# Analysis of Equivalent Flexural Stiffness of Steel-Concrete Composite Beams in Frame Structures
## 3 Parameter Analysis
### Equivalent Flexural Stiffness  
For the convenience of structural design in engineering applications, an equivalent flexural stiffness represented by the weighted average of \(EI^{+}\) and \(EI^{-}\) is frequently used to evaluate the integral stiffness of a composite beam. The weight factor of the equivalent stiffness--parameter \(\beta\) in Equation (1)--can be calculated using the theoretical model  
Figure 6: Variation in the length of the negative moment region with critical parameters \(K\) and \(\alpha\).  
presented in Section 2. The relationship between the weight factor and the rotational constraint stiffness (described by parameters \(K\) and \(\alpha\)) is presented in Figure 7.  
It can be seen that the computed value of \(\beta\) increases as parameter \(K\) increases, which indicates that a strong rotational constraint results in a small equivalent flexural stiffness. In addition, factor \(\beta\) initially increases rapidly when \(K\) is relatively small and subsequently changes gradually, approaching saturation. When parameter \(K\) is fixed, weight factor \(\beta\) increases as parameter \(\alpha\) reduces, and the difference in the equivalent flexural stiffness is larger when \(K\) ranges from 10 to 20. In Figure 7, the calculated results of weight factor \(\beta\) are also compared with the recommended values of ANSI/ASC and [PERSON] et al. It can be concluded that the fixed value of weight factor \(\beta\) cannot accurately describe the equivalent flexural stiffness of the composite beam because the effects of the boundary rotation constraints and the concrete slab cracking are not elaborately considered in most cases. In particular, for the composite beams designed with a large span and heavy loading, the rotation constraints at the beam ends are relatively weak and parameter \(K\) is small (e.g., 4-10) [27]. Fixed weight factor \(\beta\), suggested by the previously mentioned studies, significantly underestimate the integral rigidity of the composite beam. For the commonly used frame structural system arranged with equal spans, parameter \(K\) varies within the values of 10-40 and parameter \(\alpha\) is approximately 0.4. In this case, the recommended value of \(\beta\) provided in ANSI/ASC can reasonably describe the deformation mechanism and flexural stiffness of the composite beams. In comparison, the value of 0.6 suggested by [PERSON] et al. leads to a remarkable difference from the theoretical results of \(\beta\) for both the large- and common-span beams.  
Based on the parameter analysis, as prescribed, the rotational constraint condition is proved to have the dominant effect on the equivalent flexural stiffness of the composite beam. To consider the critical parameters of the rotational constraint--\(K\) and \(\alpha\)--in evaluating the equivalent flexural stiffness, a simplified formula for calculating weight factor \(\beta\) is proposed by curve fitting, as expressed in Equation (18):  
\[\beta=\frac{(0.02\alpha+0.49)K^{1.9}}{K^{1.9}+5.7\alpha^{2}+21.7\alpha+35.6}. \tag{18}\]  
In Figure 7, the change in weight factor \(\beta\) calculated using Equation (18) is represented by solid lines. It can be seen that the calculated results of weight factor \(\beta\) agree well with those obtained from the theoretical model. Taking parameter \(K\) = 0 as an example,  
Figure 7: Variation in the weight factor of equivalent stiffness with end rotational restraint stiffness.  
coefficient \(\beta\) is calculated to be 0 using Equation (18), and the equivalent flexural stiffness is \(EI^{+}\), which is in agreement with the theoretical results of a simply supported beam without the cracking effect. In addition, when parameter \(K\) approaches infinity, the limit of \(\beta\) is \(0.02\alpha+0.49\) and corresponds to the stiffness amplification coefficient, which is also in good agreement with the theoretical tendency of composite beams with a fixed support boundary.