# Dynamic Response of Slope Inertia-Based Timoshenko Beam under a Moving Load
## 5 Comparative Study
The mechanical parameters of the beam are listed in Table 1. For comparison purposes, the dynamic magnification factors of displacement and moment here are defined as: \(D_{3}=\max[y(vt,t)/y_{st}(L/2)]\), which means the displacement of the beam is evaluated at the location of the load \(P\) during the full moving time. For a Rayleigh parameter \(b=0.15\), the critical speed parameter \(c^{*}=0.958\). As can be seen from Figure 11, the normalized displacement calculated by the SIBT theory is in good agreement with that calculated by the Timoshenko beam theory (TBT). The reason is that although the SIBT takes the effect of shear inertia into account compared with the Timoshenko, the effect of shear inertia is negligible because the beam (with a small value of \(b\)) is very slender. Figure 10: Dynamic magnification factors of the beam with different slenderness ratios. Table 4 compares the results of the beam modelled according to the SIBT and Timoshenko theory. Two beam sections are considered, namely \(b\) = 0.03 and 0.15. As the moving speed of the load increases, the displacement amplification factor \(D_{3}\) also changes nonlinearly. When \(b\) = 0.03, the calculation results of the SIBT beam are very similar to the results of the Timoshenko beam in the literature. When \(b\) = 0.15, the beam is still a slender beam. Although the difference between the results from the SIBT and Timoshenko becomes obvious, the overall difference in the results is not obvious. Figure 12 further presents the dynamic magnification factors of mid-span displacement of the SIBT, Euler-Bernoulli and Timoshenko beams with different slenderness ratios. Based on Table 1, [PERSON]'s coefficient \(b\) is reset from \(\pi/32\) to \(\pi/4\), while keeping the beam length constant. As the ratio of the cross-sectional diameter to beam length increases, we can draw conclusions similar to Figure 10. For the Euler-Bernoulli beam, the dynamic magnification factors for mid-span displacement remains 1.075 and 1.706 for \(c\) = 0.11 and 0.5, respectively. The dynamic magnification factors for mid-span displacement obtained by the SIBT and Timoshenko beams are constantly increasing. Furthermore, until the ratio \(2r_{0}/L\) is greater than \(1/3\), the difference between the calculated results of the SIBT and Timoshenko beams becomes apparent. This phenomenon is attributed to the significant influence of the shear inertia in the SIBT model. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{\(D_{3}\)} \\ \cline{2-7} \(c\) & \multicolumn{4}{c}{**SIBT**} & \multicolumn{4}{c}{**Timoshenko**} \\ \cline{2-7} \(c\) & \multicolumn{2}{c}{\(b\) = 0.03} & \multicolumn{2}{c}{\(b\) = 0.15} & \multicolumn{2}{c}{\(b\) = 0.03} & \multicolumn{2}{c}{\(b\) = 0.15} \\ \cline{2-7}  & **Exact** & **FEM** & **Exact** & **FEM** & [35] & [35] \\ \hline
0.11 & 1.044 & 1.039 & 1.144 & 1.149 & 1.039 & 1.115 \\
0.3 & 1.410 & 1.412 & 1.557 & 1.557 & 1.411 & 1.555 \\
0.45 & 1.610 & 1.611 & 1.744 & 1.747 & 1.610 & 1.731 \\
0.5 & 1.602 & 1.603 & 1.712 & 1.715 & 1.602 & 1.717 \\
0.7 & 1.334 & 1.332 & 1.368 & 1.371 & 1.334 & 1.366 \\
0.9 & 1.034 & 1.034 & 1.142 & 1.143 & 1.034 & 1.134 \\
0.958 & 0.979 & 0.980 & 1.081 & 1.085 & - & - \\
0.998 & 0.946 & 0.947 & 1.043 & 1.045 & - & - \\
1.1 & 0.873 & 0.873 & 0.946 & 0.949 & 0.873 & 0.943 \\
1.3 & 0.740 & 0.739 & 0.776 & 0.776 & 0.740 & 0.777 \\
1.5 & 0.603 & 0.602 & 0.645 & 0.646 & 0.603 & 0.637 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(D_{3}\) for different cross-section and moving speeds. Figure 11: Normalized displacement of the beam.