###### Abstract  
This paper aims at providing a simplified analytical solution for functionally graded beam stress analysis and optimized material gradation on the beam deflection. The power-law (P-FGM) and exponential (E-FGM) material functions were considered for an exact solution of the normal and shear stress distributions across the beam thickness. Optimization of material function on the FGM beam deflection, which is new of its kind, was also investigated considering both simply supported and cantilever beams. It was observed that the non-dimensional normal stress and shear stress are independent of the elastic moduli values of the constituent materials but rather depends on both the ratio of the elastic moduli and the location across the beam thickness in the E-FGM material function model. This observation was first validated from available kinds of literature and through numerical simulation using ABAQUS and extended to the P-FGM stress analysis. The maximum deflection on the FGM beam occurred for a homogenous steel beam while the minimum deflection was observed on the beam with a P-FGM material function. The results of this work demonstrate that if properly designed and optimized, FGMs can provide an alternative material solution in structural applications. functionally graded material; volume fraction; normalized stress; material functions Article  
A Simplified Stress Analysis of Functionally Graded Beams and Influence of Material Function on Deflection  
for characterizing the dynamic free vibration of a functionally graded beam with material graduation axially or transversely through the thickness based on the power-law function and a finite element method to study the static behavior of Timoshenko FGM cantilever beam subjected to a concentrated load at the free end and using power law for varying material properties through-beam thickness was investigated and proposed in [5; 6]. The static behavior of functionally graded metal-ceramic beams under transverse loading using higher-order shear deformation theory, assuming power-law function to account for material variation through-beam thickness was studied in [7]. Nonlinear bending approximate solution of FGM beam was reported in [8] using higher-order shear deformation theory and based on beam physical neutral surface. The beam material properties were assumed to be temperature-dependent and vary through-thickness. The exact free vibration and buckling analysis of tapered beam-column made of FGM material and assumed the P-FGM material variation along with the cross-section's height and the vibration behavior of composite coupled conical shell structures were presented in [9; 10; 11]. FGM beam bending solution for Timoshenko beams compared with those of Euler-Bernoulli beams is also presented in [12]. Deformation analysis of functionally graded beams by using a direct approach and linear bending of FGM beams by the differential quadratic method were investigated in [13; 14]. A new insight on the potential application of FGM for an extreme environment such as for fire protection was presented in [15; 16]. There has also been a significant interest in FGM plate analysis through both analytical and finite element investigation such as in [17; 18; 19; 20]. Literature on FGM beams under arbitrarily distributed loading, crack growth, and bilayer FGM cantilever beams are also available in [21; 22; 23]. Even though there are numerous kinds of literature on FGM beam analysis, many only focus on the FE method of analysis and use only E-FGM material models in the analysis. Thus, this paper aims to provide a simplified method and solution for FGM beam normal and shear stress analysis considering two material functions, power-law (P-FGM) and exponential (E-FGM). Also, the influence of material functions on FGM beam deflection was investigated through analytical solution considering simply supported and cantilever FGM beams which exhibited a smaller deformation compared with homogenous steel beams of the same size and similar loadings.