# A Finite Element Formulation for Geometrically Non Linear Problems Using a Secant Matrix

Application to 3-D Trusses
• E. Oñate
• J. Oliver
• J. Miquel-Canet
• B. Suarez
Conference paper

## Summary

An incremental finite element formulation for the analysis of geometrically non linear problems is developed. The incremental equations are obtained via the full incremental form of the principle of virtual displacements. This leads to the obtention of a non symmetric secant stiffness matrix which allows to compute the displacement increments in a direct iterative manner. It is shown how the secant matrix yields naturally the expression of the classic tangent matrix. In the last part of the paper the formulation is applied to the analysis of 3-D trusses using simple two node elements, and an example of application to a slender 3-D truss tower is presented.

## Keywords

Finite Element Formulation Displacement Increment Virtual Displacement Standard Finite Element Tangent Matrix
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
Zienkiewicz, O.C. and Taylor R.: The finite element method. McGraw-Hill, 1985.Google Scholar
2. [2]
Yaghmai, S.: Incremental analysis of large deformations in mechanics of solids with applications to axisymmetric shells of revolution; Report N° SESM 68–17. Univ. California, Berkeley, 1968.Google Scholar
3. [3]
Larsen, P.K.: Large displacement analysis of shells of revolution including creep, plasticity and viscoelasticity. Report UC SESM 71–22, Univ. of California, Berkeley, 1971.Google Scholar
4. [4]
Bathe, K.J., Ramm, E. and Wilson, E.L.: Instability analysis of free form shells by finite elements. Int. J. Num. Meth. Engng. 9, 2, (1975), pp. 353–386.
5. [5]
Horrigmoe, G.: Non linear finite element models, in solid mechanics. Report 76–2, Norwegian Inst. Tech., Univ. Trøndheim, 1970.Google Scholar
6. [6]
Mondkar, D.P. and Powell, G.H.: Finite element analysis of non linear static and dynamic response. Int. J. Num. meth. Engng. 11, 3, (1977), pp. 499–520.
7. [7]
Frey, F.: L’analyse statique non lineaire des structures par la methode des elements finis et son application à la construction métallique. Ph. D. thesis, Univ. of Liege, 1978.Google Scholar
8. [8]
Frey, F. and Cescotto, S.: Some new aspects of the incremental total la grangian description in non linear analysis in “Finite Element in Non Linear Mechanics”, edited by P. Bergan et al., Tapir Publishers, Univ. of Trøndheim, 1978.Google Scholar
9. [9]
Malvern, L.E.: Introduction to the Mechanics of a Continuum Medium, Prentice Hall, 1969.Google Scholar
10. [10]
Bathe, K.J.: Finite Element Procedures in Non Linear Analysis, Prentice Hall, 1982.Google Scholar
11. [11]
Oñate, E.: Una formulación incremental para análisis de problemas de no linearidad geométrica en estructuras por el método de los elementos finitos. Internal Report ES-21, ETS Ing. de Caminos, Univ. Politécnica de Cataluña, Spain, 1986.Google Scholar
12. [12]
Irles, R.: Un modelo numérico para análisis de colapso en entramados metálicos. Ph. D. Thesis. Univ. Polit. Valencia, Spain, 1985.Google Scholar

## Authors and Affiliations

• E. Oñate
• 1
• J. Oliver
• 1
• J. Miquel-Canet
• 1
• B. Suarez
• 1
1. 1.E.T.S. Ingenieros de CaminosUniversidad Politécnica de CataluñaBarcelonaSpain