Abstract：The basis for the dynamic calculation of cracked members and the identification of crack damage is the element stiffness matrix with crack parameters. Current studies have mainly characterized the effect of cracking on the element stiffness by the variation of the open-cracked beam cross section, but the crack mainly affects the stress-strain distribution of the element. In this paper, for the two-node four-degree-of-freedom Euler-Bernoulli open-cracked beam element, on the basis of the cubic shape function, a step function is used to consider the influence of the crack, and a linear function is superimposed to modify the cubic shape function, and a new shape function with crack parameters is proposed, and then the stiffness matrix of the Euler-Bernoulli cracked beam element is obtained by combining the principle of virtual displacement. The simulation example shows that when the crack depth ratio is less than 0.5, the maximum relative error is 1.714% when comparing the deflection value calculated by the shape function of this paper with the finite element results, and the maximum error of the first-order nature frequency calculated by the cracked beam element stiffness matrix of this paper is 0.936%. The shape function of this paper can accurately describe the stress-strain distribution of the cracked element, and the cracked beam element stiffness matrix can be used for structural static dynamic analysis. This paper provides a new research idea to consider the effect of cracks on element stiffness.
Key words: Euler-Bernoulli crack beam; element stiffness matrix; shape function; static analysis; frequency
徐训，朱亚杉，吴浩. 基于修正形函数的Euler-Bernoulli开口裂纹梁单元刚度矩阵[J]. 振动与冲击, 2022, 41(17): 292-302.
XU Xun, ZHU Yashan, WU Hao. Element stiffness matrix of Euler-Bernoulli open crack beam based on modified shape function. JOURNAL OF VIBRATION AND SHOCK, 2022, 41(17): 292-302.
[1] Genikomsou A S, Polak M A. Finite element analysis of punching shear of concrete slabs using damaged plasticity model in ABAQUS[J]. Engineering Structures, 2015, 98:38-48.
[2] Liu G H, Su X T, Yang Z. Finite element modelling of complex 3D static and dynamic crack propagation using cohesive elements in ABAQUS[J]. Journal of Medical Genetics, 2010, 17(3):212-215.
[3] Zhou H L. Implementation of Crack Problem of Functionally Graded Materials with ABAQUS(TM)[J]. Advanced Materials Research, 2011, 284-286(1-3):297-300.
[4] Malekan M, Khosravi A, St-Pierre L. An Abaqus plug-in to simulate fatigue crack growth[J]. Engineering with Computers, 2021.
[5] Li J, Ning W. A Study of Crack Detection Algorithm[C]//Li, JB. International Conference on Instrumentation Measurement Computer Communication and Control. 2015. 1184-1187.
[6] Elshamy M, Crosby W A, Elhadary M. Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis[J]. AEJ - Alexandria Engineering Journal, 2018,57(4):3755-3766.
[7] Caddemi S, Caliò I, Cannizzaro F, et al. A procedure for the identification of multiple cracks on beams and frames by static measurements[J]. Structural Control and Health Monitoring, 2018, 25(16).
[8] Papadopoulos C A. The strain energy release approach for modeling cracks in rotors: A state of the art review[J]. Mechanical Systems and Signal Processing, 2008, 22(4): 763-789.
[9] Papadopoulos C A, Dimarogonas A D. Coupled longitudinal and bending vibrations of a rotating shaft with an open crack[J]. Journal of Sound and Vibration, 1987, 117(1):81-93.
[10] Darpe A K, Gupta K, Chawla A. Coupled bending, longitudinal and torsional vibrations of a cracked rotor[J]. Journal of Sound and Vibration, 2004, 269(1-2):33-60.
[11] Kessissoglou D. Free vibration analysis of a cracked beam by finite element method[J]. Journal of Sound and Vibration, 2004,273(4):57-75.
[12] Krawczuk M. A rectangular plate finite element with an open crack[J]. Computers and Structures, 1993, 46(3):487-493.
[13] Yazdi A K, Shooshtari A. A new two-dimensional cracked finite element for fracture mechanics[J]. Engineering Fracture Mechanics, 2015, 135:17-33.
[14] Yazdi A K, Shooshtari A, Fazelipour M. Introducing new cracked finite elements and a method for SIF calculation of cracks[J]. Mechanics Based Design of Structures and Machines, 2016, 44(3):176-188.
[15] Potirniche G P, Hearndon J, Daniewicz S R, et al. A two-dimensional damaged finite element for fracture applications[J]. Engineering Fracture Mechanics, 2008, 75(13):3895-3908.
[16] Hall K J, Potirniche G P. A three-dimensional edge-crack finite element for fracture mechanics applications[J]. International Journal of Solids and Structures, 2012, 49(2):328-337.
[17] Yazdi A K, Shooshtari A. The stiffness matrix of cracked finite elements: Introducing shortcomings in applying stiffness approach and proposing a solution[J]. Engineering Fracture Mechanics, 2018, 191:412-425
[18] 胡家顺,冯新,李昕,等.裂纹梁振动分析和裂纹识别方法研究进展[J].振动与冲击,2007(11):146-152+189.
HU Jia-shun, FENG Xin, LI Xin, et al. State-of-art of vibration analysis and crack identification of cracked beams[J]. Journal of vibration and shock,2007(11):146-152+189.
[19] Giannini O, Casini P, Vestroni F. Nonlinear harmonic identification of breathing cracks in beams[J]. Computers and Structures,2013,129:166-177.
[20] Wahab M, Roeck G D, Peeters B. Parameterization of damage in reinforced concrete structures using model updating[J]. Journal of Sound and Vibration, 1999, 228(4):717-730.
[21] Shen M , Taylor J E. An identification problem for vibrating cracked beams[J]. Journal of Sound and Vibration, 1991, 150(3):457-484.
[22] Christides S, Barr A D S. One-dimensional theory of cracked Bernoulli-Euler beams[J]. International Journal of Mechanical Sciences, 1984, 26(11-12):639-648.
[23] Sinha J K, Friswell M I, Edwards S. Simplified models for the location of cracks in beam structures using measured vibration data[J]. Journal of Sound and Vibration, 2002, 251(1):13-38.
[24] 李景涌.有限元法[M].北京：北京邮电大学出版社.1999
[25] Babuška I, Melenk J M. The Partition of unity method [J]. International Journal for Numerical Methods in Engineering, 2015, 40(4):727-758.
[26] Wu J, Cai Y. A partition of unity formulation referring to the NMM for multiple intersecting crack analysis[J]. Theoretical and Applied Fracture Mechanics, 2014, 72:28-36.
[27] Cai Y C, Lin H, Tian L G, et al. Meshless method based on Shepard function and partition of unity for two-dimensional crack problems[J]. Engineering Analysis with Boundary Elements 2016,65:126-135.
[28] Shi J P, Ma W T, Li N. Extended meshless method based on partition of unity for solving multiple crack problems[J]. Meccanica, 2013, 48(9):2263-2270.
[29] Fonseca G M, Barros F B, Oliveira T, et al. 2-D Crack propagation analysis using stable generalized finite element method with global-local enrichments[J]. Engineering Analysis with Boundary Elements, 2020, 118:70-83.
[30] Triantafyllou S P, Kakouris E G. A generalised phase‐field multiscale finite element method for brittle fracture[J]. International Journal for Numerical Methods in Engineering, 2019,121(9):1915-1945.
[31] Mazzucato A L, Nistor V, Qu Q Q. A nonconforming generalized finite element method for transmission problems[J]. SIAM Journal On Numerical Analysis, 2013, 51(1):555-576.
[32] Tian R. Extra-dof-free and linearly independent enrichments in GFEM[J]. Computer Methods in Applied Mechanics & Engineering, 2013, 266(nov.1):1-22.
[33] 冯新,刘彦辉,周晶.一种Timoshenko裂纹梁的静力挠度分析方法[J].防灾减灾工程学报,2009,29(06):652-657.
FENG Xin, LIU Yan-hui, ZHOU Jing. A method for static deflection analysis for cracked timoshenko beam[J]. Journal of Disaster Prevention and Mitigation Engineering, 2009,29(06):652-657.
[34] 马学振. 考虑混凝土损伤的钢-混双面组合梁刚度计算方法[D].石家庄市：石家庄铁道大学,2019.
[35] Tada H, Paris P, Irwin G. The Stress Analysis of Cracks Handbook [M]. Paris: Paris Productions, St.Louis, 2000.
[36] 胡家顺. 裂纹管结构的振动分析与裂纹识别[D].大连市：大连理工大学,2009.
