Buckling of cylindrical shells with small curvature

Author:
John Mallet-Paret

Journal:
Quart. Appl. Math. **35** (1977), 383-400

MSC:
Primary 73.35

DOI:
https://doi.org/10.1090/qam/478917

MathSciNet review:
478917

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Abstract: We consider the bifurcation buckling of a rectangular plate with an imperfection of magnitude under an applied lateral force of magnitude . The analysis allows the parameters to vary independently in a neighborhood of some , and describes all buckled states of small magnitude. If the plate is represented by the domain in , then the lateral force is applied to the edges , , and the imperfection is a small vertical displacement of the form , where and are fixed. Roughly, then, the plate has a small curvature in the -direction, of magnitude .

**[1]**L. Bauer and E. Reiss,*Nonlinear buckling of rectangular plates*, SIAM J.**13**, 603-626 (1965)**[2]**L. Bauer, E. Reiss and H. Keller,*Multiple eigenvalues lead to secondary bifurcations*, SIAM Review**17**, 101-122 (1975) MR**0378546****[3]**M. Berger,*On von Kármán's equations and the buckling of a thin elastic plate: I. The clamped plate*, Comm. Pure Appl. Math.**20**, 687-719 (1967) MR**0221808****[4]**M. Berger and P. Fife,*On von Kármán's equations and the buckling of a thin elastic plate: II. Plate with general edge conditions*, Comm. Pure Appl. Math.**21**, 227-241 (1968) MR**0229978****[5]**S. N. Chow, J. Hale and J. Mallet-Paret,*Applications of generic bifurcation I*, Arch. Rat. Mech. Anal.**59**, 159-188 (1975) MR**0390852****[6]**S. N. Chow, J. Hale and J. Mallet-Paret,*Applications of generic bifurcation II*, Arch. Rat. Mech. Anal.**62**, 209-235 (1975) MR**0415673****[7]**M. G. Crandall and P. H. Rabinowitz,*Bifurcation from simple eigenvalues*, J. Func. Anal.**8**, 321-340 (1971) MR**0288640****[8]**J. P. Keener,*Perturbed bifurcation theory at multiple eigenvalues*, Arch. Rat. Mech. Anal.**56**, 348-366 (1974) MR**0355710****[9]**J. P. Keener,*Secondary bifurcation and perturbed bifurcation theory*, preprint.**[10]**J. Keener and H. Keller,*Perturbed bifurcation theory*, Arch. Rat. Mech. Anal.**50**, 159-175 (1973) MR**0336479****[11]**G. H. Knightly and D. Sather,*On nonuniqueness of solutions of the von Kármán equations*, Arch. Rat. Mech. Anal.**36**, 65-78 (1970) MR**0261835****[12]**G. H. Knightly and D. Sather,*Nonlinear buckled states of rectangular plates*, Arch. Rat. Mech. Anal.**54**, 356-372 (1974) MR**0349106****[13]**G. H. Knightly,*Some mathematical problems from plate and shell theory*, in*Proceedings of the Michigan State University Conference*,*Nonlinear Functional Analysis and Differential Equations*, Marcel Dekker, Inc., New York, 1976 MR**0489242****[14]**L. D. Landau and E. M. Lifshitz,*Theory of elasticity*, Pergamon Press, 1970 MR**884707****[15]**S. E. List,*Generic bifurcation with application to the von Kármán equations*, thesis, Brown University, Providence, R. I., 1976**[16]**B. Matkowsky and L. Putnick,*Multiple buckled states of rectangular plates*, Int. J. Nonlinear Mech.**9**, 89-103 (1974)**[17]**D. H. Sattinger,*Group representation theory and branch points of nonlinear functional equations*, SIAM J. Math. Anal.**8**, 179-201 (1977) MR**0438383****[18]**D. H. Sattinger,*Group representation theory, bifurcation theory and pattern formation*, preprint. MR**493378**

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DOI:
https://doi.org/10.1090/qam/478917

Article copyright:
© Copyright 1977
American Mathematical Society