Arc‐Length Method for Passing Limit Points in Structural Calculation
Publication: Journal of Structural Engineering
Volume 118, Issue 1
Abstract
A new solution procedure is presented, based on the arc‐length method, for passing limit points (load or displacement peaks) in nonlinear finite element analysis of structures. In addition to the usual equilibrium equations, a quadratic arc‐length constraint equation is specified so that the nonlinear solution is sought on a small ellipsoidal surface in load‐deflection space. The main new feature of the proposed procedure is the resolution of the out‐of‐balance loads into a parallel and an orthogonal component with respect to the vector of applied external loads. Methods for avoiding complex roots to the quadratic constraint equation, and for selecting the appropriate root, are presented. Applications of the proposed procedure to two reinforced concrete problems give satisfactory results. The method is effective and versatile in handling both snap‐through and snap‐back problems.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Bathe, K. J., and Dvorkin, E. N. (1983). “On the automatic solution of non‐linear finite element equations.” Comput. Struct., 17(5–6), 871–879.
2.
Batoz, J. L., and Dhatt, G. (1979). “Incremental displacement algorithms for nonlinear problems.” Int. J. Numer. Meth. Engrg., 14(8), 1262–1267.
3.
Bergan, P. G. (1980). “Automated incremental‐iterative solution schemes.” Numerical methods for non‐linear problems, C. Taylor, E. Hinton, and D. R. J. Owen, (eds.), Pineridge Press, Swansea, U.K., 291–305.
4.
Bergan, P. G., and Holand, I. (1979). “Nonlinear finite element analysis of concrete structures.” Comput. Meth. in Appl. Mech. and Engrg., 17/18, 443–467.
5.
Bergan, P. G., Horrigmoe, G., Krakeland, B., and Soreide, T. H. (1978). “Solution techniques for non‐linear finite element problems.” Int. J. Numer. Meth. Engrg., 12(11), 1677–1696.
6.
Cope, R. J., and Rao, P. V. (1981). “Non‐linear finite element strategies for bridge slabs.” Advanced mechanics of reinforced concrete, IABSE, Delft, 275–290.
7.
Crisfield, M. A. (1981). “A fast incremental/iterative solution procedure that handles ‘snap‐through’,” Comput. Struct., 13(1–3), 55–62.
8.
Crisfield, M. A. (1982a). “Accelerated solution techniques and concrete cracking.” Comput. Meth. Appl. Mech. and Engrg., 33, 585–607.
9.
Crisfield, M. A. (1982b). “Local instabilities in the non‐linear analysis of reinforced concrete beams and slabs.” Proc. Inst. Civ. Engrs., 73(2), 135–145.
10.
Crisfield, M. A. (1983). “An arc‐length method including line searches and accelerations.” Int. J. Numer. Meth. Engrg., 19(9), 1269–1289.
11.
Crisfield, M. A. (1984). “Overcoming limit points with material softening and strain localisation.” Numerical methods for non‐linear problems, C. Taylor, E. Hinton, and D. R. J. Owen, eds., Pineridge Press, Swansea, U.K., 244–277.
12.
Kupfer, H., Hilsdorf, H. K., and Rusch, H. (1969). “Behaviour of concrete under biaxial stresses.” J. Am. Concr. Inst., 66(8), 656–666.
13.
Lam, W. F. (1987). “Constitutive relations for finite element analysis of tension stiffening in reinforced concrete,” thesis presented to the Univ. of Cambridge at Cambridge, England, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
14.
Morley, C. T., and Lam, W. F. (1986). “Passing load peaks in finite element calculations for reinforced concrete structures.” Proc. Int. Symp. on Fundamental Theory of Reinforced and Prestressed Concrete, (cosponsored by ASCE), Nanjing, 3, 1120–1126.
15.
Phillips, D. V., and Zienkiewicz, O. C. (1976). “Finite element non‐linear analysis of concrete structures.” Proc. Inst. Civ. Engrs., 61(2), 59–88.
16.
Ramm, E. (1981). “Strategies for tracing the nonlinear response near limit points.” Non‐linear finite element analysis in structural mechanics, W. Wunderlich, E. Stein, and K. J. Bathe, eds., Springer‐Verlag, New York, N.Y. 63–89.
17.
Riks, E. (1972). “The application of Newton's method to the problem of elastic stability.” J. Appl. Mech., 39(4), 1060–1066.
18.
Riks, E. (1979). “An incremental approach to the solution of snapping and buckling problems.” Int. J. Solids Struct., 15(7), 529–551.
19.
Wempner, G. A. (1971). “Discrete approximations related to nonlinear theories of solids.” Int. J. Solids Struct., 7(11), 1581–1599.
20.
Wright, E. W., and Gaylord, E. H. (1968). “Analysis of unbraced multistory steel rigid frames.” J. Struct. Div., ASCE, 94(5), 1143–1163.
Information & Authors
Information
Published In
Journal of Structural Engineering
Volume 118 • Issue 1 • January 1992
Pages: 169 - 185
Copyright
Copyright © 1992 ASCE.
History
Published online: Jan 1, 1992
Published in print: Jan 1992
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.