## 3 Pseudo-Static Analysis of Ultimate Bearing Capacity
### Seismic Ultimate Bearing Capacity  
Based upon the equilibrium between Equations (22) and (23), an analytical upper bound solution of the ultimate bearing capacity of strip footings in the presence of earthquake conditions is eventually derived in the following form:  
\[P_{\mathrm{e}}=\bigl{(}D_{\mathrm{int}}-W_{\mathrm{e}}-W_{\gamma}-W_{p}\bigr{)} /[v_{1}k_{\mathrm{h}}\cos(\kappa_{1}-\varphi_{t}^{*})+v_{1}(1-k_{\mathrm{v}} )\sin(\kappa_{1}-\varphi_{t}^{*})] \tag{24}\]  
For comprehensively understanding the contributions of different parameters, the normalized ultimate bearing capacity is furtherly decomposed into the following three components:  
\[p_{\mathrm{ce}}=\frac{P_{\mathrm{e}}}{B}=\frac{1}{2}\gamma B\cdot N_{\gamma \mathrm{e}}+p\cdot N_{\mathrm{pe}}+c_{t}^{*}\cdot N_{c_{t}^{*}\mathrm{e}} \tag{25}\]where \(N_{\gamma e}\), \(N_{pe}\), and \(N_{c_{t}^{*}e}\) are dimensionless functions and are known as the bearing capacity factors which separately depict the significance of the soil gravity, equivalent surcharge, and cohesion on the seismic bearing capacity. The above dimensionless functions are presented in terms of the as of yet unspecified angles \(2n+1\)\(\delta_{i}\), \(\kappa_{i}\), and \(\varphi_{t}^{*}\)  
\[N_{\gamma e}=-\frac{k_{\mathrm{h}}g_{1}+(1-k_{v})g_{2}}{\cos(\kappa_{1}-\varphi _{t}^{*})k_{\mathrm{h}}+(1-k_{v})\sin(\kappa_{1}-\varphi_{t}^{*})} \tag{26}\]  
\[N_{pe}=-\frac{k_{\mathrm{h}}g_{3}+(1-k_{v})g_{4}}{\cos(\kappa_{1}-\varphi_{t}^{ *})k_{\mathrm{h}}+(1-k_{v})\sin(\kappa_{1}-\varphi_{t}^{*})} \tag{27}\]  
\[N_{c_{t}^{*}e}=\frac{g_{5}}{\cos(\kappa_{1}-\varphi_{t}^{*})k_{\mathrm{h}}+(1- k_{v})\sin(\kappa_{1}-\varphi_{t}^{*})} \tag{28}\]  
The above dimensionless expressions \(g_{i}\)\((i=1,2,\ldots,5)\) have a specific form that is referred to in Appendix A. Note that from Equation (25), clearly the ultimate bearing capacity is determined by the \(2n\) independent angular variable \(\delta_{i}\), \(\kappa_{i}\) and one equivalent internal frictional angle \(\varphi_{t}^{*}\). Any set of these parameters will provide a certain upper-bound solution; the least one of them needs to be found by the SQP algorithm. Since the advent of MATLAB numerical software in 1984, it has gradually developed into one of the most popular scientific and technological application software due to its simple syntax rules, powerful drawing and operation capabilities, and excellent openness. Therefore, the optimization solution in this work will be realized by using the fmincon function in the built-in toolbox of MATLAB. One can call the SQP algorithm in the fmincon function. The following is a brief introduction to the use of the fmincon function. The mathematical model of the fmincon function is:  
\[\mathrm{s.t. }\left\{ \begin{array}{l}\min f(\mathbf{x})\\ C(\mathbf{x})\leq 0\\ Ceq(\mathbf{x})=0\\ A\cdot\mathbf{x}\leq\mathbf{b}\\ Aeq\cdot\mathbf{x}=\mathbf{b}\mathbf{q}\\ Ib\leq\mathbf{x}\leq\mathbf{u}\end{array} \right. \tag{29}\]  
The calling format of the fmincon function in Matlab is:  
\[\mathbf{x}=\mathrm{fmincon}(\textit{fmin},\mathbf{x_{0}},\mathbf{A},\mathbf{b},\textit{Aeq}, \textit{beq},\textit{lb},\textit{ub},\textit{nonlcon},\textit{options}) \tag{30}\]  
where _fun_ is the objective function which is defined by the actual problem; \(\mathbf{x_{0}}\) is the initial value of the variables; and \(\mathbf{A}\), \(\mathbf{b}\)_Aeq_, _beq_, _lb_, and _ub_ are the matrices that need to meet the constraint in Equation (29). Therefore, the optimization problem in this work is formulated as:  
\[ \begin{array}{l}\min\ p_{\mathrm{ce}}=f(\delta_{i},\kappa_{i},\varphi_{t}^{* })\\ \mathrm{s.t.}\left\{\begin{array}{l}\sum\limits_{i=1}^{n}\delta_{i}=\pi,\ i=1, 2,\ldots n\\ \delta_{i}+\kappa_{i}\geq\kappa_{i+1}\end{array}\right.\end{array}  \tag{31}\]  
In most previous analyses, the bearing capacity of the foundations is obtained by separately calculating Equations (26)-(28) and accumulating them.