Development of Design Charts Considering the Effect of Backfill Inclination and Wall Inclination on the Seismic Active Pressure for c-φ Soil

Abstract In this paper, the design charts have been presented to calculate the seismic active pressure for c-φ soil with surcharge. For developing the design charts, the explicit generalised equation based on pseudo-static approach is used. In the present study, the design charts have been presented for positive and negative wall inclination considering the inclined backfill. The effect of backfill inclination along with the effect of positive and negative wall inclination on the seismic active pressure has been noticed in the design charts. The design charts are very easy to use for the field.


Introduction
The design of retaining structures mainly depends on the lateral earth pressure exerted by the backfill material on the wall. Under normal conditions, almost all retaining structures are designed for active thrust from the retained backfills. The estimation of dynamic active thrust on the retaining walls is calculated by explicit generalized expression, in the field. This method is based on the pseudo-static approach. For cohesionless backfills, Okabe (1924), Mononobe and Matsuo (1929) reported this method in their study and then known as Mononobe and Okabe method Kramer (1996). But in real field condition, it doesn't work because of cohesion is present in the soil. For cohesive (c-ϕ) soil with soil cohesion, c, and angle of internal friction of soil, ϕ, Shukla et al. (2009) gives a simple expression for estimating the seismic active earth pressure. But in this study wall friction and adhesion between wall face and backfill was ignored and the wall is also assumed as vertical. Kim et al. (2010) obtained an expression for estimation of the total seismic active pressure in terms of the inclination of failure plane, but the calculation of failure plane was based on the trial and error procedure. This study has very limited applications in real design practices due to trail-error procedure. Shukla and Bathurst (2012) had included the effect of wall friction for calculating seismic active pressure on retaining wall. Shukla (2013) gave an expression to calculate seismic active pressure for sloping backfills. Shukla (2015) developed a generalized analytical expression for the dynamic active pressure on retaining wall which supports an inclined backfill of cohesive soil. The influence of many factors such as effect of wall geometry such as wall inclination and backfill inclination, cohesive/non-cohesive backfill, wall-backfill interface, tension cracks, surcharge, horizontal and the vertical ground acceleration was also examined. Using this generalized expression, for the calculation of seismic active thrust, Gupta et al. (2019) developed the design charts, showing the effect of surcharge loading. The formulation of the critical value of inclination from the horizontal of the failure plane was also developed in this study. In the present study the design charts have been presented considering the effect of soil backfill inclination along with the negative and positive wall inclination on seismic active pressure. To draw the design charts, for the calculation of seismic active pressure from c-ϕ soil backfill, the explicit generalised expression proposed by Shukla (2015) is used. The design charts reduce the calculation work and makes work effective under seismic condition.

Analytical Derivation
Considering the inclined retaining wall same as taken by Gupta et al. (2019) with height, H supporting a cohesive soil backfill with soil cohesion (c) and angle of internal friction of soil (ϕ) with an active trial failure wedge (A1A2A3) of weight W. The back-face of rigid retaining wall (A1A2) is inclined at an angle β with the horizontal. A2A3 is the assumed failure plane and it passes through the bottom of the wall and makes an angle α with the horizontal. Horizontal and vertical seismic inertial forces are considered as khW and kvW, where kh and kv are the horizontal and the vertical seismic coefficients. A surcharge q per unit surface area is considered at the top of the sloping backfill with slope angle i with horizontal. khqB and kvqB are surcharge loads along the horizontal and vertical planes and B is the length of sloping backfill A1A3. zc is the depth of tension crack from the surface of the backfill. For simplification, wall height is divided as a sum of z and h. The force F is the resultant of frictional force T and normal force N and it acts on the failure plane A2A3. Ca is the total adhesive force mobilized along the wall-backfill interface at an angle δ and C is total cohesive force. Now, taking equilibrium of forces in horizontal and vertical direction respectively, Eliminating F from equations (1) and (2) and further simplifying, Pae can be expressed as: a1, b1, c1 and a2, b2, c2 are the non-dimensional constants.
For optimising the value of dynamic active thrust Pae, the following condition should be satisfied: Or, Above equation (15) is solved for tanα to get the critical value of the inclination of the failure plane, α = αc as: 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 On substituting α = αc into equation (3); where, Kae is the coefficient of active earth pressure.

Results and Discussion
According to generalized analytical expression (Equation 18), the design charts are presented in terms of c* (non-dimensional cohesion) and q* (non-dimensional surcharge) as defined in equations (19) and (20)

Conclusion
According to design charts from present study, the following conclusions are summarized:  The value of active earth pressure coefficient (Kae) reduces with respect to increase in cohesion (c) as well as angle of shearing resistance of soil backfills and disregarding of non-dimensional backfill surcharge loading in all case.  The value of the active earth pressure coefficient (Kae) increases with the inclination of β from (75 o -105 o ) in all case. However, the percentage increment of Kae is marginally increases, when the backfill slope is 10 o .  At a given angle of backfill inclination, the value of active earth pressure coefficient (Kae) is more for the sloping backfill case. The percentage increment of Kae, also increases for the higher value of wall inclination.