# A Time Integration Method Based on Galerkin Weak Form for Nonlinear Structural Dynamics
## 5 Numerical Examples
### Cantilever Beam  
Consider the cantilever Euler-Bernoulli beam under a uniformly distributed pulse load as the example [5], which is shown in Figure 19. A linear elastic constitutive relation and degenerate Green-Lagrange strain tensor [45] are adopted to describe the finite elements, and 400 elements are used in spatial discretization. Linear Lagrange element interpolation is used for axial displacement, and cubic Hermite element interpolation is used for lateral and rotational displacements.  
Letting \(\Delta t=0.002\) s, the results of vibration displacement, velocity, and acceleration at the free end of the beam are shown in (Figures 20-22), calculated from trapezoidal rule, [PERSON] method, [PERSON] \(\theta\)-method and the proposed method with \(\alpha=0.02\), respectively. The trapezoidal rule again shows unstable performance. The Bathe method, [PERSON] \(\theta\)-method, and the proposed method all give stable responses. Both the [PERSON] method and the proposed method perform well, however, for the [PERSON] \(\theta\)-method, acceleration decays as time goes on and the numerical dissipation of the global response is a little larger, as shown in Figure 22c.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Method** & **Computational Time (s)** \\ \hline Bathe method (\(\gamma=1/2\)) & 16.34 \\ [PERSON] \(\theta\)-method (\(\theta=1.4\)) & 9.29 \\ Proposed method (\(\alpha=0.005\)) & 9.83 \\ Proposed method (\(\alpha=0.01\)) & 9.48 \\ Proposed method (\(\alpha=0.02\)) & 9.33 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Computational time of different methods.  
Figure 19: Cantilever beam model.  
Figure 20: _Cont._Figure 21: The velocity responses at the free end of the cantilever beam: (**a**) Using the proposed method, (**b**) using the trapezoidal rule, (**c**) using [PERSON] \(\theta\)-method, and (**d**) using [PERSON] method.  
Figure 20: The displacement responses at the free end of the cantilever beam: (**a**) Using the proposed method, (**b**) using the trapezoidal rule, (**c**) using [PERSON] \(\theta\)-method, and (**d**) using [PERSON] method.  
Solving the beam with different numbers of spatial elements, the corresponding computational time taken in each method is listed in Table 4, and then the variation of computational time versus the number of degrees of freedom (DOFs) is plotted in Figure 23. The computational speed of the proposed method is as good as the [PERSON] method and is obviously better than the [PERSON] method.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Method\textbackslash{}No. of DOFs** & **150** & **300** & **450** & **600** & **750** & **900** & **1050** & **1200** \\ \hline Bathe method & 90.49 & 196.38 & 300.34 & 456.37 & 656.15 & 894.85 & 1249.68 & 1759.70 \\ [PERSON] \(\theta\)-method & 53.67 & 108.31 & 175.25 & 281.07 & 373.99 & 511.82 & 696.59 & 896.19 \\ This algorithm (\(\alpha=0.02\)) & 54.73 & 107.77 & 188.43 & 294.14 & 417.34 & 530.81 & 717.74 & 935.43 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Computational times of different methods (s).  
Figure 22: The acceleration responses at the free end of the cantilever beam: (**a**) Using the proposed method, (**b**) using the trapezoidal rule, (**c**) using Wilson \(\theta\)-method, and (**d**) using Bathe method.