## 2 Classical Theory of Beams (CBT)
### Governing Equations in Terms of Displacements  
For an isotropic material the one-dimensional stress-strain relation  
\[\sigma_{xx}=E(z)\,\varepsilon_{xx}(x,z) \tag{18}\]  
We assume that the beam is graded with two material combination through the beam height according to the relation  
\[E(z)=(E_{1}-E_{2})V_{1}(z)+E_{2},\;\;V_{1}(z)=\left(\frac{1}{2}+\frac{z}{h} \right)^{n} \tag{19}\]  
where \(E_{1}\) and \(E_{2}\) are [PERSON]'s moduli of the two materials used, and \(n\) is the index that dictates the relative dominance of volume fractions \(V_{1}(z)\) and \(V_{2}(z)=1-V_{1}(z)\). We assume that [PERSON]'s ratio \(\
u\) is a constant for the FGM material.  
The stress resultants can be expressed in terms of the displacements as  
\[N_{xx} =\int_{A}\sigma_{xx}\,dA=A_{xx}\left[\frac{du}{dx}+\frac{1}{2} \bigg{(}\frac{dw}{dx}\bigg{)}^{2}\right]-B_{xx}\frac{d^{2}w}{dx^{2}} \tag{20a}\] \[M_{xx} =\int_{A}z\sigma_{xx}\,dA=B_{xx}\left[\frac{du}{dx}+\frac{1}{2} \bigg{(}\frac{dw}{dx}\bigg{)}^{2}\right]-D_{xx}\frac{d^{2}w}{dx^{2}}\] (20b) \[P_{xy} =\int_{A}M_{xy}\,dA=-A_{xy}\frac{d^{2}w}{dx^{2}} \tag{20c}\]  
where \(A_{xx}\), \(B_{xx}\), \(D_{xx}\), and \(A_{xy}\) are the extensional, extensional-bending, bending, and in-plane shear stiffness coefficients. For beams with \(E=E(x)\) (i.e., \(n=0\) or \(E_{1}=E_{2}=E\)), we have \(A_{xx}=EA_{0}\), \(B_{xx}=0\), \(D_{xx}=EI_{0}\), and \(A_{xy}=GA_{0}\ell^{2}\). For \(n\
eq 0\) (i.e., FGM beams), we have  
\[A_{xx} =\int_{A}E(z)\,dA=E_{2}A_{0}\frac{M+n}{1+n}\] \[B_{xx} =\int_{A}E(z)z\,dA=E_{2}B_{0}\frac{n(M-1)}{2(1+n)(2+n)}\] \[D_{xx} =\int_{A}E(z)z^{2}\,dA=E_{2}I_{0}\bigg{[}\frac{(6+3n+3n^{2})M+(8n +3n^{2}+n^{3})}{6+11n+6n^{2}+n^{3}}\bigg{]} \tag{21}\] \[A_{xy} =\frac{\ell^{2}}{2(1+\
u)}\int_{A}E(z)\,dA=\ell^{2}\,\frac{E_{2}A _{0}}{2(1+\
u)}\frac{M+n}{1+n}\] \[A_{0} =bh,\;\;\;B_{0}=bh^{2},\;\;\;\;I_{0}=\frac{bh^{3}}{12}\;,\;\;\;M =\frac{E_{1}}{E_{2}}.\]  
Figure 1a contains the variation of the non-dimensional axial stiffness (\(\bar{A}_{xx}=A_{xx}/E_{2}A_{0}\), \(A_{0}=bh\)) and bending stiffness \(\bar{D}_{xx}=D_{xx}/E_{2}I_{0}\), \(I_{0}=bh^{3}/12\)) as functions of the volume fraction index \(n\) for various values of the modulus ratio \(M=E_{1}/E_{2}\geq 1\) and Figure 1b contains similar plots for the non-dimensional extensional-bending coupling stiffness \(\bar{B}_{xx}=B_{xx}/E_{2}B_{0}\), \(B_{0}=bh\)). It is clear that both \(\bar{A}_{xx}\) and \(\bar{D}_{xx}\) are the maximum at \(n=0\) and decrease with increasing values of \(n\). However, \(\bar{B}_{xx}\) is zero at \(n=0\), increases to a maximum at \(n=\sqrt{2}\), and then decreases with increasing value of \(n\). Thus, beams with nonzero \(B_{xx}\) will have a response that is not monotonic with respect to \(n\).  
The equations of equilibrium can be expressed in terms of the displacements \(u\) and \(w\) using the beam constitutive relations from Equation (21). We obtain  
\[-\frac{d}{dx}\Bigg{\{}A_{xx}\Bigg{[}\frac{du}{dx}+\frac{1}{2}\bigg{(}\frac{dw}{ dx}\bigg{)}^{2}\Bigg{]}+B_{xx}\bigg{(}-\frac{d^{2}w}{dx^{2}}\bigg{)}-N_{xx}^{T} \Bigg{\}}-f=0 \tag{22}\]  
\[-\frac{d^{2}}{dx^{2}}\Bigg{\{}B_{xx}\Bigg{[}\frac{du}{dx}+\frac{1}{2}\bigg{(} \frac{dw}{dx}\bigg{)}^{2}\Bigg{]}+\big{(}D_{xx}+A_{xy}\big{)}\bigg{(}-\frac{d^{2 }w}{dx^{2}}\bigg{)}-M_{xx}^{T}\Bigg{\}}\]  
\[-\frac{d}{dx}\Bigg{\{}A_{xx}\frac{dw}{dx}\Bigg{[}\frac{du}{dx}+\frac{1}{2} \bigg{(}\frac{dw}{dx}\bigg{)}^{2}\Bigg{]}-B_{xx}\frac{dw}{dx}\frac{d^{2}w}{dx^ {2}}\Bigg{\}}-q=0 \tag{23}\]