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ISRN Soil Science
Volume 2012 (2012), Article ID 750386, 8 pages
http://dx.doi.org/10.5402/2012/750386
Research Article

Active Thrust on an Inclined Retaining Wall with Inclined Cohesionless Backfill due to Surcharge Effect

Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India

Received 13 March 2012; Accepted 8 May 2012

Academic Editor: A. P. Schwab

Copyright © 2012 D. M. Dewaikar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A method based on the application of Kötter’s equation is proposed for the complete analysis of active thrust on an inclined wall with inclined cohesionless backfill under surcharge effect. Coulomb’s failure mechanism is considered in the analysis. The point of application of active thrust is determined from the condition of moment equilibrium. The coefficient of active pressure and the point of application of the active thrust are computed and presented in nondimensional form. One distinguishing feature of the proposed method is its ability to determine the point of application of active thrust on the retaining wall. A fairly good comparison is obtained with the existing solutions.

1. Introduction

Active earth pressure evaluation is required for the design of geotechnical structures such as retaining walls, sheet piles, basements, and tunnels. Active thrust acting on a retaining wall is dependent of many parameters. The theories proposed by Coulomb [1] and Rankine [2] remain the fundamental approaches to analyze the active earth pressures. Coulomb [1] studied the earth pressure problems using the limit equilibrium method considering a triangular wedge of backfill behind a rough retaining wall with a plane failure surface and this theory is well verified for the frictional soil in active state. The point of application of active thrust is assumed at a distance one-third of the height of the wall from its base and independent of different parameters such as soil friction angle, 𝜙, angle of wall friction, 𝛿, backfill angle, 𝛽, and wall inclination angle. Coulomb’s [1] approach does not use moment equilibrium equation for the analysis, since the distribution and point of application of reaction on the failure plane are unknown. If the distribution and point of application of soil reaction on the failure plane are known, then the point of application of active thrust can be determined using moment equilibrium equation. The limit equilibrium method assuming appropriate failure surface is most frequently used to analyze static earth pressure.

A failure mechanism was proposed by Terzaghi [3] that assumes a log-spiral failure surface originating from the base of the wall, followed by a tangent, that meets the ground surface at an angle corresponding to Rankine’s [2] active earth pressure. Several experimental, analytical, and numerical studies were performed to evaluate the earth pressures on the retaining walls [414]. Some rigorous approaches such as finite element methods were attempted to determine the distribution of earth pressures on the retaining wall. Others prepared the tables for the calculation earth pressures based on log-spiral failure surface [15]. Active earth pressure coefficients for cohesionless soil were also computed using the method of slices by considering the soil mass in the state of limit equilibrium [16, 17]. The charts were prepared for the calculation earth pressures based on log-spiral failure surface [18]. Discrete element analysis is used to evaluate active and passive pressure distributions on the retaining wall [19].

Recently the active earth pressure coefficients were computed based on the lower bound theorem of plasticity [20]. Upper-bound theorem of limit analysis was used to evaluate earth pressure coefficients due to soil weight, vertical surcharge loading, and soil cohesion for the case of an inclined wall and sloping cohesionless backfill [21]. Kötter’s [22] equation was suitably used to determine the point of application of active thrust by taking moments of all the forces and reaction about the base of the retaining wall [23]. A method was proposed by Kame et al. [24] to determine the active earth pressure and its point of application on a vertical wall retaining horizontal cohesionless backfill based on log-spiral failure mechanism coupled with Kötter’s equation. Dewaikar et al. [25] used Kötter’s equation to evaluate the magnitude of active thrust on an inclined wall retaining horizontal cohesionless backfill with uniform surcharge and to find out the point of application of reaction on the failure surface.

The objective of this study is to present a mathematical approach with a high degree of simplicity for the evaluation of the active thrust and its point of application. The analysis considers the free body diagram of the retaining wall in a very simple and analytical procedure and is easier for engineering application. The validity of the proposed approach is checked with previous available results.

2. Analysis of Active Thrust

The active thrust, 𝑃𝑎, is determined from the conditions of force equilibrium and its point of application is determined from the conditions of moment equilibrium. Figure 1 shows an inclined retaining wall, with inclined cohesionless backfill subjected to a uniform surcharge intensity of q KN/m2. A trial plane failure surface [1] is considered which meets the ground surface at an angle, α with the horizontal. The trial wedge ABC is in equilibrium under the effect of three different forces: (1) the equivalent force of the surcharge, W (2) the soil reaction, R along the face trial failure BC, at an angle 𝜙 to the normal on BC and (3) the active thrust 𝑃𝑎, at an angle 𝛿 to the normal on the back.

750386.fig.001
Figure 1: Free body diagram of the trial failure wedge ABC.

The symbols used in Figure 1 are defined as follows.𝑃𝑎: active thrust.𝑊: equivalent force of the surcharge.𝑅: soil reaction on the failure wedge.𝐻: height of the retaining wall.: height of point of application of active thrust from the wall base.𝜃: inclination of the retaining wall with the horizontal.𝛿: friction angle between the wall and soil backfill.𝜙: soil friction angle.𝛼: inclination of the trial failure plane with the horizontal.𝑞: intensity of surcharge in kN/m2.

Equating all the forces in the vertical and horizontal direction, the following force equilibrium conditions are obtained.

Horizontal force equilibrium:
𝑃𝑎sin(𝜃𝛿)=𝑅sin(𝛼𝜙)(1) from which R is obtained as 𝑃𝑅=𝑎sin(𝜃𝛿).sin(𝛼𝜙)(2)

Vertical force equilibrium:
𝑃𝑎cos(𝜃𝛿)+𝑅cos(𝛼𝜙)=𝐴𝐶𝑞(3) substituting the value of R from (2) into (3) 𝑃𝑎𝑃cos(𝜃𝛿)+𝑎sin(𝜃𝛿)𝑃sin(𝛼𝜙)cos(𝛼𝜙)=𝐴𝐶𝑞,(4)𝑎cos(𝜃𝛿)+𝑃𝑎sin(𝜃𝛿)cot(𝛼𝜙)=𝐴𝐶𝑞.(5) From (5), 𝑃𝑎 is obtained as 𝑃𝑎=𝐴𝐶𝑞[].cos(𝜃𝛿)+sin(𝜃𝛿)cot(𝛼𝜙)(6) Now, applying sine rule to the triangle CAN, 𝐴𝐶=sin(180𝛼)𝐴𝑁sin(𝛼𝛽)(7) or 𝐴𝐶=𝐴𝑁sin(180𝛼).sin(𝛼𝛽)(8) Referring to Figure 1, 𝐴𝑁=𝐻(cot𝜃+cot𝛼).(9) Substituting the value of AN in (8), AC is obtained as, 𝐴𝐶=𝐻(cot𝛼+cot𝜃)sin𝛼sin(𝛼𝛽)(10) Substituting the value of AC in (6), the magnitude of active thrust 𝑃𝑎 is expressed as 𝑃𝑎=𝑞𝐻(cot𝛼+cot𝜃)sin𝛼[].sin(𝛼𝛽)cos(𝜃𝛿)+sin(𝜃𝛿)cot(𝛼𝜙)(11) The maximum value of active thrust (𝑃𝑎) is obtained when the inclination of the failure plane, BC, with the horizontal reaches the critical value, 𝛼cr.

3. Evaluation of Soil Reaction on the Failure Surface

In a cohesionless soil medium with active state of equilibrium under plane strain condition, Kötter’s [22] equation is given as (Figure 2) 𝑑𝑝𝑑𝑠2𝑝tan𝜙𝑑𝛼𝑑𝑠=𝛾sin(𝛼𝜙)𝑠,(12)

750386.fig.002
Figure 2: Reactive pressure distribution along the failure surface.

where𝑑𝑝: differential reactive pressure at a point on the failure surface,𝑑𝑠: elemental length of the failure surface,𝛼: angle made by the tangent at the point of interest with the horizontal,𝜙: soil friction angle and,𝛾: soil unit weight.

In the present analysis, only surcharge effect is taken into consideration, and therefore 𝛾 becomes zero. With this consideration (12) can be written as 𝑑𝑝𝑑𝛼=2𝑝tan𝜙(13) or 𝑑𝑝=2𝑝tan𝜙𝑑𝛼.(14) For a plane failure surface, 𝑑𝛼 is zero and the corresponding solution for the reactive pressure, 𝑝, is obtained as 𝑝=constant.(15) The above solution indicates that soil reactive pressure (𝑝) is uniformly distributed along the failure plane, BC. Therefore, the resultant soil reaction, R, acts at the mid-point of the failure plane, BC. The magnitude of soil reaction is computed after knowing the critical value, 𝛼cr, of the angle, 𝛼. Substituting the value of 𝑃𝑎 from (11) in to (2), the soil reaction R on the failure surface is obtained as 𝑅=𝑞𝐻cot𝛼cr+cot𝜃sin(𝜃𝛿)sin𝛼cr𝛼sincr𝛼𝜙sin𝑐𝑟×1𝛽𝛼cos(𝜃𝛿)+sin(𝜃𝛿)cotcr.𝜙(16)

4. Point of Application of the Active Thrust

After obtaining the value of 𝑃𝑎, the condition of moment equilibrium is applied. Figure 3 shows the free body diagram of the failure wedge ABC. The equivalent force of surcharge on the trial wedge, ABC, is 𝐴𝐶𝑞, which acts at the midpoint of AC, that is, at a distance Y from the vertical line passing through the base the wall. As the distribution of reaction (𝑅) is uniform along the failure surface, it acts at the mid-point of the failure plane BC. The moments of all forces and reactions are taken about the base of the wall, at point B.

750386.fig.003
Figure 3: Free body diagram of the failure wedge ABC.

Moment equilibrium condition:
𝑃𝑎cos𝛿𝑋+𝑞𝐴𝐶𝑌=𝑅cos𝜙𝐵𝐶2.(17) The surcharge acts at the center of AC with the distance Y given as, 𝑌=𝑃𝑄cos𝛽,(18) where 𝑃𝑄=𝐴𝐶2𝐴𝑃.(19)
From triangle AMP, 𝐴𝑃=𝐻cot𝜃.cos𝛽(20) Now, substituting the value of AC from (10) and AP from (20) in (19) (Figure 3), 𝑃𝑄 is obtained as 𝐻𝑃𝑄=cot𝜃+cot𝛼crsin𝛼cr𝛼2sincr𝛽𝐻cot𝜃cos𝛽.(21) Substituting the value of 𝑃𝑄 from (21) into (18), the point of application of equivalent surcharge force from the axis BM (Figure 3) is computed as 𝐻𝑌=cos𝛽cot𝜃+cot𝛼crsin𝛼cr𝛼2sincr𝛽𝐻cot𝜃cos𝛽.(22) The soil reaction R acts at the midpoint of the failure plane BC, with the distance, 𝑟 (Figure 3), computed as 𝑟=𝐵𝐶2.(23) Referring to Figure 3, BC is obtained as 𝐵𝐶=𝐻sin(𝜃+𝛽)𝛼sincr.𝛽sin𝜃(24) Substituting the value of BC from (24) and Y from (22) into (17), the distance X is obtained: 𝑃𝑎×𝐻cos𝛿𝑋+𝑞𝐴𝐶cos𝛽cot𝜃+cot𝛼crsin𝛼cr𝛼2sincr𝛽𝐻cot𝜃cos𝛽=𝑅cos𝜙𝐻sin(𝜃+𝛽)𝛼2sincr𝛽sin𝜃(25) or 𝑋=𝑅cos𝜙𝐻sin(𝜃+𝛽)𝛼2sincr𝛽sin𝜃𝑃𝑎𝐻cos𝛿𝑞𝐴𝐶cos𝛽cot𝜃+cot𝛼crsin𝛼cr𝛼2sincr𝛽𝐻cot𝜃cos𝛽𝑃𝑎.cos𝛿(26) The height, , of point of application of 𝑃𝑎 from the wall base is obtained as =𝑋sin𝜃.(27) The coefficient of active earth pressure (𝐾aq) for the retaining wall with the surcharge effect is obtained as 𝐾aq=𝑃𝑎/𝛾𝐻.(28)

5. Results and Discussions

One of the aims of the proposed analysis is the determination of point of application of the active thrust in a nondimensional form with the consideration of effect of various parameters. The height, , of point of application of the active thrust due to surcharge effect is expressed in a nondimensional form, 𝐻𝑟, expressed as /𝐻. The results are reported as shown in Table 1 along with figures that are described below.

tab1
Table 1: Variation of the height of point of application of active thrust, 𝐻𝑟, with angle of wall back (𝜃), angle of soil friction (𝜙), soil and wall friction (𝛿) and backfill slope (𝛽).

In Figures 4 and 5, variation of 𝐻𝑟 with the angle of wall back, 𝜃, is shown for 𝛿=2/3𝜙 and 𝛽=0.8𝜙 and 0.6𝜙 respectively. It is seen that 𝐻𝑟, decreases with increasing angle of wall back and increases with increasing soil friction angle, 𝜙.

750386.fig.004
Figure 4: Variation of 𝐻𝑟 with angle of wall back, 𝜃, for 𝛽=0.8𝜙.
750386.fig.005
Figure 5: Variation of 𝐻𝑟 with angle of wall back, 𝜃, for 𝛽=0.6𝜙.

In Figures 6, 7, and 8, variation of 𝐻𝑟 with the angle of backfill slope, β, with 𝛿=2/3𝜙 is shown for θ = 65°, 70°, and 75°, respectively. It is seen that 𝐻𝑟, increases with increasing β, with higher values for higher soil friction angle, 𝜙.

750386.fig.006
Figure 6: Variation of 𝐻𝑟 with angle of backfill slope, 𝛽, for θ = 65°.
750386.fig.007
Figure 7: Variation of 𝐻𝑟 with angle of backfill slope, 𝛽, for θ = 70°.
750386.fig.008
Figure 8: Variation of 𝐻𝑟 with angle of backfill slope, 𝛽, for θ = 75°.

In Figure 9, variation of 𝐻𝑟 with the angle of wall friction, 𝛿, is shown for 𝛽= 16° and ϕ = 40°. It is seen that 𝐻𝑟 increases with 𝛿, with higher values for higher 𝛽.

750386.fig.009
Figure 9: Variation of 𝐻𝑟 with angle of wall friction, 𝛿, for ϕ = 40°.

In Figure 10 is shown the variation of 𝐻𝑟 for a vertical wall (𝜃 = 90°) and 𝛿=2/3𝜙. It is again seen that 𝐻𝑟 increases with increase of 𝛽.

750386.fig.0010
Figure 10: Variation of 𝐻𝑟 with angle of backfill slope, 𝛽, for θ = 90°.

In Figure 11, variation of 𝐻𝑟 with angle of soil friction, 𝜙, is shown for a retaining wall with angle of wall back, θ = 90°, backfill slope, β = 0°, and angle of wall friction, δ = 0°. It is interesting to note that 𝐻𝑟 is a constant value of 0.5 for all values of angle of soil friction, 𝜙. Hence, the point of application of active thrust acts at the mid-height of a smooth vertical retaining wall with horizontal backfill.

750386.fig.0011
Figure 11: Variation of 𝐻𝑟 with angle of soil friction, 𝜙.

The other aim of the proposed analysis is to compute the values of the active earth pressure coefficient, 𝐾aq, for the various combinations of parameters considered in the analysis and compare the results with the available solution reported by Soubra and Macuh [21]. The results are presented through figures. In Figures 12, 13, and 14, the variation of active earth pressure coefficient, 𝐾aq with angle of soil friction, 𝜙 is shown for 𝛽=0.4𝜙, 0.6ϕ, and 0.8ϕ, respectively, for various values of angle of wall back, 𝜃 and 𝛿=2/3𝜙. It is seen that 𝐾aq values decrease with increasing angle of soil friction, 𝜙. It is further seen that the values show an increasing trend with decreasing values of angle of wall back, 𝜃.

750386.fig.0012
Figure 12: Variation of 𝐾aq with angle of soil friction, 𝜙, for 𝛽=0.4𝜙.
750386.fig.0013
Figure 13: Variation of 𝐾aq with angle of soil friction, 𝜙, for 𝛽=0.6𝜙.
750386.fig.0014
Figure 14: Variation of 𝐾aq with angle of soil friction,𝜙, for 𝛽=0.8𝜙.

In Figure 15 and Table 2 is shown the comparison of active earth pressure coefficients obtained from the proposed method and the method proposed by Soubra and Macuh [21]. The 𝐾aq values obtained from the proposed method are higher with the maximum difference of 7.57% for the angle of wall back of 𝜃= 80°, ϕ ≤ 40°, and 𝛿=2/3𝜙 as shown in Table 2. For lower values of soil friction angle, 𝜙, and angle of backfill slope, β, the difference between the two results is less, showing an increasing trend with increasing angle of soil friction, 𝜙, and backfill slope, 𝛽.

tab2
Table 2: Comparison of 𝐾aq with the results for the angle of wall back of 𝜃=80˚.
750386.fig.0015
Figure 15: Comparison of variation of 𝐾aq with angle of backfill slope, 𝛽, evaluated in present study with filed results.

6. Conclusions

The complete solution to a retaining wall problem is obtained only when the point of application of active thrust is known. Kötter’s [22] equation lends itself as a powerful tool in the proposed analysis to determine the reactive pressure distribution on the failure plane. The moment equilibrium condition is used effectively to compute the point of application of the active thrust. From the proposed analysis it is seen that the point of application of the active thrust depends upon a number of factors such as angle of soil friction, 𝜙, angle of wall friction, 𝛿, angle of wall back, 𝜃, and inclination of backfill, 𝛽. It shows a wide variation in the range, 0.427 to 0.814. Only for the case of a smooth vertical wall retaining horizontal backfill, the point of application of the active thrust is at the wall mid-height. The present approach is easy to adapt for retaining walls and the results obtained based on this analysis are in fairly good agreement with other results.

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