# Analysis of Equivalent Flexural Stiffness of Steel-Concrete Composite Beams in Frame Structures
## 3 Parameter Analysis  
### Rotation Constraints at Beam Ends  
For a composite beam in a frame structural system, the rotation constraints at the beam ends are closely related to the surrounding beams and columns connected to it. In the theoretical model of the composite beam established above, the rotation constraints at the beam ends are simulated by spring constraint boundaries. To accurately evaluate the restraint stiffness at the beam ends, an equal-span composite frame structural system was selected to study the calculation method of parameter \(k_{\mathrm{i}}\) (\(i=1,2\)). For the planar beam-column joint shown in Figure 3, the rotation restraint stiffness at the beam end (\(k_{\mathrm{b}}\)) and column end (\(k_{\mathrm{c}}\)) can be calculated using Equation (10):  
\[\left\{ \begin{array}{l}k_{\mathrm{b}}=k_{\mathrm{cu}}+k_{\mathrm{bl}}+k_{ \mathrm{cb}}\\ k_{\mathrm{c}}=k_{\mathrm{bl}}+k_{\mathrm{cb}}+k_{\mathrm{br}}\end{array}  \right., \tag{10}\]  
where \(k_{\mathrm{cu}}\), \(k_{\mathrm{bl}}\), \(k_{\mathrm{cb}}\), and \(k_{\mathrm{br}}\) represent the rotation restraint stiffness provided by the surrounding beams and columns. According to the mechanical principle of frame structure systems, the rotation restraint stiffness provided by a structural member to its adjacent members can be defined as a multiple of its linear stiffness:  
\[k=ci=c\frac{EI}{l}\, \tag{11}\]  
where \(c\) is the coefficient determined by the constraint condition at the other end; \(i\) is the linear stiffness of the member; \(EI\) is the flexural stiffness of the member; and \(l\) is the length of the member. To simplify the calculation, all the beams and columns are assumed to have the same linear stiffness, i.e., \(k_{\mathrm{cu}}=k_{\mathrm{cb}}\) and \(k_{\mathrm{bl}}=k_{\mathrm{br}}\). Thus, in the equal-span frame structural system, the rotation restraint stiffness at the ends of any beam or column member are the same,  
Figure 3: Composition of end rotation constraints stiffness of the beams and columns in the frame. ((**a**) Rotation constraints at beam ends; (**b**) Rotation constraints at column ends). except for the case in which the beam or column member is located in the side span or top story. Thus, Equation (10) can be rewritten as Equation (12):  
\[\left\{ \begin{array}{l}k_{b}=c_{1}i_{b}+2c_{2}i_{c}\\ k_{c}=2c_{1}i_{b}+c_{2}i_{c}\end{array} \right., \tag{12}\]  
where \(i_{b}\) and \(i_{c}\) are the linear stiffness of an adjacent beam and column, respectively, and \(c_{1}\) and \(c_{2}\) are the undetermined parameters that need to be calibrated by constraint stiffness \(k_{b}\) and \(k_{c}\). For a certain frame structural system with the given values of \(i_{b}\) and \(i_{c}\), the values of \(c_{1}\) and \(c_{2}\) can be calculated by iteration. The calculation results indicate that the values of \(c_{1}\) and \(c_{2}\) are mainly related to the linear stiffness ratio of the beam and column (\(\delta\)), as illustrated in Figure 4. In general, the value of \(\delta\) varies from 0.1 to 5 and remains not greater than 1 for a composite frame structural system [28]. Within this parameter range, typically \(c_{1}\) decreases with the increase in \(\delta\), whereas \(c_{2}\) increases. However, it is noted from the figure that the values of \(c_{1}\) and \(c_{2}\) vary very little with \(\delta\) in the given interval. Therefore, the average values of \(c_{1}\) and \(c_{2}\) (\(c_{1}=3.85\), \(c_{2}=3.60\)) are used in Equation (12) and, thus, the rotation restraint stiffness at the composite beam ends can be approximately calculated using Equation (13):  
\[k_{i}=3.85 ni_{b}+3.60 mi_{c}, \tag{13}\]  
where \(n\) and \(m\) are, respectively, the numbers of beams and columns connected to the investigated composite beam. Based on the displacement method in structural mechanics, the rotation constraint provided by a fixed supported beam to a far end is four times the linear stiffness of the member, whereas that by a hinged supported beam is thrice the linear stiffness.