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Direct evaluation of the ‘worst’ imperfection shape in shell buckling. (English) Zbl 0924.73090
Summary: For the evaluation of the imperfection-sensitivity of elastic and elastic-plastic shells a fully nonlinear finite-element-method has been developed that directly gives the ‘worst’ imperfection shape connected to the ultimate limit load. The approach uses no approximations such as asymptotical theories or computations in the deep postcritical range. The key point of the method is the description of imperfections as nodal degrees of freedom at the element level. These unknown quantities are implemented by isoparametric shape functions in a finite shell element including finite rotations and thickness stretch. The ultimate buckling load and the corresponding ‘worst’ imperfection shape is defined by two different criteria and numerically determined by an extended system of nonlinear equations. In the computation of the structurally stable collapse the deflections, the imperfection shape, the eigenvector and the lowest possible load-level are obtained. The method is illustrated by several numerical examples.

##### MSC:
 74G60 Bifurcation and buckling 74K15 Membranes 74S05 Finite element methods applied to problems in solid mechanics
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##### References:
 [1] Bauer, L.; Keller, H.B.; Reiss, E.L., Multiple eigenvalues leading to secondary bifurcation, SIAM rev., 17, 101-122, (1975) [2] Bathe, K.J.; Dvorkin, E., A formulation for general shell-elements—the use of mixed interpolation of tensorial components, J. numer. methods engrg., 22, 697-722, (1986) · Zbl 0585.73123 [3] Büchter, N.; Ramm, E., Shell theory versus degeneration—a comparison in large rotation finite element analysis, Int. J. numer. methods engrg., 34, 39-60, (1992) [4] Bushnell, D., Computerized buckling analysis of shells, (1989), Kluwer Academic Publishers Dordrecht [5] Chan, T.F., Deflated decomposition of solutions of nearly singular systems, SIAM J. numer. anal., 21, 738-754, (1984) · Zbl 0572.65012 [6] Chen, W.F.; Han, D.J., Plasticity for structural engineers, (1988), Springer New York · Zbl 0666.73010 [7] El Naschie, M.S., Stress, stability and chaos, (1990), McGraw-Hill London · Zbl 0729.73919 [8] Ho, D., Buckling load of non-linear systems with multiple eigenvalues, Int. J. solids struct., 10, 1315-1330, (1974) · Zbl 0296.73034 [9] Hutchinson, J.W., On the postbuckling behavior of imperfection-sensitive structures in the plastic range, J. appl. mech. ASME, 39, 155-162, (1972) [10] Jang, J.; Pinsky, P.M., An assumed covariant strain based 9-node shell element, Int. J. numer. methods engrg., 24, 2389-2411, (1987) · Zbl 0623.73090 [11] John, F., Refined interior shell equations, () · Zbl 0265.73076 [12] Kármán, Th.v.; Tsien, H.S., Buckling of thin cylindrical shells under axial compression, J. aeronaut. sci., 8, 302-312, (1941) · Zbl 0060.42403 [13] Koiter, W.T.; Koiter, W.T., On the stability of elastic equilibrium, (), Nasa tt f-10,833, (1967), English translation: · Zbl 0166.43705 [14] Koiter, W.T., Current trend in the theory of buckling, (), 1-16 [15] Koiter, W.T., The nonlinear buckling problem of a complete spherical shell under uniform external pressure, (), 40 · Zbl 0197.22302 [16] Lanzo, A.D.; Garcea, G., Koiter’s analysis of thin-walled structures by a finite element approach, Int. J. numer. methods engrg., 39, 3007-3032, (1996) · Zbl 0886.73066 [17] Lorenz, R., Achsensymmetrische verzerrungen in dünnwandigen hohlzylindern, Zeitschrift VDI, 52, 1706-1713, (1908) [18] Milford, R.V.; Schnobrich, W.C., Degenerated isoparametric finite elements using explicit integration, Int. J. numer. methods engrg., 23, 133-154, (1986) · Zbl 0578.73068 [19] Recke, L.; Wunderlich, W., Rotations as primary unknowns in the nonlinear theory of shells and corresponding finite element models, (), 239-258 · Zbl 0595.73084 [20] Riks, E., Some computational aspects of the stability analysis of nonlinear structures, Comput. methods appl. mech. engrg., 47, 219-259, (1984) · Zbl 0535.73062 [21] () [22] Sansour, C., A theory and finite element formulation of shells at finite deformations involving thickness change: circumventing the use of rotation tensor, Arch. appl. mech., 65, 194-216, (1995) · Zbl 0827.73044 [23] Simo, J.C.; Fox, D.D., On a stress resultant geometrically exact shell model. part 1: formulation and optimal parametrization, Comput. methods appl. mech. engrg., 72, 267-304, (1989) · Zbl 0692.73062 [24] Simo, J.C.; Fox, D.D.; Rifai, M.S., On a stress resultant geometrically exact shell model. part 3: computational aspects of the nonlinear theory, Comput. methods appl. mech. engrg., 79, 21-70, (1990) · Zbl 0746.73015 [25] Simo, J.C.; Fox, D.D.; Rifai, M.S., On a stress resultant geometrically exact shell model. part 4: variable thickness shells with through-the-thickness stretching, Comput. methods appl. mech. engrg., 81, 53-91, (1990) · Zbl 0746.73016 [26] Surana, K.S., Geometrically nonlinear formulation for the curved shell elements, Int. J. numer. methods engrg., 19, 581-615, (1983) · Zbl 0509.73082 [27] Thom, R., Structural stability and morphogenesis, (1974), W.A. Benjamin Inc., Massachusetts [28] Wagner, W., Zur behandlung von stabiltätsproblemen der elastostatik mit der methode der finiten elementen, (1991), Habilitation, Universität Hannover [29] Weingarten, V.I.; Morgan, E.J.; Seide, P., Elastic stability of thin-walled cylindrical and conical shells under axial compression, Aiaa j., 3, 500-505, (1965) [30] Wohlever, J.C.; Healy, T.J., A group theoretic approach to the global bifurcation analysis of an axially compressed cylindrical shell, Comput. methods appl. mech. engrg., 122, 315-349, (1995) · Zbl 0851.73019 [31] Wriggers, P.; Simo, J.C., A general procedure for the direct computation of turning and bifurcation points, Int. J. numer. methods engrg., 30, 155-176, (1990) · Zbl 0728.73069 [32] Wriggers, P.; Wagner, W.; Miehe, C., A quadratically convergent procedure for the calculation of stability points in finite element analysis, Comput. methods appl. mech. engrg., 70, 329-347, (1987) · Zbl 0653.73031 [33] Wunderlich, W.; Obrecht, H.; Springer, H.; Lu, Z., A semi-analytical approach to the nonlinear analysis of shells of revolution, (), 509-536 [34] Zoelly, R., Über ein knickungsproblem an der kugelschale, Dissertation, (1915)
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