=Q=fi=0, the solution leads to the following equation, which is the simple Rankin formula for the plastic critical depth 2c g zc = — tan p f - + - 4 2 - ^ (25) g Nian and Han [4] developed the Rankin theory to calculate the seismic active lateral earth pressure. They neglected the friction angle and the adhesion of the soil-wall interface and assumed that the wall is vertical (i.e., Sw=cw=Q=0). Their equation for zc is 2c(sinf(1 — tan¡3tanS) + yj(1 — tan¡3tand)2 + 4tan2 S I g (1- K )cos f (1- tan b tan S)2 + 4tan2S cos2 f (26) Also, Peng [3] developed the Mononobe-Okabe theory for the seismic case by taking into account the effect of the plastic critical depth and proposed the following equation to calculate q zc = 18. Acta Geotechnica Slovenica, 2016/1 A. Keshavarz: Evaluation of the plastic critical depth in seismic active lateral earth pressure problems using the stress-characteristics method _ c cos(0 — ß )cos f + cw sin(aa + ß )sin(aa + 0 — f) q = I — (27) ^kh2 +(1 — kv)2 cos(aa + 0)sin(aa — f — S)cosß where tana sin(f + S — 0) + m0 sin0 cos 0 + ^ 1 — m0 cos (0 + f + S) cos(0 — f — S) + m0sin 0 Cw cos(ß + S) (28) (c + cw )cos(0 — ß)cos f In addition, aa can also be calculated using the following equations: if ccos(q -b)cosf + cw sin(f + b + fi)sin($ + fi) > 0, then sin(f + fi — q) + no sin(f + fi)cos(f + fi) + yjl + no cos(q + f + fi) cos(0 — f — S) + nocos (f + S) cw cos(ß + S) (29) ccos(0 — ß)cosf + cw sinff + ß + S)sin(0 + S) if ccos(0 — b)cosf + cw sin(f + b + fi)sin(0 + fi) = 0, then aa = f + fi A comparison of the results of this paper with the results of the other methods is shown in Table 2. The percentage relative errors between the results of this paper and the other methods are shown in the table as well. In the last row of the table the average percentage errors for all the cases are written. It is clear that the least and the most errors belong to Peng's limit-equilibrium method [3] (average error 3%) and Eq. (25) (average error 38%), respectively. In addition, the depths computed from the stress-characteristics method are lower than those of the other methods. In the static case and for q=9=0, a comparison was made between the results of this study for zc and those of others, as shown in Figure 5. In this case the results of Eq. (25), Nian and Han [4] and Iskander et al. [5] are the same. It is clear that the results of this study are very close to Peng [3] (average difference is about 1.4%). Eq. (25) predicts smaller values of zc (about 15% smaller) and also the results of Lin et al. [7] are about 29% smaller than the results in this paper. To evaluate the effects of the parameters on the plastic critical depth, several parametric analyses were made. Figure 6 shows the effects of the pseudo-static horizontal (kh) and vertical (kv) coefficients on the plastic critical depth zc. The horizontal axis is kh and varies from -0.5 to 0.5. It is clear that zc increases with an increase in kh When kh is positive, its variation has more influence on increasing zc. For example, if kv=0, when kh changes from 0 to -0.5, zc decreases by about 58%, but when kh changes from 0 to 0.5, the increase in zc is about 93%. These differences in the negative values of kv are less than those of the positive values. --this study ..............................Peng [3] --------------Ref. [4-5] ------------------Lin et al. [7] H=8 m c=20 kPa cw=c/2, ^=»/3 0=0 ß= -15° q=0 Y=18 kN/m3 kh=0 K=0 20 0 (deg) 30 Figure 5. A comparison between the results of this study and those of others. m 0 a n Table 2. Comparison of the results of this study with other methods. (0=-10o, ¿3=0, Sw=0/2, cw=0.5c, kh=-0.2, kv=0.1, y=18 kN/m3). 0 (deg) c (kPa) q (kPa) Zc (m) Absolute relative error (%) Ma et al. 2012, [6] Eq. (25) Nian & Han, 2013, [4] Peng, 2012,[3] This study Ma et al. 2012, [6] Eq. (25) Nian & Han, 2013, [4] Peng, 2012, [3] 20 0 10 0 0 0 0 0 0 0 0 0 20 10 0 1.424 1.592 1.315 1.224 1.206 15 24 8 1 20 10 10 0.904 1.032 0.759 0.668 0.651 28 37 14 3 40 10 0 1.68 2.384 1.837 1.442 1.382 18 42 25 4 40 10 10 1.168 1.832 1.281 0.886 0.826 29 55 36 7 20 20 0 2.6 3.168 2.629 2.447 2.412 7 24 8 1 20 20 10 2.072 2.624 2.074 1.892 1.857 10 29 10 2 40 20 0 3.048 4.768 3.674 2.884 2.764 9 42 25 4 40 20 10 2.52 4.208 3.118 2.328 2.208 12 48 29 5 Average error: 16 38 19 3 18. Acta Geotechnica Slovenica, 2016/1 A. Keshavarz: Evaluation of the plastic critical depth in seismic active lateral earth pressure problems using the stress-characteristics method Also, increasing kv leads to an increase in zc. Similar to kh, when kv is positive, the variations are more rapid. The average changes in zc for kv= -0.5 relative to kv=0 are about 22% and 40% in the negative and positive values of kh, respectively. These changes for kv=0.5 relative to kv=0 are 39% and 144% in the negative and positive values of kh, respectively. The results of Peng [3] are also shown in Figure 6. It is clear that the values of results of this study are lower than those of Peng. For the selected parameters shown in the figure, the average relative errors between these two methods are 1% for the positive values of kh and kv and 30% for the negative values of kh and kv. The effects of the soil friction angle (0) and ground-surface slope angle (ft) values on zc are shown in Figure 7. It is clear that zc increases with an increase in 0. The results of the analyses indicate that the soil-wall interface friction angle (Sw) has very little influence on zc and its effect can be ignored. Increasing ft leads to an increase in the plastic critical depth. When ft changes from -30° to 30°, zc increases by about 17%. Figure 8 shows the influence of the soil cohesion and soil-wall interface adhesion on zc. The horizontal axis is the soil cohesion (c) and several graphs have been plotted for different values of cw/c. It is obvious that by increasing c, the plastic critical depth increases. Increasing cw/c also leads to an increase in zc, but its influence is less than that of c. For the assumed parameters indicated in the figure, the difference between the results of cw/c=0 and cw/c=1 for all values of c is about 30%. The change of zc with c is linear. The slope of this -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 kh Figure 6. The effects of the kh and kv values on the plastic critical depth. q=0 0=0 Y=18 kN/m' c=20 kPa cw=0.75c 8=0.75<|> 20 . .30 0 (deg) Figure 7. The effects of the 0 and ft values on the plastic critical depth. 20 30 c (kPa) Figure 8. The effects of the c and cw values on the plastic critical depth. 8 - 4 - q=0 *=«■/7y*/ > 9=0 / / / X : K=K=o / / / 20^ y=18 kN/m3 c=20 kPa c„=0.75c > fi„=0.75<|> ^— <|>=0' ■ 0 6 (deg) Figure 9. The effects of the 6 values on the plastic critical depth. 18. Acta Geotechnica Slovenica, 2016/1 A. Keshavarz: Evaluation of the plastic critical depth in seismic active lateral earth pressure problems using the stress-characteristics method line changes from 0.19 to 0.25 m/kPa when cw/c changes from 0 to 1. Figure 9 shows the influence of the wall angle (0) on zc. The horizontal axis shows the wall angle and changes from -30° to 30°. Different graphs have been plotted for different values of the soil friction angle. We can see that by increasing 0, the value of zc increases. zc varies more rapidly with 0 for the larger values of For example, when ^>=45°, by increasing 0 from -30° to 0, zc increases by about 202%. This increase for ^>=0, when changing 0 from -30° to 0 it is about 55% and when changing 0 from 0 to 30° it is about 34%. 4 CONCLUSIONS_ The stress-characteristics method has been used to evaluate the plastic critical depth. The most important conclusions from this study are: - The values of the results of the stress-characteristics method described in this paper for the plastic critical depth, zc, are smaller than those of the pseudo-dynamic [6], modified Rankin [4] and modified Mononobe-Okabe [3] methods and are closer to the modified Mononobe-Okabe method. The method of Lin et al. [7] under predicts the plastic critical depth. - Increasing the horizontal and vertical pseudo-static coefficients (based on their assumed directions) can increase zc. The percentage changes relative to the static case can be more than 100%. - The values of zc increase with an increase in the soil friction angle, but the soil-wall interface friction angle has very little influence on zc and can be ignored. - For the assumed direction of the ground-surface angle ft, by increasing it, zc increases. For the sample parameters selected in this paper, by changing ft from -30° to 30°, the average increase in zc is about 17%. - The cohesion of the soil (c) has a considerable effect on zc. The variation of zc with c is linear. The slope of this line increases with an increase in cw/c. - Increasing the wall angle (for the assumed direction of this study) leads to an increase in zc. This effect is greater for larger values of The influence of 0 on zc is more than that of ft. Acknowledgements The author would like to thank the reviewer for very valuable comments and suggestions that were helpful in developing the paper. 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