A. Appendix

Hermite shape functions of beam finite element:

$$N\_1 = \mathbf{x} \left( \mathbf{1} - 2\frac{\mathbf{x}}{L} + \frac{\mathbf{x}^2}{L^2} \right);\\ N\_2 = \mathbf{1} - 3\frac{\mathbf{x}^2}{L^2} + 2\frac{\mathbf{x}^3}{L^3};\\ N\_3 = \mathbf{x} \left( -\frac{\mathbf{x}}{L} + \frac{\mathbf{x}^2}{L^2} \right);\\ N\_4 = 3\frac{\mathbf{x}^2}{L^2} - 2\frac{\mathbf{x}^3}{L^3}.$$

Stiffness matrix:

$$K\_{\epsilon} = \frac{D\_{11}}{L^3} \begin{bmatrix} \mathsf{6} \left( b\_2 + b\_1 \right) & \mathsf{2} \left( b\_2 + 2b\_1 \right) L & -\mathsf{6} \left( b\_2 + b\_1 \right) & \mathsf{2} \left( 2b\_2 + b\_1 \right) L \\ & \left( b\_2 + 3b\_1 \right) L^2 & -\mathsf{2} \left( b\_2 + 2b\_1 \right) L & \left( b\_2 + b\_1 \right) L^2 \\ & & \left\mathsf{6} \left( b\_2 + b\_1 \right) & -\mathsf{2} \left( 2b\_2 + b\_1 \right) L \\ \mathrm{Sym.} & & \left( 3b\_2 + b\_1 \right) L^2 \end{bmatrix}.$$

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Mass matrix:

$$M\_{\varepsilon} = \frac{mL}{840} \begin{bmatrix} 24(3b\_2 + 10b\_1) & 2(b\_2 + b\_1)L & 54(b\_2 + b\_1) & -2(6b\_2 + 7b\_1)L \\ & (3b\_2 + 5b\_1)L^2 & 2(7b\_2 + 6b\_1)L & -3(b\_2 + b\_1)L^2 \\ & & 24(10b\_2 + 3b\_1) & -2(15b\_2 + 7b\_1)L \\ & \text{Sym.} & & (5b\_2 + 3b\_1)L^2 \end{bmatrix}.$$
