A triangular hybrid equilibrium plate element of general degree. (English) Zbl 1140.74555

Summary: Hybrid stress-based finite elements with side displacement fields have been used to generate equilibrium models having the property of equilibrium in a strong form. This paper establishes the static and kinematic characteristics of a flat triangular hybrid equilibrium element with both membrane and plate bending actions of general polynomial degree \(p\). The principal characteristics concern the existence of hyperstatic stress fields and spurious kinematic modes. The former are shown to exist for \(p > 3\), and their significance to finite element analysis is reviewed. Knowledge of the latter is crucial to the determination of the stability of a mesh of triangular elements, and to the choice of procedure adopted for the solution of the system of equations. Both types of characteristic are dependent on \(p\), and are established as regards their numbers and general algebraic forms. Graphical illustrations of these forms are included in the paper.


74S05 Finite element methods applied to problems in solid mechanics
74K20 Plates


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[1] Turner, Journal of Aeronautical Science 25 pp 805– (1956)
[2] Pereira, Computer Methods in Applied Mechanics and Engineering 136 pp 111– (1996)
[3] Ayad, International Journal for Numerical Methods in Engineering 55 pp 705– (2002)
[4] Duan, Computers and Structures 81 pp 1415– (2003)
[5] Ladeveze, Engineering Computations 8 pp 69– (1991)
[6] Pereira, Computer Methods in Applied Mechanics and Engineering 176 pp 19– (1999)
[7] Displacement and equilibrium models in the finite element method. In Stress Analysis, (eds). Wiley: New York, 1965; 145-197.
[8] Application of the dual analysis principle. In High Speed Computing of Elastic Structures, (ed.), vol. 61. Les Congres et Colloques de l’Universite de Liege, 1971; 167-207.
[9] Beckers, Universite de Liege Faculte des Sciences Appliquees Collection des Publications 41 (1973)
[10] Robinson, International Journal for Numerical Methods in Engineering 21 pp 487– (1985)
[11] Design of structural continua by finite element analysis of equilibrium models. In Engineering Software III, (ed.). Springer: Berlin, Computational Mechanics Centre, 1983.
[12] Maunder, Engineering Structures 8 pp 159– (1986)
[13] Almeida, Computers and Structures 40 pp 1043– (1991)
[14] Almeida, International Journal for Numerical Methods in Engineering 33 pp 845– (1992)
[15] MA47, A Fortran code for direct solution of indefinite sparse symmetric linear systems. Report RAL-95-001, Rutherford Appleton Laboratory, 1995.
[16] Maunder, International Journal for Numerical Methods in Engineering 39 pp 3175– (1996)
[17] Finite Element Analysis. Wiley: New York, 1991.
[18] Kaljevic, Computers and Structures 59 pp 691– (1996)
[19] Debongie, Computer Assisted Mechanics and Engineering Sciences 8 pp 261– (2001)
[20] Integrated Theory of Finite Element Methods. Wiley: London, 1973.
[21] Watwood, International Journal for Numerical Methods in Engineering 39 pp 3351– (1996) · Zbl 0918.73374
[22] Kawai, International Journal for Numerical Methods in Engineering 47 pp 275– (2000)
[23] Hybrid elements in the modelling of plates. In Finite Elements: Techniques and Developments, (ed.). Civil-Comp Press, 2000; 165-172.
[24] The Finite Element Method, vol. 1 (5th edn). Butterworth-Heinemann: Oxford, 2000.
[25] Maunder, Computer Assisted Mechanics and Engineering Sciences 10 pp 531– (2003)
[26] Nonlinear Finite Elements for Continua and Structures. Wiley: Chichester, 2000. · Zbl 0959.74001
[27] Pian, AIAA Journal 2 pp 1333– (1964)
[28] Evolution of assumed stress hybrid finite element. In Accuracy Reliability Training in FEM Technology, (ed.). Robinson and Associates: Wimborne, 1984.
[29] Strategies for error estimation using variable degree equilibrium elements. In Computational Mechanics in UK?Extended Abstracts for the 6th Conference of ACME, (ed.). School of Engineering: Exeter, 1998; 55-58.
[30] Patnaik, International Journal for Numerical Methods in Engineering 47 pp 685– (2000)
[31] Stress function approach. In B. M. Fraeijs de Veubeke Memorial Volume of Selected Papers, (ed.). Sijthoff & Noordhoff/Waterloo University, 1980; 663-715.
[32] Thamm, Periodica Polytechnica Series in Mechanical Engineering 44 pp 171– (2000)
[33] Pereira, Computers and Structures 53 pp 473– (1994)
[34] Photoelastic Stress Analysis. Wiley: London, 1974.
[35] On the equilibrium of elastic solids. In The Scientific Papers of James Clerk Maxwell (Reprinted), Transactions of the Royal Society of Edinburgh, (ed.), vol. 20. Dover Publications: New York, 1965.
[36] History of Strength of Materials. Dover Publications: New York, 1983.
[37] Quiroga, Measurement of Science and Technology 11 pp 259– (2000)
[38] Shear Deformable Beams and Plates. Elsevier: Amsterdam, 2000. · Zbl 0963.74002
[39] Continua and discontinua. In Matrix Methods in Structural Mechanics, et al. (eds). Wright-Patterson Air Force Base: Dayton, Ohio, 1965; 11-189.
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