## 4 Results and Discussion
### Influence of Material Function on FGM Beam Deflection  
The deflections along the FGM beam w(x) can be obtained by integrating both sides of the curvature equation for x and applying boundary conditions. Two boundary conditions were considered: simply supported and Cantilever FGM Beams.  
\[\kappa=\frac{\text{M(x)}}{\widetilde{\text{D}}}=\frac{\text{d}^{2}\text{w}}{ \text{dx}^{2}} \tag{37}\]  
where \(\widetilde{\text{D}}\) is the bending rigidity for the FGM beam and can be obtained as  
\[\widetilde{\text{D}}=\text{K}_{3}-\frac{\text{K}_{2}{}^{2}}{\text{K}_{1}} \tag{38}\]  
Boundary conditions for simply supported FGM beam:  
\[\text{w}(0)=0\ \ \ \ \ \text{w}(\text{L})=0 \tag{39}\]  
Boundary conditions for cantilever FGM beam:  
\[\text{w}(0)=0\ \ \ \ \frac{\text{dw}}{\text{dx}}(0)=0 \tag{40}\]  
The deflections for both simply supported and cantilever FGM beam was calculated for unit width, 0.1 m depth, and a total length of 10 m. The S-S beam deflections were analyzed for a unit distributed load while for a cantilever beam a unit concentrated load was applied at the free end. One can observe from Figures 9 and 10 that the maximum deflection occurred for a homogenous steel beam while the minimum deflection is for P-FGM with power-law index value n = 0.1 and the E-FGM beam showed the second-lowest deflection in both simply supported and cantilever beams. The reason for this  
Figure 8: Normalized Shear stress rxz distribution in simply supported P-FGM beam.  
can be explained by Young's modulus variation in Figure 1a where for n = 0.1 significant portion, about 90% of the beam thickness, is composed of a material with a higher E-value. The opposite is true for n = 5.0 where about 80% of the beam thickness is composed of a material with the smallest E-value.  
Numerical FGM Beam Deflection Example  
Deflection on the FGM beam is evaluated by double integrating Equation (37). The flexural rigidity of the FGM beam can be evaluated using Equation (38) for each material function and index parameter. For P-FGM, substituting E\({}_{1}\) = 390 GPa (ceramic) and E\({}_{2}\) = 210 GPa (steel) and solving the stiffness coefficients from Equations (34)-(36), the FGM beam rigidity can be obtained. For a uniformly distributed unit load of w = 1 kN/m, the moment at any distance from the left support can be expressed as:  
\[\mathrm{M(x)}=\frac{\mathrm{wL}}{2}\mathrm{x}-\frac{\mathrm{wx}^{2}}{2} \tag{41}\]  
Figure 10: Deflection of cantilever FGM beam.  
Figure 9: Deflection of a simply supported FGM beam.  
Deflection equation along a simply supported FGM beam can be simplified as similar to the well-known isotropic-homogenous beam deflection except for the beam rigidity.  
\[\delta(\mathrm{x})=\frac{\mathrm{w}\mathrm{x}}{24\mathrm{\SIUnitSymbolDegree}}( \mathrm{L}^{3}-2\mathrm{L}\mathrm{x}^{2}+\mathrm{x}^{3}) \tag{42}\]  
The simply supported FGM beam with E-FGM, P-FGM, and homogenous steel deflections from the unit distributed load are presented in Table 3.  
It can be noted that the maximum deflections were observed in the steel beam compared to the FGM beams. Also, amongst the material function, the P-FGM with index n = 0.1 resulted in the least deflection. This observation can lead to an optimized material function in FGM beam structural application, especially in terms of deflection.