## 2 Theoretical Development  
### Static Residual Force Algorithm  
In this section, the basic idea of the static residual force algorithm [37] is briefly reviewed. For an undamaged structure, the static equilibrium equation is expressed as follows:  
\[K\cdot u-l=0 \tag{1}\]  
where \(K\) is the (\(n\)\(\times\)\(n\)) stiffness matrix of structural FEM and \(u\) is the vector of static displacement caused by load vector \(l\). For the damaged structure, the static equilibrium equation can also be expressed as the following:  
\[K_{d}\cdot u_{d}-l=0 \tag{2}\]  
\[K_{d}=K-\Delta K \tag{3}\]  
where \(K_{d}\) is the stiffness matrix of the damaged structure, \(u_{d}\) is the corresponding displacement vector, and \(\Delta K\) is the stiffness change due to damage. Substituting Equation (3) into (2) yields the following:  
\[\Delta K\cdot u_{d}=K\cdot u_{d}-l \tag{4}\]  
letting  
\[\xi=K\cdot u_{d}-l \tag{5}\]  
Equation (4) can be rewritten as the following:  
\[\xi=\Delta K\cdot u_{d} \tag{6}\]  
where \(\xi\) is defined as the static residual force vector. Equation (6) can be further rewritten as the following:\[\left\{ \begin{array}{c}\xi_{1}\\ \xi_{2}\\ \vdots\\ \xi_{n}\end{array} \right\}=\left[ \begin{array}{c}\Delta k_{1}\\ \Delta k_{2}\\ \vdots\\ \Delta k_{n}\end{array} \right]\cdot u_{d} \tag{7}\]  
where \(\xi_{i}\) is the \(i\) th coefficient of the vector \(\xi\), and \(\Delta k_{i}\) is the \(i\) th row vector of the matrix \(\Delta K\). Noticeably, \(\Delta k_{i}\) is nonzero only if the \(i\) th degree of freedom is damaged. As a result, the static residual force vector \(\xi\) will have nonzero coefficients corresponding to the damaged DOFs. On the other hand, \(\xi\) can be calculated from Equation (5) by the intact stiffness matrix \(K\), the displacement \(u_{d}\) of the damaged structure, and the applied static load vector \(l\). In practice, \(K\) is obtained from finite element modeling, \(u_{d}\) can be measured from static displacement testing, and \(l\) can be obtained from force testing. With \(\xi\) computed by Equation (5), structural damage locations can be initially determined according to the nonzero coefficients in \(\xi\). Subsequently, damage extents can also be calculated by the nonzero coefficients of \(\xi\). The main formulas for damage quantification are derived as follows. As suggested by [PERSON] et al. [38], the damage suffered by the cable can be assumed to be properly represented by a diffused reduction of the constant axial stiffness. Therefore, the damage extent of the cable can be expressed as follows:  
\[\alpha_{j}=\frac{EA_{j}-EA_{dj}}{EA_{j}},\ 0\leq\alpha_{j}\leq 1 \tag{8}\]  
where \(EA_{j}\) and \(EA_{dj}\) are the \(j\) th cable axial stiffnesses before and after damage, respectively. \(\alpha_{j}\) is the corresponding damage extent. The value of \(\alpha_{j}\) is 0 when the \(j\) th cable is undamaged, and \(\alpha_{j}\) is 1 when the \(j\) th cable is completely damaged. For static finite element analysis, the stiffness matrix of the cable element can adopt the same form as the bar element when  
\[K_{j}^{e}=\frac{EA_{j}}{L_{j}}\left[ \begin{array}{cc}1&-1\\ -1&1\end{array} \right] \tag{9}\]  
where \(K_{j}^{e}\) is the \(j\) th elementary stiffness matrix in local coordinates. As a result, the global stiffness change \(\Delta K\) before and after damage can be expressed as follows:  
\[\Delta K=\sum_{j=1}^{N}\alpha_{j}K_{j} \tag{10}\]  
\[K_{j}=H_{j}K_{j}^{e}H_{j}^{T} \tag{11}\]  
where \(K_{j}\) is the \(j\) th elementary stiffness matrix in global coordinates, \(N\) is the total number of elements in structural FEM, \(H_{j}\) is the transformation matrix from local coordinates to global coordinates, and the superscript \(T\) denotes the matrix transposition. From Equation (10), the following results are obtained:  
\[\Delta k_{i}=\sum_{j=1}^{N}\alpha_{j}k_{i}^{j} \tag{12}\]  
where \(k_{i}^{j}\) is the \(i\) th row vector of the \(j\) th elemental stiffness matrix \(K_{j}\).