Non-symmetric Bifurcation Research Articles

Non-symmetric bifurcation of geometrically nonlinear elastic-plastic axisymmetric shells under combined loads including torsion

A finite element formulation is presented for the non-symmetric bifurcation analysis of geometrically nonlinear elastic-plastic shells of revolution. The shell may be branched and segmented. The loads are axisymmetric but may include in-plane shears (non-uniform torsion). In place of the widely used relations of Donnell, Novozhilov and Sanders, a new nonlinear shell theory is adopted which includes nonlinear strains arising from in-plane displacements. For the determination of the non-linear prebuckling load deflection path, the J 2 flow theory of plasticity is used. For the non-symmetric bifurcation analysis, three theories are provided: J 2 flow theory, J 2 deformation theory and J 2 flow theory with the shear modulus predicted by J 2 deformation theory. A new efficient and automatic solution procedure is described to determine the critical buckling mode, and hence the critical buckling stress. Several example problems are analysed and the predictions of the present analysis are compared with available theoretical and experimental results. Very close agreement is achieved. The effect of using different plasticity theories in the stability analysis is also briefly discussed.

On the calculation of limit and bifurcation points in stability problems of elastic shells

Numerical methods for the computation of singular points of nonlinear equations G(u, λ, μ) = 0 are discussed, where λ and μ are real parameters. Simple and double limit points are treated in some detail and numerical algorithms are presented and applied to elastic shell stability problems. The case of simple symmetry breaking bifurcation points is also treated with applications to nonsymmetric bifurcation from axisymmetric states of deformation of shells of revolution.

Buckling of elastic-plastic shells of revolution with discrete elastic-plastic ring stiffeners

The theory is summarized for axisymmetric prebuckling and nonsymmetric bifurcation buckling of ring-stiffened shells of revolution. The analysis is based on finite difference energy minimization in which moderately large meridional rotations, elastic-plastic effects, and primary or secondary creep are included. This theory is implemented in a computer program called BOSOR5, for the analysis of segmented and branched ring-stiffened shells of revolution of multi-material construction.Comparisons between test and theory are given for axisymmetric collapse and nonsymmetric bifurcation buckling of 69 machined ring-stiffened aluminum cylinders submitted to external hydrostatic pressure. Because most of the cylinders fail at an average stress which corresponds to the knee of the stress-strain curve, the analytical predictions are not very sensitive to modeling particulars such as nodal point density or boundary conditions. Agreement between test and theory is improved if the analytical model reflects the fact that the shell and rings intersect over finite axial lenths.

Bifurcation buckling of shells of revolution including large deflections, plasticity and creep

A summary is first presented of the conceptual difficulties and paradoxes surrounding plastic bifurcation buckling analysis. Briefly discussed are nonconservativeness, loading rate during buckling, and the discrepancy of buckling predictions with use of J2 flow theory vs J2 deformation theory. The axisymmetric prebuckling analysis, including large deflections, elastic-plastic material behavior and creep is summarized. Details are given on the analysis of nonsymmetric bifurcation from the deformed axisymmetric state. Both J2 flow theory and J2 deformation theory are described. The treatment, based on the finite-difference energy method, applies to layered segmented and branched shells of arbitrary meridional shape composed of a number of different elastic-plastic materials. Numerical results generated with a computer program based on the analysis are presented for an externally pressurized cylinder with conical heads.

Comparisons of test and theory for nonsymmetric elastic-plastic buckling of shells of revolution

Experimental and analytical buckling pressures are presented for very carefully fabricated thin cylindrical shells with 45, 60 and 75° conical heads and for cylindrical shells with torispherical heads pierced by axisymmetric cylindrical nozzles of various thicknesses and diameters. Nonsymmetric buckling occurs at pressures for which some of the material is loading plastically in the neighborhoods of stress concentrations caused by meridional slope discontinuities. The buckling pressures for the cone-cylinder vessels are predicted within 2.6 per cent and for the pierced torispherical vessels within 4.4 per cent with use of BOSOR5, a computer program based on the finite difference energy method in which axisymmetric large deflections, nonlinear material properties and nonsymmetric bifurcation buckling are accounted for. The predicted buckling pressures of the pierced torispherical specimens are rather sensitive to details of the analytical model in the neighborhood of the juncture between the nozzle and the head. The buckling pressures of the cone-cylinder vessels can be accurately predicted by treatment of the wall material as elastic, enforcement of the full compatibility conditions at the juncture in the prebuckling analysis, and release of the rotation compatibility condition in the bifurcation (eigenvalue) analysis.

Axisymmetric Postbuckling and Nonsymmetric Buckling of a Spherical Shell Compressed Between Rigid Plates

The authors have previously shown that a thin, complete spherical shell compressed between two parallel rigid plates deforms initially with the polar portion of the shell flattened against the plates and that at a critical deformation the flat region may buckle into an axisymmetric inward dimple. The present paper presents an analysis of the stresses and deflections produced during axisymmetric postbuckling and determines the deformation states at which the shell may buckle into a nonsymmetric shape. The analysis accounts for finite deflections and rotations, but assumes that the material remains linearly elastic throughout the deformation. An experiment shows that both the primary axisymmetric bifurcation point and the secondary nonsymmetric bifurcation point are stable for a shell with R/h ≃ 40.

Computer analysis of various shells of revolution.

A computer program for the buckling of shells of revolution under axisymmetrical loading has been applied in an extensive numerical analysis. The axisymmetric collapse and the nonsymmetric bifurcation buckling behavior are studied for axially compressed cylinders, cones, and spherical and toroidal segments. Comparisons are made with a large number of experimental results. A parameter study is performed in which particular emphasis is placed on the effects of eccentricity in load application and on the influence of elastic end rings. Nomenclature Cn = value of JVi due to unit axial strain E . = Young's modulus for ring G = shear modulus for ring H = see Fig. 9 Ix = moment of inertia of ring around axis perpendicular to shell axis / = torsional constant for ring Kp = ring resistance against prebuckling deformations Kb = ring resistance against buckling deformations Mi = meridional moment resultant Ni — meridional stress resultant n — number of circumferential waves in buckling pattern P = total axial load applied to shell, positive compression r = horizontal radius from axis of rotation to reference surface tr = length of side of square end ring UH = horizontal displacement uy = vertical displacement v = circumferential displacement w = normal displacement, positive outward B = meridional rotation Subscripts ( )Cr = critical = prebuckling quantity 0