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New procedure for active earth pressure calculation in retaining walls with reinforced cohesive-frictional backfill
Geotextiles and Geomembranes 27 (2009) 456–463
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New procedure for active earth pressure calculation in retaining walls with reinforced cohesive-frictional backfill M. Ahmadabadi, A. Ghanbari* Faculty of Engineering, Tarbiat Moallem University, No. 49 Mofatteh Avenue, Tehran, I.R. Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 February 2009 Received in revised form 5 June 2009 Accepted 8 June 2009 Available online 31 July 2009
A new approach is suggested to determine the active earth pressure on retaining walls with reinforced and unreinforced cohesive-frictional backfill based on the horizontal slices method. A 4n formulation for unreinforced backfill and a 5n formulation for reinforced backfill are introduced and the tensile forces of the reinforcements and angle of failure wedge are calculated. The proposed method shows that the variation of active earth pressure by the depth of the wall in cohesive-frictional soils has a non-linear distribution. Also, the point of application of the pressure always shifts to the lower two-thirds of the wall height. The angle of failure wedge for cohesive-frictional soils increases linearly with an increase in the cohesive strength of the soil. A comparison of the analytical results obtained from the proposed method with those of previous research and AASHTO method results shows a negligible difference. The analytical method presented can be used to calculate the active earth pressure, tensile force of reinforcements and angle of failure wedge for unreinforced and reinforced walls in cohesive-frictional soil. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Active earth pressure Reinforced retaining wall Horizontal slices method Limit equilibrium Pseudo-static seismic coefficient
1. Introduction The prediction of active earth pressures is essential to the design of retaining walls for cohesive and granular soils. Measuring earth pressure on retaining structures for granular soils has usually been performed using Rankine’s (1857) or Coulomb’s (1776) methods. However, Caquot and Kerisel (1948), Sokolovskii (1965), Lee and Herrington (1972) and Hua and Shen (1987) have all advanced significant procedures to estimate static lateral earth pressure in granular soils. In recent years, various research methods have been applied to the study of reinforced and unreinforced soil walls and slopes. Recent studies of reinforced soil include experimental studies of structures (Garg, 1998; Nova-Roessig and Sitar, 1999; Collin, 2001; Kazimierowicz-Frankowska, 2005; Lee and Wu, 2004; Yoo, 2004; Yoo and Jung, 2004; Chen et al., 2007; Won and Kim, 2007; El-Emam and Bathurst, 2007; Jones and Clarke, 2007; Sabermahani et al., 2008; Latha and Krishna, 2008; Yang et al., 2009) and numerical analysis (Rowe and Ho, 1998; Al-Hattamleh and Muhunthan, 2006; Rowe and Skinner, 2001; Skinner and Rowe, 2005; Hatami and Bathurst, 2000; Huang and Wu, 2006).
* Corresponding author. Tel./fax: þ98 261 456 9555/þ98 912 140 8253. E-mail addresses: [email protected] (M. Ahmadabadi), ghanbari@ tmu.ac.ir (A. Ghanbari). 0266-1144/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.geotexmem.2009.06.004
Additional analytical models include the homogenized analytical concept (Chen et al., 2000), limit analysis (Porbaha et al., 2000), limit equilibrium and the horizontal slices method (Shahgholi et al., 2001; Baker and Klein, 2004a,b; Nouri et al., 2006, 2008; Shekarian et al., 2008; Shekarian and Ghanbari, 2008; Reddy et al., 2008) and the characteristics method (Kumar and Chitikela, 2002; Jahanandish and Keshavarz, 2005). Bathurst et al. (2005) established a working stress method for the calculation of reinforcement loads in geosynthetic reinforced soil walls using a database of instrumented and monitored full-scale field and laboratory walls. The proposed method captures the essential contributions of the different wall components and properties to reinforcement loads. Research on active earth pressure on retaining walls with cohesive backfill has been done by Das and Puri (1996), Gnanapragasam (2000), Cheng (2003) and Liu et al. (2009). Das and Puri (1996) developed an improved method for calculating the static and dynamic earth pressure behind a rigid wall in consideration of the cohesion between the soil and wall. Gnanapragasam (2000) presented an analytical solution to determine the active lateral earth pressure distribution on a retaining wall for cohesive soil. Unlike Rankine’s method, his results show that for sloping cohesive backfill, the slope of the failure surface is a function of the overburden pressure and becomes shallower with depth, thus forming a curvilinear failure surface. Cheng (2003) proposed a rotation of axes as a solution to slip line equations for the determination of lateral earth pressure under general conditions. Liu et al. (2009) proposed a general tangential
M. Ahmadabadi, A. Ghanbari / Geotextiles and Geomembranes 27 (2009) 456–463
Nomenclature Ai a b c H Hi Hiþ1 hi ka ka Ni n Pa Pi Si Sv T
area of ith slice coefficient dependent upon ka, ka and a (Eq. (11)) coefficient dependent upon ka, ka and a (Eq. (8)) cohesion of soil (kPa) height of wall (m) horizontal force at top of ith slice (kN) horizontal force at bottom of ith slice (kN) height of ith slice (m) active earth pressure coefficient (dimensionless) coefficients dependent upon 4 (Eq. (6)) normal force on failure surface for ith slice (kN) number of horizontal slices active earth pressure pressure on the wall for ith slice (kPa) shear force on failure surface for ith slice (kN) vertical spacing of reinforcements (m) tensile force of the reinforcements (kN)
stress coefficient in determining active earth pressures. Their results show that, for any value of this coefficient, the active earth pressure converges to Rankine’s solution when the radius is sufficiently large compared to the depth of the wall. For cohesivefrictional soil, the critical value of this coincidental coefficient is smaller than the active earth pressure coefficient owing to the effect of the cohesive strength of the soil. In this paper, an analytical procedure based on the horizontal slices and limit equilibrium methods is used to evaluate the static earth pressure on a retaining wall considering the effect of the cohesive strength of reinforced and unreinforced soil. 2. Horizontal slices method Originally, slices were used to estimate slope stability. The conventional vertical slices method is widely used for stability of slopes. Another commonly used method to verify these slopes were introduced by Shahgholi et al. (2001). The horizontal slices method (HSM) was expanded upon by Nouri et al. (2006, 2008). This method addresses seismic acceleration at different heights of a structure. Azad et al. (2008); Shekarian and Ghanbari (2008) and Shekarian et al. (2008) employed the concept of HSM within the framework of pseudo-dynamic and pseudo-static methods to ascertain seismic active earth pressure on retaining walls. HSM
Fig. 1. Division of failure wedge into horizontal slices.
Ui Uiþ1 Vi Viþ1 Wi X Gi XVi XViþ1
457
coefficients dependent upon XVi and hi (Eq. (9)) coefficients dependent upon XViþ1 and hi (Eq. (10)) normal force at top of ith slice (kN) normal force at bottom of ith slice (kN) weight of ith slice (kN) horizontal distance of Wi from wall (m) horizontal distance of Vi from wall (m) horizontal distance of Viþ1 from wall (m)
Greek letters a inclination angle of backfill soil ( ) b angle of failure surface to horizontal plane ( ) g total unit weight (kN/m3) d friction angle between wall and backfill soil ( ) li ratio of available shear stress to shear strength (dimensionless) sf shear strength of soil (kPa) sm mobilized shear stress (kPa) f angle of internal friction of soil ( )
permits the calculation of the distribution of seismic active earth pressure and the application point of the resultant earth pressure. To determine the active earth pressure of a retaining wall with reinforced and unreinforced cohesive-frictional backfill using HSM, the present study makes the following assumptions: (1) The coordinate of the application point of the vertical interslice force is the surface center of stress distribution derived from the succeeding equations. (2) The failure surface is planar. (3) The method is limited to homogeneous masses. (4) The failure surface is assumed to pass through the heel of the wall. (5) The value of shear force between horizontal slices has been considered to be unequal (Hi s Hi1). (6) The point where Ni acts on the slice base is at the midpoint of that base. (7) The point where Pi acts is at mid-height for each slice.
3. Proposed method for unreinforced retaining walls Fig. 1 shows a retaining wall with its backfill divided into horizontal slices. The angle b forms the failure wedge of the backfill soil
Fig. 2. Force equilibrium in ith slice.
458
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Table 1 Equations and unknowns of 4n formulation to achieve lateral pressure on wall with unreinforced cohesive-frictional backfill. Unknowns
Number
Hi Inter-slice shear force Ni Normal force at base of each slice Si Shear force at base of each slice Pi Net force on wall
n n n n
Equations P Fx ¼ 0 For each slice P Fy ¼ 0 For each slice P Mo ¼ 0 For each slice Si ¼ Ni(tan 4) þ C For each slice
Number n n n n
for the limit equilibrium condition. The pressures on the ith slice of the backfill are shown in Fig. 2. If the backfill has n horizontal slices of equal height, each slice height can be derived from the following relation:
hi ¼
H n
(1)
In Fig. 2, XVi , XViþ1 and XGi are the horizontal distances of Vi, Viþ1 and Wi from the wall and are derived from the following relations:
XVi
hi ¼ þ 2 tan b
Pn
Pn
j ¼ iþ1
XViþ1 ¼
j ¼ iþ1
hj
(2)
2 tan b hj
(3)
2 tan b
Wi ¼ Ai g 1
(4)
XGi is the horizontal distance of the center of gravity from the wall and vertical stresses Viþ1 and Vi were derived from the following relation by Segrestin (1992):
2 Vi ¼ 4g
i1 X
3 hj 5 tanh ðaUi þ bÞ
(5)
j¼1
In the above relation, ka, ka, b, Ui, Uiþ1 and a are the dimensionless coefficients derived from the following relations (Segrestin, 1992):
2 6 ka ¼ 4
Fig. 4. Change in pressure on the wall by cohesive strength at different 4 values.
32
sinðp2 4Þ 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi5 p p p sinð2Þ þ sinð2Þ cosð2 4Þ sin 4
(6)
Fig. 3. Change in failure wedge angle by internal friction angle at different cohesion strengths.
ka ¼ tan2
b ¼
p 2
4
(7)
2
a log kka k ka
(8)
a
2
XV Ui ¼ Pi1 i
j¼1
(9)
hj
XV Uiþ1 ¼ Pi iþ1 j¼1
(10)
hj
a ¼ 2 tan a log
2ka ka þ ka
(11)
In Eq. (11), a is the inclination angle of the backfill soil with a horizontal axis. To analyze unreinforced retaining walls, similar to Fig. 1, there are four unknowns; Ni, Hi, Si and Pi. Thus, it is necessary to write four equations to receive these four unknowns. For this purpose, two equilibrium equations in the vertical and horizontal directions were used; a reinforcement equilibrium equation around the O point and an equation of the relation between shear stress and shear yield stress in the horizontal levels between the slices. The 4n formulations and equations are shown in Table 1. The first three equations for Table 1 are:
Fig. 5. Active earth pressure distribution for different cohesion strength based on proposed method.
M. Ahmadabadi, A. Ghanbari / Geotextiles and Geomembranes 27 (2009) 456–463 Table 2 Equations and unknowns for 5n formulation to achieve lateral pressure on retaining wall with reinforced cohesive-frictional backfill. Unknowns
Number
Hi Inter-slice shear force Ni Normal force at base of each slice Si Shear force at base of each slice Ti
n
Pi Net pressure on wall
n
X X
n n n
Equations P Fx ¼ 0 For each slice P Fy ¼ 0 For each slice P Mo ¼ 0 For each slice Si ¼ Ni(tan 4) þ c For each slice sm ¼ l (sf) For each slice
Number n n n n n
Fx ¼ 00Si cos b Ni sinðbÞ þ Pi cos d þ Hi Hiþ1 ¼ 0
(12)
Fy ¼ 00Pi sin d þ Viþ1 Vi Wi þ Ni cos b þ si sinðbÞ
(13)
P
Mo ¼ 00 Vi XVi þ Vi XViþ1 WiXGi " # 2 3 n X Ni pi cos d 4 þ hj þ hj =25 sinðbÞ j ¼ iþ1 " # " # Pn Pn þHiþ1 H h h i j ¼ iþ1 j j¼1 j ¼ 0
459
(14)
where Wi is the weight of slice i (Wi ¼ gi Ai). The fourth equation in Table 1 is the Moher–Coulomb yield criteria applicable to failure wedge points. The solution to these unknowns will be the pressure on the wall for each slice (Pi) and the horizontal force between the slices (Hi). Dividing the horizontal force on the surface between the two slices produces the average shear stress between them (sm). Since the level between the slices is not in the yield position, it is considered to be a coefficient of soil shear strength (sf) in the following way:
sm ¼ l sf
Fig. 6. Change in l by wall height for constant value of angle of internal friction.
(15)
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Fig. 9. Details of retaining wall with unreinforced soil.
wedge. Parametric study shows that for d ¼ 10 –20 , H ¼ 5–10 m and g ¼ 18–22 kN/m3, the maximum error of the results presented in Fig. 3 is about 3%. According to the proposed method, the pressure on the wall has been calculated based on the difference in the cohesion and friction angle as shown in Fig. 4. An increase in the cohesion strength of the soil reduces the pressure imposed by the soil on the wall in a linear relationship. A sample wall with cohesive-frictional backfill and the effect of cohesion strength on the distribution of active earth pressure is shown in Fig. 5. This distribution has a non-linear relationship and the tension crack zone extends as the cohesion strength of the soil increases. Fig. 7. Change in l by wall height for constant value of cohesion strength.
4. Proposed method for reinforced retaining wall
The non-dimensional coefficient li, has a value of less than one. To analyze unreinforced walls, there is no need to calculate l. However, this coefficient will be used in the next step to analyze reinforced walls. All previous researchers have analyzed retaining walls using HSM, but neglect the shear stress between the slices. Calculation of the l coefficient allows evaluation of the accuracy of this assumption. Also, in the proposed formulation for determining the failure wedge angle in cohesive-frictional soils, pressure on the walls has been calculated for different failure angles. The failure wedge angle is defined as the wedge angle having maximum pressure. Fig. 3 diagrams critical failure wedge variations in relation to the internal friction angle and cohesion of backfill soil. As can be seen, an increase in the angle of internal friction and cohesion strength of the backfill soil leads to a linear increase in the angle of failure
The tensile force of the reinforcements is added to the unknowns for reinforced walls, requiring the equation to be solved for 5n unknowns. For the above purpose, the average shear stress for each slice is assumed to be the coefficient of soil shear strength
Fig. 8. Change in l by wall height for cohesionless soils.
Fig. 10. Details of retaining wall with reinforced soil.
M. Ahmadabadi, A. Ghanbari / Geotextiles and Geomembranes 27 (2009) 456–463
461
Table 3 Comparison of results for Rankine and proposed methods. C ¼ 0 kN/m2
4 ¼ 20
b Pa
4 ¼ 25
b Pa
4 ¼ 30
b Pa
Fig. 11. Change in Pa by wall height for different vertical spacing (Sv) of reinforcements.
in a yielding condition. This coefficient is always less than one and is shown for the ith slice with li, making:
Hi ¼ ½Vi tan 4 þ cli
(16)
The proposed method assumes that li for each slice, with or without reinforcement, is equal. Thus, Eq. (16) can be considered as
C ¼ 20 kN/m2
Proposed method
Rankine method
Proposed method
Rankine method
Proposed method
Rankine method
55.0 490.3 57.5 405.9 60 333.66
55.0 490 57.5 406.0 60 333
55.0 350.2 57.5 278.4 60 218.19
55.0 350.1 57.5 279.0 60 217.76
55.0 210.2 57.5 151.0 60 102.72
55.0 210.4 57.5 151.5 60 102.34
the fifth equation calculating the amount of active pressure for reinforced soils. Table 2 shows the equations and unknowns for the complete formulation. The change in l over the height of a wall has been diagrammed where l is the percentage V tan 4 þ c for different heights or horizontal slices. The change in l by wall height for different internal friction angles, as shown in Fig. 6, is 20 –30 and the cohesion strength of the soil is 0–20 kN/m2. As can be seen, the curves are similar and the slope and length change from the principle point as c and 4 change. Changes in l by H for different 4 are shown in Fig. 7, where c ¼ kN/m2, H ¼ 10 m and d ¼ 10 . Figs. 6 and 7 show the process of change in l with an increase in cohesion strength, internal soil friction and wall height. This means that, for cohesive-frictional soils, l is different for different heights and higher points on the wall have greater l values. Therefore, disregarding the shear forces between slices is not acceptable in some cases. The change in l for cohesionless soils throughout the depth of a 10 m retaining wall is presented in Fig. 8. The equation for the l coefficient for cohesionless soils is as follows:
l ¼ Fig. 12. T and Pa distribution for a reinforced wall with cohesive backfill.
C ¼ 10 kN/m2
c1 H þ c2 H2 þ c3 H þ c4
(17)
in which c1 to c4 coefficients depend on the mechanical properties of the material and the wall height. Ten meter walls with 0.5 and 1 m vertical spacing (Sv) of reinforcements were analyzed using the proposed method and the tensile forces of the reinforcements (T) and active earth pressure distribution on the wall (Pa) were calculated. The details of the unreinforced and reinforced soil walls used in the analysis are presented in Figs. 9 and 10, respectively. The results of active earth pressure for walls with 0, 10 and 20 reinforcements and cohesionless backfill are shown in Fig. 11. The results of active earth pressure distribution and the tensile forces of the reinforcements are shown in Fig. 12 for a reinforced wall with cohesive-frictional backfill. Fig. 13 shows the change in tensile forces of reinforcements by wall height at different cohesion strengths. As can be seen, the distribution of tensile forces of the reinforcements is non-linear with a decreasing gradient in the deeper portions of the wall. Also,
Table 4 Comparison of results for Coulomb and proposed methods.
d ¼ 10
4 ¼ 20
b Pa
4 ¼ 25
b
4 ¼ 30
b
Pa Fig. 13. Change in tensile forces of reinforcements by wall height at different cohesion strengths.
Pa
d ¼ 15
d ¼ 20
Proposed method
Coulomb method
Proposed method
Coulomb method
Proposed method
Coulomb method
50.9 446.7 54.5 372.6 57.7 309.1
51.0 446.7 55.0 372.6 58.0 308.5
49.4 436.1 53.3 364.5 56.8 302.6
50.0 434.3 53 363.1 57 301.4
48.1 426.9 52.2 357.4 55.9 297.9
48.0 426.9 52.0 357.4 56.0 297.3
462
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Table 5 Comparison of results for proposed and previous methods for cohesive-frictional backfill. C ¼ 0 kN/m2
4 ( )
20
b Pa
25
b Pa
30
b Pa
C ¼ 10 kN/m2
C ¼ 20 kN/m2
Proposed method
Cheng (2003)
Das and Puri (1996)
Proposed method
Cheng (2003)
Das and Puri (1996)
Proposed method
Cheng (2003)
Das and Puri (1996)
51.0 446.7 54.4 372.6 57.7 309.08
51.0 440.0 55.0 366.9 58 303.76
51.1 446.74 54.6 372.61 57.8 308.46
52.6 318.1 55.7 255.0 58.5 201.84
53.0 313.2 56.0 251.1 59 198.17
50.6 314.36 53.2 243.79 56.5 184.62
53.7 190.4 56.6 138.1 59.2 95.04
54 187.5 57.0 136.0 59 92.94
50.6 190.48 53.1 121.39 55.7 65.64
increases in the cohesion strength of the soil and the number of reinforcements lead to decreases in active earth pressure on the wall.
Table 6 Comparison of proposed method results with AASHTO and Shekarian et al. (2008). AASHTO
5. Discussion and comparison of results To validate the design of the proposed method, the results were compared with previous studies. Tables 3–5 present comparisons of the results of the proposed method with those of the frequently used methods of Rankine, Coulomb and Cheng (2003). The results of the proposed method for the angle of failure wedge are in excellent agreement with the Rankine, Coulomb and Cheng methods. These results are also in good agreement with results presented by Das and Puri (1996), where the maximum difference is about 3.5% for higher cohesion strength values. Comparisons of results for pressure on a wall show that the proposed method is in excellent agreement with the Rankine, Coulomb and Cheng methods. However, the results of Das and Puri (1996) for higher cohesion strength values are noticeably different. The effect of cohesion strength on the distribution of active earth pressure on the wall was assessed and the curves compared with the distribution resulting from the Rankine method (Fig. 14). As can be seen, the proposed method’s distribution has a non-linear form and the greater the increase in c, the deeper tensile crack. To evaluate the validity of the proposed method for reinforced backfills, the results were compared with the results of MSEW software and the Shekarian et al. (2008) method for cohesionless soils. The comparison for d ¼ 10 , H ¼ 10 m and g ¼ 20 kN/m3 is shown in Table 6. It should be mentioned that the MSEW software provided by Leshchinsky (2006) works based on the method recommended by AASHTO. The results of the proposed method for the angle of failure wedge, tensile force of reinforcements and pressure on the wall are
4 ¼ 20 4 ¼ 25 4 ¼ 30
Shekarian et al. (2008)
Proposed method
Pa
T
b
Pa
T
b
Pa
T
b
50.3 38.6 30.1
440.0 367.0 303.8
51.6 55.9 57.9
48.7 43.0 35.4
446.9 367.4 298.3
55.0 57.9 60.2
43.0 39.2 35.5
442.7 362.9 294.8
55.0 57.6 60.0
in good agreement with the methods of Shekarian et al. (2008) and AASHTO. The maximum difference with the other methods is about 3.5% for the angle of failure wedge, 3% for tensile force of reinforcements and 7% for pressure on the wall. 6. Conclusion An analytical procedure based on the horizontal slices and limit equilibrium methods was used to evaluate the active earth pressure on a retaining wall considering the effect of the cohesive strength of reinforced and unreinforced soil that has been neglected in most previous methods. The proposed formulation for unreinforced walls has 4n equation with 4n unknowns. Using the proposed method, the active earth pressure on the wall and the failure wedge for cohesivefrictional soils were calculated. A comparison of these results with the results of the Rankine, Coulomb and Cheng (2003) and Das and Puri (1996) methods shows acceptable agreement. The results show that an increase in the angle of internal friction and cohesion strength of the backfill soil leads to a linear increase in the angle of failure wedge. For reinforced backfill, the tensile forces of the reinforcements were added to the 4n unknowns, making 5n unknowns and 5n equations for cohesive-frictional soils. The results of the proposed method were again compared with the results of methods suggested by the other researchers. The results of sample walls show that, for the proposed method, the distribution of pressure on the wall is non-linear with a negative gradient in the deeper part of the wall. Increases in the cohesion strength of the soil and in the number of reinforcements leads to a decrease in the active earth pressure on the wall. It was concluded that the analytical procedure proposed reliably calculates the resultant earth pressure, active lateral earth pressure distribution, tensile forces of the reinforcements and failure wedge angle for unreinforced and reinforced walls in cohesive-frictional soil. References
Fig. 14. Rankine (d ¼ 0) and proposed (d ¼
0–20 )
active earth pressure distribution.
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Lee, K.Z.Z., Wu, J.T.H., 2004. A synthesis of case histories on GRS bridge-supporting structures with flexible facing. Geotextiles and Geomembranes 22 (4), 181–204. Leshchinsky, D., 2006. MSEW Software Manual, version (2.0). Liu, F.Q., Wang, J.H., Zhang, L.L., 2009. Axi-symmetric active earth pressure obtained by the slip line method with a general tangential stress coefficient. Computers and Geotechnics 36, 352–358. Nouri, H., Fakher, A., Jones, C.J.F.P., 2006. Development of horizontal slices method for seismic stability analysis of reinforced slopes and walls. Geotextiles and Geomembranes 24, 175–187. Nouri, H., Fakher, A., Jones, C.J.F.P., 2008. Evaluating the effects of the magnitude and amplification of pseudo-static acceleration on reinforced soil slopes and walls using the limit equilibrium horizontal slices method. Geotextiles and Geomembranes 26 (3), 263–278. Nova-Roessig, L., Sitar, N., 1999. Centrifuge model studies of the seismic response of reinforced soil slopes, in: Proc. 2nd Int. Conf. Earthquake Geotechnical Eng., pp. 679–684. Porbaha, A., Zhao, A., Kobayashi, M., Kishida, T., 2000. Upper bound estimate of scaled reinforced soil retaining walls. Geotextiles and Geomembranes 18 (6), 403–413. Rankine, W.J.M., 1857. On the mathematical theory of the stability of earthwork and masonry. Proceedings of the Royal Society of London 8, 60–61. Reddy, G.V.N., Madhav, M.R., Reddy, E.S., 2008. Pseudo-static seismic analysis of reinforced soil wall: effect of oblique displacement. Geotextiles and Geomembranes 26 (5), 393–403. Rowe, R.K., Ho, S.K., 1998. Horizontal deformation in reinforced soil walls. Canadian Geotechnical Journal 35 (2), 312–327. Rowe, R.K., Skinner, G.D., 2001. Numerical analysis of geosynthetic reinforced retaining wall constructed on a layered soil foundation. Geotextiles and Geomembranes 19 (7), 387–412. Sabermahani, M., Ghalandarzadeh, A., Fakher, A., 2008. Experimental study on seismic deformation modes of reinforced-soil walls. Geotextiles and Geomembranes 27, 121–136. Segrestin, P., 1992. Design of sloped reinforced fill structure, in: Proc. Conference on Retaining Structures, Inst. of Civil Eng., Robinson College, Cambridge, pp. 574-584. Shahgholi, M., Fakher, A., Jones, C.J.F.P., 2001. Horizontal slice method of analysis. Geotechnique 51 (10), 881–885. Shekarian, S., Ghanbari, A., 2008. A pseudo-dynamic method to analyze retaining wall with reinforced and unreinforced backfill. Journal of Seismology and Earthquake Engineering 10 (1), 41–47. Shekarian, S., Ghanbari, A., Farhadi, A., 2008. New seismic parameters in the analysis of retaining walls with reinforced backfill. Geotextiles and Geomembranes 26, 350–356. Skinner, G.D., Rowe, R.K., 2005. Design and behavior of a geosynthetic reinforced retaining wall and bridge abutment on a yielding foundation. Geotextiles and Geomembranes 23 (3), 234–260. Sokolovskii, V.V., 1965. Statics of Granular Media. Pergamon Press, New York. Won, M.S., Kim, Y.S., 2007. Internal deformation behavior of geosynthetic-reinforced soil walls. Geotextiles and Geomembranes 25 (1), 10–22. Yang, G., Zhang, B., Lv, P., Zhou, Q., 2009. Behaviour of geogrid reinforced soil retaining wall with concrete-rigid facing, Geotextiles and Geomembranes doi:10.1016/j.geotexmem.2009.03.001. Yoo, C., 2004. Performance of a 6-year-old geosynthetic-reinforced segmental retaining wall. Geotextiles and Geomembranes 22 (5), 377–397. Yoo, C., Jung, H.S., 2004. Measured behavior of a geosynthetic-reinforced segmental retaining wall in a tiered configuration. Geotextiles and Geomembranes 22 (5), 359–376.
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New procedure for active earth pressure calculation in retaining walls with reinforced cohesive-frictional backfill M. Ahmadabadi, A. Ghanbari* Faculty of Engineering, Tarbiat Moallem University, No. 49 Mofatteh Avenue, Tehran, I.R. Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 February 2009 Received in revised form 5 June 2009 Accepted 8 June 2009 Available online 31 July 2009
A new approach is suggested to determine the active earth pressure on retaining walls with reinforced and unreinforced cohesive-frictional backfill based on the horizontal slices method. A 4n formulation for unreinforced backfill and a 5n formulation for reinforced backfill are introduced and the tensile forces of the reinforcements and angle of failure wedge are calculated. The proposed method shows that the variation of active earth pressure by the depth of the wall in cohesive-frictional soils has a non-linear distribution. Also, the point of application of the pressure always shifts to the lower two-thirds of the wall height. The angle of failure wedge for cohesive-frictional soils increases linearly with an increase in the cohesive strength of the soil. A comparison of the analytical results obtained from the proposed method with those of previous research and AASHTO method results shows a negligible difference. The analytical method presented can be used to calculate the active earth pressure, tensile force of reinforcements and angle of failure wedge for unreinforced and reinforced walls in cohesive-frictional soil. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Active earth pressure Reinforced retaining wall Horizontal slices method Limit equilibrium Pseudo-static seismic coefficient
1. Introduction The prediction of active earth pressures is essential to the design of retaining walls for cohesive and granular soils. Measuring earth pressure on retaining structures for granular soils has usually been performed using Rankine’s (1857) or Coulomb’s (1776) methods. However, Caquot and Kerisel (1948), Sokolovskii (1965), Lee and Herrington (1972) and Hua and Shen (1987) have all advanced significant procedures to estimate static lateral earth pressure in granular soils. In recent years, various research methods have been applied to the study of reinforced and unreinforced soil walls and slopes. Recent studies of reinforced soil include experimental studies of structures (Garg, 1998; Nova-Roessig and Sitar, 1999; Collin, 2001; Kazimierowicz-Frankowska, 2005; Lee and Wu, 2004; Yoo, 2004; Yoo and Jung, 2004; Chen et al., 2007; Won and Kim, 2007; El-Emam and Bathurst, 2007; Jones and Clarke, 2007; Sabermahani et al., 2008; Latha and Krishna, 2008; Yang et al., 2009) and numerical analysis (Rowe and Ho, 1998; Al-Hattamleh and Muhunthan, 2006; Rowe and Skinner, 2001; Skinner and Rowe, 2005; Hatami and Bathurst, 2000; Huang and Wu, 2006).
* Corresponding author. Tel./fax: þ98 261 456 9555/þ98 912 140 8253. E-mail addresses: [email protected] (M. Ahmadabadi), ghanbari@ tmu.ac.ir (A. Ghanbari). 0266-1144/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.geotexmem.2009.06.004
Additional analytical models include the homogenized analytical concept (Chen et al., 2000), limit analysis (Porbaha et al., 2000), limit equilibrium and the horizontal slices method (Shahgholi et al., 2001; Baker and Klein, 2004a,b; Nouri et al., 2006, 2008; Shekarian et al., 2008; Shekarian and Ghanbari, 2008; Reddy et al., 2008) and the characteristics method (Kumar and Chitikela, 2002; Jahanandish and Keshavarz, 2005). Bathurst et al. (2005) established a working stress method for the calculation of reinforcement loads in geosynthetic reinforced soil walls using a database of instrumented and monitored full-scale field and laboratory walls. The proposed method captures the essential contributions of the different wall components and properties to reinforcement loads. Research on active earth pressure on retaining walls with cohesive backfill has been done by Das and Puri (1996), Gnanapragasam (2000), Cheng (2003) and Liu et al. (2009). Das and Puri (1996) developed an improved method for calculating the static and dynamic earth pressure behind a rigid wall in consideration of the cohesion between the soil and wall. Gnanapragasam (2000) presented an analytical solution to determine the active lateral earth pressure distribution on a retaining wall for cohesive soil. Unlike Rankine’s method, his results show that for sloping cohesive backfill, the slope of the failure surface is a function of the overburden pressure and becomes shallower with depth, thus forming a curvilinear failure surface. Cheng (2003) proposed a rotation of axes as a solution to slip line equations for the determination of lateral earth pressure under general conditions. Liu et al. (2009) proposed a general tangential
M. Ahmadabadi, A. Ghanbari / Geotextiles and Geomembranes 27 (2009) 456–463
Nomenclature Ai a b c H Hi Hiþ1 hi ka ka Ni n Pa Pi Si Sv T
area of ith slice coefficient dependent upon ka, ka and a (Eq. (11)) coefficient dependent upon ka, ka and a (Eq. (8)) cohesion of soil (kPa) height of wall (m) horizontal force at top of ith slice (kN) horizontal force at bottom of ith slice (kN) height of ith slice (m) active earth pressure coefficient (dimensionless) coefficients dependent upon 4 (Eq. (6)) normal force on failure surface for ith slice (kN) number of horizontal slices active earth pressure pressure on the wall for ith slice (kPa) shear force on failure surface for ith slice (kN) vertical spacing of reinforcements (m) tensile force of the reinforcements (kN)
stress coefficient in determining active earth pressures. Their results show that, for any value of this coefficient, the active earth pressure converges to Rankine’s solution when the radius is sufficiently large compared to the depth of the wall. For cohesivefrictional soil, the critical value of this coincidental coefficient is smaller than the active earth pressure coefficient owing to the effect of the cohesive strength of the soil. In this paper, an analytical procedure based on the horizontal slices and limit equilibrium methods is used to evaluate the static earth pressure on a retaining wall considering the effect of the cohesive strength of reinforced and unreinforced soil. 2. Horizontal slices method Originally, slices were used to estimate slope stability. The conventional vertical slices method is widely used for stability of slopes. Another commonly used method to verify these slopes were introduced by Shahgholi et al. (2001). The horizontal slices method (HSM) was expanded upon by Nouri et al. (2006, 2008). This method addresses seismic acceleration at different heights of a structure. Azad et al. (2008); Shekarian and Ghanbari (2008) and Shekarian et al. (2008) employed the concept of HSM within the framework of pseudo-dynamic and pseudo-static methods to ascertain seismic active earth pressure on retaining walls. HSM
Fig. 1. Division of failure wedge into horizontal slices.
Ui Uiþ1 Vi Viþ1 Wi X Gi XVi XViþ1
457
coefficients dependent upon XVi and hi (Eq. (9)) coefficients dependent upon XViþ1 and hi (Eq. (10)) normal force at top of ith slice (kN) normal force at bottom of ith slice (kN) weight of ith slice (kN) horizontal distance of Wi from wall (m) horizontal distance of Vi from wall (m) horizontal distance of Viþ1 from wall (m)
Greek letters a inclination angle of backfill soil ( ) b angle of failure surface to horizontal plane ( ) g total unit weight (kN/m3) d friction angle between wall and backfill soil ( ) li ratio of available shear stress to shear strength (dimensionless) sf shear strength of soil (kPa) sm mobilized shear stress (kPa) f angle of internal friction of soil ( )
permits the calculation of the distribution of seismic active earth pressure and the application point of the resultant earth pressure. To determine the active earth pressure of a retaining wall with reinforced and unreinforced cohesive-frictional backfill using HSM, the present study makes the following assumptions: (1) The coordinate of the application point of the vertical interslice force is the surface center of stress distribution derived from the succeeding equations. (2) The failure surface is planar. (3) The method is limited to homogeneous masses. (4) The failure surface is assumed to pass through the heel of the wall. (5) The value of shear force between horizontal slices has been considered to be unequal (Hi s Hi1). (6) The point where Ni acts on the slice base is at the midpoint of that base. (7) The point where Pi acts is at mid-height for each slice.
3. Proposed method for unreinforced retaining walls Fig. 1 shows a retaining wall with its backfill divided into horizontal slices. The angle b forms the failure wedge of the backfill soil
Fig. 2. Force equilibrium in ith slice.
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Table 1 Equations and unknowns of 4n formulation to achieve lateral pressure on wall with unreinforced cohesive-frictional backfill. Unknowns
Number
Hi Inter-slice shear force Ni Normal force at base of each slice Si Shear force at base of each slice Pi Net force on wall
n n n n
Equations P Fx ¼ 0 For each slice P Fy ¼ 0 For each slice P Mo ¼ 0 For each slice Si ¼ Ni(tan 4) þ C For each slice
Number n n n n
for the limit equilibrium condition. The pressures on the ith slice of the backfill are shown in Fig. 2. If the backfill has n horizontal slices of equal height, each slice height can be derived from the following relation:
hi ¼
H n
(1)
In Fig. 2, XVi , XViþ1 and XGi are the horizontal distances of Vi, Viþ1 and Wi from the wall and are derived from the following relations:
XVi
hi ¼ þ 2 tan b
Pn
Pn
j ¼ iþ1
XViþ1 ¼
j ¼ iþ1
hj
(2)
2 tan b hj
(3)
2 tan b
Wi ¼ Ai g 1
(4)
XGi is the horizontal distance of the center of gravity from the wall and vertical stresses Viþ1 and Vi were derived from the following relation by Segrestin (1992):
2 Vi ¼ 4g
i1 X
3 hj 5 tanh ðaUi þ bÞ
(5)
j¼1
In the above relation, ka, ka, b, Ui, Uiþ1 and a are the dimensionless coefficients derived from the following relations (Segrestin, 1992):
2 6 ka ¼ 4
Fig. 4. Change in pressure on the wall by cohesive strength at different 4 values.
32
sinðp2 4Þ 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi5 p p p sinð2Þ þ sinð2Þ cosð2 4Þ sin 4
(6)
Fig. 3. Change in failure wedge angle by internal friction angle at different cohesion strengths.
ka ¼ tan2
b ¼
p 2
4
(7)
2
a log kka k ka
(8)
a
2
XV Ui ¼ Pi1 i
j¼1
(9)
hj
XV Uiþ1 ¼ Pi iþ1 j¼1
(10)
hj
a ¼ 2 tan a log
2ka ka þ ka
(11)
In Eq. (11), a is the inclination angle of the backfill soil with a horizontal axis. To analyze unreinforced retaining walls, similar to Fig. 1, there are four unknowns; Ni, Hi, Si and Pi. Thus, it is necessary to write four equations to receive these four unknowns. For this purpose, two equilibrium equations in the vertical and horizontal directions were used; a reinforcement equilibrium equation around the O point and an equation of the relation between shear stress and shear yield stress in the horizontal levels between the slices. The 4n formulations and equations are shown in Table 1. The first three equations for Table 1 are:
Fig. 5. Active earth pressure distribution for different cohesion strength based on proposed method.
M. Ahmadabadi, A. Ghanbari / Geotextiles and Geomembranes 27 (2009) 456–463 Table 2 Equations and unknowns for 5n formulation to achieve lateral pressure on retaining wall with reinforced cohesive-frictional backfill. Unknowns
Number
Hi Inter-slice shear force Ni Normal force at base of each slice Si Shear force at base of each slice Ti
n
Pi Net pressure on wall
n
X X
n n n
Equations P Fx ¼ 0 For each slice P Fy ¼ 0 For each slice P Mo ¼ 0 For each slice Si ¼ Ni(tan 4) þ c For each slice sm ¼ l (sf) For each slice
Number n n n n n
Fx ¼ 00Si cos b Ni sinðbÞ þ Pi cos d þ Hi Hiþ1 ¼ 0
(12)
Fy ¼ 00Pi sin d þ Viþ1 Vi Wi þ Ni cos b þ si sinðbÞ
(13)
P
Mo ¼ 00 Vi XVi þ Vi XViþ1 WiXGi " # 2 3 n X Ni pi cos d 4 þ hj þ hj =25 sinðbÞ j ¼ iþ1 " # " # Pn Pn þHiþ1 H h h i j ¼ iþ1 j j¼1 j ¼ 0
459
(14)
where Wi is the weight of slice i (Wi ¼ gi Ai). The fourth equation in Table 1 is the Moher–Coulomb yield criteria applicable to failure wedge points. The solution to these unknowns will be the pressure on the wall for each slice (Pi) and the horizontal force between the slices (Hi). Dividing the horizontal force on the surface between the two slices produces the average shear stress between them (sm). Since the level between the slices is not in the yield position, it is considered to be a coefficient of soil shear strength (sf) in the following way:
sm ¼ l sf
Fig. 6. Change in l by wall height for constant value of angle of internal friction.
(15)
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Fig. 9. Details of retaining wall with unreinforced soil.
wedge. Parametric study shows that for d ¼ 10 –20 , H ¼ 5–10 m and g ¼ 18–22 kN/m3, the maximum error of the results presented in Fig. 3 is about 3%. According to the proposed method, the pressure on the wall has been calculated based on the difference in the cohesion and friction angle as shown in Fig. 4. An increase in the cohesion strength of the soil reduces the pressure imposed by the soil on the wall in a linear relationship. A sample wall with cohesive-frictional backfill and the effect of cohesion strength on the distribution of active earth pressure is shown in Fig. 5. This distribution has a non-linear relationship and the tension crack zone extends as the cohesion strength of the soil increases. Fig. 7. Change in l by wall height for constant value of cohesion strength.
4. Proposed method for reinforced retaining wall
The non-dimensional coefficient li, has a value of less than one. To analyze unreinforced walls, there is no need to calculate l. However, this coefficient will be used in the next step to analyze reinforced walls. All previous researchers have analyzed retaining walls using HSM, but neglect the shear stress between the slices. Calculation of the l coefficient allows evaluation of the accuracy of this assumption. Also, in the proposed formulation for determining the failure wedge angle in cohesive-frictional soils, pressure on the walls has been calculated for different failure angles. The failure wedge angle is defined as the wedge angle having maximum pressure. Fig. 3 diagrams critical failure wedge variations in relation to the internal friction angle and cohesion of backfill soil. As can be seen, an increase in the angle of internal friction and cohesion strength of the backfill soil leads to a linear increase in the angle of failure
The tensile force of the reinforcements is added to the unknowns for reinforced walls, requiring the equation to be solved for 5n unknowns. For the above purpose, the average shear stress for each slice is assumed to be the coefficient of soil shear strength
Fig. 8. Change in l by wall height for cohesionless soils.
Fig. 10. Details of retaining wall with reinforced soil.
M. Ahmadabadi, A. Ghanbari / Geotextiles and Geomembranes 27 (2009) 456–463
461
Table 3 Comparison of results for Rankine and proposed methods. C ¼ 0 kN/m2
4 ¼ 20
b Pa
4 ¼ 25
b Pa
4 ¼ 30
b Pa
Fig. 11. Change in Pa by wall height for different vertical spacing (Sv) of reinforcements.
in a yielding condition. This coefficient is always less than one and is shown for the ith slice with li, making:
Hi ¼ ½Vi tan 4 þ cli
(16)
The proposed method assumes that li for each slice, with or without reinforcement, is equal. Thus, Eq. (16) can be considered as
C ¼ 20 kN/m2
Proposed method
Rankine method
Proposed method
Rankine method
Proposed method
Rankine method
55.0 490.3 57.5 405.9 60 333.66
55.0 490 57.5 406.0 60 333
55.0 350.2 57.5 278.4 60 218.19
55.0 350.1 57.5 279.0 60 217.76
55.0 210.2 57.5 151.0 60 102.72
55.0 210.4 57.5 151.5 60 102.34
the fifth equation calculating the amount of active pressure for reinforced soils. Table 2 shows the equations and unknowns for the complete formulation. The change in l over the height of a wall has been diagrammed where l is the percentage V tan 4 þ c for different heights or horizontal slices. The change in l by wall height for different internal friction angles, as shown in Fig. 6, is 20 –30 and the cohesion strength of the soil is 0–20 kN/m2. As can be seen, the curves are similar and the slope and length change from the principle point as c and 4 change. Changes in l by H for different 4 are shown in Fig. 7, where c ¼ kN/m2, H ¼ 10 m and d ¼ 10 . Figs. 6 and 7 show the process of change in l with an increase in cohesion strength, internal soil friction and wall height. This means that, for cohesive-frictional soils, l is different for different heights and higher points on the wall have greater l values. Therefore, disregarding the shear forces between slices is not acceptable in some cases. The change in l for cohesionless soils throughout the depth of a 10 m retaining wall is presented in Fig. 8. The equation for the l coefficient for cohesionless soils is as follows:
l ¼ Fig. 12. T and Pa distribution for a reinforced wall with cohesive backfill.
C ¼ 10 kN/m2
c1 H þ c2 H2 þ c3 H þ c4
(17)
in which c1 to c4 coefficients depend on the mechanical properties of the material and the wall height. Ten meter walls with 0.5 and 1 m vertical spacing (Sv) of reinforcements were analyzed using the proposed method and the tensile forces of the reinforcements (T) and active earth pressure distribution on the wall (Pa) were calculated. The details of the unreinforced and reinforced soil walls used in the analysis are presented in Figs. 9 and 10, respectively. The results of active earth pressure for walls with 0, 10 and 20 reinforcements and cohesionless backfill are shown in Fig. 11. The results of active earth pressure distribution and the tensile forces of the reinforcements are shown in Fig. 12 for a reinforced wall with cohesive-frictional backfill. Fig. 13 shows the change in tensile forces of reinforcements by wall height at different cohesion strengths. As can be seen, the distribution of tensile forces of the reinforcements is non-linear with a decreasing gradient in the deeper portions of the wall. Also,
Table 4 Comparison of results for Coulomb and proposed methods.
d ¼ 10
4 ¼ 20
b Pa
4 ¼ 25
b
4 ¼ 30
b
Pa Fig. 13. Change in tensile forces of reinforcements by wall height at different cohesion strengths.
Pa
d ¼ 15
d ¼ 20
Proposed method
Coulomb method
Proposed method
Coulomb method
Proposed method
Coulomb method
50.9 446.7 54.5 372.6 57.7 309.1
51.0 446.7 55.0 372.6 58.0 308.5
49.4 436.1 53.3 364.5 56.8 302.6
50.0 434.3 53 363.1 57 301.4
48.1 426.9 52.2 357.4 55.9 297.9
48.0 426.9 52.0 357.4 56.0 297.3
462
M. Ahmadabadi, A. Ghanbari / Geotextiles and Geomembranes 27 (2009) 456–463
Table 5 Comparison of results for proposed and previous methods for cohesive-frictional backfill. C ¼ 0 kN/m2
4 ( )
20
b Pa
25
b Pa
30
b Pa
C ¼ 10 kN/m2
C ¼ 20 kN/m2
Proposed method
Cheng (2003)
Das and Puri (1996)
Proposed method
Cheng (2003)
Das and Puri (1996)
Proposed method
Cheng (2003)
Das and Puri (1996)
51.0 446.7 54.4 372.6 57.7 309.08
51.0 440.0 55.0 366.9 58 303.76
51.1 446.74 54.6 372.61 57.8 308.46
52.6 318.1 55.7 255.0 58.5 201.84
53.0 313.2 56.0 251.1 59 198.17
50.6 314.36 53.2 243.79 56.5 184.62
53.7 190.4 56.6 138.1 59.2 95.04
54 187.5 57.0 136.0 59 92.94
50.6 190.48 53.1 121.39 55.7 65.64
increases in the cohesion strength of the soil and the number of reinforcements lead to decreases in active earth pressure on the wall.
Table 6 Comparison of proposed method results with AASHTO and Shekarian et al. (2008). AASHTO
5. Discussion and comparison of results To validate the design of the proposed method, the results were compared with previous studies. Tables 3–5 present comparisons of the results of the proposed method with those of the frequently used methods of Rankine, Coulomb and Cheng (2003). The results of the proposed method for the angle of failure wedge are in excellent agreement with the Rankine, Coulomb and Cheng methods. These results are also in good agreement with results presented by Das and Puri (1996), where the maximum difference is about 3.5% for higher cohesion strength values. Comparisons of results for pressure on a wall show that the proposed method is in excellent agreement with the Rankine, Coulomb and Cheng methods. However, the results of Das and Puri (1996) for higher cohesion strength values are noticeably different. The effect of cohesion strength on the distribution of active earth pressure on the wall was assessed and the curves compared with the distribution resulting from the Rankine method (Fig. 14). As can be seen, the proposed method’s distribution has a non-linear form and the greater the increase in c, the deeper tensile crack. To evaluate the validity of the proposed method for reinforced backfills, the results were compared with the results of MSEW software and the Shekarian et al. (2008) method for cohesionless soils. The comparison for d ¼ 10 , H ¼ 10 m and g ¼ 20 kN/m3 is shown in Table 6. It should be mentioned that the MSEW software provided by Leshchinsky (2006) works based on the method recommended by AASHTO. The results of the proposed method for the angle of failure wedge, tensile force of reinforcements and pressure on the wall are
4 ¼ 20 4 ¼ 25 4 ¼ 30
Shekarian et al. (2008)
Proposed method
Pa
T
b
Pa
T
b
Pa
T
b
50.3 38.6 30.1
440.0 367.0 303.8
51.6 55.9 57.9
48.7 43.0 35.4
446.9 367.4 298.3
55.0 57.9 60.2
43.0 39.2 35.5
442.7 362.9 294.8
55.0 57.6 60.0
in good agreement with the methods of Shekarian et al. (2008) and AASHTO. The maximum difference with the other methods is about 3.5% for the angle of failure wedge, 3% for tensile force of reinforcements and 7% for pressure on the wall. 6. Conclusion An analytical procedure based on the horizontal slices and limit equilibrium methods was used to evaluate the active earth pressure on a retaining wall considering the effect of the cohesive strength of reinforced and unreinforced soil that has been neglected in most previous methods. The proposed formulation for unreinforced walls has 4n equation with 4n unknowns. Using the proposed method, the active earth pressure on the wall and the failure wedge for cohesivefrictional soils were calculated. A comparison of these results with the results of the Rankine, Coulomb and Cheng (2003) and Das and Puri (1996) methods shows acceptable agreement. The results show that an increase in the angle of internal friction and cohesion strength of the backfill soil leads to a linear increase in the angle of failure wedge. For reinforced backfill, the tensile forces of the reinforcements were added to the 4n unknowns, making 5n unknowns and 5n equations for cohesive-frictional soils. The results of the proposed method were again compared with the results of methods suggested by the other researchers. The results of sample walls show that, for the proposed method, the distribution of pressure on the wall is non-linear with a negative gradient in the deeper part of the wall. Increases in the cohesion strength of the soil and in the number of reinforcements leads to a decrease in the active earth pressure on the wall. It was concluded that the analytical procedure proposed reliably calculates the resultant earth pressure, active lateral earth pressure distribution, tensile forces of the reinforcements and failure wedge angle for unreinforced and reinforced walls in cohesive-frictional soil. References
Fig. 14. Rankine (d ¼ 0) and proposed (d ¼
0–20 )
active earth pressure distribution.
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