Natural Frequencies of Multistep Functionally Graded Beam with Cracks

Attar M (2012) A transfer matrix method for free vibration analysis and crack identification of stepped beams with multiple edge cracks and different boundary conditions. Int J Mech Sci 57:19–33 Aydin K (2013) Free vibration of functional graded beams with arbitrary number of cracks. Eur J Mech A/Solid 42:112–124 Banerjee A, Panigrahi B, Pohit G (2015) Crack modelling and detection in Timoshenko FGM beam under transverse vibration using frequency contour and response surface model with GA. Nondestruct Test Eval. https://doi.org/10.1080/10589759.2015.1071812 Birman V, Byrd LW (2007) Modeling and Analysis of Functional Graded Materials and Structures. Appl Mech Rev 60:195–215 Chakraborty A, Gopalakrishnan S (2003) A spectrally formulated finite element for wave propagation analysis in functionally graded beams. Int J Solids Struct 40:2421–2448 Chakraborty A, Gopalakrishnan S, Reddy JN (2003) A new beam finite element for the analysis of functional graded materials. Int J Mech Sci 45:519–539 Chondros TG, Dimarogonas AD (1998) A continuous cracked beam theory. J Sound Vib 215:17–34 Chondros TG, Dimarogonas AD, Yao J (1998) Longitudinal vibration of a continuous cracked bar. Eng Fract Mech 61:593–606 Cunha J, Junior JJ (2016) Vibration analysis of Euler-Bernoulli beams in multiple steps and different shapes of cross section. J Vib Control 22(1):193–204 Eltaher MA, Alshorbagy AE, Mahmoud FF (2013) Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nano-beams. Compos Struct 99:193–201 Erdogan F, Wu BH (1997) The surface crack problem for a plate with functionally graded properties. J Appl Mech 64:448–456 Jang SK, Bert CW (1989) Free vibration of stepped beams: higher mode frequencies and effects of steps on frequency. J Sound Vib 132:164–168 Khiem NT, Kien ND, Huyen NN (2014) Vibration theory of FGM beam in the frequency domain. In: Proceedings of national conference on engineering mechanics celebrating 35th anniversary of the institute of mechanics, VAST, April 9, 2014.V.1, pp 93–98 (in Vietnamese) Kitipornchai S, Ke LL, Yang J, Xiang Y (2009) Nonlinear vibration of edge cracked functionally graded Timoshenko beams. J Sound Vib 324:962–982 Kukla S, Zamojska I (2007) Frequency analysis of axially loaded stepped beams by Green’s function method. J Sound Vib 300:1034–1041 Li QS (2001) Vibratory characteristics of multi-step beams with an arbitrary number of cracks and concentrated masses. Appl Acoust 62:691–706 Li XF (2008) A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams. J Sound Vib 318:1210–1229 Maghsoodi A, Ghadami A, Mirdamadi HR (2013) Multiple-crack damage detection in multi-step beams by a novel local flexibility-based damage index. J Sound Vib 332:294–305 Mao Q (2011) Free vibration analysis of multiple-stepped beams by using Adomian decomposition method. Math Comput Model 54:756–764 Nandwanna BP, Maiti SK (1997) Detection of the location and size of a crack in stepped cantilever beams based on measurements of natural frequencies. J Sound Vib 203(3):435–446 Panigrahi B, Pohit G (2016) Nonlinear modeling and dynamic analysis of cracked Timoshenko functionally graded beams based on neutral surface approach. J Mech Eng Sci 230(9):1486–1497 Pradhan KK, Chakraverty S (2013) Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Compos Part B 51:175–184 Sato H (1983) Free vibration of beams with abrupt changes of cross-section. J Sound Vib 89:59–64 Sina SA, Navazi HM, Haddadpour H (2009) An analytical method for free vibration analysis of functionally graded beams. Mater Design 30:741–747 Su H, Banerjee JR (2015) Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beams. Comput Struct 147:107–116 Suddoung K, Charoensuk J, Wattanasakulpong N (2014) Vibration response of stepped FGM beams with elastically end constraints using differential transformation method. Appl Acoust 77:20–28 Tsai TC, Wang YZ (1996) Vibration analysis and diagnosis of a cracked shaft. J Sound Vib 192(3):607–620 Wang XW, Wang YL (2013) Free vibration analysis of multiple-stepped beams by the differential quadrature element method. Appl Math Comput 219:5802–5810 Wattanasakulpong N, Charaensuk J (2015) Vibration characteristics of stepped beams made of FGM using differential transformation method. Meccanica 50:1089–1101 Wei D, Liu YH, Xiang ZH (2012) An analytical method for free vibration analysis of functionally graded beams with edge cracks. J Sound Vib 331:1685–1700 Yan Y, Kitipornchai S, Yang J, He XQ (2011) Dynamic behavior of edge-cracked shear deformable functionally graded beams on an elastic foundation under a moving load. Compos Struct 93:2992–3001 Yang B (2010) Exact transient vibration of stepped bars, shafts and strings carrying lumped masses. J Sound Vib 329:1191–1207 Yang J, Chen Y (2008) Free vibration and buckling analysis of functionally graded beams with edge cracks. Compos Struct 83:48–60 Yang XB, Qin YP, Zhuang Z, You XC (2008) Investigation of dynamic fracture behavior in functionally graded materials. Model Simul Mater Sci Eng. https://doi.org/10.1088/0965-0393/16/7/075004 Yang EC, Zhao X, Li YH (2015) Free vibration analysis for cracked FGM beams by means of a continuous beam model. Shock Vib. https://doi.org/10.1155/2015/197049 Yu Z, Chu F (2009) Identification of crack in functionally graded material beams using the p-version of finite element method. J Sound Vib 325:69–85 Zhong Z, Yu T (2007) Analytical solution of a cantilever functionally graded beam. Comput Sci Technol 67:481–488