A simple and efficient analytical approach is presented to determine the vibrational frequencies and mode shape functions of axially-loaded Timoshenko beams with an arbitrary number of cracks. The local compliance induced by a crack is described by a massless rotational spring model. A set of boundary conditions are used as initial parameters to define the mode shape of the segment of the beam before the first crack. Using this, the remaining set of boundary conditions and recurrence formula developed in the study, the mode shape function of vibration of the beam containing multiple cracks can be easily determined. Four different classical boundary conditions (pinned-pinned, clamped-pinned, clamped-free, and clamped-clamped) are considered. Elastically-restrained support condition with concentrated masses is also considered. Three crack depths and five axial force levels representing the conditions under service loads are used. A parametric study is carried out for each case of support conditions to investigate the effect of crack and axial load on the vibrational properties of cracked Timoshenko beams. The influence of crack on the buckling load of the beam is also studied statically. Part of the results obtained is checked against the published values. The study concludes that the crack location, crack severity, and axial force level strongly affect the eigenfrequencies.

1.
Zhang
,
W.
, and
Testa
,
R. B.
, 1999, “
Closure Effects on Fatigue Crack Detection
,”
J. Eng. Mech.
0733-9399,
125
, pp.
1125
1132
.
2.
Kisa
,
M.
,
Brandon
,
J.
, and
Topcu
,
M.
, 1998, “
Free Vibration Analysis of Cracked Beams by a Combination of Finite Elements and Component Mode Synthesis Methods
,”
Comput. Struct.
0045-7949,
67
, pp.
215
223
.
3.
Viola
,
E.
,
Federici
,
L.
, and
Nobile
,
L.
, 2001, “
Detection of Crack Location Using Cracked Beam Element Method for Structural Analysis
,”
Theor. Appl. Fract. Mech.
0167-8442,
36
, pp.
23
35
.
4.
Lele
,
S. P.
, and
Maiti
,
S. K.
, 2002, “
Modeling of Transverse Vibration of Short Beams for Crack Detection and Measurement of Crack Extension
,”
J. Sound Vib.
0022-460X,
257
, pp.
559
583
.
5.
Krawczuk
,
M.
,
Palacz
,
M.
, and
Ostachowicz
,
W.
, 2003, “
The Dynamic Analysis of Cracked Timoshenko Beams by Spectral Element Method
,”
J. Sound Vib.
0022-460X,
264
, pp.
1139
1153
.
6.
Lin
,
H. P.
, 2004, “
Direct and Inverse Methods on Free Vibration Analysis of Simply Supported Beams With a Crack
,”
Eng. Struct.
0141-0296,
26
, pp.
427
436
.
7.
Swamidas
,
A. S. J.
,
Yang
,
X.
, and
Seshadri
,
R.
, 2004, “
Identification of Cracking in Beam Structures Using Timoshenko and Euler Formulations
,”
J. Eng. Mech.
0733-9399,
130
(
11
), pp.
1297
1308
.
8.
Loya
,
J. L.
,
Rubio
,
L.
, and
Fernandez-Saez
,
J.
, 2006, “
Natural Frequencies for Bending Vibrations of Timoshenko Cracked Beams
,”
J. Sound Vib.
0022-460X,
290
, pp.
640
653
.
9.
Zheng
,
D. Y.
, and
Fan
,
S. C.
, 2001, “
Natural Frequency Changes of a Cracked Timoshenko Beam by Modified Fourier Series
,”
J. Sound Vib.
0022-460X,
246
(
2
), pp.
297
317
.
10.
Li
,
Q. S.
, 2003, “
Vibratory Characteristics of Timoshenko Beams With Arbitrary Number of Cracks
,”
J. Eng. Mech.
0733-9399,
129
(
11
), pp.
1355
1359
.
11.
Kim
,
K-H.
, and
Kim
,
J-H.
, 2000, “
Effect of a Crack on the Dynamic Stability of a Free-Free Beam Subjected to a Follower Force
,”
J. Sound Vib.
0022-460X,
233
(
1
), pp.
119
135
.
12.
Takehashi
,
I.
, 1999, “
Vibration and Stability of Non-uniform Cracked Timoshenko Beam Subjected to Follower Force
,”
Comput. Struct.
0045-7949,
71
, pp.
585
591
.
13.
Mei
,
C.
,
Karpenko
,
Y.
,
Moody
,
S.
, and
Allen
,
D.
, 2006, “
Analytical Approach to Free and Forced Vibrations of Axially Loaded Cracked Timoshenko Beams
,”
J. Sound Vib.
0022-460X,
291
, pp.
1041
1060
.
14.
Fan
,
S. C.
, and
Zheng
,
D. Y.
, 2003, “
Stability of a Cracked Timoshenko Beam Column by Modified Fourier Series
,”
J. Sound Vib.
0022-460X,
264
, pp.
475
484
.
15.
Dimarogonas
,
A. D.
, 1996, “
Vibration of Cracked Structures: A State of the Art Review
,”
Eng. Fract. Mech.
0013-7944,
55
(
5
), pp.
831
857
.
16.
Timoshenko
,
S.
,
Young
,
D.
, and
Weaver
,
W.
, 1967,
Vibration Problems in Engineering
,
Wiley
, New York, pp.
432
436
.
17.
Cheng
,
F. Y.
, 2000,
Matrix Analysis of Structural Dynamics: Applications and Earthquake Engineering
,
Marcel Dekker Inc.
, New York.
18.
Lebed
,
O. I.
, and
Karnovsky
,
I. A.
, 2000,
Formulas for Structural Dynamics: Tables, Graphs & Solutions
,
McGraw-Hill
, New York, Chap. 11.
19.
Hoit
,
M. I.
, 1994,
Computer Assisted Structural Analysis and Modeling
,
Prentice-Hall
, New Jersey.
20.
Wang
,
C. M.
,
Reddy
,
J. N.
, and
Lee
,
K. H.
, 2000,
Shear Deformable Beams and Plates, Relationships with Classical Solutions
,
Elsevier Science
, New York, Chap. 4.
You do not currently have access to this content.