# Analysis of Equivalent Flexural Stiffness of Steel-Concrete Composite Beams in Frame Structures
## 4 Validation
### Comparison with FE Analysis of Frame Structures  
To consider the bearing mechanism of a composite beam in a frame structural system, the equivalent flexural stiffness calculated by the proposed formula was further validated by conducting an FE analysis of the entire process deformation of a frame structure. The numerical model was built based on a four-story and five-span plane frame, whose detailed design parameters are provided in Figure 10. As the main variable parameter, the spans of the composite beams adopted the values of 6, 9.5, 13, 16.5, and 20 m in the numerical examples. The effective width of the concrete slab in the composite beams was calculated using stiffness amplification coefficient \(\alpha\) according to JGJ 138-2016 [33]. In addition, the sectional dimensions of the composite beams were divided into two groups. In one group, the concrete slab thickness, \(h_{\text{c}}\), was 120 mm; the span-to-steel beam height ratio, \(L/h_{\text{s}}\), was 20; and the steel beam flange width to height ratio, \(b_{\text{f}}/h_{\text{s}}\), was 0.6. In the other group, the concrete slab thickness, \(h_{\text{c}}\), was 150 mm; the span-to-steel beam height ratio, \(L/h_{\text{s}}\), was 15, and the steel beam flange width to height ratio, \(b_{\text{f}}/h_{\text{s}}\), is 1.0. For all the FE models, the parameters of steel beam flange thickness, \(t_{\text{f}}\), and web thickness, \(t_{\text{w}}\), were designed to satisfy the requirements of width-thickness ratios specified in GB 50017-2017. Based on the principle of'strong column-weak beam' in structural design, the sectional sizes of the columns changed with the span of the frame beam, and the axial compression ratio of each column was set as approximately 0.4 to ensure that the columns do not yield before the beam achieves the ultimate limit state. The FE models of the investigated frame structures were established using a modified fibre beam-column element proposed by the [PERSON] and [PERSON] teams [19], which was developed using the subroutine package, COMPONA-MARC, based on the general FE analysis software, MSC Marc. The accuracy of the modified fibre beam-column element in the non-linear analysis of composite frame systems was considerably validated [19; 37]. Based on previous research on modelling strategies [32], the length of the fibre beam-column element was suggested to be as, approximately, the depth of the cross-section. In each cross-section of the beam elements, the concrete slab was divided into 5 \(\times\) 5 fibres, i.e., 25 fibres in total; the beam flange was divided into 10 fibres; and the web plate was divided into 20 fibres. In each cross-section of the column elements, the flange and web plates were divided into 10 and 20 fibres, respectively. The constitutive law, suggested by [PERSON] [38], was adopted for the uniaxial stress-strain skeleton curve of the concrete, and  
Figure 9: Comparison of the equivalent flexural stiffness calculated by the formula proposed in this paper and other existing methods. (**(a)** Comparison of \(El_{\text{eq}}\) calculation results in large-span heavy-loading design conditions; **(b)** Comparison of \(El_{\text{eq}}\) calculation results in commonly arranged frame design conditions). the constitutive relationship suggested by [PERSON] et al. [39] was adopted for that of steel. The detailed parameters of the material constitutive model are shown in Figure 11.  
A uniformly distributed load in the vertical direction was applied to the middle and side spans of the composite frame beams located in the first and fourth stories. The applied loads started at zero and continued to increase until the frame beams completely entered the plastic stage; thus, the entire evolution process of the deflection can be traced for the considered beams. The simulated load-deflection curves of some typical frame beams are presented in Figure 12a,b.