## Finite Element Model
### Composite Beam Element Stiffness Matrix Evaluation  
When a composite beam of length \(L\) is subjected to a uniformly distributed load \(q\), the contribution to the total potential energy \(I\) from element \((e)\) of length \(L^{(e)}\) can be expressed as  
\[I^{(e)}=I_{c}^{(e)}+I_{s}^{(e)}+I_{k}^{(e)}-\int_{L^{(e)}}q^{(e)}w^{(e)}dx \tag{22}\]  
where  
\[\left\{ \begin{array}{l}I_{c}^{(e)}=\frac{1}{2}\int_{pc^{(e)}}\sigma_{c}^{(e) }\varepsilon_{c}^{(e)}dv\\ I_{s}^{(e)}=\frac{1}{2}\int_{\gamma_{\Sigma^{(e)}}}\sigma_{s}^{(e)} \varepsilon_{s}^{(e)}dv\\ I_{k}^{(e)}=\frac{1}{2}K_{T}^{(e)}\cdot s_{l}^{(e)^{2}}+\frac{1}{2}K_{T}^{(e)} \cdot s_{r}^{(e)^{2}}\end{array} \right. \tag{23}\]  
The stress-strain relationships for an isotropic elastic material and the load-displacement relationships, using Hooke's law to represent the springs, can be written as  
\[\left\{ \begin{array}{l}\sigma_{c}=E_{c}\varepsilon_{c}\\ \sigma_{s}=E_{s}\varepsilon_{s}\\ T_{kl}=K_{T}^{(e)}s_{l}\\ T_{Kr}=K_{T}^{(e)}s_{r}\end{array} \right. \tag{24}\]Substituting Equations (1), (6), (7), (20), (21) and (24) into Equation (22), the total potential energy equation for element (\(e\)) can also be written as  
\[ \begin{array}{l}I^{(e)}=\frac{1}{2}{\int_{\mathrm{pc}^{(e)}}E_{c}{\epsilon_{c} }^{2}}dv+\frac{1}{2}{\int_{\mathrm{ps}^{(e)}}E_{s}{\epsilon_{s}^{(e)}}^{2}}dv+ \frac{1}{2}K_{T}^{(e)}{\cdot s_{l}^{(e)}}^{2}+\frac{1}{2}K_{T}^{(e)}{\cdot s_{r }^{(e)}}^{2}-{\int_{L^{(e)}}qv}^{(e)}dx\\ =\frac{1}{2}{\int_{\mathrm{pc}^{(e)}}\left[\mathbf{a}^{(e)}\right]^{T}}\Big{[}\mathbf{ B}_{c}^{(e)}\Big{]}^{T}\mathbf{B}_{c}^{(e)}\Big{]}^{T}\mathbf{a}^{(e)}dv+\frac{1}{2}{ \int_{\mathrm{ps}^{(e)}}\left[\mathbf{a}^{(e)}\right]^{T}}\Big{[}\mathbf{B}_{s}^{(e)} \Big{]}^{T}\mathbf{a}^{(e)}dv+\end{array} \] \[ \begin{array}{l}\frac{1}{2}\Big{[}\mathbf{a}^{(e)}\Big{]}^{T}\Big{[} \mathbf{B}_{kl}^{(e)}\Big{]}K_{T}^{(e)}\Big{[}\mathbf{B}_{kl}^{(e)}\Big{]}^{T}\mathbf{a}^{ (e)}+\frac{1}{2}\Big{[}\mathbf{a}^{(e)}\Big{]}^{T}\Big{[}\mathbf{B}_{kl}^{(e)}\Big{]} K_{T}^{(e)}\Big{[}\mathbf{B}_{kl}^{(e)}\Big{]}^{T}\mathbf{a}^{(e)}-\int_{-1}^{1}\Big{[}\mathbf{B}_{ l}^{(e)}\Big{]}q\frac{L^{(e)}}{2}d\xi\\ =\frac{1}{2}\Big{[}\mathbf{a}^{(e)}\Big{]}^{T}\mathbf{K}_{c}^{(e)}\mathbf{a}^{(e)}+\frac{1 }{2}\Big{[}\mathbf{a}^{(e)}\Big{]}^{T}\mathbf{K}_{s}^{(e)}\mathbf{a}^{(e)}+\frac{1}{2}\Big{[} \mathbf{a}^{(e)}\Big{]}^{T}\mathbf{K}_{Tl}^{(e)}\mathbf{a}^{(e)}+\frac{1}{2}\Big{[}\mathbf{a}^ {(e)}\Big{]}^{T}\mathbf{K}_{Tl}^{(e)}\mathbf{a}^{(e)}-\int_{-1}^{1}\Big{[}\mathbf{B}_{b}^{( e)}\Big{]}q\frac{L^{(e)}}{2}d\xi\\ =\frac{1}{2}\Big{[}\mathbf{a}^{(e)}\Big{]}^{T}\mathbf{K}^{(e)}\mathbf{a}^{(e)}-\mathbf{f}\end{array}  \tag{25}\]  
in which  
\[ \begin{array}{l}\mathbf{K}^{(e)}=\mathbf{K}_{c}^{(e)}+\mathbf{K}_{s}^{(e)}+\mathbf{K}_{Tl}^{ (e)}+\mathbf{K}_{Tr}^{(e)}\\ \Big{\{}\mathbf{B}_{b}^{(e)}\Big{\}}=\Big{[}0,0,N_{3}^{(e)}(\xi),\overline{N}_{3}^{ (e)}(\xi),0,0,N_{4}^{(e)}(\xi),\overline{N}_{4}^{(e)}(\xi)\Big{]}^{T}\\ \mathbf{K}_{c}^{(e)}=\int_{-1}^{1}\Big{[}\mathbf{B}_{c}^{(e)}\Big{]}\big{(}EA\big{)}_{ c}^{(e)}\Big{[}\mathbf{B}_{c}^{(e)}\Big{]}^{T}\frac{L^{(e)}}{2}d\xi\\ \mathbf{K}_{s}^{(e)}=\int_{-1}^{1}\Big{[}\mathbf{B}_{s}^{(e)}\Big{]}\big{(}EA\big{)}_{ s}^{(e)}\Big{[}\mathbf{B}_{s}^{(e)}\Big{]}^{T}\frac{L^{(e)}}{2}d\xi\\ \mathbf{K}_{Tl}^{(e)}=\Big{[}\mathbf{B}_{kl}^{(e)}\Big{]}{\cdot}\mathbf{K}_{T}^{(e)}\Big{[} \mathbf{B}_{kl}^{(e)}\Big{]}^{T}\\ \mathbf{K}_{Tr}^{(e)}=\Big{[}\mathbf{B}_{kl}^{(e)}\Big{]}{\cdot}\mathbf{K}_{T}^{(e)}\cdot \Big{[}\mathbf{B}_{kl}^{(e)}\Big{]}^{T}\\ \mathbf{f}{=}\int_{-1}^{1}\Big{[}\mathbf{B}_{b}^{(e)}\Big{]}q\frac{L^{(e)}}{2}d\xi \end{array}