# Multiparametric Analysis of a Gravity Retaining Wall
## 2 Mathematical Model of a Gravity Retaining Wall
 \tag{1}\]  
_Geotechnical constraints and the corresponding equations:_  
\[H_{Ed}\leq H_{Rd} \tag{2}\] \[E_{a,b1}=p_{h,top}\cdot H_{0}\] (2b) \[H_{0}=H_{s}+d\] (2d) \[P_{p}=tan^{2}\left(45^{+}\ \psi+\delta_{found,d}/2\right)\cdot\gamma_{ found,k}\,d^{2}/2\] (2f) \[p_{h,bottom}=SF_{Q}\cdot K_{aqh}\cdot q_{Q,k}+SF_{C}\cdot K_{a \eta h}\cdot H_{0}\cdot\] (2h) \[\gamma_{ret,k}-K_{ach}\cdot c^{\prime}_{ret,d}\] \[K_{n}=\] \[\frac{1+\sin\varphi^{\prime}_{ret,d}\cdot\sin\left(2m_{w}+\varphi ^{\prime}_{rad}\right)}{1-\sin\varphi^{\prime}_{rot,d}\cdot\sin\left(2m_{w}+ \varphi^{\prime}_{rad}\right)}e^{2\left(m_{r}+\beta-m_{w}-\eta\right)\cdot \tan\varphi^{\prime}_{rot,d}}\] (2j) \[2m_{t}=\cos^{-1}\left(\frac{-\sin\beta_{rot,d}}{\sin\varphi^{ \prime}_{rot,d}}\right)-\varphi^{\prime}_{ret,d}-\beta\] (2l) \[\eta=\tan^{-1}(b_{b}/H_{0})\] (2n) \[E_{a,b1}=p_{h,top}\cdot H_{0}\]0 (2p) \[E_{a,b1}=p_{h,top}\cdot H_{0}\]1 (2r) \[E_{a,b1}=p_{h,top}\cdot H_{0}\]2 (2l) \[E_{a,b1}=p_{h,top}\cdot H_{0}\]3 (2r) \[E_{a,b1}=p_{h,top}\cdot H_{0}\]4 (2r) \[E_{a,b1}=p_{h,top}\cdot H_{0}\]5 (2a) \[E_{a,b1}=p_{h,top}\cdot H_{0}\]6 (2c) \[E_{a,b1}=p_{h,top}\cdot H_{0}\]7 (2 ae) \[E_{a,b1}=p_{h,top}\cdot H_{0}\]8 (3) \[E_{a,b1}=p_{h,top}\cdot H_{0}\]9 (3b) \[H_{0}=H_{s}+d\]0 (3d) \[H_{0}=H_{s}+d\]1 (3f) \[H_{0}=H_{s}+d\]2 (3h) \[H_{0}=H_{s}+d\]3 (3j) \[H_{0}=H_{s}+d\]4 (3l) \[H_{0}=H_{s}+d\]5  
\[H_{0}=H_{s}+d\]6 (2a) \[H_{0}=H_{s}+d\]7 (2c) \[H_{0}=H_{s}+d\]8 (2e) \[H_{0}=H_{s}+d\]9 (2g) \[P_{p}=tan^{2}\left(45^{+}\ \psi+\delta_{found,d}/2\right)\cdot\gamma_{ found,k}\,d^{2}/2\]0 (2i) \[P_{p}=tan^{2}\left(45^{+}\ \psi+\delta_{found,d}/2\right)\cdot\gamma_{ found,k}\,d^{2}/2\]1 (2k) \[P_{p}=tan^{2}\left(45^{+}\ \psi+\delta_{found,d}/2\right)\cdot\gamma_{ found,k}\,d^{2}/2\]2 (2k) \[P_{p}=tan^{2}\left(45^{+}\ \psi+\delta_{found,d}/2\right)\cdot\gamma_{ found,k}\,d^{2}/2\]3 (2m) \[P_{p}=tan^{2}\left(45^{+}\ \psi+\delta_{found,d}/2\right)\cdot\gamma_{ found,k}\,d^{2}/2\]4 (2o) \[P_{p}=tan^{2}\left(45^{+}\ \psi+\delta_{found,d}/2\right)\cdot\gamma_{ found,k}\,d^{2}/2\]5 (2q) \[P_{p}=tan^{2}\left(45^{+}\ \psi+\delta_{found,d}/2\right)\cdot\gamma_{ found,k}\,d^{2}/2\]6 (2s) \[P_{p}=tan^{2}\left(45^{+}\ \psi+\delta_{found,d}/2\right)\cdot\gamma_{ found,k}\,d^{2}/2\]7 (2u) \[P_{p}=tan^{2}\left(45^{+}\ \psi+\delta_{found,d}/2\right)\cdot\gamma_{ found,k}\,d^{2}/2\]8 (2w) \[P_{p}=tan^{2}\left(45^{+}\ \psi+\delta_{found,d}/2\right)\cdot\gamma_{ found,k}\,d^{2}/2\]9 (2b) \[p_{h,bottom}=SF_{Q}\cdot K_{aqh}\cdot q_{Q,k}+SF_{C}\cdot K_{a \eta h}\cdot H_{0}\cdot\]0 (2 ad) \[p_{h,bottom}=SF_{Q}\cdot K_{aqh}\cdot q_{Q,k}+SF_{C}\cdot K_{a \eta h}\cdot H_{0}\cdot\]1 (3a) \[p_{h,bottom}=SF_{Q}\cdot K_{aqh}\cdot q_{Q,k}+SF_{C}\cdot K_{a \eta h}\cdot H_{0}\cdot\]2 (3c) \[p_{h,bottom}=SF_{Q}\cdot K_{aqh}\cdot q_{Q,k}+SF_{C}\cdot K_{a \eta h}\cdot H_{0}\cdot\]3 (3e) \[p_{h,bottom}=SF_{Q}\cdot K_{aqh}\cdot q_{Q,k}+SF_{C}\cdot K_{a \eta h}\cdot H_{0}\cdot\]4 (3g) \[p_{h,bottom}=SF_{Q}\cdot K_{aqh}\cdot q_{Q,k}+SF_{C}\cdot K_{a \eta h}\cdot H_{0}\cdot\]5 (3 iThe objective function \(COST\) (see Equation (1)) includes the construction cost of the gravity retaining wall. This includes the material cost, soil excavation, fill material with compaction and the drainage system. Therefore, the optimal solution represents the minimum cost of a gravity retaining wall that satisfies all the design and stability constraints. The optimization consists of 6 conditions that must be met. The first condition states that the horizontal force acting on the retaining wall cannot exceed the sum of the  
\[p_{h,bottom}=SF_{Q}\cdot K_{aqh}\cdot q_{Q,k}+SF_{C}\cdot K_{a \eta h}\cdot H_{0}\cdot\]6 Geometry and parameters of a gravity retaining wall. resistances below the wall (see Equation (2)). It is defined by Equations (2a)-(2 ae) and must be satisfied. The eccentricity of the action from the center of the wall \(e_{B}\) (m) is bounded by the maximum allowable eccentricity \(e_{max}\). This limit is represented by the second condition (see Equation (3)) and is further defined by Equations (3a)-(3m).