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Structural Engineering and Mechanics Volume 31, Number 4, March 10 2009 , pages 383-392 DOI: https://doi.org/10.12989/sem.2009.31.4.383 |
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Convergence studies on static and dynamic analysis of beams by using the U-transformation method and finite difference method |
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Y. Yang, M. Cai and J. K. Liu
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| Abstract | ||
| The static and dynamic analyses of simply supported beams are studied by using the Utransformation method and the finite difference method. When the beam is divided into the mesh of equal elements, the mesh may be treated as a periodic structure. After an equivalent cyclic periodic system is established, the difference governing equation for such an equivalent system can be uncoupled by applying the U-transformation. Therefore, a set of single-degree-of-freedom equations is formed. These equations can be used to obtain exact analytical solutions of the deflections, bending moments, buckling loads, natural frequencies and dynamic responses of the beam subjected to particular loads or excitations. When the number of elements approaches to infinity, the exact error expression and the exact convergence rates of the difference solutions are obtained. These exact results cannot be easily derived if other methods are used instead. | ||
| Key Words | ||
| static analysis; dynamic analysis; beam; convergence rate; U-transformation; finite difference method. | ||
| Address | ||
| Y. Yang: School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, 510640, P. R. China M. Cai and J. K. Liu: Department of Mechanics, Zhongshan University, Guangzhou 510275, P. R. China | ||