## 3 Analytical Approach to FGM Beam Analysis
### The Mathematical Formulation for FGM Beam Analysis  
Consider a functionally graded beam with a rectangular cross-section shown in Figure 2 with length L, width b, and depth h. To simplify the FGM beam stress analysis, the following assumptions were made: (1) FGM beam is modeled with classical Euler-Bernoulli beam theory, where mid-plane remains undeformed; (2) Poisson ratio is assumed constant along with the beam thickness; and (3) normal stress in z-z axis is neglected. The axial and transverse displacement fields can be expressed as in Equation (8), which are derived from [3]. \[\mathrm{u}(\mathrm{x},\mathrm{z})=\mathrm{u}_{0}(\mathrm{x})+\mathrm{z}\phi( \mathrm{x})\ \ \mathrm{w}(\mathrm{x},\mathrm{z})=\mathrm{w}_{0}(\mathrm{x}) \tag{8}\]  
where \(\varnothing\) is the first derivative of the transverse displacement. The constitutive relation of the FGM beam can be assumed to obey [PERSON]'s law and can be written as  
\[\left\{\sigma\right\}=\left\{ \begin{array}{c}\sigma_{\mathrm{xx}}\\ \tau_{\mathrm{xx}}\end{array} \right\}=\left[ \begin{array}{cc}\mathrm{E}( \mathrm{z})&0\\ 0&\mathrm{G}(\mathrm{z})\end{array} \right]\left\{ \begin{array}{c}\varepsilon _{\mathrm{xx}}\\ \gamma_{\mathrm{xz}}\end{array} \right\} \tag{9}\]  
where \(\sigma_{\mathrm{xx}}\) and \(\varepsilon_{\mathrm{xx}}\) are normal stresses and normal strains in the x-direction and \(\tau_{\mathrm{xz}}\) and \(\gamma_{\mathrm{xz}}\) are the shear stress and shear strain in the x-z plane. E(z) and G(z) are the Young's and shear moduli, which are functions of the FGM beam depth z. The axial strain can be expressed using Equation (10). \[\varepsilon_{\mathrm{x}}=\varepsilon_{\mathrm{x0}}+\mathrm{z}\kappa \tag{10}\]  
where \(\varepsilon_{\mathrm{x0}}\), \(\kappa\) and z are the normal strain at mid-plain, the curvature, and the distance from the neutral axis of the FGM beam. Using the Euler-Bernoulli beam theory, the resultant axial force (N) and bending moment (M) can be evaluated using Equations (11) and (12). \[\mathrm{N}=\int\sigma_{\mathrm{x}}\mathrm{d}A \tag{11}\]  
\[\mathrm{M}=\int z\sigma_{\mathrm{x}}\mathrm{d}A \tag{12}\]  
Substituting Equations (9) and (10) into Equations (11) and (12), the resultant forces can be expressed including the material variation through the beam thickness. \[\mathrm{N}=\int\mathrm{E}(\mathrm{z})(\varepsilon_{\mathrm{x0}}+\mathrm{z} \kappa)\mathrm{d}\mathrm{z} \tag{13}\]  
Figure 2: The geometry of the functionally graded beam. Where h, b and L are beam cross-section width, depth and length respectively. \[\text{M}=\int z\text{E}(\text{z})(\epsilon_{x0}+z\kappa)\text{d}z \tag{14}\]  
Equations (13) and (14) can further be related using the matrix form below  
\[\left[ \begin{array}{c}\text{N}\\ \text{M}\end{array} \right]=\left( \begin{array}{cc}\text{K}_{1}&\text{K}_{2} \\ \text{K}_{2}&\text{K}_{3}\end{array} \right)\left\{ \begin{array}{c}\varepsilon_{ x0}\\ \kappa\end{array} \right\} \tag{15}\]  
where K\({}_{1}\), K\({}_{2}\) and K\({}_{3}\) are stiffness coefficients and can be expressed as  
\[\text{K}_{1}=\int\text{E}(\text{z})\text{d}z \tag{16}\]  
\[\text{K}_{2}=\int z\text{E}(\text{z})\text{d}z \tag{17}\]  
\[\text{K}_{3}=\int z^{2}\text{E}(\text{z})\text{d}z \tag{18}\]  
The inverse relation for Equation (15) can be written as  
\[\left\{ \begin{array}{c}\varepsilon_{x0}\\ \kappa\end{array} \right\}=\left( \begin{array}{cc}\text{K}_{1}&\text{K}_{2} \\ \text{K}_{2}&\text{K}_{3}\end{array} \right)^{-1}\left[ \begin{array}{c}\text {N}\\ \text{M}\end{array} \right]=\left( \begin{array}{cc}\text{K}_{1}{}^{*}&\text{ K}_{2}{}^{*}\\ \text{K}_{2}{}^{*}&\text{K}_{3}{}^{*}\end{array} \right)\left[ \begin{array}{c} \text{N}\\ \text{M}\end{array} \right] \tag{19}\]  
where K\({}_{1}{}^{*}\), K\({}_{2}{}^{*}\) and K\({}_{3}{}^{*}\) are the inverse stiffness coefficients.