## 2 Finite Element Formulations and Developed Procedure
### Fiber Beam Element Model with Nonlinear Interfacial Connection  
The fiber beam element model with a nonlinear interfacial connection is built with the idea of inducing the additional DOF into the conventional fiber beam element to describe the interfacial slip deformation. In contrast to the methods in [33; 34; 35; 36], only two additional DOFs describing the relative slip deformation are introduced at both beam ends, and a two-node 8-DOF fiber beam element is proposed, as shown in Figure 2. For each node, four DOFs are defined to describe the kinematics of the composite beam, including the axial displacement \(u\), vertical displacement \(v\), rotation \(\theta\) and interfacial relative slip \(u_{\text{cs}}\) (introduced DOF compared with the conventional fiber beam element). The cross-section is defined by discrete fibers, including concrete fibers, steel fibers and reinforcement fibers. Figure 1: Externally prestressed composite beam and the proposed model components. The numbers of element nodes are named \(i\) and \(j\). The element local coordinate axial \(x\) is defined along the element line and the \(yOz\) plane is defined in the cross-section, as shown in Figure 2. The displacement vector \(\mathbf{u}\) and resistance vector \(\mathbf{F}\) of the element are defined in Equations (1) and (2), respectively. The axial force \(N\), shear force \(V\), moment \(M\) and interfacial shear force \(F_{cs}\) are defined as the resistance force concerning \(u\), \(v\), \(\theta\) and \(u_{cs}\). The subscript \(i\) and \(j\) denote the corresponding nodal number. \[\mathbf{u}=\left[ \begin{array}{cccccccc}u_{i}&v_{i}&\theta_{i}&u_{csi}&u_{j} &v_{j}&\theta_{j}&u_{csj}\end{array} \right]^{T} \tag{1}\]  
\[\mathbf{F}=\left[ \begin{array}{cccccccc}N_{i}&V_{i}&M_{i}&F_{csi}&N_{j}&V_{j} &M_{j}&F_{csj}\end{array} \right]^{T} \tag{2}\]  
Eliminating the rigid body displacements, the main deformation modes of the element can be labeled axial deformation, rotational deformation and slip deformation, as shown in Figure 3. The basic deformation vector \(\mathbf{u}^{b}\) can be expressed as Equation (3). The transformation relation between \(\mathbf{u}^{b}\) and \(\mathbf{u}\) is shown in Equation (4). \(c_{x}=(L+\Delta u_{x})/L_{n}\), \(c_{x}=\Delta u_{y}/L_{n}\). \(L\) is the initial element length and \(L_{n}\) is the deformed element length. \(\Delta u_{x}\) and \(\Delta u_{y}\) denote relative displacement along the local \(x\) and \(y\) axes, respectively. \[\mathbf{u}^{b}=\left[ \begin{array}{cccccccc}\Delta u&\theta^{\prime}&\theta^ {\prime}&u_{csi}&u_{csj}\end{array} \right]^{T} \tag{3}\]  
\[\mathbf{u}^{b}=\mathbf{P}\mathbf{u} \tag{4}\] \[\mathbf{P}=\left[ \begin{array}{cccccccc}-c_{x}&-c_{y}&0&0&c_{x}& c_{y}&0&0\\ -c_{y}/L_{n}&c_{x}/L_{n}&1&0&c_{y}/L_{n}&-c_{x}/L_{n}&0&0\\ -c_{y}/L_{n}&c_{x}/L_{n}&0&0&c_{y}/L_{n}&-c_{x}/L_{n}&1&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&0&0&0&1\end{array} \right]\]  
With the basic deformation vector \(\mathbf{u}^{b}\), the deformation field inside the element can be denoted by the Hermite polynomial interpolation method, as shown in Equation (5), in which \(\xi=x/L\), \(x\) is the local coordinate in the element, and \(0\leq\xi\leq 1\). The axial and slip deformations are interpolated by linear equations and the rotation deformation is interpolated by a quadratic polynomial. Figure 2: Fiber beam element and fiber section definition. The generalized strain tensor \(\mathbf{E}(x)\) of the section at location \(x\) can be expressed as Equation (6) by derivation. \(\varepsilon_{s}(x)\) and \(\varepsilon_{c}(x)\) denote the sectional axial strain of the steel and the concrete beam, respectively. \(\varphi(x)\) denotes the sectional curvature and \(u_{cs}(x)\) denotes the interfacial slip displacement at location \(x\).