Differential Quadrature Method Based Study of Vibrational Behaviour of Inclined Edge Cracked Beams

The study of vibration behaviour of cracked system is an important area of research. In the present work we present a mathematical model to study the effect of inclination, location and size of the crack on the vibrational behavior of beam with different boundary conditions. The model is based on the assumption that the equivalent flexible rigidity of the cracked beam can be written in terms of the flexible rigidity of the uncracked beam, based on the energy approach as proposed by earlier researchers. In the present work the Differential Quadrature Method (DQM) is used to solve equation of motion derived by using Euler’s beam theory. The primary interest of the paper is to study the effect of inclined crack on natural frequency. We have also studied the beam vibration with and without vertical edge crack as a special case to validate the model. The DQM results for the natural frequencies of cracked beams agree well with other literature values and ANSYS solutions.


Introduction
The behavior of structures containing cracks is the interesting area of research in the light of potential developments in automatic monitoring of structure quality.The study of influence on eigen-frequencies and modes shapes of the structure due to crack is important in many aspects.A number of research has been reported their work in this area.A crack introduces a local flexibility in a system which is a function of crack depth.The dynamical behavior of the system and its stability characteristics changes due to the flexibility.Here, in this work, we have taken the beam structure, specifically the Bernoulli-Euler beam is of our interest with appropriate boundary conditions.In this paper, the assumption and formulation of the model has been discussed in Section 2. Section 3 deals with the methodology used to solve the governing equation.Case studies are reported in Section 4, and the concluding remarks are given in Section 5.

Model
The variation of the equivalent bending stiffness and depth (along the beam length) for a cracked beam are obtained using an energy-based model as proposed by Yang et.al.[1] to investigate the influence of cracks on structural dynamic characteristics during the vibration of a beam with open crack.transverse vibration are obtained for a rectangular beam containing cracks.Here, we have extended the model for inclined crack (Figure 1).

Assumptions for the model
x At the location of the crack the local stiffness got reduced due to crack.x The change in strain energy due to crack under constant load assumption is computed using energy balance approach.x The equivalent bending stiffness and equivalent depth of the beam is obtained by modeling strain energy variation along the beam length.x Crack is always open during vibration.Energy needed for a crack growth of length 'a' is given by where, G is the strain energy release rate.

Vertical edge crack
For vertical edge crack G is given as, where, K 1 is the stress intensity factor of the first mode and is given by, where, M is the bending moment of the beam, and where, a/h < 0.6

Inclined edge crack
Let the kink angle is defined by α and crack is assumed to be inclined on an angle β [2], where, The coefficients C ij are defined by, (2.9) Further, if EI c is the bending stiffness of the cracked beam, strain energy in the cracked beam can be written as: For the transverse vibration of the long beam, the crack is mainly subjected to the direct bending stresses and the shear stresses can be neglected; therefore, only the first mode crack exists.While the inclined crack includes both Mode I and Mode II (mixed type) crack.The model defined in section 2 can be defined for vertical edge crack by using the strain energy release rate as shown in equation (2.3).For inclined crack the equivalent stored energy G is the function of both K I and K II (equation(2.6)).Equation 2.16-2.19gives the model of equivalent approach for vertical edge crack.

Figure 2. Effect of crack angle on strain energy [2].
A similar model has been used for the inclined edge crack with the modification in strain energy release rate as discussed above.

Energy expression
The stresses/strains are highly concentrated around the crack tip, and reach the nominal stress at a location far away from the crack.So it can be assumed that the increase of strain energy due to crack growth, under constant applied moment, is concentrated mainly around the crack region.The energy consumed for crack growth along the beam defined by [3][4] where, the terms A(a,c) and k(a) are determined over the beam length for which where, c is location of the crack from one end of the beam.Now with the help of equations 2.2-2.5 and 2.14, we can obtain,

S
(2.19) Also, at the position when x=c, we have the following relation: where, ‫ܫܧ‬ is given by (2.16).Equation (2.22) can be rewritten as, 0 2

Methodology (DQM)
In this section a brief overview of DQM has been discussed [5].It is assumed that a function W(x) is sufficiently smooth over the whole domain.The n th order derivative of the function W(x) with respect to x at m number of grid points x i , is approximated by a linear sum of all the functional values in the whole domain, that is, where, c ij represent the weighting coefficients, and n is the number of grid points in the whole domain.Equation 1

Grid point Distribution
The selection of number and type of grid points has a significant effect on the accuracy of the DQM results.It is found that the optimal selection of the sampling points in the vibration problems is the normalized Chebyshev-Gauss-Lobatto points [5].

Case Studies
The Model discussed in this paper has been used for different cracks and boundary conditions.The governing equation of motion (equation(2.23))has been transformed into a discrete eigen value problem with the help of DQM as given in equations 3.1-3.5.Also the ANSYS models has been used to compare the solution.Figure 6 shows an ansys model of actual vertical edge cracked beam, while figure 7 depicts the equivalent cracked model of the beam.The inclined edge cracked model of the beam is shown in Figure 8. Table 1-2 shows the numerical results obtained for uncracked beams by using our model as a special case.While in table 3 the effect of crack location on frequency has been shown.Tables 4 and 5 compares the effect of crack angle on frequency parameters for clamped and fixed boundary conditions.In table 6 the effect of orientation of crack has been shown for clamped beam.

Figure 1 .
Figure 1.Geometry of the inclined edge cracked beam.For an uncracked beam the strain energy is given by 1 and k 2 are the local stress intensity factor at the tip of the kink and K I and K II are the stress intensity factors for the main tilted crack given by equation (2.8-2.9),(figure(2))

Figure 3 .
Figure 3. Variation of energy function with crack location for a=0.8. and by using 2.1, 2.10 and 2.14, together with the assumption that the final strain energy in the cracked beam is c c E U U , we get the following relations: Modified bending stiffness and height (figures 4and 5) of the beam is given by:

3
Equation of motionFor an Euler beam the transverse vibration equation of cracked beam is given by, is density and A is the cross-sectional area of the beam.

Figure 7 .
Figure 7. Equivalent vertical edge cracked in beam ANSYS.

Table 6 .
Effect of crack orientation about the crack tip on frequency in fixed beam for E=2GPa, ρ=2700kg/m 3 , l=10m, b=h=1m, c/l=0.5, a/h=0.4ʋ=0.3.The model discussed in this paper has been used to find the effect of inclined crack on beam vibrations.The governing equation of motion is solved by using Differential Quadrature Method using Chebyshev's collocation points.It has been observed that the model gives optimum results for different types of boundary conditions.The applicability of the model has been obtained by the comparing the results with ANSYS actual crack model and ANSYS equivalent crack model for two types of crack (i) inclined crack and (ii) vertical crack.The results are also compared with the uncracked beam.The comparative study justify and validate the model.