## 4 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(\mathbf{L_{p}}\) & & \(\mathbf{N=11}\) & \(\mathbf{N=21}\) & \(\mathbf{N=31}\) & \(\mathbf{N=41}\) & \(\mathbf{N=51}\) & \(\mathbf{N=61}\) & \(\mathbf{N=103}\) & **ABAQUS** \\ \hline \multirow{2}{*}{\(L\)} & \(c\)* & 0.06780 & 0.06744 & 0.06745 & 0.06745 & 0.06745 & 0.06745 & 0.06745 & 0.06745 & 0.06745 \\  & \(-c\)* & 0.06770 & 0.06735 & 0.06735 & 0.06735 & 0.06735 & 0.06735 & 0.06735 & 0.06735 \\ \hline \multirow{2}{*}{\(0.5L\)} & \(c\) & 0.10603 & 0.10577 & 0.10576 & 0.10575 & 0.10575 & 0.10575 & 0.10575 & 0.10575 \\  & \(-c\) & 0.10583 & 0.10556 & 0.10556 & 0.10556 & 0.10556 & 0.10556 & 0.10556 & 0.10556 \\ \hline \multirow{2}{*}{\(0.2L\)} & \(c\) & 0.12740 & 0.12736 & 0.12724 & 0.12725 & 0.12726 & 0.12725 & 0.12725 & 0.12725 \\  & \(-c\) & 0.12692 & 0.12681 & 0.12676 & 0.12676 & 0.12676 & 0.12676 & 0.12676 & 0.12676 \\ \hline \multirow{2}{*}{\(0.1L\)} & \(c\) & 0.13146 & 0.13266 & 0.13288 & 0.13287 & 0.13285 & 0.13285 & 0.13285 & 0.13285 & 0.13285 \\  & \(-c\) & 0.13090 & 0.13172 & 0.13183 & 0.13183 & 0.13183 & 0.13183 & 0.13183 & 0.13183 \\ \hline \multirow{2}{*}{\(0.01L\)} & \(c\) & 0.13287 & 0.13471 & 0.13537 & 0.13553 & 0.13561 & 0.13564 & 0.13567 & 0.13568 \\  & \(-c\) & 0.13228 & 0.13356 & 0.13387 & 0.13386 & 0.13387 & 0.13387 & 0.13387 & 0.13387 \\ \hline \multirow{2}{*}{\(0.001L\)} & \(c\) & 0.13288 & 0.13473 & 0.13540 & 0.13556 & 0.13564 & 0.13567 & 0.13571 & 0.13572 \\  & \(-c\) & 0.13230 & 0.13358 & 0.13389 & 0.13388 & 0.13389 & 0.13389 & 0.13389 & 0.13389 \\ \hline \multirow{2}{*}{Point load} & \(c\) & 0.13288 & 0.13473 & 0.13540 & 0.13556 & 0.13564 & 0.13567 & 0.13571 & 0.13572 \\  & \(-c\) & 0.13230 & 0.13358 & 0.13389 & 0.13388 & 0.13389 & 0.13389 & 0.13389 \\ \hline \hline \end{tabular}
* \(c\) represents \(w(c,0)\); \(-c\) represents \(w(-c,0)\).  
\end{table} 
Table 1: Convergence study of the HDQ (harmonic differential quadrature) center deflection \(w(\pm c,0)/w_{n}\) for the C-C sandwich panels under locally distributed load (\(M=9\)). Figure 4 shows the comparison of the core displacement \(w/w_{n}\) obtained by the HDQM and ABAQUS for the ABAQUS for the C-C sandwich beam under the same loading (\(L_{p}\) = 0.1\(L\)). Again the scale in the thickness direction is enlarged for clarity. It is seen that the HDQ results agree well with the ones of ABAQUS. The through-thickness distribution of the core stress (\(\sigma_{xx}/q\)) varies along the axial direction due to fixed ends and locally distributed load. High value of stress is observed at the fixed boundary and the sign change of the curvature near the fixed edge areas causes the larger gradient of core axial stress, since the fixed end bending moment quickly reduces to zero at the inflection point close to the fixed end. At the middle of the sandwich panel, lower part of the core has higher axial stress level than the upper part of the core, although the upper part is closer to the loading area. Figure 6 shows the comparisons of iso-stress (\(\sigma_{zz}/q\)) obtained by the HDQM and ABAQUS for the C-C sandwich beam under the same loading (\(L_{p}\) = 0.1\(L\)). It is seen that the HDQ results agree well with the ones of ABAQUS. The through-thickness distribution of the core stress (\(\sigma_{zz}/q\)) is also varies along the axial direction due to fixed ends and locally distributed load. The stress at the fixed boundary and loading areas is much higher than other areas. Different from the axial stress distribution, at the  
Figure 4: Comparisons of iso-displacement (\(w/w_{n}\)) diagram for sandwich panels under locally distributed loads (\(L_{p}\) = 0.1\(L\)): (**a**) HDQM; and (**b**) ABAQUS. Figure 3: Comparisons of iso-displacement (\(u/w_{n}\)) diagram for sandwich panels under locally distributed loads (\(Lp\) = 0.1\(L\)): (**a**) HDQM (harmonic differential quadrature method); and (**b**) ABAQUS. middle of the sandwich panel, the upper part of the core sheet, closer to the loading area, has higher transverse stress then the lower part.