# Analysis of Equivalent Flexural Stiffness of Steel-Concrete Composite Beams in Frame Structures
## 1 Introduction  
Composite frame structural systems have been widely employed in high-rise buildings, girder bridges, and other structures with large span and heavy loads, because of their favourable structural performance and economic benefit [1, 2, 3, 4, 5]. Compared with simply supported beams, the continuous steel-concrete frame beams can obtain a significant increase in structural stiffness, such that they have the advantages of higher span ratio, less deflection, and fundamental frequency of vibration. However, for a continuous composite beam in frame structures, the concrete slab over the intermediate supports suffers a tension condition and may be cracked when in service. Due to the cracking of the concrete slab under negative bending, the sectional flexural stiffness values of the continuous composite beams in the positive and negative moment areas are different. The varied sectional flexural stiffness along the member length induces a remarkable reduction in the integral rigidity of the composite frame beams and should be considered in their limit-state analysis [6]. To date, many researchers have focused on the deflection analysis and design method of the composite beams with full and partial interactions [7, 8, 9, 10]. Moreover, for a frame beam under vertical loads, the distribution of the bending moment and the cracking region areclosely related to the deformation capacity of the beams and the columns connected to it, i.e., the rotational constraint condition at the frame beam end. This complex boundary condition increases the difficulty in evaluating the flexural stiffness of the continuous composite beam. The mechanism behaviour of steel-concrete composite beams has been investigated in previous researches [11; 12; 13]. Moreover, in order to facilitate the design of the composite frame structures, extensive experimental and numerical researches have also been conducted to focus on the flexural behaviour of the steel-concrete composite beams. [PERSON] [14] was the first to develop an elastic analysis theory of composite beams, including the slip that occurs in the concrete slab-steel beam interface. The basic equilibrium and compatibility equations for a beam element were established to calculate the deflection and analyse the stresses distribution of composite beams. Considering the slip effect at the steel beam-concrete slab interface, analytical models and design formulae were presented by [PERSON] et al. [15] to predict the effective stiffness of composite beams. [PERSON] et al. [16] and [PERSON] et al. [17; 18] investigated the flexure behaviour of composite beams through experimental and numerical studies on simply supported composite beams under the condition of both positive and negative bending, and also verified the calculation method proposed by [PERSON] and [PERSON] [19] to estimate the flexural capacity of composite beams. Based on the realistic discrete shear force distribution, [PERSON] et al. [20] modelled the shear study of composite beams by using a set of nonlinear springs, such that the interaction between the steel beam and concrete slab could be elaborately simulated. [PERSON] et al. [21] worked on the long-term behaviour of the composite beam and carried out a numerical study to predict long-term deformation, which focused on the effects of concrete creep and shrinkage. [PERSON] et al. [22] proposed a formulation for the evaluation of composite beam response under varied monotonic load levels, including both short- and long-term loading. To improve the bending capacity and anti-slip performance of composite beams, [PERSON] et al. [23] proposed new assembled monolithic steel-prestressed concrete composite beams and demonstrated their superiority in flexure capacity by experimental investigations.