Conservative semidiscrete difference schemes for Timoshenko systems. (English) Zbl 1442.74220

Summary: We present a parameterized family of finite-difference schemes to analyze the energy properties for linearly elastic constant-coefficient Timoshenko systems considering shear deformation and rotatory inertia. We derive numerical energies showing the positivity, and the energy conservation property and we show how to avoid a numerical anomaly known as locking phenomenon on shear force. Our method of proof relies on discrete multiplier techniques.


74S05 Finite element methods applied to problems in solid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Full Text: DOI


[1] Han, S. M.; Benaroya, H.; Wei, T., Dynamics of transversely vibrating beams using four engineering theories, Journal of Sound and Vibration, 225, 5, 935-988 (1999) · Zbl 1235.74075
[2] Timoshenko, S. P., On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, 6, 744-746 (1921)
[3] Huang, T. C., The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions, Journal of Applied Mechanics, 28, 579-584 (1961) · Zbl 0102.19005
[4] Krieg, R. D., On the behavior of a numerical approximation to the rotatory inertia and transverse shear plate, Journal of Applied Mechanics, 40, 4, 977-982 (1973) · Zbl 0272.73036
[5] Belytschko, T.; Mindle, W. L., Flexural wave propagation behavior of lumped mass approximations, Computers and Structures, 12, 6, 805-812 (1980) · Zbl 0442.73027
[6] Mindle, W. L.; Belytschko, T., A study of shear factors in reduced-selective integration mindlin beam elements, Computers and Structures, 17, 3, 339-344 (1983) · Zbl 0512.73068
[7] Wright, J. P., A mixed time integration method for Timoshenko and Mindlin type elements, Communications in Applied Numerical Methods, 3, 3, 181-185 (1987) · Zbl 0611.73083
[8] Wright, J. P., Numerical stability of a variable time step explicit method for Timoshenko and Mindlin type structures, Communications in Numerical Methods in Engineering, 14, 2, 81-86 (1998) · Zbl 0906.73078
[9] Arnold, D. N., Discretization by finite elements of a model parameter dependent problem, Numerische Mathematik, 37, 3, 405-421 (1981) · Zbl 0446.73066 · doi:10.1007/BF01400318
[10] Hughes, T. J. R.; Taylor, R. L.; Kanoknukulchai, W., A simple and efficient finite element method for plate bending, International Journal for Numerical Methods in Engineering, 11, 10, 1529-1543 (1977) · Zbl 0363.73067
[11] Prathap, G.; Bhashyam, G. R., Reduced integration and the shear-flexible beam element, International Journal for Numerical Methods in Engineering, 18, 2, 195-210 (1982) · Zbl 0473.73084
[12] Li, L. K., Discretization of the Timoshenko beam problem by the \(p\) and the h-p versions of the finite element method, Numerische Mathematik, 57, 4, 413-420 (1990) · Zbl 0683.73041 · doi:10.1007/BF01386420
[13] Loula, A. F. D.; Hughes, T. J. R.; Franca, L. P., Petrov-Galerkin formulations of the Timoshenko beam problem, Computer Methods in Applied Mechanics and Engineering, 63, 2, 115-132 (1987) · Zbl 0645.73030 · doi:10.1016/0045-7825(87)90167-8
[14] Loula, A. F. D.; Hughes, T. J. R.; Franca, L. P.; Miranda, I., Mixed Petrov-Galerkin methods for the Timoshenko beam problem, Computer Methods in Applied Mechanics and Engineering, 63, 2, 133-154 (1987) · Zbl 0607.73076 · doi:10.1016/0045-7825(87)90168-X
[15] Reddy, J. N., On locking-free shear deformable beam finite elements, Computer Methods in Applied Mechanics and Engineering, 149, 1-4, 113-132 (1997) · Zbl 0918.73131
[16] Zuazua, E., Propagation, observation, and control of waves approximated by finite difference methods, SIAM Review, 47, 2, 197-243 (2005) · Zbl 1077.65095 · doi:10.1137/S0036144503432862
[17] Infante, J. A.; Zuazua, E., Boundary observability for the space semi-discretizations of the 1-D wave equation, Mathematical Modelling and Numerical Analysis, 33, 2, 407-438 (1999) · Zbl 0947.65101
[18] LeVeque, R. J., Numerical Methods for Conservation Laws (1992), Birkhäuser · Zbl 0847.65053 · doi:10.1007/978-3-0348-8629-1
[19] Muñoz Rivera, J. E.; Racke, R., Mildly dissipative nonlinear Timoshenko systems—global existence and exponential stability, Journal of Mathematical Analysis and Applications, 276, 1, 248-278 (2002) · Zbl 1106.35333 · doi:10.1016/S0022-247X(02)00436-5
[20] Muñoz Rivera, J. E.; Racke, R., Global stability for damped Timoshenko systems, Discrete and Continuous Dynamical Systems B, 9, 6, 1625-1639 (2003) · Zbl 1047.35023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.