## 2 Free Vibration Equation for CBGCSW  
### Strain Energy of CBGCSW Considering Shear Lag and Accordion Effect  
Since the corrugated steel webs form a folding pattern in the longitudinal direction, even under a small axial load, there is significant axial deformation. This is referred to as the accordion effect, which significantly reduces the effective elastic modulus of the corrugated webs in the longitudinal direction to approximately 1/100 or 1/1000 of that of the general steel webs [21]. Thus, in order to reduce the calculation difficulty, the axial rigidity of corrugated web plates was assumed to be negligible, i.e., the effective elastic modulus of corrugated webs in longitudinal direction \(E_{e}=0\). The vertical compressions, transverse normal strains, and transverse bending of the concrete slabs and the corrugated steel webs were assumed to be negligible [22; 23]. On a CBGCSW section, assuming that the longitudinal displacement of any point is the superposition of the longitudinal warping displacement caused by the shear lag effect and the longitudinal displacement according to the plane-section assumption, the longitudinal displacement expression is [24]  
\[u_{i}(x,y,z,t)=g_{i}-z\theta\ \ \ \ \ (i=1,2,3,4) \tag{1}\]  
where \(\theta\) is the sectional rotation angle of a CBGCSW; \(u_{i}(x,y,z,t)(i=1,2,3,4)\), respectively, are the longitudinal displacement functions of various points on the top plate, cantilever plate, low plate, and corrugated webs of a CBGCSW; \(g_{i}(x,y,t)(i=1,2,3)\), \(g_{4}(x,z,t)\), respectively, are functions for the longitudinal warping displacement of various points on the top plate, cantilever plate, low plate, and corrugated webs of a CBGCSW. Assuming that the longitudinal warping distribution patterns of various points on the top plate, cantilever plate, low plate, and corrugated webs section of a CBGCSW are parabolas, they can be respectively expressed as [25; 26]  
\[g_{i}=\psi_{i}(y)U(x,t)\ \ \ \ (i=1,2,3,4) \tag{2}\]  
\[\psi_{i}(y)=\alpha_{i}\Big{(}y^{2}/b_{i}^{2}-1\Big{)}+D_{w} \tag{3}\]  
\[\alpha_{1}=1,\ \alpha_{2}=b_{2}^{2}/b_{1}^{2},\ \alpha_{3}=z_{3}b_{3}^{2}/ \Big{(}b_{1}^{2}z_{1}\Big{)},\ \alpha_{4}=0 \tag{4}\]  
On the basis of self-balance of the warping stresses, Equations (2)-(4) reduce to  
\[D_{w}=2(\alpha_{1}A_{1}+\alpha_{2}A_{2}+\alpha_{3}A_{3})/(3A_{0}) \tag{5}\]  
where \(i=2\), \(y\) is replaced by \(\overline{y}=b_{1}+b_{2}-y\); \(2b_{1}\), \(b_{2}\), and \(2b_{3}\), respectively, are the widths of the top plate, cantilever plate, and low plate, as shown in Figure 1; \(b_{4}\) is the height of the corrugated web plate; \(t_{1}\), \(t_{2}\), \(t_{3}\), and \(t_{4}\), respectively, are the thicknesses of the top plate, cantilever plate, low plate, and corrugated web plate; \(z_{1}\), \(z_{2}\), and \(z_{3}\) are the coordinates of the top-plate, cantilever-plate, and low-plate centroids in the \(z\)-direction, respectively; \(U(x,t)\) is the function for longitudinal warping amplitude; \(D_{w}\) is the longitudinal warping displacement of the CBGCSW section; \(A_{1}=2b_{1}t_{1}\), \(A_{2}=2b_{2}t_{2}\), \(A_{3}=2b_{3}t_{3}\), \(A_{s}=2b_{4}t_{4}\); \(A_{0}=A_{1}+A_{2}+A_{3}+n_{0}A_{s}\); \(n_{0}=E_{e}/E_{c}=0\); \(E_{e}\) is the steel elastic modulus; and \(E_{c}\) is the concrete elastic modulus. The shear modulus \(G_{s}\) of general steel webs is greater than the effective shear modulus \(G_{e}\) of corrugated webs because of the folding performance of corrugated webs. Thus, the effective shear modulus \(G_{e}\) can be approximately expressed as \(G_{e}=G_{s}(B+D)/(B+C)\)[3], where \(B\), \(C\), and \(D\) are, respectively, the flat-plate section length, inclined-plate section length, and the longitudinal projected length of the inclined plate section, as shown in Figure 2.  
Figure 1: Dimension and coordinate system of Composite box girder bridges with corrugated steel web (CBGCSW) cross-section.