# Analysis of Equivalent Flexural Stiffness of Steel-Concrete Composite Beams in Frame Structures
## 5 Simplified Design Method
### Simplified Design Formula for Calculating Vertical Deflection  
To utilize Equation (18) to calculate the vertical deflection of a composite beam, two critical parameters, i.e., \(K\) and \(\alpha\), need to be calibrated first. By substituting Equation (29) into the definition formula of \(K\), the ratio of the rotation restraint stiffness and the linear stiffness of the composite beam after the concrete slab cracking can be easily obtained as  
\[K=\frac{k}{i^{-}}=\frac{6\alpha\big{(}2\alpha_{\text{cr}}-\alpha_{\text{cr}} {}^{2}\big{)}}{(2-6\alpha_{\text{cr}}+3\alpha_{\text{cr}}{}^{2})}. \tag{31}\]  
Figure 14 illustrates the relationship between parameter \(K\) and negative moment region factor \(\alpha_{\text{cr}}\) for different values of parameter \(\alpha\). Generally, when \(\alpha\) is fixed, parameter \(K\) increases as the negative moment region increases. Note that when \(\alpha_{\text{cr}}=0\), the composite frame beam degenerates to a simply supported beam and parameter \(K=\text{zero}\). When \(\alpha_{\text{cr}}\) is  
close to its theoretical maximum value of \(1-\sqrt{3}/3\), parameter \(K\) approaches infinity. For the case when \(\alpha_{\text{cr}}\) is fixed, parameter \(K\) increases as the stiffness amplification coefficient \(\alpha\) becomes larger, which indicates that a large flexure stiffness of the composite beam leads to strong rotation constraints at the beam ends. In practical design, critical parameter \(K\) can be calculated using coefficients \(\alpha\) and \(\alpha_{\text{cr}}\) in Equation (31). The weight factor of the equivalent stiffness, \(\beta\), can also be calculated using this simplified method, and the relationship between parameters \(\beta\) and \(\alpha_{\text{cr}}\) is presented in Figure 15.  
To facilitate the utilization of this simplified method, a vertical deflection correction factor \(\alpha_{\Delta}\) is introduced and defined as follows:  
\[\alpha_{\Delta}=\frac{\Delta_{\text{co}}}{\Delta} \tag{32}\]  
Figure 14: Relationship between parameter \(K\) and negative moment region factor \(\alpha_{\text{cr}}\). Figure 15: Variation in equivalent stiffness weight coefficient \(\beta\) with parameters \(\alpha\) and \(\alpha_{\text{cr}}\). where \(\Delta_{\rm co}\) is the vertical deflection of the composite frame beam considering the effect of the concrete slab cracking and \(\Delta\) is the vertical deflection of the composite beam neglecting the effect of the concrete slab cracking. Based on the relationship between the flexural stiffness and the vertical deflection of the composite beam during the elastic stage, the definition of the vertical deflection correction factor, \(\alpha_{\Delta}\), can be rewritten as  
\[\alpha_{\Delta}=\frac{EI^{+}}{E_{\rm Ieq}}=\frac{\alpha}{\beta+(1-\beta)\alpha} \tag{33}\]  
For composite frame beams with different cross-section properties and rotation constraints, the theoretical value of \(\alpha_{\Delta}\) can be calculated using Equation (33). Figure 16 illustrates the relationship of \(\alpha_{\Delta}\) with the negative moment region factor, \(\alpha_{\rm cr}\), for different values of the stiffness amplification coefficient, \(\alpha\). As expressed in Equation (33), \(\alpha_{\Delta}\) is determined using the equivalent flexural stiffness weight coefficient, \(\beta\), and the stiffness amplification coefficient, \(\alpha\), where \(\beta\) is determined by critical parameters \(K\) and \(\alpha\). In addition, if the elastic analysis results of the internal force of a frame structure are available, parameter \(K\) can be further expressed using \(\alpha\) and \(\alpha_{\rm cr}\). Therefore, it is reasonable to use the stiffness amplification coefficient, \(\alpha\), and the negative moment region factor, \(\alpha_{\rm cr}\), as independent variables when determining the vertical deflection correction factor, \(\alpha_{\Delta}\).