# Derivation and Verification of Laminated Composite T-Beam Theory
## 2 Derivation of the Formula for Laminated T beam
### Centroid Position in 2D Theoretical Analysis
#### 2.3.1. Symmetrical Laminated Composite T-Beam Center Position  
* Narrow flange center position  
As shown in Figure 2, under the lateral load, the resultant lateral force \(N_{y}=0\) and the bending moment \(M_{y}=0\). The relationship between the strain and resultant force and the curvature and resultant moment of the narrow-flange laminated T-beam are as follows:  
\[\left\{ \begin{matrix}e_{x}^{0}\\ e_{y}^{0}\\ e_{xy}^{0}\end{matrix} \right\}=\left[A^{-1}\right]\left\{ \begin{matrix}N_{x} \\ 0\\ 0\end{matrix} \right\},\left\{ \begin{matrix}\kappa_{x}\\ x_{y}\\ x_{xy}\end{matrix} \right\}=\left[D^{-1}\right]\left\{ \begin{matrix}M_{x}\\ 0\\ 0\end{matrix} \right\} \tag{26}\]  
By simplifying Equation (26) to Equation (27):  
\[N_{x}=\frac{1}{A_{11}^{-1}}\varepsilon_{x}^{0}=\frac{Z_{1}}{A_{11}^{-1}}\kappa _{x}\text{ and }M_{x}=\frac{1}{D_{11}^{-1}}\kappa_{x} \tag{27}\]  
The stress of the T-beam is zero at the neutral axis, as shown in Figure 1. Therefore, the sum of the stresses of the web and flange is zero Equation (28). \[N_{x,w}+N_{x,f}=0 \tag{28}\]  
The resultant stresses of the web and flange are derived from Equation (27), and shown in Equations (29) and (30):  
\[N_{x,w}=\int_{-\frac{1}{2}+Z_{c}}^{\frac{1}{2}+Z_{c}}N_{x}dz=\int_{-\frac{1}{2 }+Z_{c}}^{\frac{1}{2}+Z_{c}}\frac{z}{A_{11,w}^{-1}}\kappa_{x}dz=\frac{hZ_{c}}{A _{11,w}^{-1}}\kappa_{x} \tag{29}\]\[N_{x,f}=bN_{x}=b\Bigg{(}\frac{1}{A_{11,f}^{-1}}\varepsilon_{x}^{0} \Bigg{)}=b\Bigg{(}\frac{Z_{1}}{A_{11,f}^{-1}}\Bigg{)}\kappa_{x}=b\left(\frac{- \frac{h}{2}+Z_{c}-t}{A_{11,f}^{-1}}\right)\kappa_{x} \tag{30}\] \[\text{where }Z_{1}=\left|\frac{-h}{2}+Z_{c}-t\right| \tag{31}\]  
By substituting Equations (29) and (30) into Equation (28):  
\[\left(\frac{hZ_{c}}{A_{11,w}^{-1}}+\frac{b\left(-\frac{h}{2}+Z_{c}-t\right)}{ A_{11,f}^{-1}}\right)\kappa_{x}=0 \tag{32}\]  
where \(t\) is the distance from the flange centroid, \(o_{f}\) to the outermost layer of the flange, \(Z_{1}\) is the distance from the centroid \(c\) to \(o_{f}\), \(Z_{c}\) is the distance from \(c\) to the centroid of the web. The \(o_{w}\) can be calculated for the laminated narrow flange composite T-beam shown in Equation (33). \[Z_{c}=\frac{\frac{h}{2}+t}{1+\frac{hA_{11,f}^{-1}}{bA_{11,w}^{-1}}} \tag{33}\]  
Wide flange center position  
Under the assumption of a wide symmetrical flange, \([B]=0\). Transverse resultant force \(N_{y}=0\), \(N_{xy}=0\), and transverse curvature \(\kappa_{y}=0\), \(\kappa_{xy}=0\). The relationship between the strain and resultant force and the curvature and resultant moment of the wide-flange laminated T-beam becomes:  
\[\left\{ \begin{array}{c}N_{x}\\ 0\\ 0\\ M_{x}\\ M_{y}\\ M_{xy}\end{array} \right\}=\left[ \begin{array}{cccccc}A_{11}&A_{12}&A_{13}&0& 0&0\\ A_{21}&A_{22}&A_{23}&0&0&0\\ A_{31}&A_{32}&A_{33}&0&0&0\\ 0&0&0&D_{11}&D_{12}&D_{13}\\ 0&0&0&D_{21}&D_{22}&D_{23}\\ 0&0&0&D_{31}&D_{32}&D_{333}\end{array} \right]\left\{ \begin{array}{c}\varepsilon _{x}^{0}\\ \varepsilon_{y}^{0}\\ \gamma\gamma_{xy}^{0}\\ \kappa_{x}\\ 0\end{array} \right\} \tag{34}\]  
Figure 2: The A-A section of laminated T-beam. where \(A\) and \(D\) are laminate-stiffness matrices. Equation (34) can be simplified as follows:  
\[N_{x}=\frac{1}{A_{11}^{-1}}\epsilon_{x}^{0}=\frac{Z_{1}}{A_{11}^{-1}}x_{x}\text { and }M_{x}=\frac{1}{D_{11}^{-1}}\kappa_{x} \tag{35}\]  
The stress of the T-beam is zero at the neutral axis, as shown in Figure 1. Therefore, the sum of the stresses in the web and flange is zero Equation (36). \[N_{x,w}+N_{x,f}=0 \tag{36}\]  
The resultant stresses of the web and flange are derived from Equation (35) and are shown in Equations (37) and (38).