## 3 First-Order Theory of Beams (TBT)
### Exact Solutions
#### 3.5.2 Pinned-Pinned Beams  
The exact solution of a beam pinned at both ends (\(u(0)=0\), \(w(0)=0\), \(M_{xx}(0)=0\), \(u(L)=0\), \(w(L)=0\), and \(M_{xx}(L)=0\)) is  
\[D_{xx}^{*}\,u(\xi) =B_{xx}\frac{q_{0}L^{3}}{12}\Big{(}\xi-3\xi^{2}+2\xi^{3}\Big{)}, \tag{69}\] \[D_{xx}^{*}\,\phi_{x}(\xi) =\frac{B_{xx}^{2}}{D_{xx}}\frac{q_{0}L^{3}}{24}(1-2\xi)\] \[\quad-A_{xx}\frac{q_{0}L^{3}}{24}\Big{(}1-6\xi+4\xi^{3}\Big{)},\] (70) \[D_{xx}^{*}\,w(\xi) =-\frac{B_{xx}^{2}}{D_{xx}}\frac{q_{0}L^{4}}{24}\Big{(}\xi-\xi^{2 }\Big{)}\] \[\quad+A_{xx}\frac{q_{0}L^{4}}{24}\Big{(}\xi-2\xi^{3}+\xi^{4} \Big{)}+\frac{D_{xx}^{*}}{S_{xz}}\frac{q_{0}L^{2}}{2}\Big{(}\xi-\xi^{2}\Big{)},\] (71) \[\bar{M}_{xx} =\frac{q_{0}L^{2}}{2}\Big{(}\xi-\xi^{2}\Big{)},\ \ N_{xz}=\frac{q_{0}L}{2}(1-2\xi),\ \ N_{xx}=0. \tag{72}\]  
It is clear that all functions except the transverse deflection are the same as those predicted by the classical beam theory (CBT). The transverse deflection has an additional positive term that _adds_ to the value predicted by the CBT. Thus, the first-order shear deformation theory (TBT) deflection \(w\) is larger than those predicted by the CBT (i.e., the CBT underpredicts \(w\)).  
The numerical results obtained by the TBT are either the same or very close to those obtained using the CBT. Because of the fact that the beam considered is a thin beam with length-to-height ratio of \(L/h=100\), the effect of the shear deformation is not seen. Table 1 shows the numerical results obtained with the two theories for the data:  
\[E_{1}=30\times 10^{6}\ \text{psi}\ (210\ \text{GPa}),\ \ E_{2}=3\times 10^{6}\ \text{psi}\ (21\ \text{GPa}),\ \ \ v=0.3,\ \ K=\frac{5}{6}\]  
Transverse deflections obtained with the CBT and TBT for three different length-to-height ratios, \(L/h=50\), \(L/h=20\), and \(L/h=10\) are presented in Table 2. It is clear that when the beam is moderately thick (\(L/h=20\)) to very thick (\(L/h=10\)), the CBT under predicts the deflections, although the difference between the two solutions may not be significant.  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \(\boldsymbol{n}\) & \(\boldsymbol{\bar{u}}\)**-CBT** & \(\boldsymbol{\bar{u}}\)**-TBT** & \(\boldsymbol{w}\)**-CBT** & \(\boldsymbol{w}\)**-TBT** & \(\boldsymbol{\theta}_{x}\) & \(\boldsymbol{\phi}_{x}\) \\ \hline
0.0 & 0.00000 & 0.00000 & 0.5208 & 0.5210 & 0.01657 & 0.01657 \\
1.0 & 0.09973 & 0.09973 & 1.0014 & 1.0016 & 0.03062 & 0.03062 \\
2.0 & 0.20118 & 0.20118 & 1.2635 & 1.2638 & 0.03722 & 0.03722 \\
3.0 & 0.26301 & 0.26301 & 1.4261 & 1.4265 & 0.04148 & 0.04148 \\
4.0 & 0.29297 & 0.29297 & 1.5440 & 1.5445 & 0.04495 & 0.04495 \\
5.0 & 0.30523 & 0.30523 & 1.6415 & 1.6420 & 0.04806 & 0.04806 \\
6.0 & 0.30850 & 0.30850 & 1.7286 & 1.7292 & 0.05097 & 0.05097 \\
10.0 & 0.29367 & 0.29367 & 2.0305 & 2.0312 & 0.06131 & 0.06131 \\
20.0 & 0.24302 & 0.24302 & 2.6047 & 2.6056 & 0.08080 & 0.08080 \\ \hline \end{tabular}
\end{table}
Table 1: Numerical results obtained with the classical (CBT) and the shear deformation (TBT) beam theories for the displacements \(\boldsymbol{\bar{u}}=u(0.25L)\times 10^{2}\) and \(w(0.5L)\) and slopes \(\theta_{x}(L)=-(dw/dx)(L)\) and \(\phi_{x}(L)\) of a _pinned–pinned_ FGM beams under a uniformly distributed load (all results are normalized with the load).