## 2 Fundamentals of Limit Analysis  
A structure collapses if the dissipation due to permanent deformation increases in time indefinitely. The dissipation increment can be produced by proportionally increasing loads, hence, proper proportional collapse occurs. With no difference, the dissipation can increase in time alternating periods when no dissipation occurs and periods when dissipation gets further increase. In this latter case, it is possible that either the displacement becomes excessive for functionality and safety, or the total dissipation reaches a limit that produces the material failure. The first is the ratcheting phenomenon, the second the plastic fatigue rupture (also known as low cycle number fatigue) [16]. To evaluate the load collapse multiplier, the static collapse theorem states that if under actual loads, there exists any equilibrated stress that definitely belongs to the elastic domain, the permanent strain rate vanishes whenever time increasing, hence collapse does not occur [1]. In other words, the structure does stabilize, i.e., the structure behaves as to be elastic after that a certain amount of permanent strain has been accumulated. Moreover, a limiting equation exists that gives an upper bound of the dissipated energy until stabilization occurs [14; 17]. The present work has described a numerical procedure that implemented the Melan theorem. For the sake of completeness, the Melan theorem has been reported hereafter briefly. Let us consider a structure that occupies a domain \(\Omega\) with the boundary \(\partial\Omega\) (Figure 1). The structure is in equilibrium under prescribed body forces, \(b_{i}\), in the interior of the solid, \(\overset{o}{\Omega}\), and prescribed tractions, \(t_{i}\), on the part of the boundary \(\partial\Omega_{f}\); both forces vary in time within the prescribed limit with absolutely random time law. On the part of the boundary, \(\partial\Omega_{u}=\partial\Omega\backslash\partial\Omega_{f}\), where the symbol \(\backslash\) denotes the complement of the manifold \(\partial\Omega\) with respect to \(\partial\Omega_{f}\), the displacement is prescribed [18]. Due to the applied loads and displacements, the solid presents the stress, \(\alpha_{ij}\) and the strain, \(\varepsilon_{hk}\), in the interior and reactive tractions, \(t_{i}^{s}\), on the boundary \(\partial\Omega_{u}\). In the following, indicial notation is adopted hence a subscript (\(0_{i}\) have denoted the \(i^{\text{th}}\) component of a vector in the reference frame, two subscripts have referred to the two components of a second-order tensor and so on; moreover, subscripts following the comma indicate partial derivatives with respect to the coordinate directions. Self-equilibrated stress, \(\zeta_{ij}\), is a stress field that satisfies the homogeneous balance equations and the corresponding homogeneous natural boundary conditions on the loaded boundary \(\partial\Omega_{f}\). Only on the constrained boundary \(\partial\Omega_{u}\) tractions can be non-null:  
\[ \begin{array}{l}\zeta_{ij,j}=0 in\Omega\\ \zeta_{ij}n_{j}=0 on\partial\Omega_{f}\\ \zeta_{ij}n_{j}=t_{i}^{s}on\partial\Omega_{u}\end{array}  \tag{1}\]  
where in Equation (1), \(n_{j}\) are the components of the outward normal unit vector to \(\partial\Omega\). The material constituting the structure is elastic-perfectly plastic, so the stress-strain law is  
\[\sigma_{ij}=S_{ijhk}(\varepsilon_{hk}-p_{hk}) \tag{2}\]  
where \(S_{ijhk}\) is the fourth-order elastic tensor. The permanent strain, \(p_{hk}\), results from the integration during the load process of the permanent strain rate, \(\dot{p}_{hk}\), that is related to the stress through the constitutively associated flow rule:  
\[\left\{ \begin{array}{l}f(\sigma_{hk})=0\\ \frac{\partial f}{\partial\sigma_{hk}}\dot{\sigma}_{hk}=0\end{array} \right.