Scheme for Elasticas With Snap‐Back and Looping
Publication: Journal of Engineering Mechanics
Volume 115, Issue 10
Abstract
A simple mixed method is formulated for elasticas. The total displacements are decomposed into large rigid‐body motions and small elastic deformation in total Lagrangian approach. The elementwise linear moment field and the rigid‐body rotations of straight elements are introduced as independent variables in the mixed functional. The compatibility equations for nodal slopes consequently become linear in terms of these rotational parameters. The nodal bending moments will then be eliminated from the discretized elastica equations. The number of the rotational degrees of freedom is finally equal to the number of elements. An integration procedure proposed by Shinohara is applied to solve the nonlinear governing equations. The major interest is to develop a nonlinear scheme that is able to trace the snap‐back and the looping behavior in load‐deflection history. Numerical examples suggest that the mixed method in the present study is quite accurate despite its simplicity. Shinohara's procedure also proved to be efficient, and may be applied to nonlinear plate and shell analysis.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Batoz, J.‐L., and Dhatt, G. (1979). “Incremental displacement algorithms for nonlinear problems.” Int. J. for Numer. Methods. in Engrg., 14, 1262–1266.
2.
Crisfield, M. A. (1981). “A fast incremental/iterative solution procedure that handles snap‐through.” Comput. Struct., 13, 55–62.
3.
Crisfield, M. A. (1983). “An arc‐length method including line searches and acceleration.” Int. J. for Numer. Methods in Engrg., 19, 1269–1289.
4.
DaDeppo, D. A., and Schmidt, R. (1975). “Instability of clamped‐hinged circular arches subjected to a point load.” Trans. of American Society of Mechanical Engineers, New York, N.Y., 894–896.
5.
Fujii, F. (1981a). “Discrete and non‐discrete mixed methods for plate bending analysis.” Int. J. for Numer. Methods in Engrg., 17, 1843–1859.
6.
Fujii, F. (1981b). “Spline‐funktionen mit mehrfachen knoten in der gemischten plattenberechnung.” Ingenieur‐Arckiv. 50, 365–375 (in German).
7.
Fujii, F. (1983). “A simple mixed formulation for elastica problems.” Comput. Struct., 17(1), 79–88.
8.
Fujii, F., and Gong, S. X. (1987). “Field transfer matrix for nonlinear curved beams.” Proc., ASCE, (3), 675–692.
9.
Haisler, W. E., and Stricklin, J. A. (1977). “Displacement incrementation in nonlinear structure analysis by the self‐correcting method.” Int. J. for Numer. Methods in Engrg., 11, 3–10.
10.
Harrison, H. B. (1978). “Post‐buckling behavior of elastic circular arches.” Proc., Institution of Civil Engineers, Part 2, (65), 283–298.
11.
Harrison, H. B. (1982). “In‐plane stability of parabolic arches.” Proc., ASCE, 108(1), 195–205.
12.
Hsiao, K. M., and Hou, F. Y. (1987). “Nonlinear finite element analysis of elastic frames,” Comput. Struct., 26(4), 693–701.
13.
Lee, S. L., Manuel, F. S., and Rosow, E. D. (1968). “Large deflections and stability of elastic frames.” (2), ASCE, 521–547.
14.
Noor, A. K. (1983). “Multifield (mixed and hybrid) finite element models.” 127–162.
15.
Noor, A. K., and Pilkey, W. D. (1983). “State‐of‐the‐art surveys on finite element technology.” American Society of Mechanical Engineers.
16.
Powell, G., and Simons, J. (1981). “Improved iteration strategy for nonlinear structures.” Int. J.for Numer. Methods in Engrg., 17, 1455–1467.
17.
Rheinboldt, W. C., and Riks, E. (1983). “Solution techniques for nonlinear finite element equations.” 183–223.
18.
Riks, E. (1979). “An incremental approach to the solution of snapping and buckling problems.” Int. J. Solids and Struct., 15, 524–551.
19.
Riks, E. (1984). “Some computational aspects of the stability analysis of nonlinear structures.” Comput. Methods in Appl. Mech. and Engrg., 47, 219–259.
20.
Sanbir, A. B., and Lock, A. C. (1973). “Large deflexion, geometrically nonlinear finite element analysis of circular arches.” Int. J. Mech. Sci., 15, 37–47.
21.
Schweizerhof, K. H., and Wriggers, P. (1986). “Consistent linearization for path following methods in nonlinear FE analysis.” Comput. Methods in Applied Mech. and Engrg., 59, 261–279.
22.
Shinohara, Y. (1969). “A geometric method of numerical solution of nonlinear equations and error estimation by Urabe's Proposition.” Publications of Res. Inst. for Math. Sci., Kyoto Univ., Kyoto, Japan, Series A, 5(1).
23.
Shinohara, Y. (1972). “A geometric method for the numerical solution of nonlinear equations and its application to nonlinear oscillations.” Publications of the Res. Inst. for Math. Sci., Kyoto Univ., Kyoto, Japan, 8(1).
24.
Wempner, G. A. (1971). “Discrete approximations related to nonlinear theories of solids.” Int. J. of Solids and Struct., 7, 1581–1599.
25.
Wood, R. D., and Zienckiewicz, O. C. (1977). “Geometrically nonlinear finite element analysis of beams, frames, arches and axisymmetric shells.” Comput. Struct., 7, 725–735.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 115 • Issue 10 • October 1989
Pages: 2166 - 2181
Copyright
Copyright © 1989 ASCE.
History
Published online: Oct 1, 1989
Published in print: Oct 1989
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.