# Multiparametric Analysis of a Gravity Retaining Wall
## 3 Parametric Analysis of a Gravity Retaining Wall  
The optimization model presented above was developed in a general form so that an optimal design for a gravity retaining wall can be obtained for any project data (e.g., soil properties, terrain configuration, loads). A series of optimizations of a gravity retaining wall were performed for a combination of different parameters, i.e., different values of retaining wall heights \(H_{5}\), soil shear angle \(\varphi_{k}\), soil-wall interaction coefficient \(k\), slope angle \(\beta\) and surcharge load \(q_{Qk}\). The soil-wall interaction coefficient \(k\) (\(\delta/\varphi\)) was used to determine the angle of interface friction between the soil and the wall (\(\delta\)) based on the soil friction angle (\(\varphi\)). Input data, variables and a cost objective function subjected to (in)equality geotechnical and design constraints were included in the model. The optimal cost and dimensions of the gravity retaining wall were determined for each optimization. An RCGA was used to solve the optimization problems. A parametric optimization was performed for all 567 combinations of the following different design parameters:The input data used in the optimization model are shown in Table 2 and remain constant in the multiparametric analysis. Note that in the multiparametric analysis, the shear angle of the retained ground and the shear angle of the foundation soil are assumed to be equal (\(\varphi_{ret,k}=\varphi_{found,k}=\varphi_{k}\)). The same assumption was used for the soil-wall interaction coefficient (\(k_{net}=k_{found}=k\)). The subscript \"_ret_\" denotes the properties of retaining soil, while the subscript \"_found_\" denotes the properties of the foundation soil. The partial safety factors correspond to design approach 1 (combination 2) defined in Eurocode standard 7 [1]. Figure 2 shows the optimal results for a gravity retaining wall with a soil-wall interaction coefficient \(k=2/3\), slope angle \(\beta=0^{\circ}\) and variable surcharge load \(q_{Qk}=0\) kPa. With the help of these diagrams, it is possible to determine the optimal wall dimension (\(b_{f}\), \(b\), \(B\) and \(d\)) and which geotechnical constraints have the highest degree of utilization. Note that nonsmooth graphs of optimal wall dimensions were obtained while the wall dimensions were defined as discrete variables. The optimum construction costs and weight of the gravity retaining wall can also be determined. It can be clearly seen that for weak soil (\(\varphi_{k}=30^{\circ}\)), the sliding failure and the bearing capacity failure are decisive, while for strong soil (\(\varphi_{k}=40^{\circ}\)), the eccentricity condition determines the optimal design. In all 567 combinations, the optimal width of the rear wall section \(b_{b}\) was found to be 0 m. A retaining wall with a height \(H_{s}=4\) m, supporting the ground with a shear angle \(\varphi_{k}=35^{\circ}\), has optimal dimensions of \(b_{f}=1.3\) m, \(B=1.8\) m, \(b=0.5\) m and \(d=0.6\) m. For the optimal design, the construction cost and weight of the wall are approximately 850 EUR/m and 150 kN/m, respectively. The bearing capacity condition has the highest utilization rate of about 96%. Using the procedure described above, 567 individual optimizations were performed for all 567 combinations of the various defined parameters, yielding 567 different optimal results. These results are shown graphically in the diagrams in Figures 3-11. These figures allow the engineer to obtain optimum design parameters of a gravity retaining wall for various project data. For example, a gravity wall with a height \(H_{s}=2.5\) m and other project data such as \(k=0.5\), \(\beta=20^{\circ}\), \(\gamma=18\) kN/m\({}^{3}\) and q\({}_{Qk}=4\) kPa has an optimal design that includes \(b_{f}=2.1\) m, \(d=1.6\) m and \(b=0.6\) m, as shown in Figure 3.