Keywords: cable structures, non-linear dynamic analysis, modal analysis, accuracy.

Cables which are used to span suspension, cable stayed bridges and large roofs, have geometrical non-linear behaviour. For dynamic analysis two methods are usually used, step by step integration method and modal superposition method. Modal superposition method has been used by many researches to solve non-linear problems. In modal superposition method, the effects of the combination of higher modes are neglected. So the smallest modes which have more effect on analysis can be used. The reason is to reduce the number of the equation from , to , where is the number of chosen smallest modes and is the total degree of freedom of structures (). In this research modal superposition method was used for non-linear analysis and was observed that choosing the number of required modes to get very good results depended on the degree of freedom of the structure. It was realized that "(5/8) structure's degrees of freedom" the lowest modes gave very accurate results for non-linear analysis.

As a result of cables geometrical non-linear behaviour, a variety of iterative methods known as 'geometrically non-linear solution techniques' have been used. One such method is the 'Newton-Raphson technique', which has proved to be accurate, efficient and applicable to the majority of cable structures.

Mode superposition method has been used extensively by many researches for linear dynamic analysis. M.T.H. Elkatt [1] has given a survey, on linear dynamic analysis using the mode superposition method.

A number of researchers, such as N.F. Morris [2], have applied the mode superposition method to calculate the non-linear dynamic response of structures. T.J.A. Agar [3] modelled a suspension bridge by the finite element method and used modal analysis to predict the flutter speed. He used the 30 lowest frequencies and mode shapes to obtain a reasonably accurate model of the dynamic behaviour. T. Kumarasena et al. [4] used data based on on-site wind and vibration measurements, and the results from a finite element analysis of a suspension bridge to define with some accuracy, a number of its prominent modes to predict the wind response of flexible bridges. They conclude that although the vibration spectral amplitudes found for the frequency components corresponding to the higher modes are much smaller relative to the lower modes, some acceleration spectra indicate that the amplitudes of certain accelerations are comparatively more significant at higher frequencies. Thus, for those design considerations in which accelerations may play a prominent role, the inclusion of higher modes in the analysis may be important. J.M.W. Brownjohn et al. [5] considered eight mode shapes to analyse a suspension bridge.

On the number of mode shapes for cable systems, J.W. Leonard [6] wrote that those systems, because of their flexibility and capability to deform easily in each of the modes, many modes are required to be included for convergence.

M.T.H. Elkatt and M.A. Millar [7] developed an algorithm for the dynamic analysis of non-linear structural truss and cable systems using the normal mode superposition technique. A small number of the lower mode shapes and frequencies were used and a correction was made to include the effects of all higher modes based on static analysis. An iterative procedure under static effects was applied.

In this research different problems such as guyed towers, simple cable and cable net structures were solved. It was observed that using only a few of the lowest modes in structural dynamics analyses, reasonably accurate displacements can be obtained while the internal loads (the stress resultants) are often unacceptable. The results showed that "(5/8) × unrestrained degrees of freedom of structure" the lowest modes gave very accurate results for non-linear analysis.

1

M.T.H. Elkatt, "Non-linear dynamic analysis of structures", PhD Thesis, UMIST, 1984.

2

N.F. Morris, "The use of modal superposition in non-linear dynamics", Computers and Structures, 7, 65-72, 1977. doi:10.1016/0045-7949(77)90061-X

3

T.J.A. Agar, "Aerodynamic flutter analysis of suspension bridges by a modal technique", Engineering Structures, 11, 75-82, 1989. doi:10.1016/0141-0296(89)90016-3

4

T. Kumarasena et. al., "Deer Isle bridge: field and computed vibrations", Journal of structural engineering, 115(9), 2313-2328, 1989. doi:10.1061/(ASCE)0733-9445(1989)115:9(2313)

5

J.M.W. Brownjohn et. al., "Ambient vibration measurements of the Humber suspension bridge and comparison with calculated characteristics", Proc. Institution., Civil Engineers, Part 2, (83), 561-600, 1987.

6

J.W. Leonard, "Tension structures, behaviour and analysis", McGraw Hill, 1988.

7

M.T.H. Elkatt and M.A. Millar, "On the accuracy of a mode superposition method in non-linear dynamics", Proceedings of the Fourth International Conference on Civil and Structural Engineering Computing, 2, 37-46, Civil-Comp Press, Edinburgh, 1989. doi:10.4203/ccp.9.11.2

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