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.     

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.     

VIBRO-ACOUSTIC BEHAVIOR OF SUBMERGED CYLINDRICAL SHELLS: ANALYTICAL FORMULATION AND NUMERICAL MODEL

We propose both an analytical formulation and a numerical model to study the hydroelastic or vibroacoustic behaviour of cylindrical thin shells immersed in an unbounded, inviscid and heavy fluid. The analytical solution allows us to calculate the dynamic response and the pressure radiated in the far field by a baffled cylinder. This formulation uses the truncated modal basis of the dry structure to expand the displacements of the submerged shell. The analytical model is used as a reference in order to validate a numerical model which couples the finite element method (FEM) to the boundary element method (BEM). As opposed to the analytical formulation which is dedicated to baffled circular cylinders only, the numerical model allows us to treat any axisymmetric shell, such as cylindrical and spherical shells, or more complex shells of revolution. The structure is idealized by two-node ring finite elements and the boundary equation is solved using the method of singularities.     

Necking of biaxially stretched elastic-plastic circular plates

The necking of an elastic-plastic circular plate under uniform radial tensile loading is investigated both within the framework of the three-dimensional theory and within the context of the plane-stress approximation. Attention is restricted to axisymmetric deformations of the plate. The material behavior is described by two different constitutive laws. One is a finite-strain version of the simplest flow-theory of plasticity and the other is a finite-strain generalization of the simplest deformationtheory, which is employed as a simple model of a solid with a vertex on its yield surface. For an initially uniform plate made of an incompressible material, bifurcation from the uniformly stretched state is studied analytically. The regimes of stress and moduli where the governing axisymmetric three-dimensional equations are elliptic, parabolic or hyperbolic are identified. The plane-stress local-necking mode emerges as the appropriate limiting mode from the bifurcation modes available in the elliptic regime. In the elliptic regime, the main qualitative features of the bifurcation behavior are revealed by the plane-stress analysis, although three-dimensional effects delay the onset of necking somewhat. For the deformation theory employed here, the first bifurcation modes are encountered in the parabolic regime if the hardening-rate is sufficiently high. These bifurcations are not revealed by a plane-stress analysis. For a plate with an initial inhomogeneity, the growth of an imperfection is studied by a perturbation method, by a plane-stress analysis of localized necking, and by numerical computations within the framework of the three-dimensional theory. When bifurcation of the corresponding perfect plate takes place in the elliptic regime, the finite element results show that the plane-stress analysis gives reasonably good agreement with the numerical results. When bifurcation of the corresponding perfect plate first occurs in the parabolic regime, then a bifurcation of the imperfect plate is encountered, that is, the finite element stiffness matrix ceases to be positive definite.     

Initial postbuckling behavior of cylindrical composite shells under axisymmetric deformation

A solution is obtained that describes the postbuckling behavior of cylindrical shells in the case of axisymmetric buckling. The basis for this solution is Koiter’s asymptotic method and the nonlinear equations of the third-order Timoshenko theory of shells. It is shown that the bifurcation point in this case is a symmetrically unstable one. The effect of the initial axisymmetric deflections on the buckling loads is weaker when buckling is axisymmetric. The results obtained by Koiter’s special theory evidence this __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 4, pp. 108–118, April 2006.     

Experimental creep buckling of aluminum cylinders in axial compression

Creep-buckling tests were conducted on aluminum alloy 2024-0 circular cylinders in axial compression at 500° F having nominalR/t values of 90 and 50. Creep-buckling times for a variety of applied creep-stress values were compared with theoretical predictions of Gerard's unified theory of creep buckling of columns, plates and shells. In this theory, creep-buckling solutions are analogous to plastic-buckling solutions, provided that the material parameters used in the theoretical relation are developed from constant-strain-rate stress-strain data derived by a graphical process from compressive-creep data. The theoretical data were evaluated using appropriate classical plastic-buckling theory and previously obtained creep data on the 2024-0 aluminum material at 500° F. End shortening of the cylinders was autographically recorded during the tests and creep-buckling times were obtained from an analysis of the end-shortening record. A comparison of theory and test data indicated that the theory was somewhat conservative in predicting creep-buckling times. The discrepancy may have been due, in part, to the uncertainty in determining the precise time at which the experimental cylinders buckled. The cylinders withR/t～90 buckled in the axisymmetric mode for the lower creep stresses while, forR/t～50, all buckling occurred in the axisymmetric mode.     

Theoretical and Experimental Results on the Post-Buckling of Ring-Stiffened Cylinders

ABSTRACT

The post-buckling behavior of ring-stiffened cylinders subjected to lateral pressure is studied theoretically and experimentally. The theoretical analysis uses the energy method and terms of up to and including fourth order of the buckling displacements are considered. The results of experiments on ten machined ring-stiffened cylinders subjected to lateral pressure and used to verify the theoretical analysis. The theory shows the post-buckling behavior to be stable-symmetric for both the shell and general instability modes. The experimental results are in reasonable agreement with the theory, although they indicate, in general, a lower post-buckling stiffness. For the range of parameters considered, no instability or imperfection sensitivity is found in either the theoretical or the experimental studies.     

Effect of thickness inhomogeneities in internally pressurized elastic-plastic spherical shells

The behaviour of elastic-plastic spherical shells under internal pressure is investigated numerically for thickness-to-radius ratios ranging from cases of thin shells to very thick shells. The shells under consideration are made of strain-hardening elastic-plastic material with a smooth yield-surface. Attention is restricted to axisymmetric deformations, and results are presented for initial thickness inhomogeneities in various axisymmetric shapes. For smooth thickness-variations in the shape of the critical bifurcation mode, the reduction in maximum pressure is studied together with the distribution of deformations in the final collapse mode. Also, the possibility of flow localization due to more localized, initially thin regions on a spherical shell is investigated.     

The parameters affecting the differences between the solutions corresponding to two different definitions of the reference surface of deformed rubber-like shells of revolution

The parameters affecting the differences between the solutions to problems related with incompressible rubber—like shells of revolution, undergoing axisymmetric finite strains and rotations, corresponding to two different definitions of the reference surface of deformed rubber-like shells of revolution are obtained through an asymptotic analysis.     

A new approach of predicting bifurcation points of elastic-plastic buckling of plates and shells

The paper proposes a new approach of predicting the bifurcation points of elastic-plastic buckling of plates and shells, which is obtained from the natural combination of the Lyaponov's dynamic criterion on stability and the modified adaptive Dynamic Relaxation (maDR) method developed recently by the authors. This new method can overcome the difficulties in the applications of the dynamic criterion. Numerical results show that the theoretically predicted bifurcation points are in very good agreement with the corresponding experimental ones. The paper also provides a new means for further research on the plastic buckling paradox of plates and shells.     

Nonsymmetric buckle patterns in progressive plastic buckling

This paper presents the results of a series of experiments on the progressive plastic buckling of cylindrical shells under axial compressive load. It shows that, for shell bodies with anR/t less than 100, the normal axisymmetric ring buckling will develop into nonsymmetric patterns. We demonstrate that there exists also a class of shells within this thickness-radius range for which nonsymmetric plastic buckling always occurs without the prior formation of a ring. It appears from the limited number of tests made that, for a particularR, R/t, material and rate of loading, there is a critical value ofL, above which there is a high probability of the buckle pattern developing in a nonsymmetric fashion. It seems probable, too, that there are bands ofR/t for a particularL/R, R, material and rate of loading for which the buckle number will be constant. The experiments tend to indicate that the postbuckling efficiency of the shell decreases with increasing buckle number. The nonsymmetric patterns demonstrated appear to be inextensional deformations. They are very similar in character to the Yoshimura pattern which occurs as the limiting case for thin shells in axial compression and, under impact loading. Load-displacement histories are presented for some of the various modes of failure demonstrated.     

Traveling waves vibrations and buckling of rotating anisotropic shells of revolution by finite elements

A quasi-analytical finite element procedure is developed which can obtain the frequency and buckling eigenvalues of prestressed rotating anisotropic shells of revolution. In addition to the usual centrifugal forces, the rotation effects treated also include the contribution of Coriolis forces. Furthermore, since a nonlinear version of Novoshilov's shell theory is employed to develop the element formulation, the effects of moderately large prestress deflection states can be handled. Due to the generality of solution procedure developed, the axisymmetric prestress states treated can also consist of torque loads. In order to illustrate the procedures capabilities, as well as the significant effects of Coriolis forces, torque prestress and material anisotropy, several numerical experiments are presented.     

Nonlinear long-term behaviour of spherical shallow thin-walled concrete shells of revolution

The nonlinear long-term buckling behaviour (creep buckling) of spherical shallow, thin-walled concrete shells of revolution (including domes) subjected to sustained loads is investigated herein. A thorough understanding of their nonlinear time-dependent behaviour, as well as the development of comprehensive analytical models for their analysis, has hitherto not been fully established and further studies are required. A nonlinear axisymmetric theoretical model, which accounts for the effects of creep and shrinkage, and which considers the ageing of the concrete material and the variation of the internal stresses and geometry in time, is developed for this purpose. The governing field equations are derived using variational principles, equilibrium requirements, and integral-type constitutive relations. A systematic step-by-step procedure is used for the solution of the integral-type governing equations. First, the nonlinear short-term behaviour is studied to provide a benchmark for the long-term analysis. Different theories for the analysis of the shell structure are examined for this purpose and compared with results obtained by the finite element method. A numerical study, which highlights the capabilities of the nonlinear theoretical model and which provides insight into the nonlinear long-term behaviour of shallow concrete domes, is presented. The results show that long-term effects are critical for the design and structural safety of shallow, thin-walled concrete domes, and so these effects need to be fully understood and quantifiable.     

Elastic instability of a uniformly compressed annular plate with axisymmetric initial deflection

This paper presents a theoretical study of the elastic instability of a uniformly compressed, thin, circular annular plate with axisymmetric initial deflection. The dynamic version of the nonlinear Marguerre plate theory is used, and the linear free vibration problems around the axisymmetric finite deformation of the plate are solved by a finite difference method. By examining the frequency spectrum with various asymmetric modes, the critical compressive load under which the axisymmetric additional deformation of the plate becomes unstable due to the bifurcation buckling is determined, which is found to depend severely on the magnitude of the axisymmetric initial deflection.     

On buckling of axisymmetric thin elastic-plastic shells

A buckling criterion for shells with an axisymmetric middle surface and subjected to edge loads and hydrostatic surface pressure loading is formulated starting from Hill's three-dimensional continuum theory for uniqueness of deformation of inelastic solids. It turns out that a physically consistent two-dimensional set of equations may be derived for a quite general class of strain-hardening elastic-plastic solids, the only essential restriction being that of a smooth yield function. The intrinsic errors inherent in the derived rate equations, being an integral part of an eigenvalue problem, are discussed in relation to a circular cylinder under axial compression. Analytical results are given which illustrate the influence of the constitutive properties and the boundary contraints on the magnitude of the critical load.     

Stability of long transversely-isotropic elastic cylindrical shells under bending

The present paper investigates buckling of cylindrical shells of transversely-isotropic elastic material subjected to bending, considering the nonlinear prebuckling ovalized configuration. A large-strain hypoelastic model is developed to simulate the anisotropic material behavior. The model is incorporated in a finite-element formulation that uses a special-purpose “tube element”. For comparison purposes, a hyperelastic model is also employed. Using an eigenvalue analysis, bifurcation on the prebuckling ovalization path to a uniform wrinkling state is detected. Subsequently, the postbuckling equilibrium path is traced through a continuation arc-length algorithm. The effects of anisotropy on the bifurcation moment, the corresponding curvature and the critical wavelength are examined, for a wide range of radius-to-thickness ratio values. The calculated values of bifurcation moment and curvature are also compared with analytical predictions, based on a heuristic argument. Finally, numerical results for the imperfection sensitivity of bent cylinders are obtained, which show good comparison with previously reported asymptotic expressions.     

Postbuckling analysis of elastic shells of revolution considering mode switching and interaction

The postbuckling response of shells is known to exhibit complex phenomena including mode switching and interaction, particularly in the advanced postbuckling range. The existing literature contains many initial postbuckling analyses as well as advanced postbuckling analyses for a single buckling mode, but little work is available on the advanced postbuckling analysis of shells of revolution considering mode switching and interaction. In this paper, a numerical method for the advanced postbuckling analysis of thin shells of revolution subject to torsionless axisymmetric loads is presented, in which such mode switching and interaction are properly captured. Numerical results obtained using the present method for several typical problems not only demonstrate the capability of the method, but also lead to significant observations concerning the postbuckling behavior of thin shells of revolution. In particular, the results show that strong interaction between different harmonic modes may exist and the transition of deformation mode from one to another is gradual. Consequently, the conventional approach of finding the postbuckling path of a shell as the lower festoon curve of postbuckling paths of individual harmonic modes is not valid and is at best a convenient approximation.     

Buckling of elastic-plastic cylindrical panel under axial compression

The buckling behaviour is investigated for an axially compressed elastic-plastic cylindrical panel of the type occurring in stiffened shells. The bifurcation stress is determined analytically and an asymptotically exact expansion is obtained for the initial post-bifurcation behaviour in the plastic range. For panels with small initial imperfections the behaviour is analysed asymptotically on the basis of the hypoelastic theory that results from neglecting the effect of elastic unloading. The imperfection-sensitivity of an elastic-plastic panel is also computed numerically by a linear incremental method, and the results are compared with the results of the asymptotic analysis. For a low hardening material the panel is found to be imperfection-sensitive in the whole range of curvatures considered, whereas for a high hardening material the panel is only imperfection-sensitive if the curvature exceeds a certain value.     

Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects

This paper presents an analytical approach to investigate the non-linear axisymmetric response of functionally graded shallow spherical shells subjected to uniform external pressure incorporating the effects of temperature. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and compatibility equations for shallow spherical shells are derived by using the classical shell theory and specialized for axisymmetric deformation with both geometrical non-linearity and initial geometrical imperfection are taken into consideration. One-term deflection mode is assumed and explicit expressions of buckling loads and load-deflection curves are determined due to Galerkin method. Stability analysis for a clamped spherical shell shows the effects of material and geometric parameters, edge restraint and temperature conditions, and imperfection on the behavior of the shells.     

外压对环肋圆柱壳轴向弹塑性动力屈曲的影响

采用增量理论,借助增量数值解法研究了复合加载（轴向流－固冲击载荷＋径向均匀外压）条件下环肋圆柱壳的弹塑性动力屈曲,采用类似B－R准则和Southwell方法来确定临界载荷,讨论了径向均匀外压对结构动力性态及抗轴冲击能力的影响。