JAMP  Vol.4 No.4 , April 2016
New Formula for Geometric Stiffness Matrix Calculation
Abstract: The standard formula for geometric stiffness matrix calculation, which is convenient for most engineering applications, is seen to be unsatisfactory for large strains because of poor accuracy, low convergence rate, and stability. For very large compressions, the tangent stiffness in the direction of the compression can even become negative, which can be regarded as physical nonsense. So in many cases rubber materials exposed to great compression cannot be analyzed, or the analysis could lead to very poor convergence. Problems with the standard geometric stiffness matrix can even occur with a small strain in the case of plastic yielding, which eventuates even greater practical problems. The authors demonstrate that amore precisional approach would not lead to such strange and theoretically unjustified results. An improved formula that would eliminate the disadvantages mentioned above and leads to higher convergence rate and more robust computations is suggested in this paper. The new formula can be derived from the principle of virtual work using a modified Green-Lagrange strain tensor, or from equilibrium conditions where in the choice of a specific strain measure is not needed for the geometric stiffness derivation (which can also be used for derivation of geometric stiffness of a rigid truss member). The new formula has been verified in practice with many calculations and implemented in the RFEM and SCIA Engineer programs. The advantages of the new formula in comparison with the standard formula are shown using several examples.
Cite this paper: Němec, I. , Trcala, M. , Ševčík, I. and Štekbauer, H. (2016) New Formula for Geometric Stiffness Matrix Calculation. Journal of Applied Mathematics and Physics, 4, 733-748. doi: 10.4236/jamp.2016.44084.

[1]   Bathe, K.-J. (1996) Finite Element Procedures. Prentice Hall, Upper Saddle River.

[2]   Belytschko, T., Liu, W.K., Moran, B. and Elkhodary, K. (2000) Nonlinear Finite Elements for Continua and Structures. John Wiley & Sons, Hoboken.

[3]   Bonet, J. and Wood, R.D. (2008) Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge.

[4]   Cook, R.D., Malkus, D.S., Plesha, M.E. and Witt, R.J. (2002) Concepts and Applications of Finite Element Analysis. John Wiley & Sons, Hoboken.

[5]   Němec I., et al. (2010) Finite Element Analysis of Structures: Principles and Praxis. Shaker-Verlag, Aachen.

[6]   Reddy, J.N. (2004) Nonlinear Finite Element Analysis. Oxford University Press, Oxford.

[7]   Zienkiewicz, O.C. and Taylor, R.L. (2000) The Finite Element Method. Butterworth Heinemann, Oxford.

[8]   de Souza Neto, E.A., Periæ, D. and Owen, D.R.J. (2008) Computational Methods for Plasticity: Theory and Applications. John Wiley & Sons, Hoboken.

[9]   Simo, J.C. & Hughes, T.J.R. (2008) ComputationalInelasticity. Springer, New York, 392 p.

[10]   Wriggers, P. (2008) Nonlinear Finite Element Methods. Springer-Verlag, New York.

[11]   Lacarbonara, W. and Pacitti, A. (2008) Nonlinear Modeling of Cables with Flexural Stiffness. Mathematical Problems in Engineering, 2008, Article ID: 370767.

[12]   Xiao, H. and Chen, L.S. (2002) Hencky’s Elasticity Model and Linear Stress-Strain Relations in Isotropic Finite Hyperelasticity. Acta Mechanica, 157, 51-60.

[13]   Xiao, H., Bruhns, O.T. and Meyers, A. (1998) Objective Corotational Rates and Unified Work-Conjugacy Relation between Eulerian and Lagrangean Strain and Stress Measures. Archives of Mechanics, 50, 1015-1045.

[14]   Meyers, A. (1999) On the Consistency of Some Eulerian Strain Rates. Zeitschrift fur Angewandte Mathematik und Mechanik, 79, 171-177.<171::AID-ZAMM171>3.0.CO;2-6

[15]   Simo, J.C. and Pister, K.S. (1984) Remarks on Rate Constitutive Equations for Finite Deformation Problem: Computational Implications. Computer Methods in Applied Mechanics and Engineering, 46, 201-215.

[16]   Curnier, A. and Rakotomanana, L. (1991) Generalized Strain and Stress Measures: Critical Survey and New Results. Engineering Transactions, 39, 461-538.

[17]   Chiskis, A. and Parnes, R. (2000) Linear Stress-Strain Relations in Nonlinear Elasticity. Acta Mechanica, 146, 109-113.

[18]   Farahani, K. and Naghdabadi, R. (2000) Conjugate Stresses of the Seth-Hill Strain Tensors. International Journal of Solids and Structures, 37, 5247-5255.

[19]   Darijani, H. and Naghdabadi, R. (2010) Constitutive Modeling of Solids at Finitede Formation Using a Second-Order Stress-Strain Relation. International Journal of Engineering Science, 48, 223-236.

[20]   Hill, R. (1978) Aspects of Invariance in Solid Mechanics. Advances in Applied Mechanics, 18, 1-75.

[21]   Farahani, K. and Bahai, H. (2004) Hyper-Elastic Constitutive Equations of Conjugate Stresses and Strain Tensors for the Seth-Hill Strain Measures. International Journal of Engineering Science, 42, 29-41.