# Adaptive Salp Swarm Algorithm for Optimization of Geotechnical Structures
## 5 Foundation Optimization
### Design Constraints  
While optimizing and designing a reinforced concrete footing, both structural and geotechnical limit states should be considered. Two different geotechnical limit states are the bearing capacity of the surrounding geo-material and the permitted settlement of the footing. The shear capacity of the footing (one- and two-way shear), flexural capacity, and reinforcement limitation are all structural limit states. The structural limit states are investigated using ACI 318-11 specifications [59]. Service loads are commonly used to satisfy geotechnical limit states. Even so, factored loads can be used for structural limit states. Table 8 provides a list of both structural and geotechnical limit states.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Symbol** & **Unit** & **Unit Cost (USD)** \\ \hline Excavation & \(C_{e}\) & m\({}^{3}\) & 25.16 \\ Formwork & \(C_{f}\) & m\({}^{2}\) & 51.97 \\ Reinforcement & \(C_{s}\) & kg & 2.16 \\ Concrete & \(C_{c}\) & m\({}^{3}\) & 173.96 \\ Backfill & \(C_{b}\) & m\({}^{3}\) & 3.97 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Spread footing assembly unit cost [60].  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Failure Mode** & **Constraint** \\ \hline \hline Bearing capacity & \(q_{max}\leq\frac{4\omega}{1S}\) \\ \hline Settlement of foundation & \(\delta\leq\delta_{all}\) \\ \(k_{z}=-0.0017\Big{(}\frac{Y_{2}}{Y_{1}}\Big{)}^{2}+0.0597\Big{(}\frac{Y_{2}}{Y _{1}}\Big{)}+0.9843\) \\ \hline Eccentricity & \(e\leq Y_{1}/6\) \\ \hline One-way (wide beam) shear & \(V_{u}\leq\frac{1}{6}\varOmega_{v}\sqrt{f_{t}}bd\) \\ \hline Two-way (punching) shear & \(\underset{min}{min}\Big{\{}\Big{(}1+\frac{2}{\varPhi_{v}}\Big{)}/6,\Big{(} \frac{\text{bd}}{\text{bd}}+2\Big{)}/12,\frac{1}{3}\Big{\}}\varOmega_{v}\sqrt {f_{t}}bd\) \\ \hline Bending moment & \(M_{u}\leq\varOmega_{M}A_{s}f_{y}\big{(}d-\frac{\bar{q}}{2}\big{)}\) \\ \hline Minimum and maximum reinforcements & \(\rho_{min}bd\leq A_{s}\leq\rho_{max}bd\) \\ \hline Limitation of depth of embedment & \(0.5\leq Y_{4}\leq 2\) \\ \hline \hline \end{tabular}
\end{table}
Table 8: Design constraints of spread footing.  
Figure 4: Design variables of the footing.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Definition** \\ \hline \hline \(q_{all}\) & ultimate bearing capacity of the foundation soil \\ \(q_{max}\) & maximum contact pressure at the interface between the bottom of a foundation and the underlying soil \\ \(\delta_{all}\) & allowable settlement of foundation \\ \(\delta\) & immediate settlement of foundation \\ \(\delta_{V}\) & shear strength reduction factor equal to 0.75 \\ \(f_{c}\) & compression strength of concrete \\ \(b_{0}\) & perimeter of critical section taken at \(d/2\) from face of column \\ \(b\) & width of the section \\ \(\bar{p}_{c}\) & ratio of long side to short side of column section \\ \(a_{c}\) & is equal to 40 for interior columns \\ \(M_{w}\) & bending moment \\ \(\phi_{M}\) & flexure strength reduction factor equal to 0.9 \\ \(A_{s}\) & cross-sectional area of steel reinforcement \\ \(f_{y}\) & yield strength of steel \\ \(\rho_{min}\) & minimum reinforcement ratio \\ \(\rho_{max}\) & maximum reinforcement ratio \\ \hline \hline \end{tabular}
\end{table}
Table 9: Definition of parameters of Table 7.