Tracing Secondary Equilibrium Paths of Elastic Framed Structures
This article has been corrected.
VIEW CORRECTIONPublication: Journal of Engineering Mechanics
Volume 125, Issue 12
Abstract
For a framed structure that is subjected to bifurcation buckling, it may be useful to trace its secondary equilibrium path to gauge its sensitivity to geometric imperfections or to study the nature of load shedding from the buckled structure. For this purpose, a substantial number of branch-switching algorithms for tracing the secondary equilibrium paths of elastic structures have been proposed in the literature. However, virtually all of the published algorithms have heavy mathematical overtones that are not readily appreciated by practicing structural engineers. This paper presents a simple and efficient branch-switching algorithm that is explained in more easily understood terms. The proposed algorithm is demonstrated through numerical examples to be effective in tracing the secondary equilibrium paths of various framed structures with different types of postbuckling behaviors.
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References
1.
Allman, D. J. (1989). “On the general theory of the stability of equilibrium of discrete conservative systems.” Aeronautical J., January, 29–35.
2.
Bazant, Z. P., and Cedolin, L. (1989). “Initial postcritical analysis of asymmetric bifurcation in frames.”J. Struct. Engrg., ASCE, 115(11), 2845–2857.
3.
Bergan, P. G., Horrigmoe, G., Krakeland, B., and Soreide, T. H. (1978). “Solution techniques for non-linear finite element problems.” Int. J. Numer. Methods in Engrg., 12, 1677–1696.
4.
Chajes, A. (1983). “Post-buckling behavior.”J. Struct. Engrg., ASCE, 109(10), 2450–2462.
5.
Chan, S. L. (1988). “Geometric and material nonlinear analysis of beam columns and frames using the minimum residual displacement method.” Int. J. Numer. Methods in Engrg., 26, 2657–2669.
6.
Chen, H., and Blandford, G. E. (1993). “Work-increment-control method for nonlinear analysis.” Int. J. Numer. Methods in Engrg., 36, 909–930.
7.
Choong, K. K., and Hangai, Y. (1993). “Review on methods of bifurcation analysis for geometrically nonlinear structures.” Bulletin of the IASS, 34, 133–149.
8.
Clarke, M. J., and Hancock, G. J. (1990). “A study of incremental-iterative strategies for nonlinear analyses.” Int. J. Numer. Methods in Engrg., 29, 1365–1391.
9.
Crisfield, M. A. (1981). “A fast incremental/iterative solution procedure that handles snap through.” Comp. and Struct., 13, 55–62.
10.
Crisfield, M. A. (1991). Non-linear finite element analysis of solids and structures. Vol. 1, Wiley, New York.
11.
Crisfield, M. A., and Shi, J. ( 1991). “A review of solution procedures and path-following techniques in relation to the nonlinear finite element analysis of structures.” Nonlinear computational mechanics—State of the art, P. Wriggers and W. Wagner, eds., Springer, Berlin, 47–68.
12.
Crisfield, M. A., and Willis, J. (1988). “Solution strategies and softening materials.” Comp. Meth. Appl. Mech. Engrg., 66, 267–289.
13.
Dutta, A., and White, D. (1997). “Automated solution procedures for negotiating abrupt nonlinearities and branch points.” Engrg. Computations, 14, 31–56.
14.
Eriksson, A. (1989). “On linear constraints for Newton-Rhapson corrections and critical point searches in structural FE problems.” Int. J. Numer. Methods in Engrg., 28, 1317–1334.
15.
Fujii, F., and Choong, K. K. (1992). “Branch switching in bifurcation of structures.”J. Engrg. Mech., ASCE, 118(8), 1578–1596.
16.
Fujii, F., and Okazawa, S. (1997). “Pinpointing bifurcation points and branch-switching.”J. Engrg. Mech., ASCE, 123(3), 179–189.
17.
Fujikake, M. (1985). “A simple approach to bifurcation and limit point calculations.” Int. J. Numer. Methods in Engrg., 21, 183–191.
18.
Huddleston, J. V. (1968). “Finite deflection and snap-through of high circular arches.” J. Appl. Mech., 35, 763–769.
19.
Kani, I. M., and McConnel, R. E. (1987). “Collapse of shallow lattice domes.”J. Struct. Engrg., ASCE, 113(8), 1806–1819.
20.
Kassimali, A., and Abbasnia, R. (1991). “Large deformation analysis of elastic space frames.”J. Struct. Engrg., ASCE, 117(7), 2069–2087.
21.
Kouhia, R., and Mikkola, M. (1989). “Tracing the equilibrium path beyond simple critical points.” Int. J. Numer. Methods in Engrg., 28, 2932–2941.
22.
Kouhia, R., and Tuomala, M. (1993). “Static and dynamic analysis of space frames using simple Timoshenko type elements.” Int. J. Numer. Methods in Engrg., 36, 1189–1221.
23.
Meek, J. L., and Loganathan, S. (1989). “Geometrically non-linear behaviour of space frame structures.” Comp. and Struct., 31, 35–45.
24.
Nishino, F., and Hartono, W. (1989). “Influential mode of imperfection on carrying capacity of structures.”J. Engrg. Mech., ASCE, 115(10), 2150–2165.
25.
Onate, E. O., and Matias, W. T. (1996). “A critical displacement for predicting structural instability.” Comp. Meth. Appl. Mech. Engrg., 134, 135–161.
26.
Papadrakakis, M., and Ghionis, P. (1986). “Conjugate gradient algorithms in nonlinear structural analysis problems.” Comp. Meth. Appl. Mech. Engrg., 59, 11–27.
27.
Pecknold, D. A., Ghaboussi, J., and Healey, T. J. (1985). “Snap-through and bifurcation in a simple structure.”J. Engrg. Mech., ASCE, 111(7), 909–922.
28.
Powell, G. H., and Simons, J. (1981). “Improved iteration strategy for nonlinear structures.” Int. J. Numer. Methods in Engrg., 17, 1455–1467.
29.
Remseth, S. N. (1979). “Nonlinear static and dynamic analysis of framed structures.” Comp. and Struct., 10, 879–897.
30.
Rheinboldt, W. C. (1981). “Numerical analysis of continuation methods for nonlinear structural problems.” Comp. and Struct., 13, 103–113.
31.
Riks, E. (1972). “The application of Newton's method to the problem of elastic stability.” J. Appl. Mech., 39, 1060–1066.
32.
Riks, E. (1979). “An incremental approach to the solution of snapping and buckling problems.” Int. J. Solids and Struct., 15, 529–551.
33.
Roorda, J. (1965). “Stability of structures with small imperfections.”J. Engrg. Mech. Div., ASCE, 91, 87–106.
34.
See, T., and McConnel, R. E. (1986). “Large displacement elastic buckling of space structures.”J. Struct. Engrg., ASCE, 112(5), 1052–1069.
35.
Seydel, R. (1979). “Numerical computation of branching points in nonlinear equations.” Numerische. Mathematik, Berlin, 33, 339–352.
36.
Shi, J., and Crisfield, M. A. (1992). “A simple indicator and branch switching technique for hidden unstable equilibrium paths.” FE Anal. Des., 12, 303–312.
37.
Shi, J., and Crisfield, M. A. (1994). “A semi-direct approach for the computation of singular points.” Comp. and Struct., 51, 107–115.
38.
Teh, L. H., and Clarke, M. J. (1998). “Co-rotational and Lagrangian formulations of elastic three-dimensional beam finite elements.” J. Constr. Steel Res., 48, 123–144.
39.
Teh, L. H., and Clarke, M. J. (1999). “Symmetry of tangent stiffness matrices of 3D elastic frame.”J. Engrg. Mech., ASCE, 125(2), 248–251.
40.
Thompson, J. M. T., and Hunt, G. W. (1973). A general theory of elastic stability. Wiley, New York.
41.
Thurston, G. A., Brogan, F. A., and Stehlin, P. (1986). “Postbuckling analysis using a general-purpose code.” AIAA J., 24, 1013–1020.
42.
Wagner, W., and Wriggers, P. (1988). “A simple method for the calculation of postcritical branches.” Engrg. Computations, Swansea, Wales, 5, 103–110.
43.
Waszczyszyn, Z. (1983). “Numerical problems of nonlinear stability analysis of elastic structures.” Comp. and Struct., 17, 13–24.
44.
Widjaja, B. R. (1998). “Path-following technique based on residual energy suppression for nonlinear finite element analysis.” Comp. and Struct., 66, 201–209.
45.
Williams, F. S. (1964). “An approach to the nonlinear behaviour of the members of a rigid jointed plane framework with finite deflections.” Quart. J. Mech. Appl. Math., 17, 451–469.
46.
Wriggers, P., and Simo, J. C. (1990). “A general procedure for the direct computation of turning and bifurcation points.” Int. J. Numer. Methods in Engrg., 30, 155–176.
47.
Zhou, Z., and Murray, D. W. (1994). “An incremental solution technique for unstable equilibrium paths of shell structures.” Comp. and Struct., 55, 749–759.
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Received: Feb 24, 1999
Published online: Dec 1, 1999
Published in print: Dec 1999
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