## 4 Numerical Example: The Axial Compression on the Part Boundary of the Circular Shaft  
The part boundary of the circular shaft is subjected to the axial compression. The computational model is shown in Figure 2. The radius \(R\) of the circular shaft is 1 dm, \(2\alpha=7.2^{\circ}\), \(p_{n}=100\) MPa, elastic modulus \(E=2200\) MPa, and Poisson ratio \(\
u=0.1\). The numerical example is viewed as 2D stress problem.  
The discretization of the circular shaft in Figure 3: 200 real elements are evenly distributed on the real boundary \(\Gamma\); the number of virtual boundary elements, namely, 100, is taken as the general number of real boundary elements, and the circular virtual boundary \(S\) (\(r=1.4\)) is evenly divided. The number of virtual elements and the number of virtual nodes are identical. Virtual nodes are positioned at the midpoint of their virtual elements. The construction of the virtual source function utilizes eight virtual nodes. Each virtual or real element adopts four Gauss points for integration by using RBFCS.  
Here, the stresses \(\sigma_{x}\) and \(\sigma_{y}\) at points \((0,y)\) and \((x,0)\) are calculated, respectively.  
Some scholars also calculate the numerical example. [PERSON] and [PERSON]. [16] employ the complex function method to obtain the analytical solution (Ass).  
The analytical solutions of stresses along the y-axis (\(\rho=y/R\)):  
\[\sigma_{x}=\frac{-2p_{n}}{\pi}\bigg{[}\frac{\left(1-\rho^{2} \right)\sin 2\kappa}{\left(1-2\rho^{2}\cos 2\kappa+\rho^{4}\right)}-\arctan \left(\frac{1+\rho^{2}}{1-\rho^{2}}\tan\alpha\right)\bigg{]}\] \[\sigma_{y}=\frac{2p_{n}}{\pi}\bigg{[}\frac{\left(1-\rho^{2} \right)\sin 2\kappa}{\left(1-2\rho^{2}\cos 2\kappa+\rho^{4}\right)}+\arctan \left(\frac{1+\rho^{2}}{1-\rho^{2}}\tan\alpha\right)\bigg{]}\] \[\tau_{xy}=0\]  
The analytical solutions of stresses along the x-axis (\(\rho=x/R\)):  
\[\sigma_{x}=\frac{-2p_{n}}{\pi}\bigg{[}\frac{\left(1-\rho^{2} \right)\sin 2\kappa}{\left(1+2\rho^{2}\cos 2\kappa+\rho^{4}\right)}-\arctan \left(\frac{1-\rho^{2}}{1+\rho^{2}}\tan\alpha\right)\bigg{]}\] \[\sigma_{y}=\frac{2p_{n}}{\pi}\bigg{[}\frac{\left(1-\rho^{2} \right)\sin 2\kappa}{\left(1+2\rho^{2}\cos 2\kappa+\rho^{4}\right)}+\arctan \left(\frac{1-\rho^{2}}{1+\rho^{2}}\tan\alpha\right)\bigg{]}\] \[\tau_{xy}=0\]  
Figure 3: Discretization of the axial compression on the part boundary of the circular shaft.  
Figure 2: The axial compression on the part boundary of the circular shaft.  
[PERSON], [PERSON], and [PERSON]. [17] apply the boundary element method (BEM), virtual stress method (VSM), and displacement discontinuity method (DDM) to achieve the results. [PERSON] et al. take the quarter of the circular shaft as the computational model, taking into account the symmetry of this problem; namely, the quarter of circular boundary and two radial boundaries are equally partitioned into 50, 20, and 20 elements, and 200 elements on the whole circular boundary; however, the virtual element number is 100 in this paper, on account of the virtual load as the unknown physical quantity. The stresses \(\sigma_{x}\) (MPa) and \(\sigma_{y}\) (MPa) at points (0, \(y\)) and (\(x\), 0) are shown in Tables 1 and 2, respectively. \(x\) and \(y\) are dm. The computational results show that the problem can be solved by the method in this paper and the computational accuracy is proved. The stress contour maps of \(\sigma_{x}\), \(\sigma_{y}  
Figure 4: \(\sigma_{x}\) contour map of the axial compression on the part boundary of the circular shaft.  
Figure 5: \(\sigma_{y}\) contour map of the axial compression on the part boundary of the circular shaft.