# Cable Force Identification of Two Bending Cable Networks with Arbitrary Boundary Conditions
## 4 Application of Cable Force Identification Method
### A Two-Cable Network with Mixed Boundary Conditions

Table 8: Estimated axial stiffness of the cross-tie. Since the supports of Cable 1 in this case study are fixed-end supports, the measured frequencies of the cable network in this case study are slightly higher than those in the previous study case. Step 4: Since the cable forces of single cables are determined in Step 1, these values are beneficial for calculating the cable network's predicted range of cable forces. Then, we can expect the cable forces from 1705.91 kN to 2085.00 kN (1895.45 kN \(\pm\) 10%) for Cable 1 and from 1436.87 kN to 1756.17 kN (1596.52 kN \(\pm\) 10%) for Cable 2. Step 5: In this step, the cable network's measured frequencies from mode 1 to mode 10 and the cable forces \(T_{1}\) varying from 1705.91 kN to 2085.00 kN, and \(T_{2}\) varying from 1436.87 kN to 1756.17 kN are used to create numerous combinations, such as (1744.02 kN, 1755.80 kN, 1.3498 Hz), (1705.91 kN, 1680.99 kN, 2.7178 Hz), (1833.19 kN, 1627.74 kN, 5.5268 Hz), etc. These combinations are substituted into Equation(16) to find the combinations that satisfy the condition \(F(T_{1},T_{2}f_{m})=0\). Thirty examples in numerous satisfied combinations are listed in Table 9 for reference. Step 6: Cable forces \(T_{1}\) and \(T_{2}\) in numerous selected combinations obtained in Step 5 are introduced in Equation (18) to calculate the cable network frequencies. The _MAPE_ values of thirty satisfying combinations are shown in Table 10 for reference. The combination with \(T_{1}\) = 1892.02 kN and \(T_{2}\) = 1597.78 kN has the minimum _MAPE_ of the computed frequencies at 0.07%. Hence, the cable tensions, namely \(T_{1}\) = 1892.02 kN and \(T_{2}\) = 1597.78 kN with errors of 0.42% and 0.01%, respectively, are identified as the cable forces of the cable network. Noticeably, the relative errors are calculated by comparing the obtained cable forces in this step and the known cable forces in Step 1. Although the error of the estimated \(K_{c}\) is more than 10.0%, the error of the identified cable forces is less than 1.0%. To evaluate the effect of the error size of the estimated \(K_{c}\) on the correctness of the cable force identification, the authors will vary the error of the estimated \(K_{c}\) in a wide range from \(-\)40% to 50%. The relationship between the relative error of \(K_{c}\) and the relative errors of \(T_{1}\) and \(T_{2}\) is shown in Figure 11. Though the relative error of the estimated \(K_{c}\) is significant, the effect on the correctness of the cable force identification is minor. The relative error of the estimated \(K_{c}\) does not affect the relative error of the cable force identification of Cable 1. Nevertheless, the relative error of the estimated \(K_{c}\) has a slightly inverted influence on the cable force identification of Cable 2. Since this study uses the equivalent cable network configuration, the intersection point of lines \(T_{1}\) and \(T_{2}\) occurs at \(-\)13.43% of \(K_{c}\) error. For the cable network with all hinged-end supports, the intersection point of lines \(T_{1}\) and \(T_{2}\) will occur at 0.0% of \(K_{c}\) error.