## 4 Results and Discussion  
Dimensions of the sandwich panel shown in Figure 1 are \(f_{l}=f_{b}=2\) mm, \(c=8\) mm and \(L=400\) mm. The elastic modulus of the graphite/epoxy faces is \(E_{1}^{L}=E_{1}^{b}=181\) GPa. The material properties of glass phenolic honeycomb core are \(E_{xx}=32\) MPa, \(E_{zz}=300\) MPa, \(G_{xz}=48\) MPa and \(v_{zx}=0.25\). These data are directly taken from [4] for comparison purpose. The normalized deflection \(w_{n}\) is defined as  
\[w_{n}=\frac{3 PL^{3}}{2\pi^{4}E_{1}^{l}f_{t}^{3}} \tag{26}\]  
where \(P=qL_{p}\) is the total amount of applied load shown in Figure 1. The non-dimensional displacements are the corresponding displacements divided by \(w_{n}\) and the non-dimensional stresses are the corresponding stresses divided by \(q=P/L\). In the literature, the aspect ratio of rectangular plate investigated by using the DQM or HDQM is usually less than 4 and \(N=M\) to achieve best accuracy of the solution. For present investigation, the aspect ratio of the core reaches 25, thus \(N>>M\). To investigate the effect of the number of grid points on the solution accuracy, static analysis of sandwich beams under locally distributed loads is performed by the HDQM with different number (\(N\)) of grid points in \(x\) direction and \(M=9\) (fixed). Seven loads are considered, including uniformly distributed load (\(L_{p}/L=1\)), five locally distributed loads (\(L_{p}/L=0.5\), 0.2, 0.1, 0.01, 0.001), and the point load. Table 1 lists the non-dimensional deflections of top (denoted by \(c\)) and bottom (denoted by \(-c\)) face sheets at \(x=0\). \(N\) varies from 11 to 103 and \(M=9\). With the self-written FORTRAN code and using a personal computer, the required CPU (central processing unit) time ranges from several seconds to 2.7 mins. It is believed that with better optimized code and a better computer, less CPU time will be taken. Taking \(M=9\) is based on the fact of that the variation of displacement in \(z\) direction is rather small. Larger \(M\) is also tried and the results remain unchanged. Finite element data obtained by ABAQUS (Version 6.13-1, Dassault Systemes Simulia Corporation, Providence, RI, USA) with very fine meshes (\(2000\times 50\)) are included in the table for comparisons. The entire sandwich panel is modeled by 100,000 rectangular elements named CPS8R (the 8-node bi-quadratic plane stress element with reduced integration). It is seen that the rate of convergence decrease with the decrease of \(L_{p}/L\). However, the difference between the HDQM data and finite element data is small when \(N=61\). When \(N=103\), the results obtained by the HDQM are almost the same as the ones by ABAQUS. It is also observed that no difference between the locally distributed load (\(L_{p}/L=0.001\)) and the point load. In other words, the two loading cases are equivalent as was reported early in the buckling analysis of rectangular plates [20; 21]. From data presented in Table 1, both formulations and developed programs are verified. Figure 3 shows the comparison of the core displacement \(u/w_{n}\) obtained by the HDQM and ABAQUS for the C-C sandwich beam under the locally distributed loading (\(L_{p}=0.1L\)). The data shown in Figure 3 and other figures followed are obtained by using \(N=103\) to ensure the solution accuracy. Note that the scale in the thickness direction of the plot is enlarged for clarity. In other words, the ratio of unit length in \(x\)-direction and \(z\)-direction in Figure 3 does not equal to 25. The number of iso-displacement and the color bar may also be slightly different, since they are drawn using different software. It is seen that the HDQ results agree well with the ones of ABAQUS. The through-thickness distribution of the core displacement \(u/w_{n}\) is very complicated due to fixed ends and locally distributed load. It should be pointed out that the magnitude of the axial displacement is very small.