# Analysis of Equivalent Flexural Stiffness of Steel-Concrete Composite Beams in Frame Structures
## 2 Theoretical Analysis
The distribution function of the vertical deflection can be further derived by integrating Equation (5) as follows:  
\[EI_{i}w_{i}(x)=\int\theta_{i}(x)dx=-\frac{1}{6}F_{1}x^{3}+\frac{1}{2}k_{1} \theta_{1}x^{2}+\frac{1}{24}qx^{4}+C_{1i}x+C_{2i}, \tag{6}\]  
where \(C_{2i}\) is the integration constant, which is determined by the boundary conditions of the composite beam. The boundary conditions of the composite beam in the frame structural system are as follows:  
\[\left\{ \begin{array}{c}M(x_{1})=M(x_{2})=M_{0}\\ \theta_{1}(0)=\theta_{1}\\ \theta_{3}(l)=\theta_{2}\\ w_{1}(0)=0\\ w_{3}(l)=0\\ \theta_{1}(x_{1})=\theta_{2}(x_{1})\\ \theta_{2}(x_{2})=\theta_{3}(x_{2})\\ w_{1}(x_{1})=w_{2}(x_{1})\\ w_{2}(x_{2})=w_{3}(x_{2})\end{array} \right., \tag{7}\]  
where \(x_{1}\) and \(x_{2}\) represent the coordinates positions where the bending moments reach the value of cracking moment \(M_{0}\) on the left and right sides of the composite beam, respectively. Using the first boundary condition function expressed in Equation (7), the rotation angles at the beam ends--\(\theta_{1}\) and \(\theta_{2}\)--can be expressed as follows:  
\[\left\{ \begin{array}{c}\theta_{1}=\frac{-qx_{1}x_{2}+2M_{0}}{2k_{1}}\\ \theta_{2}=\frac{-l^{2}q+lqx_{1}+lqx_{2}-qx_{1}x_{2}+2M_{0}}{2k_{2}}\end{array}  \right.. \tag{8}\]  
By substituting Equation (7) into Equations (5) and (6), all the integration constants (\(C_{1i}\) and \(C_{2i}\), \(i=1\), \(2\), \(3\)) can be eliminated, and thus, a non-linear equation system of \(x_{1}\) and \(x_{2}\) is obtained. \[\left\{ \begin{array}{c}k_{2}\big{[}6 EI^{-}I^{+}qx_{1}x_{2}+I^{-}k_{1}q(x_{1} ^{3}-3x_{1}^{2}x_{2})+I^{+}k_{1}q\big{(}-x_{1}^{3}+3x_{1}^{2}x_{2}\big{)}+12M_{ 0}\big{(}-EI^{-}I^{+}+I^{-}k_{1}x_{1}-I^{+}k_{1}x_{1}\big{)}\big{]}\\ =k_{1}\bigg{[}6 EI^{-}I^{+}q(l-x_{1})(x_{2}-l)+I^{-}k_{2}q(x_{2}^{3}-3x_{1}x_{2} ^{2})+I^{+}k_{2}q\big{(}-x_{2}^{3}+3x_{1}x_{2}^{2}-6 lx_{1}x_{2}+3l^{2}(x_{1}+ x_{2})-2l^{3}\big{)}+\\ 12M_{0}\big{(}EI^{-}I^{+}+I^{-}k_{2}x_{2}+I^{-}k_{2}l-l^{+}k_{2}x_{2}\big{)} \end{array} \right] \tag{9}\]  
The non-linear system of equations can be solved using the Levenberg-Marquardt method; thus, all the parameters in Equation (9) can be determined by the given boundary conditions in practical cases. When the values of \(x_{1}\) and \(x_{2}\) are determined, the vertical deflection distribution as well as the equivalent flexural stiffness can be calculated by the following steps:\(x_{1}\) and \(x_{2}\) are substituted into Equation (7), and the rotation angles at the beam ends (\(\theta_{1}\) and \(\theta_{2}\)) are solved;  
\(\theta_{1}\) and \(\theta_{2}\) are substituted into Equations (2) and (3) to solve the bending moment distribution function, \(M\)(x);  
\(M\)(x) is integrated using Equations (5) and (6), and the boundary condition functions (Equation (7)) are used to determine the integration constants (\(C_{1i}\) and \(C_{2i}\), \(i=1,2,3\)). Therefore, the vertical deflection distributions of each beam segment (\(wi(x)\)) and the entire composite beam (\(w(x)\)) can be obtained.