# Static Analysis

The static analysis is concerned with the solution of the linear system of equations for the displacements, $\mathbf{u}$, given the applied loads. The applied loads give the load or force vector, $\mathbf{f}$. The elements contribute stiffness, $\mathbf{K}$, so the system of equations is

$\mathbf{Ku} = \mathbf{f}$

When nonlinear elements such as ties and struts are introduced the analysis is no-longer linear and become iterative. GSA uses initial stiffness method to avoid creation of global stiffness matrix at each iteration. The system of equation to be solved at each iteration

$\Delta \mathbf{u}^i = \mathbf{K}^{-1}\mathbf r^{i-1}$
$\mathbf{u}^{i} = \mathbf{u}^{i - 1} + \Delta \mathbf{u}^{i}$

Where $\mathbf{r}^{i - 1}$ is residual force from the previous iteration

$\mathbf{r}^{i - 1} = \mathbf{f} - \mathbf{f}_{int}^{i - 1}$

and $\Delta \mathbf{u}^{i}$ is the change in displacement for current iteration. This method sometimes requires a large number of iterations to converge to the solution and this may offset the advantage of constant stiffness. To improve the convergence speed acceleration scheme is applied to the solution strategy

$\mathbf{u}^i = \mathbf{u}^{i-1}+\alpha \Delta \mathbf{u}^1$
$\alpha = \frac{\Delta u^{i - 1}}{\left( \Delta u^{i - 1} - \Delta u^{i} \right)}$

The acceleration scheme estimates the ratio of the original tangent stiffness to local secant stiffness using the successive change in the displacements. To avoid inaccurate estimate the ratio, the acceleration scheme is only applied at every alternative iteration.