## 2 Radial Point Interpolation Method
In this equation, \(\mathbf{K}\) represents the global stiffness matrix, which can be expressed as follows:  
\[\mathbf{K}=\int_{\Omega}\mathbf{B}^{\mathbf{T}}\cdot\mathbf{c}\cdot\mathbf{B} \,d\Omega=\sum_{i=1}^{n_{Q}}\mathbf{B}(\mathbf{x}_{t})^{\mathbf{T}}\cdot \mathbf{c}\cdot\mathbf{B}(\mathbf{x}_{t})\widehat{w}_{t}, \tag{16}\]  
being \(n_{Q}\), the number of integration points discretising the domain and \(\widehat{w_{l}}\) their corresponding integration weight. The deformability matrix \(\mathbf{B}\) can be defined as follows:  
\[\mathbf{B}= \begin{bmatrix}\frac{\partial\varphi_{1}(\mathbf{x}_{t})}{\partial x }&0&0&\cdots&\frac{\partial\varphi_{n}(\mathbf{x}_{t})}{\partial x}&0&0\\ 0&\frac{\partial\varphi_{1}(\mathbf{x}_{t})}{\partial y}&0&\cdots&0&\frac{ \partial\varphi_{n}(\mathbf{x}_{t})}{\partial y}&0\\ 0&0&\frac{\partial\varphi_{1}(\mathbf{x}_{t})}{\partial x}&\cdots&0&0&\frac {\partial\varphi_{n}(\mathbf{x}_{t})}{\partial x}\\ \frac{\partial\varphi_{1}(\mathbf{x}_{t})}{\partial y}&\frac{\partial\varphi _{1}(\mathbf{x}_{t})}{\partial x}&0&\cdots&\frac{\partial\varphi_{n}(\mathbf{ x}_{t})}{\partial y}&\frac{\partial\varphi_{n}(\mathbf{x}_{t})}{\partial x}&0\\ 0&\frac{\partial\varphi_{1}(\mathbf{x}_{t})}{\partial x}&\frac{\partial \varphi_{1}(\mathbf{x}_{t})}{\partial y}&\cdots&0&\frac{\partial\varphi_{n}( \mathbf{x}_{t})}{\partial x}&\frac{\partial\varphi_{n}(\mathbf{x}_{t})}{ \partial y}\\ \frac{\partial\varphi_{1}(\mathbf{x}_{t})}{\partial z}&0&\frac{\partial \varphi_{1}(\mathbf{x}_{t})}{\partial x}&\cdots&\frac{\partial\varphi_{n}( \mathbf{x}_{t})}{\partial z}&0&\frac{\partial\varphi_{n}(\mathbf{x}_{t})}{ \partial x}\end{bmatrix} . \tag{17}\]  
From Equation (14), it is possible to express the body force vector (denoted by \(\mathbf{f_{b}}\)) and the external force vector (denoted by \(\mathbf{f_{f}}\)) as follows:  
\[\mathbf{f_{b}}=\int_{\Omega}\mathbf{H}^{\mathbf{T}}\cdot\mathbf{b}\;d\Omega= \sum_{i=1}^{n_{Q}}\mathbf{H}(\mathbf{x}_{t})^{\mathbf{T}}\cdot\mathbf{b}\cdot \widehat{w_{l}}, \tag{18}\]  
\[\mathbf{f_{t}}=\int_{\mathbb{T}}\mathbf{H}^{\mathbf{T}}\cdot\mathbf{\tilde{t} }\,d\Gamma=\sum_{i=1}^{n_{Q}^{*}}\mathbf{H}(\mathbf{x}_{t})^{\mathbf{T}}\cdot \mathbf{f}\cdot\widehat{w_{l}^{*}}, \tag{19}\]  
where \(n_{Q}^{*}\) denotes the number of integration points defining the boundary where this force is applied to, and \(\widehat{w_{l}^{*}}\) their corresponding integration weight. Finally, \(\mathbf{H}\) represents the interpolation matrix:  
\[\mathbf{H}= \begin{bmatrix}\varphi_{1}(\mathbf{x}_{l})&0&0&\cdots&\varphi_{n }(\mathbf{x}_{l})&0&0\\ 0&\varphi_{1}(\mathbf{x}_{l})&0&\cdots&0&\varphi_{n}(\mathbf{x}_{l})&0\\ 0&0&\varphi_{1}(\mathbf{x}_{l})&\cdots&0&0&\varphi_{n}(\mathbf{x}_{l})\end{bmatrix} . \tag{20}\]  
The Kronecker delta property inherent to RPI shape functions allows for the direct imposition of essential boundary conditions on the stiffness matrix, \(\mathbf{K}\). Thus, the same numerical techniques applied to standard FEM can be used with RPIM. If plane stress is assumed, the problem reduces to a 2D analysis. The presented 3D formulation is still valid; however, all components associated with the \(Oz\) direction are removed. Thus, regarding the stress and strain vectors,  
\[\mathbf{\sigma}= \begin{Bmatrix}\sigma_{xx}&\sigma_{yy}&\tau_{xy}\end{Bmatrix} ^{T}, \tag{21}\]  
\[\mathbf{\epsilon}= \begin{Bmatrix}\epsilon_{xx}&\epsilon_{yy}&\gamma_{xy}\end{Bmatrix} ^{T}. \tag{22}\]  
The constitutive matrix is reduced to  
\[\mathbf{c}=\mu_{1} \begin{bmatrix}1&v&0\\ v&1&0\\ 0&0&\mu_{2}\end{bmatrix}  \tag{23}\]and the deformability and interpolation matrices become  
\[\mathbf{B}=