# Elastoplastic Analysis of Frame Structures Using Radial Point Interpolation Meshless Methods
## 3 Elastoplastic Model
### Non-Linear Solution Algorithm  
The use of incremental methods [31] has the intent of achieving an approximate solution to a certain problem with the application of successive load increments, as it is shown in Figure 5a.  
The displacement, \(u\), for each load stage \(\delta f\) is given by  
\[\delta u=\left(\frac{df}{du}\right)^{-1}\delta f=K_{T}^{-1}\delta f \tag{28}\]  
where \(K_{T}\) is the tangent stiffness.  
The biggest issue with the incremental method is the accumulation of errors as the load increments are applied. As it can be seen in Figure 5a, considering point 2 of the diagram, the solution given by the method drifts from the equilibrium curve of the solution. As the load increases, these errors will accumulate. As the nonlinearity of the solution increases, \(\frac{df}{du}\to 0\), the number of cumulative errors will increase as well. As a way to overcome this fact, an iterative method is required. One of the most commonly used is the Newton-Raphson incremental scheme. In short, the Newton-Raphson method only provides the displacement, \(u\), of a pre-defined load, \(f\), as it is shown in Figure 5b.  
The iterative changes, \(j\), are defined as  
\[\partial u^{j}=\left(\frac{dq^{j}}{du}\right)^{-1}q^{j}(u^{j}) \tag{29}\]  
Figure 5: Incremental methods. (**a**) Incremental (Euler) scheme. (**b**) Newton–Raphson method.  
and the displacements are updated using  
\[u^{j}=u^{j-1}+\partial u^{j} \tag{30}\]  
The incremental and iterative methods can be combined (Figure 6), allowing one to obtain the problem solution and a reduction in the lack of equilibrium presented in the incremental process.  
Analyzing Figure 6, the incremental methods act as a predictor agent. The process starts with an initial solution, \(u_{i}^{0}\)--it can be the elastic solution for an arbitrary set of loads--and \(u_{i}^{j}\) represents the displacement of iteration \(j\) for a load increment \(i\). The process described in Figure 6 is the full Newton-Raphson method, used to compute the non-linear solutions of the next sections of this paper. In this method, the stiffness matrix of the problem is calculated at the beginning of each increment, and the calculation is repeated for all iterations of said increment. Thus, the stiffness matrix is updated regularly considering the newly updated elastoplastic constitutive matrix. In the same algorithm, a standard backward-Euler scheme is implemented [5] in order to ensure that the stresses that are developed remain inside the yield surface or, at least, very close to it. Hence, the backward-Euler scheme is applied when the stresses of the Gauss points are located outside the yield surface [31]. Thus, the following condition is ensured:  
\[f(\sigma)\leq\sigma_{Y}^{*} \tag{31}\]  
where \(f(\sigma)\) is the yield surface, and \(\sigma_{Y}^{*}\) is the updated yield stress, given by  
\[\sigma_{Y}^{*}=\sigma+[E_{T}\cdot\overline{\varepsilon_{i-1}^{p}}] \tag{32}\]  
\(\overline{\varepsilon_{i-1}^{p}}\) is the accumulated effective plastic strain from the previous increment, \((i-1)\). A detailed mathematical explanation of this non-linear algorithm can be found in [32].