An identification method for short hanger tension of arch bridges

The hanger tension is crucial in arch bridges and the fundamental frequencies of short hangers are significantly affected by the flexural rigidity. In order to measure the hanger tension considering the flexural rigidity, the Additional Mass Method (AMM) is proposed in this paper. The tension and flexural rigidity can be calculated by the measured frequencies without and with the additional mass. The difficult inverse calculation among the frequency, tension and stiffness is solved by the genetic algorithm (GA). The proposed method is verified against a numerical example and an experimental bridge, and meanwhile its feasibility in simultaneously identifying hanger tension and flexural rigidity is proved. Pei Yuan, Liangfeng Sun and Xu Xie 515 Figure 1 : Finite element model of hanger 2.2 The additional mass method (AMM) In order to simultaneously identify the flexural rigidity and the hanger tension, the additional mass method (AMM) has been adopted in this study. Fig.2 sketches the principle of identifying the hanger tension using the AMM. By adding a known mass to the hanger and knowing the mass location, we may obtain some frequencies corresponding to different mass locations. Then the hanger tension T and the stiffness EI can be identified by using the GA. Figure 2 : The principle of the additional mass method 3 IDENTIFICATION AND VERIFICATION OF THE TENSION BASED ON GA 3.1 Identification of the hanger tension and the flexural rigidity The key point to identify the hanger tension and the flexural rigidity is calculating the structural parameters according to measured natural frequencies. But the problem consists in the high difficulty in the inverse calculation. To solve this problem, this paper attempts to employ the genetic algorithms based on the principles of biological evolution. Such algorithm takes groups of random samples as seeds and replaces samples with lower fitness by those with higher fitness. According to the biological evolutionism, the newly-generated samples have higher possibilities of obtaining high quality. And we will obtain relatively accurate solutions eventually by constantly repeating above calculating process. Fig.3 shows the flowchart of applying the GA and the main steps are described as follows: (1) Input the measured natural frequencies and other known parameters Input the measured natural frequencies of the hanger and other known parameters. This paper assumes that the mass per unit length (m), the cross-sectional area (A), the elastic modulus (E), the 516 ARCH’10 – 6th International Conference on Arch Bridges length (l) and the additional mass position (li) are known beforehand. To avoid the influence of measuring errors, this study develops a program which can input multiple groups of measured parameters. And to obtain the flexural rigidity and the tension, the input data should not be less than the amount of 2 groups. (2) Encoding Establish the initial sample groups within the set range and then calculate the frequency of each sample. Although binary systems are commonly used in the GA, it is not applicable to hanger parameters. Therefore, we adopt real numbers to denote the chromosome with the tension T and the flexural rigidity EI included. Besides, as the flexural rigidity and the tension are of a wide value range, it needs a large amount of samples while calculating in the field of positive real numbers. In order to avoid the tedious calculation, we used some relative quantities to narrow the range of the chromosomes: 1 2 , o s EI s EI T s T = = (2) where, Ts is the approximate tension on the basis of the string vibration theory; s1 and s2 are the parameters of the rigidity and the tension, respectively. By experience, s1 ranges from 0.3 to 1.1 and s2 from 0.3 to 1.0. Of course, the ranges of above parameters are relevant to the specific conditions of the hangers. In general, above data can satisfy the engineering requirements. Figure 3 : Calculation flowchart of the GA. (3) Error and applicability calculation According to the frequencies obtained from the eigenvalue calculation, errors can be defined as: