## 3 Lower Bound Limit Analysis  
The shakedown is the phenomenon where under randomly repeated loads, the structure undergoes elastic behavior _definitely_ for increasing time. In other words, the permanent strain rate \(\dot{p}_{lk}\) approaches zero for t\(\rightarrow\)\(\infty\), hence the permanent strain is bounded in time. In the present formulation, the time is solely the measure of the evolution of the phenomenon, because of the rate of the quantities involved must be considered as their increment, no dynamics effects are considered. The shakedown is ruled by [PERSON]'s theorem. The theorem is formulated in term of self-equilibrated stress of the form \(\zeta_{ij}\). The line of reasoning of [PERSON]'s theorem is suitable as well for the application to the monotone load-induced collapse analysis. In the following formulation the \(\zeta_{ij}\) is obtained from the \(p_{hk}\) as the resulting stress from the application of the \(p_{hk}\) as [PERSON]'s dislocation. It is possible to solve the elastic problem of the structure subjected by [PERSON]'s dislocation, \(p_{hk}\), at a point \(x\) and calculate, at any point \(y\), the resulting stress \(\zeta_{ij}\). The linearity of the elastic problem allows writing the \(\zeta_{ij}\) in terms of the linear operator \(Z_{ijhk}\). In numerical application a discrete approximation of the operator \(Z_{ijhk}\) can be obtained using the boundary integral equation method [20]  
Finally, the relationship between \(p_{hk}\) and \(\zeta_{ij}\) can be written as follows:  
\[\zeta_{ij}=Z_{ijhk}\,p_{hk} \tag{11}\]  
The condition of shakedown is stated by [PERSON]'s theorem in the following form: stabilization occurs under a load multiplier \(k\) if and only if there exists a self-equilibrated stress \(\zeta_{ij}\), time-independent eigenstress, such that:  
\[f\big{(}kC_{ijhk}\varepsilon_{hk}^{E}+\zeta_{ij}\big{)}\leq 0 \tag{12}\]  
at any time where \(C_{ijhk}\varepsilon_{hk}^{E}\) is the actual stress under the applied loads when the structure was supposed indefinitely elastic. By considering Equation (11), [PERSON]'s theorem can be rewritten in terms of permanent strain \(p_{hk}^{r}\) such that  
\[f\big{(}kC_{ijhk}\varepsilon_{hk}^{E}+Z_{ijhk}p_{hk}^{r}\big{)}<0 \tag{13}\]  
The optimization program that searches for the load limit multiplier, \(\mathrm{s_{c}}\), is summarized in  
\[s_{\alpha}=\underset{p_{hk}^{r}}{\mathrm{sup}}\Bigg{|}f\big{(}kC_{ijhk} \varepsilon_{hk}^{E}+Z_{ijhk}p_{hk}^{r}\big{)}<0 \tag{14}\]  
Program (14) is a constrained optimization program that can be solved using a standard linear programming routine, namely, the simplex method, if the constraints are a set of linear inequalities. This occurs when von Mises or Drucker-Prager limit-domains describe the material behavior. The solution of the program (14) is the limit collapse multiplier that is either the proportional collapse multiplier, \(\alpha=c\), or the shakedown limit, \(\alpha=s\), provided the elastic stress corresponds to monotone or variable loads. Moreover, the program furnishes a set of permanent time-independent strain \(\vec{p}_{hk}^{r}\) satisfying the theorem. As a corollary of [PERSON]'s theorem, see [PERSON] and [PERSON] [1, 16, 21], an upper bound on total dissipated energy until stabilization is obtained as a function of the elastic strain energy corresponding to [PERSON]'s residual strain, \(\vec{p}_{hk}^{r}\). \[\int_{\Omega}\sigma_{ij}p_{ij}dV\leq\frac{m}{2(m-1)}\int_{\Omega}S_{hkij}\, \vec{p}_{hk}^{r}\vec{p}_{hk}dV \tag{15}\]  
where the term on the left-hand side of Equation (15) is a lower bound of the total dissipation up to the load level, having the safety factor \(m>1\) with respect to the limit at the incoming collapse. \[\mathcal{E}^{\mathrm{D}}\int_{\Omega}\int_{0}^{\tau}\sigma_{ij}\dot{p}_{ij} dtdV\geq\int_{\Omega}\sigma_{ij}p_{ij}dV \tag{16}\]  
The inequality at the leftmost side of Equation (16) holds if the material of the structures follows von Mises or Drucker-Prager associate constitutive law.