1. Dynamics Characteristics of Cracked Members New Formulation Axial MembersBy
Dr. A. Ranjbaran, Associate Professor
Department of Civil Engineering, Shiraz University, Shiraz Iran
Bahman, 1387 (February 2009)

2. Available Finite Element Model Number of Equations > 10000
MOTIVATION
Available Finite Element Model
Number of Equations > 10000
Desired Finite Element Model
Number of Equations <20
Finite

3. Review of Literature (1)
Rice, J.R. & Levy, N., in Journal Of Applied Mechanics 39 (1972) , demonstrated that:
A mass less spring can be used to represent the compliance due to both longitudinal and transverse motion of beams.

4. Review of Literature (2)
Chondros , et. Al., in Engineering Fracture Mechanics 61 (1998) , developed:
A continuous cracked bar vibration theory for longitudinal vibration of rods with an edge crack. They derived the differential equation and boundary conditions based on the Hu-Washizu-Bar variational formulation.

5. Review of Literature (3)
Ruotolo R. & Surace C., Journal of Sounds and Vibration 272 (2004)
They used the smooth function method for calculation of longitudinal natural frequencies of a bar with transverse open cracks.
They used the Heaviside step function and Dirac’s delta distribution in their formulation.

6. Review of Literature (5)
Previous Methods:
The Transformation Matrix Method (TMM) is used very frequently.
All the other methods lead to a system of equations (for the best case with n+2 unknown) for n cracks.

7. Review of Literature (6)
Short Comes of Previous work.
Cumbersome for multi-cracked case.
Not suitable for development of the finite element method.
Not suitable for determining the exact solution.

8. Governing Equations
In Fracture mechanics a crack is modeled by a spring.
The spring introduces a jump in axial displacement as, see Crack model 1:
(1)
Taking derivative w.r.to x leads to:
(2)
In which H is the Heaviside step function and δ is Dirac’s delta function.
To pave the way for new formulation, select new variable:
(3)

9. The boundary conditions are changed as follows: (4) (5) Then
Governing Equations
The boundary conditions are changed as follows:
(4)
(5)
Then
Boundary Conditions
Fixed →
Free

10. Governing Equations See crack model 2.
In modeling, the cracked bar is assumed to be composed of intact and cracked contributions.
See crack model 2.
For each part the governing equations are derived.
The GE for cracked bar is determined by combining the two.

11. Governing Equations (6)
The curvature at a crack position is composed of crack and intact contribution, i.e. as follows:
(6)
The Governing Equation (GE) for free vibration is:
(7)
Substitute From (6) into (7) to obtain:
(8)
The ordinary differential equation 8 is the GE of cracked member.

12. Finite Element Formulation
Using the Weighted Residual Method (WRM), the error on the domain (For eigen value problem natural B.CS are zero) is distributed as follows:
(9)
Use the chain rule of differentiation to write:
(10)
Use 10 into 9 to obtain the weak form of GE:
(11)

13. Finite Element Formulation
Define the weight and essential parameters in terms of the same shape functions, N, as (Galerkin Method):
(12)
Insert equation 12 into 11 and simplify to obtain:
(13)
Equation 13 is the Finite Element Equation of the problem.

14. Finite Element Formulation
In which the stiffness matrix , the mass matrix and the equivalent mass matrix of cracks are defined as follows:
(14)
The Einstein’s summation convention (Indicial notation) is used throughout. In this convention the Greek letters are summed over the number of element’s node and j is summed over the number of cracks.

15. Superiorities of the Present Formulation
There is no need to model singular element.
There is no restriction on the number of cracks.
The effect of cracks is considered in mass matrix.
Computation of equivalent mass matrix is simple.
One dimensional (bar) elements may be used for solution.
Analysis of structures including cracked bar is possible.
The exact solution for the GE may be derived.

16. Exact Solution The Laplace Transform (LT) of GE is written as: (15)
And the LT of the displacement is:
(16)
By taking inverse transform the solution is:
(17)
In which
(18)

17. Exact Solution
Applying the boundary conditions to the general solution (equation 17 and its derivative) the natural frequencies and mode shapes are obtained.
The actual mode shapes is determined by differentiating from that of W.
The derived exact solution is the most simple and general solution available in the literature to date.

18. Verification
To verify the present work it is used in several examples.
In each example the result of the present work is compared with that of commercial software.
The agreement of the results verify the work.
Comparison of number of equations needed and the computer time used show the high superiority of the present work.

19. Example 1
A fixed free bar with rectangular section of length L crack position b and crack depth a and the following properties is selected for study.
The present numerical result is denoted by Finite Element Method (FEM) and the exact result by EXACT.
In FEM the bar is divided into 10 elements!

20. Example 1
The result of analysis by commercial software is denoted by ANSYS. The mesh used is shown in Figure 1. More than 6000 elements is used.
In FEM the bar is divided into 10 elements.
Using the exact solution the following equation is obtained for computing the Eigen value

21. Example 1 And the mode shape is computed as
The variation of first natural frequency ratio (cracked/intact) is compared in Figure 2.
The error is shown in Figure 3.
Excellent agreement of the results verified the work. Number of equation in ANSYS is 10s of thousands times of the present work.

22. Example 2
A fixed free bar with rectangular section of length L with two cracks is considered. The following property is selected for study.
Using the exact solution the natural frequencies are obtained as:

23. Example 2 The mesh used for analysis by ANSYS is shown in Figure 4.
The variation of first natural frequency ratio (cracked/intact) is compare in Figure 5.
The error is shown in Figure 6.
The first mode shape is shown in Figure 7.

24. Example 2
Excellent agreement of the results verified the work. The number of equations in ANSYS is 10s of thousands times of the present work.

25. Conclusions An ordinary differential equation, as GE, is developed.
A new and novel formulation for vibration of cracked bars is derived.
An ordinary differential equation, as GE, is developed.
A n exact solution is presented.
A novel finite element formulation is introduced.

26. Conclusions
The singularity of cracks is considered in mass matrix in an efficient manner.
For the same accuracy the efficiency of the presented method is 10s of 1000 more than others.
The presented formulation is novel, innovative and robust.

27. THANKS
Thank You

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50. Figure 4
Mesh Used by ANSYS

51. Variation of Frequency Ratio
Figure 4
Variation of Frequency Ratio

52. Comparison of Results (error)
Figure 4
Comparison of Results (error)

53. Figure 4
Mesh Used by ANSYS

54. Variation of Frequency Ratio
Figure 5
Variation of Frequency Ratio

55. Comparison of results (error)
Figure 6
Comparison of results (error)

56. The First Mode Shape of the double Cracked Bar
Figure 7
The First Mode Shape of the double Cracked Bar

57. THANKS FOR YOUR ATTENTION
END
THANKS FOR YOUR ATTENTION

58. Models
Cracked Bar Model 2

59. Crack Model 1
Properties of Spring