Modelling of a Beam with a Breathing Edge Crack and Some Observations for Crack Detection

Abstract

The breathing behaviour of closing cracks has been adequately simulated as a small-displacement, frictionless contact problem. The problem of a beam with an edge crack subjected to harmonic loading has been considered as a plane problem and an attempt is made to solve it by using finite elements employing eight-node plane isoparametric elements. The proposed model allows the crack size and position to be varied. Another physically important problem of a cantilever beam held between two heavy jaws at the top and bottom, which are not equally flushed, is considered. This beam is also excited by a harmonic load at the tip. The contact model and a simple single degree of freedom model are used to solve the problem. Both the above problems (cracked beam and beam in offset jaws) show presence of integral multiples of the forcing frequency in their frequency spectra. An important observation regarding cracked beams and beams with imperfect support is made. If the forcing frequency is such that it coincides or is close to any one of the integral sub-multiples (1/ n) of the first natural frequency of the system, then the nth harmonic of the forcing frequency will considerably shoot up. This effect is highly pronounced for the case n = 2 and this observation may be used to detect cracks in beams as small as 2.5% of the depth. For cracked beams, the even harmonics are considerably stronger than the odd ones. As the crack size decreases, the odd harmonics become weaker. For a 2.5% crack only the second and fourth harmonics are visible in an 80 dB scale, with the former being the stronger. However, it is important to note that cracked beams and beams with imperfect support have closely similar spectral characteristics and so due caution must be exercised during crack detection.