# Dynamic Responses of Liquid Storage Tanks Caused by Wind and Earthquake in Special Environment
## 3 Structure Control Equations  
The structure equation of motion is  
\[\mathbf{M}_{ss}\ddot{\mathbf{u}}_{s}+\mathbf{C}_{ss}\dot{\mathbf{u}}_{s}+ \mathbf{K}_{ss}\mathbf{u}_{s}=\mathbf{F}_{ss} \tag{5}\]  
where \(\mathbf{M}_{ss}\), \(\mathbf{C}_{ss}\), and \(\mathbf{K}_{ss}\) are mass, damping and stiffness matrices of structures, respectively; \(\mathbf{F}_{ss}\) is load vector acting on structure, which includes liquid pressure; \(\ddot{\mathbf{u}}_{s}\), \(\dot{\mathbf{u}}_{s}\), and \(\mathbf{u}_{s}\) are vectors of acceleration, velocity, and displacement of structure, respectively.  
Newmark method is used to solve the dynamic Equation (5), and the first assumption is  
\[\dot{\mathbf{u}}_{s(i+1)}=\dot{\mathbf{u}}_{s(i)}+\Big{[}(1-\beta)\ddot{ \mathbf{u}}_{s(i)}+\beta\ddot{\mathbf{u}}_{s(i+1)}\Big{]}\Delta t \tag{6}\]\[\mathbf{u}_{s(i+1)}=\mathbf{u}_{s(i)}+\dot{\mathbf{u}}_{i}\Delta t+\left[\left( \frac{1}{2}-\gamma\right)\ddot{\mathbf{u}}_{i}+\gamma\ddot{\mathbf{u}}_{i+1} \right]\Delta t^{2} \tag{7}\]  
where \(\beta\) and \(\gamma\) are adjustment coefficients for accuracy and stability.  
The incremental forms \(\Delta\dot{\mathbf{u}}_{s}\) and \(\Delta\mathbf{u}_{s}\) of velocity \(\dot{\mathbf{u}}_{s}\) and displacement \(\mathbf{u}_{s}\) can be obtained from Equations (6) and (7), respectively  
\[\Delta\dot{\mathbf{u}}_{s(i)}=\dot{\mathbf{u}}_{s(i+1)}-\dot{\mathbf{u}}_{s(i) }=\left(\ddot{\mathbf{u}}_{s(i)}+\beta\Delta\ddot{\mathbf{u}}_{s(i)}\right)\Delta t \tag{8}\]  
\[\Delta\mathbf{u}_{s(i)}=\mathbf{u}_{s(i+1)}-\mathbf{u}_{s(i)}=\dot{\mathbf{u}} _{s(i)}\Delta t+\frac{1}{2}\ddot{\mathbf{u}}_{s(i)}\Delta t^{2}+\gamma\Delta \ddot{\mathbf{u}}_{s(i)} \tag{9}\]  
Acceleration increment \(\Delta\ddot{\mathbf{u}}_{i}\) can be obtained by transforming Equation (9), then taking \(\Delta\ddot{\mathbf{u}}_{i}\) into Equation (8)  
\[\Delta\ddot{\mathbf{u}}_{i}=\frac{1}{\gamma\Delta t^{2}}\Delta\mathbf{u}_{i}- \frac{1}{\gamma\Delta t}\dot{\mathbf{u}}_{i}-\left(\frac{1}{2\gamma}-1\right) \ddot{\mathbf{u}}_{i} \tag{10}\]  
\[\Delta\dot{\mathbf{u}}_{i}=\frac{\beta}{\gamma\Delta t}\Delta\mathbf{u}_{i}+ \left(1-\frac{\beta}{\gamma}\right)\dot{\mathbf{u}}_{i}+\left(1-\frac{\beta} {2\gamma}\right)\Delta t\ddot{\mathbf{u}}_{i} \tag{11}\]  
The incremental form corresponding to Equation (5) is  
\[\mathbf{M}_{ss}\Delta\ddot{\mathbf{u}}_{s(i)}+\mathbf{C}_{ss}\Delta\dot{ \mathbf{u}}_{s(i)}+\mathbf{K}_{ss}\Delta\mathbf{u}_{s(i)}=\Delta\mathbf{F}_{ ss(i)} \tag{12}\]  
Taking Equations (9)-(11) into Equation (12)  
\[\mathbf{\overline{K}}\Delta\mathbf{u}_{s(i)}=\mathbf{\overline{F}} \tag{13}\]  
where \(\mathbf{\overline{K}}=\mathbf{K}+\frac{1}{\gamma\Delta t^{2}}\mathbf{M}+\frac {\beta}{\gamma\Delta t}\mathbf{C}\); \(\mathbf{\overline{F}}=\Delta\mathbf{F}_{ss(i)}+\mathbf{M}\left[\frac{1}{ \gamma\Delta t}\dot{\mathbf{u}}_{i}+\left(\frac{1}{2\gamma}-1\right)\ddot{ \mathbf{u}}_{i}\right]+\mathbf{C}\left[\left(\frac{\beta}{\gamma}-1\right) \dot{\mathbf{u}}_{i}+\left(\frac{\beta}{2\gamma}-1\right)\Delta t\ddot{ \mathbf{u}}_{i}\right]\).  
The displacement increment \(\Delta\mathbf{u}_{s(i)}\) can be obtained by Equation (13), velocity increment \(\Delta\dot{\mathbf{u}}_{s(i)}\) can be obtained by substituting displacement increment \(\Delta\mathbf{u}_{s(i)}\) into Equation (11). As a result, the displacement \(\mathbf{u}_{s(i+1)}\) and velocity \(\dot{\mathbf{u}}_{s(i+1)}\) of \(i+1\) time step can be obtained  
\[\dot{\mathbf{u}}_{s(i+1)}=\mathbf{u}_{s(i)}+\Delta\mathbf{u}_{s(i)} \tag{14}\]  
\[\dot{\mathbf{u}}_{s(i+1)}=\dot{\mathbf{u}}_{s(i)}+\Delta\dot{\mathbf{u}}_{s(i)} \tag{15}\]  
The acceleration \(\ddot{\mathbf{u}}_{s(i+1)}\) of time step \(i+1\) can be obtained by substituting Equations (14) and (15) into Equation (5)  
\[\ddot{\mathbf{u}}_{s(i+1)}=\mathbf{M}_{ss}{}^{-1}\cdot\left[\mathbf{F}_{ss}- \mathbf{C}_{ss}\cdot\dot{\mathbf{u}}_{s(i+1)}-\mathbf{K}_{ss}\cdot\mathbf{u}_{ s(i+1)}\right] \tag{16}\]