# Seismic Response Study of L-Shaped Frame Structure with Magnetorheological Dampers
## 3 Modeling Theory of the L-Shaped Frame Structure
### Theory of the Shell Element  
For the shell element, as shown in Figure 5, each node also has 6 DOFs, and the DOFs of the shell element node are the same as those of the spatial beam element node. The numbers in brackets for Figure 5 denote the DOF number.  
The stiffness [32] of the shell element consists of the membrane stiffness \(k_{e}^{m}\) and the bending stiffness \(k_{e}^{b}\). \(k_{e}^{m}\), given in Equation (9), is used to deal with shell element membrane effects, whose corresponding DOFs are \(u\), \(v\) and \(\theta_{z}\); \(k_{e}^{b}\) given in Equation (10) is used to deal with shell element bending effects, whose corresponding DOFs are \(w\), \(\theta_{x}\), and \(\theta_{y}\); \(k_{ij}^{m}\) and \(k_{ij}^{b}\) are the submatrices with the dimension of \(3\times 3\). Then, the shell element stiffness matrix \(k_{e}\) is obtained by combining Equations (9) and (10), the dimension of which is \(24\times 24\), as shown in Equation (11).  
\[k_{e}^{m}=\left[\begin{array}{cccc}k_{11}^{m}&k_{12}^{m}&k_{13}^{m}&k_{14}^ {m}\\ k_{21}^{m}&k_{22}^{m}&k_{23}^{m}&k_{24}^{m}\\ k_{31}^{m}&k_{32}^{m}&k_{33}^{m}&k_{34}^{m}\\ k_{41}^{m}&k_{42}^{m}&k_{43}^{m}&k_{44}^{m}\end{array}\right] \tag{9}\]  
Figure 5: Node displacement and DOF number of the shell element.  
\[k_{e}^{b}=\left[\begin{array}{cccc}k_{11}^{b}&k_{12}^{b}&k_{13}^{b}&k_{14}^{b} \\ k_{21}^{b}&k_{22}^{b}&k_{23}^{b}&k_{24}^{b}\\ k_{31}^{b}&k_{32}^{b}&k_{33}^{b}&k_{34}^{b}\\ k_{41}^{b}&k_{42}^{b}&k_{43}^{b}&k_{44}^{b}\end{array}\right] \tag{10}\]  
\[k_{e}=\left[\begin{array}{cccc}k_{11}&k_{12}&k_{13}&k_{14}\\ k_{21}&k_{22}&k_{23}&k_{24}\\ k_{31}&k_{32}&k_{33}&k_{34}\\ k_{41}&k_{42}&k_{43}&k_{44}\end{array}\right] \tag{11}\]  
In which  
\[\mathbf{k}_{ij}=\left[\begin{array}{cccc}k_{ij}^{m}(1,1)&k_{ij}^{m}(1,2)&\mathbf{0}& k_{ij}^{m}(1,3)\\ k_{ij}^{m}(2,1)&k_{ij}^{m}(2,2)&\mathbf{0}&k_{ij}^{m}(2,3)\\ \mathbf{0}&\mathbf{0}&\mathbf{k}_{ij}^{b}&\mathbf{0}\\ k_{ij}^{m}(3,1)&k_{ij}^{m}(3,2)&\mathbf{0}&k_{ij}^{m}(3,3)\end{array}\right]\quad i,j=1,2,3,4 \tag{12}\]  
Analogously, the shell element mass matrix [32] is derived in an identical method as the stiffness matrix. The membrane mass matrix \(m_{e}^{m}\) given in Equation (13) is used to deal with shell element membrane effects, whose corresponding DOFs are \(u\), \(v\) and \(\theta_{z}\); the bending mass matrix \(m_{e}^{b}\) given in Equation (14) is used to deal with shell element bending effects, whose corresponding DOFs are \(w\), \(\theta_{x}\), \(\theta_{y}\). \  
where the transformation matrix \(\mathbf{T}\) can be given as follow:  
\[\mathbf{T}=\left[\begin{array}{ccc}\mathbf{t}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\ddots&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{t}\end{array}\right]_{\mathbf{a}\times\mathbf{a}} \tag{18}\]  
in which a is equal to 4 and 8 for the spatial beam element and the shell element, respectively. \(\mathbf{t}\) can be derived according to the direction cosine relationship between the global coordinate system and the local coordinate system.  
\[\mathbf{t}=\left[\begin{array}{ccc}\cos(\mathrm{X},\,\mathrm{x})&\cos(\mathrm{X },\,\mathrm{y})&\cos(\mathrm{X},\,\mathrm{z})\\ \cos(\mathrm{Y},\,\mathrm{x})&\cos(\mathrm{Y},\,\mathrm{y})&\cos(\mathrm{Y}, \,\mathrm{z})\\ \cos(\mathrm{Z},\,\mathrm{x})&\cos(\mathrm{Z},\,\mathrm{y})&\cos(\mathrm{Z}, \,\mathrm{z})\end{array}\right] \tag{19}\]