# Derivation and Verification of Laminated Composite T-Beam Theory
## 2 Derivation of the Formula for Laminated T beam
### Centroid Position in 2D Theoretical Analysis
#### 2.3.2 Centroid Position of Nonsymmetrical Laminated Composite T-Beams  
* Narrow flange centroid position The [B] matrices are part of the non-symmetric matrices; therefore, the F matrix is used to represent the inverse of the ABD matrix, as shown in Equation (42): \[[F]\equiv \begin{bmatrix}A&B\\ B&D\end{bmatrix} ^{-1}\] (42)  
Using the assumption of a narrow beam: \(M_{y}=M_{xy}=0\); therefore, only the axial resultant force \(N_{x}\) and the resultant bending moment \(M_{x}\) exist. The relationships between the strain and resultant force and the curvature and resultant moment of the nonsymmetric narrow flange T-beam are expressed as follows:  
\[\left\{ \begin{array}{c}\epsilon_{x}^{0}\\ \epsilon_{y}^{0}\\ \gamma_{xy}^{0}\\ \kappa_{x}\\ \kappa_{y}\end{array} \right\}=\left[ \begin{matrix}F_{11}&F_{12}&F_{12}&F_{14 }&F_{15}&F_{16}\\ F_{21}&F_{22}&F_{23}&F_{24}&F_{25}&F_{26}\\ F_{31}&F_{32}&F_{33}&F_{34}&F_{35}&F_{36}\\ F_{41}&F_{42}&F_{43}&F_{44}&F_{45}&F_{46}\\ F_{51}&F_{52}&F_{53}&F_{54}&F_{55}&F_{56}\\ F_{61}&F_{62}&F_{63}&F_{64}&F_{65}&F_{66}\\ \end{matrix} \right]\left\{ \begin{array}{c}N_{x}\\ 0\\ 0\\ 0\\ 0\\ 0\end{array} \right\} \tag{43}\]By expanding and simplifying Equation (43), \(N_{x}\) and \(M_{x}\) are deduced as follows:  
\[ \begin{array}{c}N_{x}=\frac{F_{44}}{\Delta}\varepsilon_{x}^{0}-\frac{F_{44}}{ \Delta}\kappa_{x},\ M_{x}=\frac{-F_{44}}{\Delta}\varepsilon_{x}^{0}+\frac{F_{ 14}}{\Delta}\kappa_{x}\\ \Delta\equiv F_{11}F_{44}-F_{14}F_{41}\end{array}  \tag{44}\]  
The resultant stresses of the web and flange are derived from Equation (44), as shown in Equation (45) and Equation (46), respectively,  
\[N_{x,w}=\int_{-\frac{h}{2}+Z_{c}}^{\frac{h}{2}+Z_{c}}N_{x}dz=\int_{-\frac{h}{2 }+Z_{c}}^{\frac{h}{2}+Z_{c}}\frac{z}{A_{11,w}^{-1}}\kappa_{x}dz=\frac{hZ_{c}} {A_{11,w}^{-1}}\kappa_{x} \tag{45}\]  
\[N_{x,f}=bN_{x}=b\bigg{(}\frac{F_{44}}{\Delta}Z_{1}\kappa_{x}-\frac{F_{14}}{ \Delta}\kappa_{x}\bigg{)}=b\bigg{[}\bigg{(}\frac{F_{44}}{\Delta}\bigg{(}\frac {-h}{2}+Z_{c}-t\bigg{)}-\frac{F_{14}}{\Delta}\bigg{)}\bigg{]}\kappa_{x} \tag{46}\]  
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, as shown in Equation (47). \[N_{x,w}+N_{x,f}=0 \tag{47}\]  
Equations (45) and (46) are substituted into Equation (47):  
\[\bigg{[}\frac{hZ_{c}}{A_{11,w}^{-1}}+\frac{bF_{44}}{\Delta}\bigg{(}\frac{-h}{2 }+Z_{c}-t\bigg{)}-\frac{bF_{14}}{\Delta}\bigg{]}\kappa_{x}=0 \tag{48}\]  
The \(o_{w}\) can be obtained for the laminated narrow-flange composite T beam shown in Equation (49). \[Z_{c}=\frac{\frac{h}{\Delta}\bigg{(}\frac{hF_{44}}{2}+tF_{44}+F_{14}\bigg{)}} {\bigg{(}\frac{h}{A_{11,w}^{-1}}+\frac{bF_{44}}{\Delta}\bigg{)}} \tag{49}\]  
* Wide flange centroid position  
Using the assumption of \(\kappa_{y}=\kappa_{xy}=0\) for a wide beam, the relationships between the strain and resultant force and the curvature and resultant moment of the nonsymmetric wide flange T-beam are as follows:  
\[\left\{ \begin{array}{c}N_{x}\\ 0\\ 0\\ M_{x}\\ M_{y}\\ M_{xy}\end{array} \right\}=\left[ \begin{array}{cccccc}A_{11}&A_{12}&A_{13}&B _{11}&B_{12}&B_{13}\\ A_{21}&A_{22}&A_{23}&B_{21}&B_{22}&B_{23}\\ A_{31}&A_{32}&A_{33}&B_{31}&B_{32}&B_{33}\\ B_{11}&B_{12}&B_{13}&D_{11}&D_{12}&D_{13}\\ B_{21}&B_{22}&B_{23}&D_{21}&D_{22}&D_{23}\\ B_{31}&B_{32}&B_{33}&D_{31}&D_{32}&D_{33}\end{array} \right]\left\{ \begin{array}{c} \varepsilon_{x}^{0}\\ \varepsilon_{y}^{0}\\ \gamma_{xy}^{0}\\ \kappa_{x}\\ 0\\ \end{array} \right\} \tag{50}\]  
The inverse matrix of the ABD matrix Equation (50), represented by the F matrix, becomes Equation (51):  
\[\left\{ \begin{array}{c}\varepsilon_{x}^{0}\\ \varepsilon_{y}^{0}\\ \gamma_{xy}^{0}\\ \kappa_{x}\\ 0\\ \end{array} \right\}=\left[ \begin{array}{cccccc}F_{11}&F_{12}&F_{12}&F_{14}&F_ {15}&F_{16}\\ F_{21}&F_{22}&F_{23}&F_{24}&F_{25}&F_{26}\\ F_{31}&F_{32}&F_{33}&F_{34}&F_{35}&F_{36}\\ F_{41}&F_{42}&F_{43}&F_{44}&F_{45}&F_{46}\\ F_{51}&F_{52}&F_{53}&F_{54}&F_{55}&F_{56}\\ F_{61}&F_{62}&F_{63}&F_{64}&F_{65}&F_{66}\end{array}