Abstract
The fractal two-level finite element method is extended to the free vibration behavior of cracked beams for various end boundary conditions. A cracked beam is separated into its singular and regular regions. Within the singular region, infinite number of finite elements are virturally generated by fractal geometry to model the singular behavior of the crack tip. The corresponding numerous degrees of freedom are reduced to a small set of generalized displacements by fractal transformation technique. The solution time and computer storage can be remarkably reduced without sacrifying accuracy. The resonant frequencies and mode shapes computed compared well with the results from a commercial program.
Highlights
In many structures such as plates and beams, cracks may be developed as a results of corrosion or cyclic loading
It is well known that cracks in a structure may introduce considerable change in its natural frequencies and mode shapes
Dynamic analyses of cracked structures are important in order to detect cracks in the structures (Davies and Mayes [6] and Hu and Liang [5])
Summary
In many structures such as plates and beams, cracks may be developed as a results of corrosion or cyclic loading. The fractal two-level finite element method developed by Leung and Su [9] (F2LFEM) will be extended to determine the resonant frequencies and mode shapes of two-dimensional cracked structures. In order to avoid the troublesome mesh refinement around the crack tip and retain the advantage of agility of conventional finite element method, a cracked structure is separated into its singular (enclosing the crack) and regular regions. Infinite number of finite elements are virturally generated by fractal geometry to model the singular behavior of the crack tip. The regular region is modeled by conventional finite elements By this approach, the computer storage and the solution time for the eigenvalue problems can be effectively reduced. The results are in close agreement with those obtained from the commerical finite element package COSMOS/M but the present method requires only 10% of the computing memory
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