## 2 Structural Assessment. Limit States Calculation Methods, Design Situation, and Loads
### Loads
#### 2.4.1 Lateral Earth Pressure  
Checking the lateral pressure from the backfill, two methods have been considered among the possibilities currently available [27]: the Coulomb method (Figure 1a), in case of static conditions, and the Mononobe-Okabe method (Figure 1b) for dynamic conditions from seismic events. * Static case: Coulomb Method Because friction between the wall and the soil is considered, [PERSON]'s theory is applied in order to determine the resultant force (\(P_{AS}\)) in case of active earth pressure (Figure 1a). This method assumes that the failure surface is two-dimensional and passes through the heel of the wall, whose inclination to the horizontal plane (\(\alpha_{A}\)) is unknown. This failure plane in conjunction with wall surface closer to the backfill defines a wedge of soil, which can slide over these two planes [28]. For this particular case of active earth pressure, this wedge moves away from the backfill and downward. By posing the equilibrium of this wedge as a rigid body in terms of the angle (\(\alpha_{A}\)) and solving an optimization problem, a maximum value for \(P_{AS}\) is obtained. The force obtained thereby is located at one-third of the height of backfill from the bottom and acts at an angle (\(\delta\)) with the normal to the back of the wall, which ranges from \(1/3\) to \(2/3\) times the soil friction  
Figure 1: Schemes for determining the lateral earth thrust in retaining walls: (**a**) Coulomb theory [28] (**b**) Mononobe-Okabe theory [29]. angle (\(\phi^{\prime}\)). The magnitude of this force is obtained by applying Equation (42), where \(K_{AS}\) is defined by Equation (43). \[P_{AS}=\frac{1}{2}\;\gamma\;H^{2}\;K_{AS} \tag{42}\]  
\[K_{AS}=\frac{\cos^{2}(\phi^{\prime}_{d}-\theta)}{\cos^{2}\;\theta\cos(\theta+ \delta_{d})\left[1+\sqrt{\frac{\text{sen}\left(\phi^{\prime}_{d}+\delta_{d} \right)\text{sen}\left(\phi_{d}-\beta\right)}{\cos(\delta_{d}+\theta)\cos( \beta-\theta)}}\;\right]^{2}} \tag{43}\]  
\[\phi^{\prime}_{d}=\tan^{-1}\left(\frac{\tan\phi^{\prime}}{\gamma^{\prime}_{ \phi}}\right) \tag{44}\]  
\[\delta_{d}=\tan^{-1}\left(\frac{\tan\delta}{\gamma^{\prime}_{\phi}}\right) \tag{45}\]  
In the case of the wall supporting a uniform surcharge (q), the resultant force (\(P_{AS}\)) must also include the lateral earth pressure of the soil skeleton as a result of the imposed load. For this scenario, it is necessary to define a new value of unit weight for the wedge \((\gamma^{*})\) to take the surcharge into account. Its magnitude can be obtained according to Equation (46):  
\[\gamma^{*}=\gamma\left[1+\frac{2q\cos(\theta)}{\gamma H\cos(\beta-\theta)}\right] \tag{46}\]  
By using [PERSON]'s theory it is also possible to determine the magnitude of the resultant force (\(P_{PS}\)) in case of passive earth pressure (Figure 1a). In this case, the wedge defined by the horizontal plane (\(\alpha_{P}\)) moves towards the backfill and upward. As previously, this force is located at one-third of the height of the backfill from the bottom and acts at an angle (\(\delta\)) with the normal to the back of the wall, taking a zero value. The magnitude of the resultant force is now obtained by applying Equation (47), where \(K_{PS}\) is defined by Equation (48). \[P_{PS}=\frac{1}{2}\;\gamma\;H^{2}\;K_{PS} \tag{47}\]  
\[K_{PS}=\frac{\cos^{2}\left(\phi^{\prime}_{d}+\theta\right)}{\cos^{2}\;\theta \cos(\delta_{d}-\theta)\left[1-\sqrt{\frac{\text{sen}\left(\phi^{\prime}_{d} +\delta_{d}\right)\text{sen}\left(\phi^{\prime}_{d}+\beta\right)}{\cos(\delta _{d}-\theta)\cos(\beta-\theta)}}\;\right]^{2}} \tag{48}\]  
\(\bullet\) Dynamic case: Mononobe Okabe Method  
An estimation of the resultant force during earthquakes is performed using the Mononobe-Okabe method [29; 30; 31], which is considered to be an extension of the above-mentioned [PERSON]'s theory. Seismic loads are applied to the soil wedge by using a pseudo-static method, so that horizontal and vertical inertia forces are introduced into the model using coefficients (Figure 1b).