## 2 Basic Equations  
The mechanical properties of engineering structures including novel bridge girders with CSWs are closely related to mechanics, especially the elastic response of bridges. For thin-walled shells used in engineering, the force distribution diagram of an orthotropic open cylindrical shell element is illustrated in Figure 2; the diagram shows a thin-walled structure considering the membrane effect. From the classical elasticity theory, the equilibrium equation, geometric equation and constitutive relation of an orthotropic open circular cylindrical shell (OOCCS) can be obtained:  
Equilibrium differential equations:  
\[\frac{\partial N_{x}}{\partial x}+\frac{\partial N_{yx}}{\partial y}=0 \tag{1}\]  
\[\frac{\partial N_{xy}}{\partial x}+\frac{\partial N_{y}}{\partial y}=0 \tag{2}\]  
\[\frac{\partial Q_{x}}{\partial x}+\frac{\partial Q_{y}}{\partial y}+N_{x}( \frac{1}{R}+\frac{\partial^{2}w}{\partial x^{2}})+N_{y}\frac{\partial^{2}w}{ \partial y^{2}}+N_{xy}\frac{\partial^{2}w}{\partial x\partial y}+N_{yx}\frac{ \partial^{2}w}{\partial x\partial y}=0 \tag{3}\]  
Figure 2: Force distribution diagram of an orthotropic open cylindrical shell element. OOCCS: orthotropic open circular cylindrical shell.  
\[\frac{\partial M_{x}}{\partial x}+\frac{\partial M_{yx}}{\partial y}-Q_{x}=0 \tag{4}\]  
\[\frac{\partial M_{xy}}{\partial x}+\frac{\partial M_{y}}{\partial y}-Q_{y}=0 \tag{5}\]  
Deformation geometric equations:  
\[\varepsilon_{x}=\frac{\partial u}{\partial x}-\frac{1}{R}w+\frac{1}{2}{( \frac{\partial w}{\partial x})}^{2},\varepsilon_{y}=\frac{\partial v}{ \partial y}+\frac{1}{2}{(\frac{\partial w}{\partial y})}^{2},\gamma_{xy}= \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}+\frac{\partial w }{\partial x}\frac{\partial w}{\partial y} \tag{6}\]  
Orthotropic constitutive equations:  
\[\varepsilon_{x}=\frac{\sigma_{x}}{E_{x}}-v_{y}\frac{\sigma_{y}}{E_{y}}, \varepsilon_{y}=\frac{\sigma_{y}}{E_{y}}-v_{x}\frac{\sigma_{x}}{E_{x}},\gamma _{xy}=\frac{2(1+v)}{E}\tau_{xy} \tag{7}\]  
where \(xyz\) denotes the cylindrical coordinate system, and the \(x\)-, \(y\)- and \(z\)-axes are the circumferential, vertical and radial directions, respectively. \(\sigma_{x},\sigma_{y}\) and \(\tau_{xy}\) are the normal and shear stresses, respectively. \(\varepsilon_{x}\) and \(\varepsilon_{y}\) are the normal strains in the \(x\)-direction and \(y\)-direction, respectively, and \(\gamma_{xy}\) is the shear strains. \(u\), \(v\) and \(w\) are the circumferential, vertical and radial displacements, respectively. \(R\) is the radius of curvature. \(E_{x}\) and \(E_{y}\) are the [PERSON]'s moduli in the \(x\)-direction and \(y\)-direction, respectively, and \(v_{x}\) and \(v_{y}\) are the [PERSON]'s ratios in the \(x\)- and \(y\)-direction, respectively. \(Q_{x}\) and \(Q_{y}\) are the radial shear forces per unit length in the \(x\)-direction and \(y\)-direction, respectively. \(M_{x}\) and \(M_{y}\) are the bending moments per unit length in the \(x\)- and \(y\)-direction, respectively. \(N_{x}\) and \(N_{y}\) are the normal forces per unit length in the \(x\)-direction and \(y\)-direction and are defined as \(N_{x}=t\sigma_{x}\) and \(N_{y}=t\sigma_{y}\), respectively. \(M_{xy}\) and \(M_{yx}\) are the twisting moments per unit length that satisfy the reciprocal theorem of torsion moment, that is, \(M_{xy}=M_{yx}\). \(N_{xy}\) and \(N_{yx}\) are the membrane shear forces in the \(x\)-\(y\) plane per unit length that meet the shear stress reciprocal principle, namely, \(N_{xy}=N_{yx}=t\tau_{xy}=t\tau_{yx}\), where \(t\) is the plate thickness. \(w=w(x,y)\) is a function depending on \(x\) and \(y\). The relationships between the elastic constants of orthotropic materials are \(v_{x}/v_{y}=E_{x}/E_{y}\), \(E=\sqrt{E_{x}E_{y}}\) and \(v=\sqrt{v_{x}v_{y}}\). In addition, \(u=-z\frac{\partial w}{\partial x}\) and \(v=-z\frac{\partial w}{\partial y}\).