## 5 Conclusions  
A simplified analytical formulation for functionally graded beam stresses analysis and optimization of material function for deflection were presented. The material properties in the FGM beam were assumed to vary in the thickness direction using power-law (P-FGM) and exponential function (E-FGM) material functions. The FE model was also conducted using ABAQUS and some of the results obtained were validated by comparing with literature. The following observations were drawn from the results of the analysis:  
* The non-dimensional normal stress and shear stress are independent of the actual elastic moduli values but rather depend on the ratio of E1/E2 and the location across the beam thickness (z) for the E-FGM model.
* The maximum normalized shear stress is located at the neutral axis locations for each E1/E2 considered. Moreover, the normalized shear stress is 1.5 times the average shear stress for E1 = E2, which is consistent with the classical beam theory. However, this maximum ratio is 1.68, 1.55, 1.49, and 1.62 for E1/E2 = 10, E1/E2 = 5, E1/E2 = 0.5, and E1/E2 = 0.1, respectively.
* That the maximum shear stress for each power-law index value considered corresponds at the neural axis location along with the beam thickness. The maximum and minimum normalized shear stress are observed for n = 0.1 and n = 2.0 with 1.45 and 1.14 times the average shear, respectively.
* The maximum deflection on the FGM beam occurred for a homogenous steel beam while the minimum deflection is for P-FGM with power-law index value n = 0.1 while E-FGM showed the second-lowest deflection in both simply supported and cantilever beams.  
The observations and findings in this paper considered only E-FGM and P-FGM material functions. Future studies need to include the S-FGM material function to derive a similar simplified stress analysis solution, perform more parametric studies including  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Length (m)** & **Steel** & **E-FGM** & **n = 0.1** & **n = 1** & **n = 5** \\ \hline
0 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
1 & \(-\)0.0023 & \(-\)0.0016 & \(-\)0.0013 & \(-\)0.0017 & \(-\)0.0019 \\
2 & \(-\)0.0044 & \(-\)0.0031 & \(-\)0.0025 & \(-\)0.0032 & \(-\)0.0036 \\
3 & \(-\)0.0061 & \(-\)0.0042 & \(-\)0.0034 & \(-\)0.0044 & \(-\)0.0050 \\
4 & \(-\)0.0071 & \(-\)0.0049 & \(-\)0.0040 & \(-\)0.0051 & \(-\)0.0058 \\
5 & \(-\)0.0074 & \(-\)0.0052 & \(-\)0.0042 & \(-\)0.0054 & \(-\)0.0061 \\
6 & \(-\)0.0071 & \(-\)0.0049 & \(-\)0.0040 & \(-\)0.0051 & \(-\)0.0058 \\
7 & \(-\)0.0061 & \(-\)0.0042 & \(-\)0.0034 & \(-\)0.0044 & \(-\)0.0050 \\
8 & \(-\)0.0044 & \(-\)0.0031 & \(-\)0.0025 & \(-\)0.0032 & \(-\)0.0036 \\
9 & \(-\)0.0023 & \(-\)0.0016 & \(-\)0.0013 & \(-\)0.0017 & \(-\)0.0019 \\
10 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simply supported FGM beam deflection.  
different loading types, boundary conditions, and optimization of material functions for both stress and deflection requirements of FGM beams.  
Conceptualization, E.A.; methodology, E.A.; software, E.A.; validation, F.A., writing--original draft preparation, E.A.; writing--review and editing, F.A. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.