## Theoretical Modeling
### Element Stiffness and Mass Matrices  
Different from the traditional Euler-Bernoulli beams, the coupling among axial, bending and torsional effects is added into the element stiffness and mass matrices in the present framework. These effects are included by considering the flap-torsion (FT), lag-torsion (LT) and axial-torsion (AT) cross-stiffness properties in the stiffness matrix, and dynamic coupling terms between torsion and bending in the mass matrix. As illustrated in Figure 1, there is a basic reference coordinate system \(O_{xyz}\), whose origin is chosen arbitrarily on the cross-section. The \(x\) axis lies parallel to the beam axis, and the \(y\) and \(z\) axes correspond, respectively, to the lagwise movement direction (or flapwise bending axis) and the flapwise one (or lagwise bending axis), which in general will not be the principal bending directions. The local reference system for the stiffness matrix \(T_{xyz}\) is parallel to the basic one with origin at the tension center (T) of the cross-section, and the \(x\) axis will be the neutral axis of the beam. It can be noted that T is generally different from the shear center (C). The local reference system for mass matrix \(G_{xyz^{\prime}z^{\prime}}\) is obtained by translating the basic system to the center of gravity (G) of the cross-section, and then rotating with an angle \(\alpha_{m}\) to coincide with the principal axes of inertia \(\{y^{\prime},z^{\prime}\}\). The element stiffness matrix in the basic coordinate system is given as:  
\[\mathbf{k}_{o}=\mathbf{T}_{k}^{T}\mathbf{k}_{el}\mathbf{T}_{k} \tag{28}\]  
Figure 1: Beam reference coordinate system. where  
\[\mathbf{T}_{k}=\left[ \begin{array}{cc}\mathbf{t}_{k}&\mathbf{0}_{6\times 6}\\ \mathbf{0}_{6\times 6}&\mathbf{t}_{k}\end{array} \right]\quad\mathbf{t}_{k}= \left[ \begin{array}{cccccc}1&0&0&0&z_{t}&-y_{t}\\ 0&1&0&-z_{c}&0&0\\ 0&0&1&y_{c}&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array} \right]\]  
The element stiffness matrix in local axes [44] can be expressed as:  
\[\mathbf{k}_{el}=\left[ \begin{array}{cc}\mathbf{A}&\mathbf{B}^{T}\\ \mathbf{B}&\mathbf{C}\end{array} \right] \tag{29}\]  
in which  
\[\mathbf{A}=\left[ \begin{array}{cccccc}\frac{EA}{I}&0&0&\frac{AT}{I}&0&0\\ 0&\frac{12 EI_{z}}{I^{3}}&\frac{12 EI_{yz}}{I^{3}}&0&-\frac{6 EI_{yz}}{I^{2}}& \frac{6 EI_{z}}{I^{2}}\\ 0&\frac{12 EI_{yz}}{I^{3}}&\frac{12 EI_{y}}{I^{3}}&0&-\frac{6 EI_{y}}{I^{2}}& \frac{6 EI_{yz}}{I^{2}}\\ \frac{AT}{I}&0&0&\frac{GI}{I}&\frac{FT}{I}&-\frac{LT}{I}\\ 0&-\frac{6 EI_{yz}}{I^{2}}&-\frac{6 EI_{y}}{I^{2}}&\frac{FT}{I}&\frac{4 EI_{y}}{I} &-\frac{4 EI_{yz}}{I}\\ 0&\frac{6 EI_{z}}{I^{2}}&\frac{6 EI_{yz}}{I^{2}}&-\frac{LT}{I}&-\frac{4 EI_{yz}}{I }&\frac{4 EI_{z}}{I}\end{array} \right]\]  
\[\mathbf{B}=\left[ \begin{array}{cccccc}\frac{-EA}{I}&0&0&\frac{-AT}{I}&0&0\\ 0&\frac{-12 EI_{z}}{I^{3}}&\frac{-12 EI_{yz}}{I^{3}}&0&\frac{6 EI_{yz}}{I^{2}}& \frac{-6 EI_{z}}{I^{2}}\\ 0&\frac{-12 EI_{yz}}{I^{3}}&\frac{-12 EI_{y}}{I^{3}}&0&\frac{6 EI_{y}}{I^{2}}& \frac{-6 EI_{yz}}{I^{2}}\\ -\frac{AT}{I}&0&0&\frac{-GI}{I}&-\frac{FT}{I}&\frac{LT}{I}\\ 0&\frac{-6 EI_{yz}}{I^{2}}&\frac{-6 EI_{y}}{I^{2}}&\frac{-FT}{I}&\frac{2 EI_{y}}{ I}&-\frac{-2 EI_{yz}}{I}\\ 0&\frac{6 EI_{z}}{I^{2}}&\frac{6 EI_{yz}}{I^{2}}&\frac{LT}{I}&\frac{-2 EI_{yz}}{ I}&\frac{2 EI_{z}}{I}\end{array} \right]\]  
\[\mathbf{C}=\left[ \begin{array}{cccccc}\frac{EA}{I}&0&0&\frac{AT}{I}&0&0\\ 0&\frac{12 EI_{z}}{I^{3}}&\frac{12 EI_{y}}{I^{3}}&0&\frac{6 EI_{yz}}{I^{2}}&-\frac {6 EI_{z}}{I^{2}}\\ 0&\frac{12 EI_{y}}{I^{3}}&\frac{12 EI_{y}}{I^{3}}&0&\frac{6 EI_{y}}{I^{2}}&-\frac {6 EI_{yz}}{I^{2}}\\ \frac{AT}{I}&0&0&\frac{GI}{I}&\frac{FT}{I}&-\frac{LT}{I}\\ 0&\frac{6 EI_{yz}}{I^{2}}&\frac{6 EI_{y}}{I^{2}}&\frac{FT}{I}&\frac{4 EI_{y}}{I}& -\frac{4 EI_{yz}}{I}\\ 0&-\frac{6 EI_{z}}{I^{2}}&-\frac{6 EI_{yz}}{I^{2}}&-\frac{LT}{I}&-\frac{4 EI_{yz}}{ I}&\frac{4 EI_{z}}{I}\end{array}