# Analysis of Equivalent Flexural Stiffness of Steel-Concrete Composite Beams in Frame Structures
## 1 Introduction
With a growing demand in engineering applications of composite beam systems, researches on performance of composite beams with varied types of configurations--such as high-strength steel concrete composite beams [24], composite beams with different slab materials and shear transfer mechanisms [25], prefabricated composite beams with bolt shear connectors [26], among others--also have been conducted. Despite the success of the foregoing studies in the prediction of the flexural behaviour of composite beams by incorporating various factors, such as interlayer slip, cracked concrete, material nonlinearity, loading conditions, and member configurations, most of the aforementioned methods focus on the independent composite beam without considering the realistic boundary conditions at the beams ends. In the calculation of the flexure stiffness for composite frame beams, the exiting methods generally take the boundary constraints of the beam component as fixed ends, and thereby lead to an inaccurate evaluation of the cracking effect in the negative moment region. For example, when the stiffness ratio of a frame beam to the adjacent beam and column is relatively small, particularly for composite beams employed under large-span and heavy loading conditions as discussed by [PERSON] et al. [27], the cracking area of the concrete slab and its influence on the flexural stiffness would be overestimated. As for the consideration of the flexural stiffness reduction in the negative moment region for a composite beam, two methods are frequently adopted in the structural design procedure. One of the most commonly used methods is dividing the entire continuous composite beam into several segments with different section stiffness. Specifically, these are negative moment segments with reduced section stiffness (\(EI^{-}\)) at the beam ends and a positive moment segment with normal section stiffness (\(EI^{+}\)) in the middle. However, the division into beam segments is subjective, and the ranges of positive and negative moment regions are determined roughly. For example, considering the Chinese Code [28] and Eurocode 4 [29], at each end, 15% of the length of a beam is suggested as one of the negative moment segments, and the remainder of the span is defined as the positive moment segment. Another simplified method to evaluate the equivalent flexural stiffness of a continuous composite beam is employing the weighted average of \(EI^{+}\) and \(EI^{-}\), which is calculated using Equation (1):  
\[EI_{eq}=\beta EI^{-}+\ (1-\beta)EI^{+}, \tag{1}\]  
where \(\beta\) is the weight factor, which is not greater than 1. In American codes ANSI/AISC [30], \(\beta\) is recommended as 0.4 for a composite frame beam under vertical loads. [PERSON] et al. [31] showed that the elastic behaviour of a composite frame beam can be simulated well when \(\beta\) = 0.6. Both of the above methods are based on a fundamental assumption that the length ratio of the negative moment region and the entire continuous composite beam is constant. However, this assumption evidently does not hold when considering the actual boundary constraint conditions of the continuous beams in a practical composite frame structural system. As shown in Figure 1, the distribution of the internal force and the cracking region of a composite beam are closely related to the deformation capacity of the beams and columns connected to it, i.e., the rotational constraint condition at the frame beam end. If the rotational restraint stiffness at the beam ends is sufficiently strong, the frame beam can be considered similar to a fixed beam, and thus, the cracking region of the concrete slab is relatively large. Conversely, when the rotational restraint stiffness is relatively weak, such as in the case of composite beams designed with large stiffness for large-span and heavy loading conditions, the frame beam can be considered as a simply supported beam. Consequently, the cracking region of the concrete slab is small.