Vibratory Characteristics of Axially-Loaded Timoshenko Beams With Arbitrary Number of Cracks

Kamil Aydin

doi:10.1115/1.2731411

Journal of Vibration and Acoustics | Volume 129 | Issue 3 | TECHNICAL PAPERS

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Vibratory Characteristics of Axially-Loaded Timoshenko Beams With Arbitrary Number of Cracks

Kamil Aydin

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Kamil Aydin

Erciyes University, Engineering Faculty, Department of Civil Engineering, 38039 Kayseri, Turkeykaydin@erciyes.edu.tr

J. Vib. Acoust 129(3), 341-354 (Jan 15, 2007) (14 pages) doi:10.1115/1.2731411 History: Received July 26, 2006; Revised January 15, 2007

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Abstract

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.

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References

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Krawczuk, M., Palacz, M., and Ostachowicz, W., 2003, “The Dynamic Analysis of Cracked Timoshenko Beams by Spectral Element Method,” J. Sound Vib. [CrossRef], 264 , pp. 1139–1153.

Lin, H. P., 2004, “Direct and Inverse Methods on Free Vibration Analysis of Simply Supported Beams With a Crack,” Eng. Struct., 26 , pp. 427–436.

Swamidas, A. S. J., Yang, X., and Seshadri, R., 2004, “Identification of Cracking in Beam Structures Using Timoshenko and Euler Formulations,” J. Eng. Mech. [CrossRef], 130 (11), pp. 1297–1308.

Loya, J. L., Rubio, L., and Fernandez-Saez, J., 2006, “Natural Frequencies for Bending Vibrations of Timoshenko Cracked Beams,” J. Sound Vib., 290 , pp. 640–653.

Zheng, D. Y., and Fan, S. C., 2001, “Natural Frequency Changes of a Cracked Timoshenko Beam by Modified Fourier Series,” J. Sound Vib., 246 (2), pp. 297–317.

Li, Q. S., 2003, “Vibratory Characteristics of Timoshenko Beams With Arbitrary Number of Cracks,” J. Eng. Mech. [CrossRef], 129 (11), pp. 1355–1359.

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Takehashi, I., 1999, “Vibration and Stability of Non-uniform Cracked Timoshenko Beam Subjected to Follower Force,” Comput. Struct. [CrossRef], 71 , pp. 585–591.

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Fan, S. C., and Zheng, D. Y., 2003, “Stability of a Cracked Timoshenko Beam Column by Modified Fourier Series,” J. Sound Vib., 264 , pp. 475–484.

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Timoshenko, S., Young, D., and Weaver, W., 1967, "Vibration Problems in Engineering", Wiley, New York, pp. 432–436.

Cheng, F. Y., 2000, "Matrix Analysis of Structural Dynamics: Applications and Earthquake Engineering", Marcel Dekker Inc., New York.

Lebed, O. I., and Karnovsky, I. A., 2000, "Formulas for Structural Dynamics: Tables, Graphs & Solutions", McGraw-Hill, New York, Chap. 11.

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Figures

(a) Timoshenko beam with n number of cracks, (b) an element of the beam showing the bending and shear slopes, (c) free body diagram of the element of length dx

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Figure 1

(a) Timoshenko beam with n number of cracks, (b) an element of the beam showing the bending and shear slopes, (c) free body diagram of the element of length dx

Buckling loads of simple-simple beam with (a) single crack (x1 is varied), (b) two cracks (x1 is fixed, x2 is varied), (c) three cracks (x1 and x2 are fixed, x3 is varied). The curves are obtained for crack depth ratios of 0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 from top to bottom, respectively, in each plot.

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Figure 2

Buckling loads of simple-simple beam with (a) single crack (x1 is varied), (b) two cracks (x1 is fixed, x2 is varied), (c) three cracks (x1 and x2 are fixed, x3 is varied). The curves are obtained for crack depth ratios of 0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 from top to bottom, respectively, in each plot.

Buckling loads of fixed-fixed beam with (a) single crack (x1 is varied), (b) two cracks (x1 is fixed, x2 is varied), (c) three cracks (x1 and x2 are fixed, x3 is varied). The curves are obtained for crack depth ratios of 0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 from top to bottom respectively in each plot.

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Figure 3

Buckling loads of fixed-fixed beam with (a) single crack (x1 is varied), (b) two cracks (x1 is fixed, x2 is varied), (c) three cracks (x1 and x2 are fixed, x3 is varied). The curves are obtained for crack depth ratios of 0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 from top to bottom respectively in each plot.

Buckling loads of fixed-hinged beam with (a) single crack (x1 is varied), (b) two cracks (x1 is fixed, x2 is varied), (c) three cracks (x1 and x2 are fixed, x3 is varied). The curves are obtained for crack depth ratios of 0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 from top to bottom, respectively, in each plot.

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Figure 4

Buckling loads of fixed-hinged beam with (a) single crack (x1 is varied), (b) two cracks (x1 is fixed, x2 is varied), (c) three cracks (x1 and x2 are fixed, x3 is varied). The curves are obtained for crack depth ratios of 0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 from top to bottom, respectively, in each plot.

Buckling loads of fixed-free beam with (a) single crack (x1 is varied), (b) two cracks (x1 is fixed, x2 is varied), (c) three cracks (x1 and x2 are fixed, x3 is varied). The curves are obtained for crack depth ratios of 0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 from top to bottom respectively in each plot.

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Figure 5

Buckling loads of fixed-free beam with (a) single crack (x1 is varied), (b) two cracks (x1 is fixed, x2 is varied), (c) three cracks (x1 and x2 are fixed, x3 is varied). The curves are obtained for crack depth ratios of 0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 from top to bottom respectively in each plot.

First, second and third vibrational frequencies of axially-loaded hinged-hinged Timoshenko beam with crack depth ratios of (a)a1∕h=0.1, (b)a1∕h=0.3, and (c)a1∕h=0.5. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

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Figure 6

First, second and third vibrational frequencies of axially-loaded hinged-hinged Timoshenko beam with crack depth ratios of (a)a1∕h=0.1, (b)a1∕h=0.3, and (c)a1∕h=0.5. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

First, second, and third vibrational frequencies of axially-loaded hinged-hinged Timoshenko beam having two cracks with crack depth ratios of (a)a1∕h=a2∕h=0.3 and (b)a1∕h=a2∕h=0.5 for x1∕L=0.25 and varying x2∕L. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

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Figure 7

First, second, and third vibrational frequencies of axially-loaded hinged-hinged Timoshenko beam having two cracks with crack depth ratios of (a)a1∕h=a2∕h=0.3 and (b)a1∕h=a2∕h=0.5 for x1∕L=0.25 and varying x2∕L. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

First, second, and third vibrational frequencies of axially-loaded hinged-hinged Timoshenko beam having three cracks with crack depth ratios of a1∕h=a2∕h=a3∕h=0.3 for x1∕L=0.25, x2∕L=0.65, and varying x3∕L. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.3 from top to bottom, respectively, in each plot.

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Figure 8

First, second, and third vibrational frequencies of axially-loaded hinged-hinged Timoshenko beam having three cracks with crack depth ratios of a1∕h=a2∕h=a3∕h=0.3 for x1∕L=0.25, x2∕L=0.65, and varying x3∕L. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.3 from top to bottom, respectively, in each plot.

First, second, and third vibrational frequencies of axially-loaded fixed-fixed Timoshenko beam with a crack depth ratio of (a)a1∕h=0.1, (b)a1∕h=0.3, and (c)a1∕h=0.5. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

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Figure 9

First, second, and third vibrational frequencies of axially-loaded fixed-fixed Timoshenko beam with a crack depth ratio of (a)a1∕h=0.1, (b)a1∕h=0.3, and (c)a1∕h=0.5. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

First, second, and third vibrational frequencies of axially-loaded fixed-fixed Timoshenko beam having two cracks with crack depth ratios of (a)a1∕h=a2∕h=0.3 and (b)a1∕h=a2∕h=0.5 for x1∕L=0.25 and varying x2∕L. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

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Figure 10

First, second, and third vibrational frequencies of axially-loaded fixed-fixed Timoshenko beam having two cracks with crack depth ratios of (a)a1∕h=a2∕h=0.3 and (b)a1∕h=a2∕h=0.5 for x1∕L=0.25 and varying x2∕L. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

First, second, and third vibrational frequencies of axially-loaded fixed-fixed Timoshenko beam having three cracks with crack depth ratios of a1∕h=a2∕h=a3∕h=0.3 for x1∕L=0.25, x2∕L=0.65, and varying x3∕L. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.3 from top to bottom, respectively, in each plot.

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Figure 11

First, second, and third vibrational frequencies of axially-loaded fixed-fixed Timoshenko beam having three cracks with crack depth ratios of a1∕h=a2∕h=a3∕h=0.3 for x1∕L=0.25, x2∕L=0.65, and varying x3∕L. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.3 from top to bottom, respectively, in each plot.

First, second, and third vibrational frequencies of axially-loaded fixed-hinged Timoshenko beam with a crack depth ratio of a1∕h=0.3. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

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Figure 12

First, second, and third vibrational frequencies of axially-loaded fixed-hinged Timoshenko beam with a crack depth ratio of a1∕h=0.3. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

First, second, and third vibrational frequencies of axially-loaded fixed-free Timoshenko beam with a crack depth ratio of a1∕h=0.3. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

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Figure 13

First, second, and third vibrational frequencies of axially-loaded fixed-free Timoshenko beam with a crack depth ratio of a1∕h=0.3. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

(a) Guided-pinned beam, (b) elastically restrained beam with a concentraed mass at the right end

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Figure 14

(a) Guided-pinned beam, (b) elastically restrained beam with a concentraed mass at the right end

First, second, and third vibrational frequencies of axially-loaded guided-pinned Timoshenko beam with a crack depth ratio of a1∕h=0.3. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

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Figure 15

First, second, and third vibrational frequencies of axially-loaded guided-pinned Timoshenko beam with a crack depth ratio of a1∕h=0.3. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

First, second, and third vibrational frequencies of axially-loaded and elastically-restrained Timoshenko beam with a crack depth ratio of a1∕h=0.5. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

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Figure 16

First, second, and third vibrational frequencies of axially-loaded and elastically-restrained Timoshenko beam with a crack depth ratio of a1∕h=0.5. The curves are obtained for axial load levels P∕Pcr of −0.2, −0.1, 0.0, 0.1, and 0.2 from top to bottom, respectively, in each plot.

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Aydin K. Vibratory Characteristics of Axially-Loaded Timoshenko Beams With Arbitrary Number of Cracks. ASME. J. Vib. Acoust. 2007;129(3):341-354. doi:10.1115/1.2731411.

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Vibratory Characteristics of Axially-Loaded Timoshenko Beams With Arbitrary Number of Cracks

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