# Analysis of Equivalent Flexural Stiffness of Steel-Concrete Composite Beams in Frame Structures
## 2 Theoretical Analysis  
After the concrete slab cracking in the negative moment area, the effective flexural stiffness reduces and the composite beam performs an inelastic deformation. To discuss the inelastic bending deformation of composite frame beams, a theoretical model is established for a composite beam subject to a uniformly distributed vertical load. As shown in Figure 2, the composite beam is divided into three segments with different section flexural stiffness depending on the cracking of its concrete slab. For the middle segment of the beam, the concrete slab remains intact under a positive moment, and the section stiffness is defined as the normal value (\(EI^{+}\)). However, the segments at the two ends of the beam suffer cracking in the concrete slab, and hence, the reduced section stiffness (\(EI^{-}\)) is chosen. The spring constraint boundaries are also included in the proposed model to simulate in detail the rotation constraint imposed by the adjacent beams and columns in the frame structural system. As the full shear connection is assumed for all composite beams discussed in this study, the slip effect at the steel-concrete interfaces is neglected in the theoretical model as well as in the numerical model discussed in the following sections. This simplified treatment in modelling is validated to be reasonable by previous studies [3; 32], especially when considering the slip constraint provided by the beam-column joints. First, the internal force distribution of the composite beam is derived by the theoretical model. The reaction force at the left support is obtained from the equilibrium condition as  
\[F_{1}=\frac{1}{2}ql+\ (k_{1}\theta_{1}+k_{2}\theta_{2})/l, \tag{2}\]  
where \(q\) is the uniformly distributed vertical load applied on the composite beam; \(l\) is the length of the beam span; \(k_{1}\) and \(k_{2}\) represent the rotation restraint stiffness of the spring boundaries at the left and right ends of the beam, respectively; and \(\theta_{1}\) and \(\theta_{2}\) are the rotation angles at the two ends of the composite beam under the vertical load, respectively. The moment distribution, \(M(x)\), of the composite beam can be expressed by the following equation:  
\[M(x)=F_{1}x-k_{1}\theta_{1}-\frac{1}{2}qx^{2}, \tag{3}\]  
where \(x\) is the distance of the calculated point to the left end support. Based on the theory of material mechanics, the bending moment and vertical deflection of the composite beam satisfy the following relationship:  
\[EI_{i}\frac{d^{2}w_{i}(x)}{dx^{2}}=-M(x), \tag{4}\]  
where subscript \(i\) denotes the three segments of the composite beam, which are numbered from left to right as 1, 2, and 3, respectively; \(w_{i}(x)\) represents the distribution of the vertical deflection for the corresponding beam segment; and \(El_{i}\) is the bending stiffness of the \(i\)th beam segment, being \(EI^{-}\) or \(EI^{+}\) depending on the segment. The contribution of reinforced  
Figure 2: Theoretical model of the composite frame beam under uniform vertical loads. ((**a**) Boundary constraints and deformation; (**b**) Moment and section stiffness distribution). steel rebars in the concrete slab is assumed to be negligible [3], and is thereby ignored in the calculation of \(EI^{-}\) and \(EI^{+}\). The rotation angle distribution of the composite beam can be derived by integrating Equation (4) as follows:  
\[EI_{i}\theta_{i}(x)=-\int M(x)dx=-\frac{1}{2}F_{1}x^{2}+k_{1}\theta_{1}x+\frac{ 1}{6}qx^{3}+C_{1i} \tag{5}\]  
where \(\theta_{i}(x)\) is the rotation angle distribution function of the corresponding beam segment; and \(C_{1i}\) is the integration constant, which is determined by the boundary conditions of the composite beam.