# Dynamic Response of Slope Inertia-Based Timoshenko Beam under a Moving Load
## 5 Comparative Study  
Firstly, the dynamic response of a Euler-Bernoulli beam with a square cross section studied in reference [7; 39; 40; 41] is revisited. The magnitude of the moving load \(P\) is set to \(4.448\ N\), and the physical properties of the beam are listed in Table 2. Similarly, for the SIBT model, the critical speed parameter \(c^{*}=0.993\). Figure 9 shows the comparison in terms of dynamic responses of the beam modeled according to the SIBT and Euler-Bernoulli with \(c=0.25\). It can be seen that time histories of the normalized mid-span displacement and bending moment obtained according to the SIBT model are virtually the same as those calculated by the Euler-Bernoulli model [7]. This is expected in view of the fact that the height-span ratio of the beam is 1:16, which is very small. Consequently, for a slender beam, the effect of shear deformation and shear inertia captured by the SIBT model is negligible. Figure 8: Moment–time graph at each beam node. Figure 7: Displacement–time graph at each beam node. Table 3 further compares the dynamic amplification factors of the SIBT beam and Euler-Bernoulli beam. As can be seen from Table 3, and referring to Figure 6, with the increase of the moving speed of the load, the displacement and moment amplification factors both change nonlinearly. In addition, although the calculation results of the SIBT beam are generally larger than the calculation results of the Euler-Bernoulli beam, the differences are very small, due to the slenderness of the beam. To further compare the dynamic responses of the SIBT, Euler-Bernoulli and Timoshenko beams with different slenderness ratios, based on Table 2, the ratio of the cross-sectional dimension to beam length is reset from \(1/16\) to \(1/2\). One can find from Figure 10 that, as the ratio of the cross-sectional height h to beam length L increases, the dynamic magnification factors for mid-span displacement obtained by the Euler-Bernoulli always remain \(1.266\) and \(1.706\) when \(c=0.25\) and \(0.5\), respectively. In contrast, the dynamic magnification factors for mid-span displacement obtained by the SIBT and Timoshenko beams become larger and larger than that obtained by the Euler-Bernoulli beams. Furthermore, \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Beam length \(L\) & \(0.1016\) m \\ Young’s modulus \(E\) & \(2.07\times 10^{11}\) N/m\({}^{2}\) \\ Shear modulus \(G\) & \(7.76\times 10^{10}\) N/m\({}^{2}\) \\ Shear correction coefficient \(k\) & \(5/6\) \\ Density \(\rho\) & \(10,663\) kg/m\({}^{3}\) \\ Square cross-sectional area \(A\) & \(4.03\times 10^{-5}\) m\({}^{2}\) \\ Area moment of inertia \(I\) & \(1.35\times 10^{-10}\) m\({}^{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters and properties of the beam. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(c\)} & \multicolumn{6}{c}{\(\boldsymbol{D_{1}}\)} & \multicolumn{6}{c}{\(\boldsymbol{D_{2}}\)} \\ \cline{2-11}  & \multicolumn{2}{c}{**SIBT**} & \multicolumn{2}{c}{**Euler-Bernoulli**} & \multicolumn{4}{c}{**SIBT**} & \multicolumn{2}{c}{**Euler-Bernoulli**} \\ \cline{2-11}  & **Exact** & **FEM** & [7] & [39] & [40] & [41] & **Exact** & **FEM** & [7] & [41] \\ \hline \(0.125\) & \(1.137\) & \(1.151\) & \(1.121\) & \(1.112\) & - & \(1.122\) & \(1.038\) & \(1.048\) & \(1.027\) & \(1.031\) \\ \hline \(0.250\) & \(1.275\) & \(1.284\) & \(1.258\) & \(1.251\) & \(1.258\) & \(1.25 until the \(h/L\) is greater than \(1/3\), the difference between the calculated results of the SIBT and Timoshenko beams is apparent. Taking \(h/L=1/2\) as an example, when \(c=0.25\), the dynamic magnification factors for the mid-span displacement obtained by the Timoshenko model is 2.652, and that by the SIBT is 2.766. When \(c=0.5\), the dynamic magnification factors for mid-span displacement obtained by the Timoshenko model is 2.772, and that by the SIBT is 2.638. Figure 11 shows the dynamic response of the same beam modeled based on the SIBT and Timoshenko models.