# Cable Force Identification of Two Bending Cable Networks with Arbitrary Boundary Conditions
## 1 Introduction  
Cable forces in cable-stayed bridges may vary in design values due to the adjustment of the cable tension during construction to fit geometric designs. Furthermore, the cable tension may be changed in the service stage due to bridge modifications, such as the change of pavement layers and barriers. Moreover, the cable loading capacity could decrease due to fatigue, anchor damage, and corrosion. If bridge maintenance engineers cannot determine the current cable conditions, failure may occur, such as the collapse of the Nanfangao bridge in Taiwan in 2019. Hence, diagnosing the current cable conditions, including tendon force, anchor rigidity, and bending rigidity, is crucial to bridge safety evaluation at certain confidence levels. Among the known NDT methods for cable force measurements, the vibration technique is simple, fast, and economical. Thus, the vibration technique was used by many researchers, such as [PERSON] [1], [PERSON] and [PERSON] [2], [PERSON] et al. [3], [PERSON] [4], and [PERSON] et al. [5]. For practical applications, [PERSON] and [PERSON] [6], [PERSON] and [PERSON] [7], [PERSON] et al. [8], and [PERSON] et al. [9] introduced empirical formulas to calculate cable tensions by considering the bending stiffness, the cable length, and the boundary conditions. Nevertheless, these functional equations are derived for a single cable and, therefore, inappropriate for cable networks where flexible or rigid cross-ties interconnect two or more cables. Suppose the measured frequencies of the cable networks are inputted in these equations to obtain the cable forces of the cable network. In that case, the estimated cable forces are significantly different from the cable network's actual cable forces because the natural frequencies of the cable networks are much different from those of the single cables. For instance, a cable network is created from two simply supported cables AB and CD, with cable lengths \(L_{1}\) and \(L_{2}\), cable tensions \(T_{1}\) and \(T_{2}\), and interconnected by a rigidcross-tie at nodes N\({}_{1}\) and N\({}_{2}\) at location \(l_{11}\) and \(l_{21}\) from the left restraints of cable AB and CD, respectively, as shown in Figure 1. (\(x_{11}\), \(v_{11}\)), (\(x_{12}\), \(v_{12}\)), (\(x_{21}\), \(v_{21}\)), and (\(x_{22}\), \(v_{22}\)) are the local axes of cable segments 1-1, 1-2, 2-1, and 2-2, respectively. Axis (\(x\), \(z\)) is the global coordinate of the cable system. The comparison of the natural frequencies of the cable network and the single cable is shown in Figure 2. The natural frequencies \(f_{N}\) at point \(S_{1}\) and \(S_{2}\) on cables AB and CD of the cable network are identical but different from the natural frequency \(f_{S}\) at point \(S_{1}\) on single cable 1. Noticeably, the subscript \(N\) denotes the cable network and the subscript \(S\) denotes the single cable. Since many terms of \"frequency\" are used in this paper, three terms are defined to classify their uses. The term \"computed frequency\" is used for the cable frequency computed using backward analysis. The term \"natural frequency\" is used for the cable frequency obtained from solving the eigenvalue problem. The term \"measured frequency\" is used for the cable frequency obtained from the field measurements or the finite element models. To apply existing vibration techniques for the cable networks, engineers have to release the cross-ties to separate the cable networks into single cables. Each cable is excited under free vibration to identify the cable tensile force. However, removing the cross-ties will increase the inspection cost and not reflect the cable network's actual behaviors. Thus, developing a new method to identify the cable forces of a cable network without removing the cross-ties to preserve the accuracy of cable-force identification and reduce the inspection cost is worthy of attention. Nevertheless, most existing research about cable networks uses forward analysis.