## 4 Results and Discussion
### Exponential Function FGM Beam
#### 4.1.2 Shear Stress Distribution in E-FGM Beam  
Similarly, the shear stress distribution in the FGM beam was evaluated using Equation (30). From the normalized shear stress distribution, as shown in Figure 4, it is evident that the maximum normalized shear stress is in the neutral axis locations for each E\({}_{1}\)/E\({}_{2}\) considered. Moreover, the normalized shear stress is 1.5 times the average shear stress for E\({}_{1}\) = E\({}_{2}\), which is consistent with the classical beam theory. However, this maximum ratio is 1.68, 1.55, 1.49, and 1.62 for E\({}_{1}\)/E\({}_{2}\) = 10, E\({}_{1}\)/E\({}_{2}\) = 5, E\({}_{1}\)/E\({}_{2}\) = 0.5, and E\({}_{1}\)/E\({}_{2}\) = 0.1, respectively. This observation implies that an FGM beam could withstand more shear stress than a homogenous and isotropic beam.  
To validate the results presented above for E-FGM beam axial stress, a numerical simulation was performed using ABAQUS and compared with results from [2] and [26]. The FGM beams were modeled in ABAQUS with a step-wise material gradation from ceramic at top of the beam (Ec = 390 MPa) to steel (Es = 210 MPa) using 16-layers (6.25 mm  
Figure 4: Normalized Shear stress \(\propto\)xz distribution in simply supported E-FGM beam.  
Figure 3: Normalized Axial stress \(\propto\)xx distribution in simply supported E-FGM beam.  
thick each layer) across the 10 cm thickness of the beam where each layer is assumed to be isotropic and homogenous based on the volume fraction. The ABAQUS model is shown in Figure 5.  
An eight-node linear brick with a reduced integration element type (C3D8R) with a mesh size of 5 mm was used for discretization. A simply-supported boundary condition was considered in the model and a uniformly distributed load of 1 KN/m was applied on the beam top face. Mesh sensitivity was first performed for mesh sizes of 10 mm, 7.5 mm, 5.0 mm, and 2.0 mm with corresponding stress ratios the top and bottom beam flexural stresses shown in Table 1.  
It can be seen from Figure 6 that the analytical solution for stress distribution presented here for E-FGM agrees well with numerical simulation and with similar literature as shown in Table 2. The comparison was made by considering the ratio of the top and bottom beam normalized flexural stresses along the beam longitudinal axis.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Mesh Size** & \multirow{2}{*}{\(\mathbf{E_{1}/E_{2}=10}\)} & \multirow{2}{*}{\(\mathbf{E_{1}/E_{2}=0.1}\)} \\
**(mm)** & & \\ \hline
2.0 & \(-\)0.25 & \(-\)4.65 \\
5.0 & \(-\)0.244 & \(-\)4.60 \\
7.5 & \(-\)0.241 & \(-\)4.14 \\
10 & \(-\)0.237 & \(-\)4.20 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mesh size sensitivity.  
Figure 5: Numerical modeling of FGM beam in ABAQUS.  
Figure 6: Numerical models for normal stress distribution in E-FGM beam (**a**) \(\mathrm{E_{1}/E_{2}=10}\), (**b**) \(\mathrm{E_{1}/E_{2}=1.0}\), and (c) \(\mathrm{E_{1}/E_{2}=0.1}\).  
The analytical solution well agreed with available literature results in all cases, while a slight difference in the numerical model was observed. The reasons for such differences are, a stepwise material distribution is used in the ABAQUS model, as opposed to continuous variation in the integral of the analytical solution. Furthermore, current commercial software, such as ABAQUS, concentrates on approximate modeling using piecewise homogenous material assignments. Another reason is the number of layers used (sixteen-layers) in a 3D model simulation makes its error susceptible. However, the overall stress distribution in all numerical model cases agrees with the distribution from the analytical solution.