# Damage Analysis of Reinforced Concrete Columns Based on Fiber Beam-Column Element
## 2 Analysis Model of Reinforced Concrete Columns
### Fiber Beam-Column Element Model  
The RSAPS platform includes a three-dimensional two-node fiber beam-column element model based on the stiffness method. The model is established based on the classical Euler-Bernoulli beam theory and the plane section assumption. The element displacement field is described using the following formula:  
\[\mathbf{u}(x)=\left\{u(x),v(x),w(x)\right\}^{\mathrm{T}}=\mathbf{N}(x)\cdot \mathbf{U} \tag{1}\]  
where, \(u(x)\) is the axial displacement of the element calculated using Lagrange linear interpolation; \(v(x)\) and \(w(x)\) are the transverse displacements of the two orthogonal directions of the element, respectively, calculated using cubic Hermite polynomial interpolation; \(\mathbf{N}(x)\) is the matrix of the interpolation function; and \(\mathbf{U}\) is the displacement vector of the node.  
The relationship between the section deformation vector and the node displacement vector is as follows:  
\[\mathbf{s}(x)=\left\{\varepsilon(x),\varphi_{z}(x),\varphi_{y}(x)\right\}^{T} =\partial\mathbf{u}(x)=\mathbf{B}(x)\cdot\mathbf{U} \tag{2}\]  
where, \(\varepsilon(x)\) is the axial strain of the \(x\) section; \(\varphi_{y}(x)\) and \(\varphi_{z}(x)\) are the curvatures of the x section about the \(y\)-axis and the \(z\)-axis, respectively; and \(\mathbf{B}(x)=\partial\mathbf{N}(x)\) is the geometric transformation matrix.  
After the section deformation is determined, the strain of the \(i\)th fiber in the section, \(\varepsilon_{i}=\varepsilon(x)-y_{i}\varphi_{z}(x)+z_{i}\varphi_{y}(x)\), can be determined according to the plane section assumption, where, \(y_{i}\) and \(z_{i}\) represent the geometric coordinates of the \(i\)th fiber in the section, respectively. After that, the tangent stiffness \(E_{i}\) and stress \(\sigma_{i}\) of the \(i\)th fiber can be obtained according to the constitutive relationship of the fiber material. According to the contribution of all fiber stiffness in the section to the stiffness matrix of the section \(\mathbf{k}(x)\) and the contribution of fiber stress to the resistance vector of the section \(\mathbf{D}(x)\), the following equations can be obtained:  
\[\mathbf{k}(x)=\sum_{i}^{n}\begin{bmatrix}E_{i}A_{i}&E_{i}A_{i}z_{i}&-E_{i}A_{i}y _{i}\\ E_{i}A_{i}z_{i}&E_{i}A_{i}z_{i}^{2}&-E_{i}A_{i}y_{i}z_{i}\\ -E_{i}A_{i}y_{i}&-E_{i}A_{i}y_{i}z_{i}&E_{i}A_{i}y_{i}^{2}\end{bmatrix} \tag{3}\]  
\[\mathbf{D}(x)=\left\{\begin{matrix}\sum_{i}^{n}\sigma_{i}A_{i}&\sum_{i}^{n} \sigma_{i}A_{i}z_{i}&-\sum_{i}^{n}\sigma_{i}A_{i}y_{i}\end{matrix}\right\}^{T} \tag{4}\]  
where, \(A_{i}\) is the cross-sectional area of the \(i\)th fiber.  
According to the principle of virtual displacement, the resistance vector of the element \(\mathbf{F}\) is derived as follows:  
\[\mathbf{F}=\int_{0}^{L}\mathbf{B}(x)^{T}\cdot\mathbf{D}(x)dx \tag{5}\]  
where, \(L\) is the element length.  
According to Equation (5) and the section constitutive relation \(\Delta\mathbf{D}(x)=\mathbf{k}(x)\cdot\Delta\mathbf{s}(x)\), the stiffness matrix of the element \(\mathbf{K}\) can be obtained as:  
\[\mathbf{K}=\int_{0}^{L}\mathbf{B}(x)^{T}\cdot\mathbf{k}(x)\cdot\mathbf{B}(x)dx \tag{6}\]  
Equations (5) and (6) can be solved using the numerical integration method. The Gauss-Lobatto numerical integration method has integral points at the end of the element, which can consider the force of the end section of the element. Therefore, the Gauss-Lobatto numerical integration method was used in this paper. Four integral points were selected in the calculation, which meet the accuracy requirements.