The Mechanics of a Cantilever Beam with an Embedded Horizontal Crack Subjected to an End Transverse Force, Part A: Modelling

<- Back to II. Mechanical Engineering & Physics Vol. 5

Cite the paper

Panos G. Charalambides & Xiaomin Fang (2016). The Mechanics of a Cantilever Beam with an Embedded Horizontal Crack Subjected to an End Transverse Force, Part A: Modelling. Mechanics, Materials Science & Engineering, Vol 5. doi:10.13140/RG.2.1.5051.5605

Authors: Panos G. Charalambides, Xiaomin Fang

ABSTRACT. This study addresses the mechanics of a cracked cantilever beam subjected to a transverse force applied at it’s free end. In this Part A of a two Part series of papers, emphasis is placed on the development of a four-beam model for a beam with a fully embedded horizontal sharp crack. The beam aspect ratio, crack length and crack centre location appear as general model parameters. Rotary springs are introduced at the crack tip cross sections as needed to account for the changes in the structural compliance due to the presence of the sharp crack and augmented load transfer through the near-tip transition regions.

Guided by recent finite element findings reported elsewhere, the four-beam model is advanced by recognizing two key observations, (a) the free surface and neutral axis curvatures of the cracked beam at the crack center location match the curvature of a healthy beam (an identical beam without a crack under the same loading conditions), (b) the neutral axis rotations (slope) of the cracked beam in the region between the applied load and the nearest crack tip matches the corresponding slope of the healthy beam. The above observations led to the development of close form solutions for the resultant forces (axial and shear) and moment acting in the beams above and below the crack. Axial force and bending moment predictions are found to be in excellent agreement with 2D finite element results for all normalized crack depths considered. Shear force estimates dominating the beams above and below the crack as well as transition region length estimates are also obtained. The model developed in this study is then used along with 2D finite elements in conducting parametric studies aimed at both validating the model and establishing the mechanics of the cracked system under consideration. The latter studies are reported in the companion paper Part B-Results and Discussion.

Keywords: cantilever, beam, model, embedded, horizontal, crack, mechanics, modelling

DOI 10.13140/RG.2.1.5051.5605

References

[1] Thomson, T.W. & Dahleh, M.D. (1993). Theory of Vibration with Applications, 5th Ed. Prentice Hall.

[2] Stanbridge, A.B., Martarelli, M. & Ewins, D.J. (1999). Measuring Area Mode Shapes with a Scanning Laser Doppler Vibrometer. Proceedings of International Modal Analysis Conference, 1999, Kissimmee, FL.

[3] Ewins, D.J. (2000). Modal Testing: Theory, Practice and Application, 2nd edition. Research Studies Press Ltd., Baldock, Hertfordshire, England: 422-427. 37.

[4] Stanbridge, A.B., Ewins, D.J. & Khan, A.Z. (2000). Modal Testing Using Impact Excitation and a Scanning LDV. Shock and Vibration, 7, 91-100.

[5] Vignola, J.F., Judge, J.A. & Kurdila, A.J. (2009). Shaping of a System’s Frequency Response Using an Array of Subordinate Oscillators. Journal of the Acoustical Society of America, 126(1), 129-139.

[6] Rizos, P.F., Aspragathos, N. & Dimarogonas, A.D. (1990). Identification of Crack Locations and Magnitude in a Cantilever Beam from Vibration Modes. J. of Sound and Vibration, 138(3), 381-388.

[7] Wong, C.N., Zhu, W.D. & Xu, G.Y. (2004). On an Iterative General-Order Perturbation Method for Multiple Structural Damage Detection. J. Sound and Vibration, 273, 363-386.

[8] Xu, G.Y., Zhu, W.D. & Emory, B.H. (2007). Experimental and Numerical Investigation of Structural Damage Detection Using Changes in Natural Frequencies. J. Vibration and Acoustics, 129(6), 686-700.

[9] He, K. & Zhu, W.D. (2011). A Vibration-based Structural Damage Detection Method and Its Applications to Engineering Structures. International Journal of Smart and Nano Materials, 2(3), 194-218, doi: 10.1080/19475411.2011.594105

[10] Xu, Y.F. & Zhu, W.D. (2011). Operational Modal Analysis of a Rectangular Plate Using Noncontact Acoustic Excitation. Proceedings of the International Modal Analysis Conference, 2011, Jacksonville, FL, doi: 10.1007/978-1-4419-9428-8_30

[11] He, K. & Zhu, W.D. (2010). Detection of Damage and Loosening of Bolted Joints in Structures Using Changes in Natural Frequencies. ASNT Material Evaluation, June, 2010, 721-732, doi: 10.1088/1742-6596/305/1/012054

[12] He, K. & Zhu, W.D. (2011). Damage Detection of Space Frame Structures with L-shaped Beams and Bolted Joints Using Changes in Natural Frequencies. Proceedings of the 23rd ASME Biennial Conference on Mechanical Vibration and Noise, Washington, DC, Aug. 28-31, 2011, doi:10.1115/DETC2011-48982

[13] He, K. & Zhu, W.D. (2011). Detecting Loosening of Bolted Connections in a Pipeline Using Changes in Natural Frequencies. Proceedings of the 23rd ASME Biennial Conference on Mechanical Vibration and Noise, Washington, DC, Aug. 28-31, 2011.

[14] He, K. & Zhu, W.D. (2011). Finite element modeling of structures with L-shaped beams and bolted joints. ASME Journal of Vibration and Acoustics, Vol. 133(1), doi:10.1115/1.4001840

[15] He, K. & Zhu, W.D. (2009). Modeling of Fillets in Thin-walled Beams Using Shell/Plate and Beam Finite Elements. ASME Journal of Vibration and Acoustics, 131, 051002 (16 pages), doi:10.1115/1.3142879

[16] Lin, R.J. & Cheng, F.P. (2008). Multiple crack Identification of a Free-free Beam with Uniform Material Property Variation and Varied Noised Frequency. Engineering Structures, 30, 909-929.

[17] Ganeriwala, S.N., Yang, J. & Richardson, M. (2011). Using Modal Analysis for Detecting Cracks in Wind Turbine Blades. Sound and Vibration., 45(5), 10-13.

[18] Das, P. (2012). Detection of Cracks in Beam Structures Using Modal Analysis. Applied Mechanics and Materials, 105-107, 689-694.

[19] Swamidas, A.S., Yang, X. & Seshadri, R. (2004). Identification of cracking in beam structures using Timoshenko and Euler formulations. J. Eng. Mech., 130(11), 1297–1308.

[20] Caddemi, S. & Calio, I. (2009). Exact closed-form solution for the vibration modes of the Euler-Bernoulli Beam with multiple open cracks. J. Sound and Vibration, 327, 473-489.

[21] Rubio, L. (2011). An Efficient Method for Crack Identification in Simply Supported Euler-Bernoulli Beams. Journal of Vibrations and Acoustics, 131, 051001-1 to 051001-6.

[22] Krawczuk, M. & Ostachowicz, W.M. (1995). Modeling and Vibration Analysis of a Cantilever Composite Beam with a Transverse open crack. J. Sound and Vibration, 183(1), 69-89.

[23] Dimarogonas, A.D. (1996). Vibration of Cracked Structures: A State of the Art Review. Engineering Fracture Mechanics, 55(5), 831-857.

[24] Wang, K., Inman, D.J. & Farrar, C.R. (2005). Modeling and Analysis of a Cracked Composite Beam Vibrating in Coupled Bending and Torsion. J. Sound and Vibration, 284, 23-49.

[25] Zhang, X.Q., Han, Q. & Li, F. (2010). Analytical Approach for Detection of Multiple Cracks in a Beam. J. Eng. Mech., 136(3), 345-357.

[26] Chatterjee, A. (2011). Nonlinear Dynamics and Damage Assessment of a Cantilever Beam with Breathing Edge Crack. J. Vibration and Acoustics, 133(5), 051004-1 to 051004-6, doi:10.1115/1.4003934

[27] Liu, J., Zhu, W.D., Charalambides, P.G., et al. (2016). Four-beam Model for Vibration Analysis of a Cantilever Beam with an Embedded Horizontal Crack. Chinese Journal of Mechanical Engineering, 29(1), 163-179. doi: 10.3901/CJME.2015.0901.108

[28] Broek, D. (1974). Elementary Engineering Fracture Mechanics. Springer Press.

[29] Kanninen, M.F. & Popelar, C.H. (1986). Advanced Fracture Mechanics. Oxford Press.

[30] Anderson, T.L. (1991). Fracture Mechanics: Fundamentals and Applications, Second Editions. CRC Press Inc.

[31] Data, H., Paris, P.C. & Irwin, G.R. (2000). The Stress Analysis of Cracks Handbook, 3rd Edition. American Society of Mechanical Engineers Press.

[32] Erdogan, E. (2000). Fracture Mechanics. International Journal of Solids and Structures, 27, 171-183.

[33] Fang, X. (2013). The Mechanics of an Elastically Deforming Cantilever Beam with an Embedded Sharp Crack Subjected to an End Transverse Load. Ph.D. Dissertation. Department of Mechanical Engineering, The University of Maryland, Baltimore County. December 2013.

[34] Fang, X. & Charalambides, P.G. (2015). The Fracture Mechanics of Cantilever Beams with an Embedded Sharp Crack under End Force Loading. Engineering Fracture Mechanics, Vol. 149, 1-17, doi:10.1016/j.engfracmech.2015.09.039

[35] Aladzyeu, V. (2013). The Dynamic Response of an Elastic Structure with an Embedded Sharp Crack for Damage Detection. Ph.D. Dissertation. Department of Mechanical Engineering, The University of Maryland, Baltimore County. December 2013.

[36] Sokolnikoff, I.S. (1956). Mathematical Theory of Elasticity, 2nd Edition. McGraw-Hill Press.

[37] Timoshenko, S.P. (1921). On the correction factor for shear of the differential equation for transverse vibrations of bars of uniform cross-section. Philosophical Magazine, p. 744.

[38] Cowper, G.R. (1966). The Shear Coefficient in Timoshenko’s Beam Theory. ASME, J. Appl. Mech., 33, 335-340.

[39] Riley, W.F., Sturges, L.D. & Morris, D.H. (2008). Mechanics of Materials, Sixth Edition. John Wiley & Sons, Inc.

[40] Rice, J.R. (1968). A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks. J. Appl. Mech., 35, 379-386.

[41] Rice, J.R. (1988). Elastic Fracture Mechanics Concepts for Interfacial Cracks. J. Appl. Mech., 55, 98-103.

[42] Charalambides, P.G. (1990). Fiber Debonding in Residually Stressed Brittle Matrix Composites. J. Am. Ceram. Soc., 73(6), 1674-1680.

[43] Charalambides, P.G., Lund, J., Evans, A.G. & McMeeking, R.M. (1989). A Test Specimen for Determining the Fracture Resistance of Bimaterial Interfaces. J. Appl. Mech., 56(1), 77-82.

[44] Charalambides, P.G. (1991). Steady-State Delamination Cracking in Laminated Ceramic Matrix Composites. J. Am. Ceram. Soc., 74(12), 3066-3080.

[45] Hutchinson, J.W. & Suo, Z. (1992). Mixed-mode cracking in layered materials. Advances in Applied Mechanics, 29, 63-191.

[46] Pandey, A.K., Biswas, M. & Samman, M.M. (1991). Damage Detection from Changes in Curvature Mode Shapes. J. Sound and Vibration, 145(2), 321-332.

[47] Ratcliffe, C.P. (2000). A Frequency and Curvature Based Experimental Method for Locating Damage in Structures. J. Vibration and Acoustics, 122(3), 324-329.

[48] Ratcliffe, C.P. & Crane, R.M. (2005). Structural Irregularity and Damage Evaluation Routine (SIDER) for Testing of the ½-Scale Corvette Hull Section Subjected to UNDEX Testing. NSWCCD-65-TR-2005/24.

[49] Crane, R.M., Ratcliffe, C.P., Gould, R. & Forsyth, D.S. (2006). Comparison Of The Structural Irregularity And Damage Evaluation Routine (SIDER) Inspection Method With Ultrasonic And Thermographic Inspections To Locate Impact Damage On An A-320 Vertical Stabilizer. Proceedings of the 33rd Annual Review of Progress in Quantitative Nondestructive Evaluation, July 30 – Aug 4, 2006, Portland, OR.

[50] Irwin. G. (1957). Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate. J. Appl. Mech., 24, 361-364.

Creative Commons Licence
Mechanics, Materials Science & Engineering Journal by Magnolithe GmbH is licensed under a Creative Commons Attribution 4.0 International License.
Based on a work at www.mmse.xyz.