## 4 Results and Discussion
Figure 7 shows the comparison of iso-shear-stress (\(\tau_{xz}/q\)) obtained by the HDQM and ABAQUS for the C-C sandwich beam under locally distributed loading (\(L_{p}=0.1L\)). It is seen that the HDQ results agree well with the ones of ABAQUS. The through-thickness distribution of core shear stress (\(\tau_{xz}/q\)) is simple and close to linear. The larger gradient is observed only at the boundary and loading areas, while the shear stress is almost constant in other areas of the core. The comparisons of core displacements and stresses, shown in Figures 3-7, verify further the correctness of the modeling strategy, the HDQM and the written program. Figure 5: Comparisons of iso-stress (\(\sigma_{xx}/q\)) diagram for sandwich panels under locally distributed loads (\(L_{p}=0.1L\)): (**a**) HDQM; and (**b**) ABAQUS. Figure 6: Comparisons of iso-stress (\(\sigma_{xz}/q\)) diagram for sandwich panels under locally distributed loads (\(L_{p}=0.1L\)): (**a**) HDQM; and (**b**) ABAQUS. Figure 7: Comparisons of iso-displacement (\(\tau_{xz}/q\)) diagram for sandwich panels under locally distributed loads (\(L_{p}=0.1L\)): (**a**) HDQM; and (**b**) ABAQUS. The through-thickness variations of three in-plane stresses with different loading at the section \(x=-0.0464L\) are shown in Figures 8-10. In the figures, the locally distributed loading for \(L_{p}/L=0.001\) and the point load are not included since they almost coincide with the ones of \(L_{p}/L=0.01\). Finite element (FE or FEM) data for \(L_{p}/L=0.01\) and \(0.2\) are included for verifications. It is seen that the HDQ data agree well with FE data. For the normalized axial stress \(\sigma_{xx}/q\), both the distribution pattern and the location of maximum value depend on the size of the loading area, i.e., the value of \(L_{p}/L\). The through-thickness variations of \(\sigma_{xx}/q\) for loading case of \(L_{p}/L=0.01\) is non-linear and approximately linear for all remaining loadings. Both the transverse normal stress and shear stress have similar through-thickness distributions with different size of loading area. The through-thickness variation of \(\sigma_{zz}/q\) is approximately linear and the through-thickness variation of \(\tau_{xz}/q\) also seems linear but is actually non-linear if smaller scale is used. The maximum of \(\sigma_{zz}/q\) is always located at top of the core (\(z=+c\)). The Maximum of \(\sigma_{xx}/q\) may locate at \(z=+c\) when \(L_{p}/L\) is large, i.e., \(0.5\) and \(1.0\), or locate at \(z=-c\) when loading is limited to a moderate area, i.e., \(L_{p}/L=0.1\) and \(L_{p}/L=0.2\), or locate at the center of the core when the distributed load is applied at an extremely small area, i.e., \(L_{p}/L=0.01\) and \(0.001\) and even when the point load is applied. To show the similarity and difference of the through-thickness distributions more clearly for different loadings, much smaller scale than the one in Figure 10 is used in Figures 11 and 12. Figure 11 shows the through-thickness variation of \(\tau_{xz}/q\) at section \(x=-0.0464L\) which is close to the loading area. Two locally distributed loading (\(L_{p}/L=0.01\) and \(0.001\)) and the point load are considered. It is seen that the through-thickness variation of \(\tau_{xz}/q\) is non-linear and small difference between locally distributed load (\(L_{p}/L=0.01\)) and (\(L_{p}/L=0.001\)) is observed. However, the small difference cannot be shown on the larger scale in Figure 10. The through-thickness variations of \(\tau_{xz}/q\) for the locally distributed load (\(L_{p}/L=0.001\)) and point load coincide with each other, even in such a small scale. In other words, the two loading cases are identical. Figure 8: Comparison of stress (\(\sigma_{xx}/q\)) of sandwich panels at \(x=-0.0464L\). Figure 9: Comparison of stress (\(\sigma_{zz}/q\)) of sandwich panels at \(x=-0.0464L\). Figure 12 shows the through-thickness variation of \(\tau_{xz}/q\) at section \(x=-0.4726L\) which is close to the fixed end. Four loading cases are considered.