## 2 Stability Analysis Relating to the Design of RC-Retaining Walls  
The profile of a standard T shaped retaining wall is given in Figure 1a. In Figure 1 the height of RC-retaining wall stem, depth of excavation, depth of embedment, the thickness of the wall foundation, the thickness of the wall stem at the top, the thickness of the wall stem at the bottom, the length of the toe, and the length of the heel are abbreviated by \(H\), \(h\), \(d\), \(x_{5}\), \(x_{3}\), \(x_{4}\), \(x_{2}\), and \(x_{1}\), respectively. The mobilized stresses according to the excavation process can be defined as active and passive stress states and the distribution of the mentioned stresses are illustrated in Figure 1b. Generally, RC cantilever retaining walls at the planning stage can be designed for the active soil pressure case when the wall is sufficiently flexible [20]. Keeping this situation in mind, only active pressures are taken into consideration for calculations. Lateral earth pressures are calculated by Rankine Earth Pressure Theory with the use of Equations (1) and (2) for active (\(K_{a}\)) and passive (\(K_{p}\)) states [2]. \[K_{a}=tan^{2}(45-\frac{\varnothing}{2}) \tag{1}\]  
\[K_{p}=tan^{2}(45+\frac{\varnothing}{2}) \tag{2}\]  
\(P_{sa}\) and \(P_{sp}\) are the horizontal soil reaction forces for active and passive sides and can be calculated by Equations (3) and (4). The terms \(\gamma_{s1}\), \(K_{a1}\), \(c_{s1}\), and \(\gamma_{s2}\), \(K_{a2}\), \(c_{s2}\). \(\gamma\) represents the unit weight of soil, \(K_{a}\) represents active lateral earth pressure coefficient, and \(c_{s}\) represents the cohesion of the soil at the active site and the passive side of the wall, respectively. In Figures 1b and 2b, it is supposed that a surcharge (\(q_{a}\)) is applied to the retaining system from the active side. \[P_{sa}=\frac{1}{2}\gamma_{s1}H^{2}K_{a1}-2c_{s1}H\sqrt{K_{a1}} \tag{3}\]  
\[P_{sp}=\frac{1}{2}\gamma_{s2}d^{2}K_{p2}+2c_{s2}d\sqrt{K_{p2}} \tag{4}\]  
Figure 1: **(a)** The profile of the reinforced concrete (RC) cantilever retaining wall; **(b)** the loads and stresses acting on the RC cantilever retaining wall. This surcharge is used to determine the acting horizontal load (\(P_{ga}\)) by the use of \(K_{a1}\). The horizontal value of distributed load (\(P_{ga}\)) that is affected to the wall section from the active site of the wall can be determined by the multiplication of vertical surcharge (\(q_{a}\)) with the height of the wall (H) and with the active lateral earth pressure. \(W_{us}\) and \(W_{uf}\) represent the weight of the stem and foundation base. \(W_{sa}\) represents the backfill soil weight existing on the foundation base. \(q_{max}\) and \(q_{min}\) values are the base pressures. The sections of the wall have to be satisfied with both the equilibrium of forces and moment. Unbalanced external and internal loads can make possible modes of failure like sliding, overturning, and/or overstress in soil. The combination of lateral forces can cause the wall to slide through its base. The ratio of the total lateral resisting forces (\(\Sigma F_{R}\)) to the total lateral sliding forces (\(\Sigma F_{s}\)) gives the factor of safety value for sliding (Equation (5)). The resisting forces can be combined by the weights of the wall sections, the foundation soil friction resistance, and the lateral components of passive soil forces. Sliding forces include the lateral components of active soil forces and the influence of the loads acting from the backfill side. \[FoS_{s}=\frac{\Sigma F_{R}}{\Sigma F_{S}} \tag{5}\]  
Another mode of failure consists of the overturning of the wall system about its toe point. The ratio of the total moments of resisting forces (\(\Sigma M_{R}\)) to total moments of sliding forces (\(\Sigma M_{S}\)) gives the factor of safety value for overturning (Equation (6)). Resisting moments are generated by the passive pressure of the soil, the self-weight of the overall wall, and the weight of backfill.