Single Cracked Cantilever Under Tension Loads

Abstract

As cracks may change significantly the behaviour of the whole structure, the development of reliable models of the mechanical behaviour is especially important. A detailed discretisation of the crack and its surrounding can be achieved with an appropriate mesh of finite elements, but such an approach is not recommended in inverse problems where a model suitable for intensive crack's location and position modification is required. Therefore, in such cases simplified models are needed, and one of the most simple and popular models is the model where the crack itself is replaced by a rotational linear spring connecting uncracked parts of the structure that are modelled as elastic elements. This elementary model neglects the fact that in the case of a single side crack an centric tensile load causes also transverse displacements that do not appear when the crack in double sided. This deficiency can be efficiently avoided by the introduction of an additional virtual bending moment at the crack location and a new appropriate definition of rotational spring stiffness. The paper presents the fundamentals for the computation with a simplified model and compares the finite elements approach with the solution of the differential equation. For illustrative purposes a simple cracked cantilever under concentrated axial force is presented and the results of both methods are compared. With such a model it is possible to compute transverse displacements of a single cracked element due to a axial load by solving a differential equation rather than implementing huge and time consuming finite element meshes. Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 420 Computational Methods and Experimental Measurements 1 The problem: introduction Consider a single cracked cantilever beam subjected to a horizontal tensile force H=3 MN at the free end (Figure 1). The transverse crack of uniform depth is located at the distance 0.8 in from the clamped edge at the upper surface of the beam. The beam has a rectangular cross section and a modulus of elasticity of E=30 GPa. The objective of the example is to compute the vertical displacements of the central point of the cross section at the free end for various crack depths. H 0.8 m Q().2 m 0.1 m L=2.0 m Figure 1: Cracked cantilever beam 2 The finite element method solution For the finite element analysis with 2D finite elements the commercial program COSMOS/M (registered trademark of Structural Research and Analysis Corporation) was used. The cantilever was discretisied using 4000 2D 8-node Quadrilateral elements with Poisson's ratio v=0.3. Figure 2: Deformed shape of the cantilever for the crack depth d=0.1 m (6=0.5) computed with COSMOS/M Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X Computational Methods and Experimental Measurements 421 The number of used nodes was varying from 12281 to 12362, depending on the depth of the introduced crack (Figure 2). Two translational degrees of freedom per node were considered for structural analysis. The solution was obtained by solving up to 24480 equations (depending on the depth of the crack), which were solved for each different crack depth. Table 1 displays the vertical displacements of the central point at the free end for various crack depths. It is evident from the results that vertical displacements increase with the crack depth and that despite the exclusively horizontal load also a non neglective transverse displacements occur. Table 1: Transverse displacements at the free end for various crack depths. Crack depth [m] 0.0 0.01 0.02 0.03 0.04 0.05 Displacements [m] -6.22273-10-' -2.72074W -1.22983-10 -2.94142 10' -5.55145-10 -9.28964-10' Crack depth [m] 0.06 0.07 0.08 0.09 0.10 Displacements [m] -1.44976 10 -2. 1 6763-1 0'-3.15673-10-4.52910-10-6.45904 103 Simplified model for inclusion of transverse displacements The simplified model must allow the inclusion of transverse displacements due to the tensile concentrated axial load N at cracked elements. It can be obtained from the following scheme (Figure 3). Figure 3: Torque due to tensile axial concentrated load Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 422 Computational Methods and Experimental Measurements At its location the crack of the depth d causes a shift of the neutral axis lower into the uncracked zone, thus creating a torque of inner and outer forces. A consequence is a concentrated moment denoted as IVI . It can be mathematically formulated as: