13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Theoretical and Numerical Study of Symmetric, In-plane, Free Vibration of Timoshenko Portal Frame with Open Crack Nikam M Satyavan, Sandeep Kumar and S.M. Murigendrappa* Department of Mechanical Engineering, National Institute of Technology Karnataka, Surathkal-575025, India *Corresponding author: smm@nitk.ac.in Abstract The local flexibility introduced by cracks changes the vibration behaviour of the structure and by examining this change, crack severity can be identified. This paper presents the natural frequencies of symmetric, in-plane free-vibrations of Timoshenko portal frame with and without open crack for different boundary conditions. Cracked segment is modelled as two segments connected by a massless torsional spring. Considering appropriate compatibility requirements at the crack section in any one of segments and at the junction of two segments, the characteristic equations are established for corresponding boundary conditions and solved for natural frequencies by numerically. Crack location ranging from 20% to 70% of length of segment and crack size ranging from 20% to 60% of depth have been considered. Results obtained analytically are compared numerically using standard commercially available finite element software. The frame has been modelled by using quadratic quadrilateral shell elements and quarter-point singular elements are employed around the crack-tip. It is observed that as expected, with increase in crack depth the change in frequencies of the frame with and without crack increases. The maximum difference between the analytical and numerical results is 7.09% for all the cases considered, which proves usefulness of the data. Keywords Timoshenko Portal Frame, Open Crack, Massless Torsional Spring, In-plane Free Vibration, FEM 1. Introduction The problem of Timoshenko portal frames with defect is of importance in many fields of engineering. Defects are almost unavoidable in such frames and their existences will decrease stiffness, strength and safety. Although, a number of accurate, effective and reliable on-line damage detection methods based on either X-ray, ultrasonic tests etc., are available, their adoption require scanning of the whole length of frame. This process is a very time consuming, labour-intensive and expensive. In view of these limitations there is a need to develop Non-Destructive Testing methods which can detect damages in a component from the measurement of vibration responses, which may be collected from at a single point, or at the most, a few points, on the component. The most significant vibration parameter applied in damage identification methods is change in natural frequencies of vibrations of structures caused by the crack. Hence, it may be possible to predict the presence of a crack from the measurements of natural frequencies of the damaged component. A wide variety of beam structures modelled by Euler-Bernoulli or Timoshenko beam theory have been considered for crack detection by representing the crack with massless torsional spring[1-9] etc. Experimental results, though not very exhaustive, are also reported. Most of the studies on frames consider them to be free from defects [10-16]. A few investigators have reported inverse problem of determination of crack details from the natural frequencies or mode shapes (e.g.,[17-19]) for frame modelled by Euler-Bernoulli beam theory. Frames modelled by Timoshenko beam theory with crack have not yet been studied. This paper presents, a method of solving a forward problem i.e., determination of natural frequencies knowing the crack details of symmetric, in-plane vibrations of Timoshenko portal frame with crack. Associated cracked segment in the portal frame is modelled as two segments connected by a massless torsional spring. The characteristic equations are established using boundary conditions, compatibility conditions at
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