# Analysis of Equivalent Flexural Stiffness of Steel-Concrete Composite Beams in Frame Structures
## 3 Parameter Analysis
### Identification of Critical Parameters  
Using the theoretical model described in Section 2, an extensive parametric analysis was first conducted for the composite beam with equal rotation constraints at both ends, i.e., for the case of \(k_{1}=k_{2}\) (the unequal constraint case will be discussed in a subsequent section). The vertical deflection of the composite beam was proved to be primarily related to only  
Figure 4: Variation in \(c_{1}\) and \(c_{2}\) with beam-to-column line stiffness ratio. (**(a)**) Relationship between \(c_{1}\) and \(\delta\); (**b**) Relationship between \(c_{2}\) and \(\delta\)).  
three key parameters: (1) the linear stiffness of the composite beam before the concrete slab cracking (denoted as \(i^{+}=EI^{+}/l\)); (2) the linear stiffness of the composite beam after the concrete slab cracking (denoted as \(i^{+}=EI^{+}/l\)); and (3) the rotation constraint stiffness of the composite beam at both ends (denoted as \(k=k_{1}=k_{2}\)). In addition, it was observed that, when these three factors change proportionally, i.e., increase or decrease simultaneously by the same multiple, the structural responses--vertical deflection, internal force, and equivalent flexural stiffness--of the concerned composite frame beam remained invariant. Therefore, two dimensionless parameters were identified as the dominant factors for the vertical deformation of composite beams. These are (1) the ratio of the rotation restraint stiffness and the linear stiffness of the composite beam after the concrete slab cracking (denoted as \(K=k/l^{-}\)), and (2) the ratio of the linear stiffness of the composite beam before and after the concrete slab cracking (denoted as \(\alpha=i^{+}/l^{-}\)).  
To facilitate the detailed parameter analysis of the equivalent flexural stiffness of composite beams, the regular ranges of critical parameters \(K\) and \(\alpha\) were determined based on the engineering practice. Parameter \(K\) represents the effect of the rotation constraints at the beam ends on the calculation of the equivalent flexural stiffness. This is discussed for two scenarios: common frame structures designed with equal-span arrangements and composite beams designed with large spans and heavy loads. For the case of common frame structures, the linear stiffness ratio of the composite beam and the column should range from 0.2 to 1, and thus, the range of parameter \(K\) was calculated to be 10-40 using Equation (13). For the case of composite beams designed with large spans and heavy loads, the range of parameter \(K\) was 4-10 based on relevant practical engineering examples [27].  
Parameter \(\alpha\) reflects the effect of the concrete slab cracking on the equivalent flexural stiffness of a composite beam, and it is also referred to as the stiffness amplification coefficient in structure design codes. According to the JGJ 138-2016 Chinese Code for Design of Composite Structure [33], the stiffness amplification coefficient, \(\alpha\), is calculated using Equation (14),  
\[\alpha=\frac{2.2}{(I_{\mathrm{s}}/I_{\mathrm{c}})^{0.3}-0.5}+1, \tag{14}\]  
where \(I_{\mathrm{c}}\) is the equivalent moment of inertia of the concrete slab section and \(I_{\mathrm{s}}\) is the moment of inertia for a pure steel beam section.  
[PERSON] and co-workers [34; 35] have completed numerous numerical studies on composite beams that considered the cracking effect of the concrete slab. Based on their analysis results, the value of parameter \(\alpha\) is found to range from 1.1 to 7, which covers most engineering application cases of composite beams.