Keywords: cracked elements, finite-element model, transverse displacements, simplified computational model, crack, linear springs, rotational spring.

As an alternative to the huge finite-element model solution, simple physical models must be implemented in some cases to adequately describe the behaviour of the structure. Such models are usually required in engineering problems where only a limited amount of the data is available or required. A typical example of such problems are inverse problems where the simplified model has to be capable of intensive modification of crack location because a detailed discretisation of the crack and its surrounding with an appropriate detailed mesh of finite elements has severe practical limitations.

To model beams with transverse cracks various approaches can be used. From the inverse identification point of view huge meshes of finite elements are not convenient and thus simple models, as for example the model where the crack itself is replaced by a rotational spring, seem to be more promising.

The key to the efficient implementation of an idealised model certainly lies in the appropriate stiffness definitions of linear springs that is utilised to model the crack and therefore the paper presented a numerical comparison of various definitions for linear springs stiffness implemented for transverse displacement computation.

In order to numerically validate the definitions presented, the transverse displacements were computed with a commercial finite element program COSMOS/M. In the computational model 4000, 2D 8 noded quadrilateral elements were implemented with 12300 nodal points and the discrete crack approach was utilised for the crack description. Several types of structures with different geometrical dimensions, positions and depths of the crack were considered. The obtained results were compared with the results obtained with the simplified model where the governing differential equations were solved for each structure. The applied transverse load consisted of a combination of a uniform load over the whole length of the element with a single concentrated load. The ratio of the magnitudes of these two loads was also altered.

From the cases presented it can be concluded that some definition overestimate and some underestimate the displacements. However, the definition presented by Sundermayer et. al [1] yields valid engineering results regardless the crack depth as long as it can be considered that the structure behaves within the linear theory (i.e. small displacements) that is usually a correct assumption for most civil engineering structures. These two definitions yield acceptable agreement with the results obtained with huge finite elements solutions and despite a drastic difference in computational effort the difference between the best results obtained was within 2%.

1

Sundermayer, J.N., Weaver, R.L., "On crack identification and characterization in a beam by nonlinear vibration analysis", Theoretical and applied mechanics, TAM Report No. 743, UILU-ENG-93-6041, 1993.

2

Hasan, W.M., "Crack detection from the variation of the eigenfrequencies of a beam on elastic foundation", Engineering Fracture Mechanics 52(3), 409-421, 1995. doi:10.1016/0013-7944(95)00037-V

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