Non Uniform Rational B Spline (NURBS) Based Non-Linear Analysis of Straight Beams with Mixed Formulations

Document Type : Research Paper


1 School of Aerospace and Mechanical Engineering, 865 Asp Avenue, Norman, OK, 73019, USA

2 Department of Mechanical Engineering, 3123 TAMU, College Station, TX, USA


Displacement finite element models of various beam theories have been developed traditionally using conventional finite element basis functions (i.e., cubic Hermite, equi-spaced Lagrange interpolation functions, or spectral/hp Legendre functions). Various finite element models of beams differ from each other in the choice of the interpolation functions used for the transverse deflection w, total rotation , and/or shear strain , as well as the variational method used (e.g., collocation, weak form Galerkin, or least-squares). When nonlinear shear deformation theories are used, the displacement finite element models experience membrane and shear locking. The present study is concerned with development of alternative beam finite elements using both uniform and non-uniform rational b-splines (NURBS) to eliminate shear and membrane locking in an hpk finite element setting for both the Euler-Bernoulli beam and Timoshenko beam theories. Both linear and non-linear analysis are performed using mixed finite element models of the beam theories studied. Results obtained are compared with analytical (series) solutions and non-linear finite element and spectral/hp solutions available in the literature, and excellent agreement is found for all cases.


[1] Reddy J.N., Wang C., Lee K., 1997, Relationships between bending solutions of classical and shear deformation beam theories, International Journal of Solids and Structures 34(26): 3373-3384.
[2] Reddy J.N., 2014, An Introduction to Nonlinear Finite Element Analysis: with Applications to Heat Transfer Fluid Mechanics, and Solid Mechanics, OUP Oxford.
[3] Reddy J.N., Wang C., Lam K., 1997, Unified finite elements based on the classical and shear deformation theories of beams and axisymmetric circular plates, Communications in Numerical Methods in Engineering 13(6): 495-510.
[4] Severn R., 1970, Inclusion of shear deflection in the stiffness matrix for a beam element, The Journal of Strain Analysis for Engineering Design 5(4): 239-241.
[5] Reddy J.N., 1997, On locking-free shear deformable beam finite elements, Computer Methods in Applied Mechanics and Engineering 149(1): 113-132.
[6] Oden J.T., Reddy J.N., 2012, Variational Methods in Theoretical Mechanics, Springer Science & Business Media.
[7] Arciniega R., Reddy J.N., 2007, Large deformation analysis of functionally graded shells, International Journal of Solids and Structures 44(6): 2036-2052.
[8] Karniadakis G., Sherwin S., 2013, Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford University Press.
[9] Melenk J.M., 2002, On condition numbers in hp-fem with gauss-Lobatto-based shape functions, Journal of Computational and Applied Mathematics 139(1): 21-48.
[10] Ranjan R., Feng Y., Chronopolous A., 2016, Augmented stabilized and Galerkin least squares formulations, Journal of Mathematics Research 8 (6): 1-12.
[11] Ranjan R., Chronopolous A., Feng Y., 2016, Computational algorithms for solving spectral/hp stabilized incompressible flow problems, Journal of Mathematics Research 8(4): 1-19.
[12] Ranjan R., Reddy J.N., 2009, Hp-spectral finite element analysis of shear deformable beams and plates, Journal of Solid Mechanics 1(3): 245-259.
[13] Ranjan R., 2011, Nonlinear finite element analysis of bending of straight beams using hp-spectral approximations, Journal of Solid Mechanics 3(1): 96-113.
[14] Da Veiga L.B., Lovadina C., Reali A., 2012, Avoiding shear locking for the timoshenko beam problem via isogeometric collocation methods, Computer Methods in Applied Mechanics and Engineering 241: 38-51.
[15] Tran L.V., Ferreira A., Nguyen-Xuan H., 2013, Isogeometric analysis of functionally graded plates using higher-order shear deformation theory, Composites Part B: Engineering 51: 368-383.
[16] Ranjan R., 2010, Hp-spectral Methods for Structural Mechanics and Fluid Dynamics Problems, Texas A&M University, Ph.D. Thesis.
[17] Cottrell J.A., Hughes T.J., Bazilevs Y., 2009, Isogeometric Analysis: Toward Integration of CAD and FEA, John Wiley & Sons.
[18] 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.
[19] Weeger O., Wever U., Simeon B., 2013, Isogeometric analysis of nonlinear Euler-Bernoulli beam vibrations, Nonlinear Dynamics 72(4): 813-835.
[20] Thai C.H., Nguyen-Xuan H., Bordas S., Nguyen-Thanh N., Rabczuk T., 2015, Isogeometric analysis of laminated composite plates using the higher-order shear deformation theory, Mechanics of Advanced Materials and Structures 22(6): 451-469.
[21] Kapoor H., Kapania R., 2012, Geometrically nonlinear nurbs isogeometric finite element analysis of laminated composite plates, Composite Structures 94(12): 3434-3447.
[22] Reddy J.N., 2006, Theory and Analysis of Elastic Plates and Shells, CRC press.