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Journal of Applied Mathematics
Volume 2012, Article ID 543828, 16 pages
http://dx.doi.org/10.1155/2012/543828
Research Article

Modified and Simplified Sectional Flexibility of a Cracked Beam

Department of Applied Mathematics, National Chung Hsing University, Taichung 402, Taiwan

Received 25 April 2011; Revised 20 July 2011; Accepted 27 September 2011

Academic Editor: George Jaiani

Copyright © 2012 Chih-Shiung Wang and Lin-Tsang Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents a new sectional flexibility factor to simulate the reduction of the stiffness of a single-edge open cracked beam. The structural model for crack of the beam is considered as a rotational spring which is related to the ratio of crack depth to the beam height, a/h. The mathematical model of this single-edge open crack beam is considered as an Euler-Bernoulli beam. The modified factor, f(a/h), derived in this paper is in good agreement with previous researchers' results for crack depth ratio a/h less than 0.5. The natural frequencies and corresponding mode shapes for lateral vibration with different types of single-edge open crack beams can then be evaluated by applying this modified factor f(a/h). Using the compatibility conditions on the crack and the analytical transfer matrix method, the numerical solutions for natural frequencies of the cracked beam are obtained. The natural frequencies and the mode shapes with crack at different locations are obtained and compared with the latest research literature. The numerical results of the proposed cracked beam model obtained by this method can be extended to construct frequency contour. The natural frequencies measured from field can be used in solving the inverse problem to identify cracks in structures.