A continuous model for the transverse vibrations of cracked beams including the effect of shear deformation is derived. Partial differential equations of motion and associated boundary conditions are obtained via the Hu-Washizu-Barr variational principle, which allows simultaneous and independent assumptions on the displacement, stress and strain fields. The stress and strain concentration caused by the presence of a crack are represented by so-called crack disturbance functions, which modify the kinematic assumptions used in the variational procedure. For the shear stress/strain fields, a quadratic distribution over the beam depth is assumed, which is a refinement of the typical constant shear stress distribution implicit in the Timoshenko model for uncracked beams. The resulting equations of motion are solved by a Galerkin method using local B-splines as test functions. As a numerical verification, natural frequencies of the linear, open-crack model are computed and the results are compared to analytical results from similar models based on Euler-Bernoulli assumptions and experimental results found in the literature. For short beams, results from a 2-D finite element model are used to confirm the advantages of the proposed model when compared with previous formulations.

1.
Rytter, A., 1993, “Vibration Based Inspection of Civil Engineering Structures;” Ph.D. thesis, Aalborg University, Denmark.
2.
Dimarogonas
,
A. D.
,
1996
, “
Vibration of Cracked Structures: a State-of-the-Art Review
,”
Eng. Fract. Mech.
,
55
(
5
), pp.
831
857
.
3.
Gudmundson
,
P.
,
1983
, “
The Dynamic Behavior of Slender Structures with Cross-Sectional Cracks
,”
J. Mech. Phys. Solids
,
31
(
4
), pp.
329
345
.
4.
Ostachowicz
,
W. M.
, and
Krawczuk
,
M.
,
1990
, “
Vibration Analysis of a Cracked Beam
,”
Comput. Struct.
,
36
(
2
), pp.
245
250
.
5.
Wauer
,
J.
,
1990
, “
On the Dynamics of Cracked Rotors: a Literature Survey
,”
Appl. Mech. Rev.
,
43
(
1
), pp.
13
17
.
6.
Abraham
,
O. N. L.
, and
Brandon
,
J. A.
,
1995
, “
The Modelling of the Opening and Closure of a Crack
,”
ASME J. Vibr. Acoust.
,
117
, pp.
370
377
.
7.
Brandon
,
J. A.
,
1999
, “
Towards a Nonlinear Identification Methodology for Mechanical Signature Analysis
,”
Key Eng. Mater.
, pp.
167–168
, pp.
265
272
.
8.
Christides
,
S.
, and
Barr
,
A. D. S.
,
1984
, “
One-Dimensional Theory of Cracked Bernoulli-Euler Beams
,”
Int. J. Mech. Sci.
,
26
(
11/12
), pp.
639
648
.
9.
Shen
,
M-H. H.
, and
Pierre
,
C.
,
1990
, “
Natural Modes of Bernoulli-Euler Beams with Symmetric Cracks
,”
J. Sound Vib.
,
138
(
1
), pp.
115
134
.
10.
Shen
,
M-H. H.
, and
Pierre
,
C.
,
1994
, “
Free Vibrations of Beams with a Single Edge Crack
,”
J. Sound Vib.
,
170
(
2
), pp.
237
259
.
11.
Chondros
,
T. G.
,
Dimarogonas
,
A. D.
, and
Yao
,
J.
,
1998
, “
A Continuous Cracked Beam Vibration Theory
,”
J. Sound Vib.
,
215
(
1
), pp.
17
34
.
12.
Barr
,
A. D. S.
,
1966
, “
An Extension of the Hu-Washizu Variational Principle in Linear Elasticity for Dynamic Problems
,”
ASME J. Appl. Mech.
,
33
(
2
), p.
465
465
.
13.
Armon
,
D.
,
Ben-Haim
,
Y.
, and
Braun
,
S.
,
1994
, “
Crack Detection in Beams by Rank-Ordering of Eigenfrequency Shifts
,”
Mech. Syst. Signal Process.
,
8
(
1
), pp.
81
91
.
14.
Banks
,
H. T.
,
Inman
,
D. J.
,
Leo
,
D. J.
, and
Wang
,
Y.
,
1996
, “
An Experimentally Validated Damage Detection Theory in Smart Structures
,”
J. Sound Vib.
,
191
(
5
), pp.
859
880
.
15.
Reddy, J. N., 1984, Energy and Variational Methods in Applied Mechanics, John Wiley & Sons, New York.
16.
Carneiro, S. H. S., 2000, “Model-Based Vibration Diagnosis of Cracked Members in the Time Domain,” Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.
17.
Wang, Y., 1991, “Damping Modeling and Parameter Estimation in Timoshenko Beams,” Ph.D. thesis, Brown University, Providence, RI.
You do not currently have access to this content.