An Approximate Solution of Functionally Graded Timoshenko Beam Using B-Spline Collocation Method

Document Type : Research Paper


1 BRSMCAET, IGKV, Mungeli, India

2 Department of Mechanical Engineering, NIT Raipur, India



Collocation methods are popular in providing numerical approximations to complicated governing equations owing to their simplicity in implementation. However, point collocation methods have limitations regarding accuracy and have been modified upon with the application of B-spline approximations. The present study reports the stress and deformation behavior of shear deformable functionally graded cantilever beam using B-spline collocation technique. The material grading is along the beam height and varies according to power law. Poisson’s ratio is assumed to be a constant. The equations are derived using virtual work principle in the framework of Timoshenko beams to obtain a unified formulation for such beams. A sixth order basis function is used for approximation and collocation points are generated using Greville abscissa. Deformation and stresses; bending (axial) stresses and transverse (shear) stresses, and position of neutral axis are studied for a wide range of power law index values. The results are reported along the beam cross-section and beam length.                                 


Main Subjects

[1] Miyamoto Y., Kaysser W.A., Rabin B.H., Kawasaki A., Ford R.G., 1999, Functionally Graded Materials, Materials Technology Series, Springer US, Boston.
[2] Suresh S., Mortensen A., 1998, Fundamentals of Functionally Graded Materials, London.
[3] Rasheedat M.M., Akinlabi E.T., 2012, Functionally graded material: An overview, Proceedings of the World Congress in Engineering, London, UK.
[4] Udupa G., Rao S.S., Gangadharan K.V., 2014, Functionally graded composite materials: An overview, Procedia Materials Science 5: 1291-1299.
[5] Kieback B., Neubrand A., Riedel H., 2003, Processing techniques for functionally graded materials, Materials Science and Engineering 362: 81-106.
[6] Birman V., Byrd L.W., 2007, Modeling and analysis of functionally graded materials and structures, Applied Mechanics Reviews 60: 195.
[7] Reddy J.N., Chin C.D., 1998, Thermomechanical analysis of functionally graded cylinders and plates, Journal of Thermal Stresses 21: 593-626.
[8] Reddy J.N., 2000, Analysis of functionally graded plates, International Journal for Numerical Methods in Engineering 47: 663-684.
[9] Sankar B.V., 2001, An elasticity solution for functionally graded beams, Composites Science and Technology 61: 689-696.
[10] Sankar B.V., Tzeng J.T., 2002, Thermal stresses in functionally graded beams, AIAA Journal 40: 1228-1232.
[11] Zhu H., Sankar B.V., 2004, A combined fourier series–Galerkin method for the analysis of functionally graded beams, Journal of Applied Mechanics 71: 421.
[12] Chakraborty A., Gopalakrishnan S., Reddy J.N., 2003, A new beam finite element for the analysis of functionally graded materials, International Journal of Mechanical Sciences 45: 519-539.
[13] Aydogdu M., Taskin V., 2007, Free vibration analysis of functionally graded beams with simply supported edges, Materials & Design 28: 1651-1656.
[14] Zhong Z., Yu T., 2007, Analytical solution of a cantilever functionally graded beam, Composites Science and Technology 67: 481-488.
[15] Kadoli R., Akhtar K., Ganesan N., 2008, Static analysis of functionally graded beams using higher order shear deformation theory, Applied Mathematical Modelling 32: 2509-2525.
[16] Benatta M.A., Mechab I., Tounsi A., Adda Bedia E.A., 2008, Static analysis of functionally graded short beams including warping and shear deformation effects, Computational Materials Science 44: 765-773.
[17] Sina S.A., Navazi H.M., Haddadpour H., 2009, An analytical method for free vibration analysis of functionally graded beams, Materials & Design 30: 741-747.
[18] Şimşek M., 2010, Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories, Nuclear Engineering and Design 240: 697-705.
[19] Giunta G., Belouettar S., Carrera E., 2010, Analysis of FGM beams by means of classical and advanced theories, Mechanics of Advanced Materials and Structures 17: 622-635.
[20] Tahani M., Torabizadeh M.A., Fereidoon A., 2006, Nonlinear analysis of functionally graded beams, Journal of Achievements in Materials and Manufacturing Engineering 18: 315-318.
[21] Reddy J.N., 2011, Microstructure-dependent couple stress theories of functionally graded beams, Journal of the Mechanics and Physics of Solids 59: 2382-2399.
[22] Zhang D.-G., 2013, Nonlinear bending analysis of FGM beams based on physical neutral surface and high order shear deformation theory, Composite Structures 100: 121-126.
[23] Yaghoobi H., Fereidoon A., 2010, Influence of neutral surface position on deflection of functionally graded beam under uniformly distributed load, World Applied Sciences Journal 10: 337-341.
[24] Mohanty S.C., Dash R.R., Rout T., 2011, Parametric instability of a functionally graded Timoshenko beam on Winkler’s elastic foundation, Nuclear Engineering and Design 241: 2698-2715.
[25] Mohanty S.C., Dash R.R., Rout T., 2012, Static and dynamic stability analysis of a functionally graded Timoshenko beam, International Journal of Structural Stability and Dynamics 12: 1250025.
[26] Li X.-F., 2008, A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams, Journal of Sound and Vibration 318: 1210-1229.
[27] Kadalbajoo M.K., Yadaw A.S., 2011, Finite difference, finite element and b-spline collocation methods applied to two parameter singularly perturbed boundary value problems1, Jnaiam 5: 163-180.
[28] Chawla T.C., Leaf G., Chen W., 1975, A collocation method using b-splines for one-dimensional heat or mass-transfer-controlled moving boundary problems, Nuclear Engineering and Design 35: 163-180.
[29] Chawla T.C., Chan S.H., 1979, Solution of radiation-conduction problems with collocation method using b-splines as approximating functions, International Journal of Heat and Mass Transfer 22: 1657-1667.
[30] Bert C.W., Sheu Y., 1996, Static analyses of beams and plates by spline collocation method, Journal of Engineering Mechanics 122: 375-378.
[31] Sun W., 2001, B-spline collocation methods for elasticity problems, Scientific Computing and Applications 2001: 133-141.
[32] Hsu M.-H., 2009, Vibration analysis of non-uniform beams resting on elastic foundations using the spline collocation method, Tamkang Journal of Science and Engineering 12: 113-122.
[33] Hsu M.-H., 2009, Vibration analysis of pre-twisted beams using the spline collocation method, Journal of Marine Science and Technology 17: 106-115.
[34] Wu L.-Y., Chung L.-L., Huang H.-H., 2008, Radial spline collocation method for static analysis of beams, Applied Mathematics and Computing 201: 184-199.
[35] Provatidis C., 2014, Finite element analysis of structures using C 1 -continuous cubic b-splines or equivalent hermite elements, Journal of Structural 2014: 1-9.
[36] Cottrell J.A., Hughes T.J.R., Bazilevs Y., 2009, Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, Chichester.
[37] Auricchio F., Da Veiga L.B., Hughes T.J.R., Reali A., Sangalli G., 2010, Isogeometric collocation methods, Mathematical Models and Methods in Applied Sciences 20: 2075-2107.
[38] Reali A., Gomez H., 2015, An isogeometric collocation approach for Bernoulli–Euler beams and Kirchhoff plates, Computer Methods in Applied Mechanics and Engineering 284: 623-636.
[39] Patlashenko I., 1993, Cubic b-spline collocation method for non-linear static analysis of panels under mechanical and thermal loadings, Computers & Structures 49: 89-96.
[40] Patlashenko I., Weller T., 1995, Two dimensional spline collocation method for nonlinear analysis of laminated panels, Computers & Structures 57: 131-139.
[41] Mizusawa T., Kito H., 1995,Vibration of cross ply laminated cylindrical panels by spline strip method, Computers & Structures 57: 253-267.
[42] Mizusawa T., 1996, Vibration of thick laminated cylindrical panels by spline strip method, Computers & Structures 61: 441-457.
[43] Akhras G., Li W., 2011, Stability and free vibration analysis of thick piezoelectric composite plates using finite strip method, Journal of Mechanical Sciences 53: 575-584.
[44] Loja M.A.R., Mota Soares C.M., Barbosa J.I., 2013, Analysis of functionally graded sandwich plate structures with piezoelectric skins, using b-spline finite strip method, Computers & Structures 96: 606-615.
[45] Provatidis C.G., 2017, B-splines collocation for plate bending eigen analysis, Journal of Mechanics of Materials and Structures 12: 353-371.
[46] Johnson R.W., 2005, Higher order b-spline collocation at the Greville abscissae, Applied Numerical Mathematics 52: 63-75.