# Derivation and Verification of Laminated Composite T-Beam Theory
## 2 Derivation of the Formula for Laminated T beam
### Stress and Strain of Nonsymmetric Laminated Composite T-Beam
#### 2.5.2 Wide Flange  
The assumption used for the nonsymmetric wide flange was that the resultant transverse force \(N_{y}=0\) and transverse curvature \(\kappa_{y}=0\); therefore, the relationships between the strain and resultant force and the curvature and resultant moment of the T-beam are as follows:  
\[\left\{ \begin{array}{c}N_{x}\\ 0\\ 0\\ M_{x}\\ M_{y}\\ M_{xy}\end{array} \right\}=\left[ \begin{array}{cccccc}A_{11}&A_{12}&A_{13}&B_{11} &B_{12}&B_{13}\\ A_{21}&A_{22}&A_{23}&B_{21}&B_{22}&B_{23}\\ A_{31}&A_{32}&A_{33}&B_{31}&B_{32}&B_{33}\\ B_{11}&B_{12}&B_{13}&D_{11}&D_{12}&D_{13}\\ B_{21}&B_{22}&B_{23}&D_{21}&D_{22}&D_{23}\\ B_{31}&B_{32}&B_{33}&D_{31}&D_{32}&D_{33}\end{array} \right\}\left\{ \begin{array} []{c}\epsilon_{x}^{0}\\ \epsilon_{y}^{0}\\ \epsilon_{y}^{0}\\ \gamma_{xy}\\ \kappa_{x}\\ 0\\ 0\end{array} \right\} \tag{79}\]  
The F matrix was used to represent the inverse of the ABD matrix; thus Equation (79) becomes:  
\[\left\{ \begin{array}{c}\epsilon_{x}^{0}\\ \epsilon_{y}^{0}\\ \gamma_{xy}^{2}\\ \kappa_{x}\\ 0\\ 0\end{array} \right\}=\left[ \begin{array}{cccccc}F_{11}&F_{12}&F_{12}&F_{14} &F_{15}&F_{16}\\ F_{21}&F_{22}&F_{23}&F_{24}&F_{25}&F_{26}\\ F_{31}&F_{32}&F_{33}&F_{34}&F_{35}&F_{36}\\ F_{41}&F_{42}&F_{43}&F_{44}&F_{45}&F_{46}\\ F_{51}&F_{52}&F_{53}&F_{54}&F_{55}&F_{56}\\ F_{61}&F_{62}&F_{63}&F_{64}&F_{65}&F_{66}\end{array} \right]\left\{ \begin{array} []{c}N_{x}\\ 0\\ 0\\ M_{y}\\ M_{xy}\end{array} \right\} \tag{80}\]  
Equation (80) is simplified to Equation (81). The new simplified matrix is called the G matrix. \[\left\{ \begin{array}{c}\epsilon_{x}^{0}\\ \kappa_{x}\\ 0\\ 0\end{array} \right\}=\left[ \begin{array}{cccc}F_{11}&F_{14}&F_{15}&F_{16} \\ F_{41}&F_{44}&F_{45}&F_{46}\\ F_{51}&F_{54}&F_{55}&F_{56}\\ F_{61}&F_{64}&F_{65}&F_{66}\end{array} \right]\left\{ \begin{array}{c}N_{x }\\ M_{x}\\ M_{y}\\ M_{xy}\end{array} \right\}=\left[ \begin{array}{cccc}G_{11}&G_{12}&G_{13}&G_{ 14}\\ G_{21}&G_{22}&G_{23}&G_{24}\\ G_{31}&G_{32}&G_{33}&G_{34}\\ G_{41}&G_{42}&G_{43}&G_{44}\end{array} \right]\left\{ \begin{array}{c}N_{x} \\ M_{x}\\ M_{y}\\ M_{xy}\end{array} \right\} \tag{81}\]  
\[\left\{ \begin{array}{c}N_{x}\\ M_{x}\\ M_{y}\\ M_{xy}\end{array} \right\}=\left[ \begin{array}{cccc}G_{11}&G_{12}&G_{13}&G_{ 14}\\ G_{21}&G_{22}&G_{23}&G_{24}\\ G_{31}&G_{32}&G_{33}&G_{34}\\ G_{41}&G_{42}&G_{43}&G_{44}\end{array} \right]\left\{ \begin{array}{c}\epsilon _{x}^{0}\\ \kappa_{x}\\ 0\\ 0\end{array} \right\} \tag{82}\]  
Subsequently, the stress, moment, and curvature of the flange are expressed as Equations (83)-(86)  
\[N_{x}=G_{11}^{-1}\epsilon_{x}^{0}+G_{12}^{-1}\kappa_{x}=\left(Z_{1}G_{11}^{-1} +G_{12}^{-1}\right)\kappa_{x} \tag{83}\]  
\[M_{x}=G_{21}^{-1}\epsilon_{x}^{0}+G_{22}^{-1}\kappa_{x}=\left(Z_{1}G_{21}^{-1} +G_{22}^{-1}\right)\kappa_{x} \tag{84}\]  
\[M_{y}=G_{31}^{-1}\epsilon_{x}^{0}+G_{32}^{-1}\kappa_{x}=\left(Z_{1}G_{31}^{-1} +G_{32}^{-1}\right)\kappa_{x} \tag{85}\]  
\[M_{xy}=G_{41}^{-1}\epsilon_{x}^{0}+G_{42}^{-1}\kappa_{x}=\left(Z_{1}G_{41}^{-1 }+G_{42}^{-1}\right)\kappa_{x} \tag{86}\]  
The total moment of the flange is  
\[M_{flange}=b(N_{x}Z_{1}+M_{x})=b\Big{[}\Big{(}G_{11}^{-1}Z_{1}^{2}+G_{12}^{-1} Z_{1}\Big{)}+\left(G_{21}^{-1}Z_{1}+G_{22}^{-1}\Big{)}\Big{]}\;\kappa_{x} \tag{87}\]  
Consequently, the total moment of the beam can be expressed as:  
\[M_{beam}=M_{web}+M_{flange}=\left[\left(\frac{h^{3}}{12A_{11,w}^{-1}}+\frac{Z_ {c}^{2}h}{A_{11,w}^{-1}}\right)+b\Big{(}\Big{(}G_{11}^{-1}Z_{1}^{2}+G_{12}^{-1} Z_{1}\Big{)}+\left(G_{21}^{-1}Z_{1}+G_{22}^{-1}\Big{)}\right)\right] \kappa_{x}=EI_{eff}\kappa_{x} \tag{88}\]  
The effective stiffness of the wide flange can be expressed as:  
\[EI_{eff}=\left(\frac{h^{3}}{12A_{11,w}^{-1}}+\frac{Z_{c}^{2}h}{A_{11,w}^{-1}} \right)+b\Big{[}G_{11}^{-1}Z_{1}^{2}+\Big{(}G_{12}^{-1}+G_{21}^{-1}\Big{)}Z_{1}+ G_{22}^{-1}\Big{]} \tag{89}\]The strain of the flange at each laminate is:  
\[\left\{ \begin{array}{c}\varepsilon_{x}\\ \varepsilon_{y}\\ \gamma_{xy}\end{array} \right\}=\left\{