Simplified prediction of creep buckling of cylinders under external pressure. Part 1: finite element validation

In this paper, a simplified method is proposed for the prediction of creep buckling. This simplified approach relies upon a model which yields an analytical evaluation of creep buckling times for cylinders under external pressure. This model is fully developed herein, and a ‘closed-form’ solution is given for the evaluation of the critical creep collapse time. The collapse mechanism is assumed to be due to the formation of a plastic hinge which induces an unstable post-buckling of the ring. The analytical ‘closed-form’ creep collapse time is then compared to finite element buckling predictions using the quasi-axisymmetric COMU shell element in the INCA code of the CASTEM system. The model is then applied to four different cylinders under external pressure and compared to finite element predictions; the cylinders' radius-to-thickness ratio varies between 50 and 550. It is shown that the proposed model performs well for this type of prediction: in all cases, the times to failure predicted by the model are lower than the finite element predictions. These predictions prove to be rather conservative for thicker cylinders. It is shown that creep buckling is a very dangerous failure mode. If the shape of the structure is observed as a function of time, nothing seems to happen during a very long ‘incubation’ period; when the initial imperfection reaches some critical value, buckling then suddenly occurs. This phenomenon is shown by the two methods of evaluation presented herein.     

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.     

Collapse of thick-walled cylinders under external pressure

Hydrostatic collapse tests performed on thick-walled capped cylinders are described. Finite-element predictions which incluce the effects of end-cap stiffening, cross-section ovalities and material strain hardening are compared to experimental results. The analyses correctly predict the sequence of events leading to collapse, but experimental failure pressures are significantly below predictions. It is concluded that the von Mises yield criterion used in the analysis did not accurately represent the yield behavior of the 1018 steel tubing material of the test-cylinders for the triaxial-stress conditions of interest.     

Equilibrium and buckling of combined shells under uniform external pressure

Nonlinear strain is used to formulate the energy functional of combined structure with several kind of shells. The nonlinear finite element method (N.F.E.M.) is proposed for calculating bending and buckling of the structure subjected to external hydrostatic pressure. The numerical results are found to be in good agreement with experimental ones.     

Experimental buckling of internal integral ring-stiffened cylinders

Twenty aluminum cylinders with internal, integral tee-stiffener rings were tested under combinations of axisymmetrical axial load and external lateral pressure to determine buckling characteristics. Seven geometric types were tested; the primary variables were the ratios of cylinder radius to shell thickness, stiffener spacing to shell thickness, and stiffener spacing to stiffener depth. An eighth type, which had variable stiffener spacing and depth, was tested under a lateral pressure varying linearly in the axial direction. Strain-gage data were obtained to aid in evaluation of results. The test results agree well with the theoretical work used for the design.Paper was presented at 1966 SESA Annual Meeting held in Pittsburgh, Pa., on November 6–9.     

Axisymmetrical buckling of an initially deformed shallow spherical shell under external pressure

Axisymmetrical snap buckling of a clamped shallow spherical shell with initial deformation and subject to uniform pressure is analysed based on the finite-deformation theory. As a refinement to analyses given in the author's previous papers, a variable term in the deformation pattern is introduced and Galerkin's method is applied to the non-linear simultaneous differential equations, so that upper buckling pressures are obtained as a function of the geometrical parameter α. Non-uniqueness of solution is shown in the region of larger values of α, where the higher order of deformation mode takes place. Stability of a number of possible equilibrium states is checked by use of the second variation of total potential energy to clarify the actual buckling process. The remark- able decrease of buckling pressure is reasonably explained by introducing a small initial deflection, which explains the discrepancy between the classical buckling pressure and experimental results.     

Static and dynamic buckling modes of a cylindrical shell under external pressure

We consider the problem of static and dynamic buckling modes of thin shells under external hydrostatic pressure. If the statement of the problem uses the linearized equations of motion obtained in the moderately large bending theory of shells according to the classical or refined model, then part of terms related to the external load in these equations are assumed to be conservative, and the other terms are assumed to be nonconservative. In this connection, we study four statements of the elastic stability problem for a cylindrical shell with hinged faces. The first of them is the statement of the static boundary value problem in the sense of Euler, where the action of external pressure is assumed to be conservative. The second statement is used to study small vibrations near the static equilibrium by a dynamic method for the same conservative load. The third and fourth statements of the problem correspond to the action of a nonconservative load and are similar to the first and second statements, respectively. They use the linearized equations of equilibrium and motion constructed earlier in a consistent version on the basis of a Timoshenko type model and allowing one to reveal all classical and nonclassical shell buckling modes.     

A boundary layer theory for the buckling of thin cylindrical shells under external pressure

Based on the boundary layer theory for the buckling of thin elastic shells suggested in ref. [14]. the buckling and postbuckling behavior of clamped circular cylindrical shells under lateral or hydrostatic pressure is studied applying singular perturbation method by taking deflection as perturbation parameter. The effects of initial geometric imperfection are also considered. Some numerical results for perfect and imperfect cylindrical shells are given. The analytical results obtained are compared with some experimental data in detail, which shows that both are rather coincident.     

Experimental analysis of perforated torispherical shells under external pressure

Load tests were conducted on torispherical shells at room temperature for the purpose of predicting mechanical behavior of a reactor-vessel head at 1200°F. These shells were models of a calandria-vessel head designed for use in the sodium-reactor experiment. Principles of dimensional analysis were used in designing the tests and relating experimental results to behavior of the full-scale structure. External pressure and point loads were imposed to simulate service conditions caused by submersion of the calandria in sodium and transient thermal contraction of process tubes. Deflections and strains were monitored at several locations throughout various loading sequences. The stability limit of one 36-in. diam shell was reached at a strain number of:

$$\frac{{p(external{\text{ }}pressure)}}{{E(modulus{\text{ }}of{\text{ }}elasticity)}} = 3.07 \times 10^{ - 6} $$     

An experimental investigation into the buckling of cylindrical shells of variable-wall thickness under radial external pressure

Cylindrical shells, with wall thickness-to-diameter ratios varying in steps from 1:2000 to 1:500 are used in large floating-roof storage tanks, where buckling under partial vacuum is a potential risk. To examine the validity of several methods of analysis, aluminum models with a nominal diameter of 150 mm were tested. Some models exhibited a sudden buckling, at the pressure predicted by one of the methods of analysis, others deformed at increased rate as the critical pressure was approached. A gradual change in, modal-buckling pattern, under decreasing pressure, was also observed.     

Plate and shell buckling by the Karman scheme under creep conditions

The problem of stability and buckling of plates under creep conditions has been studied in the fundamental papers [1–4]. In the papers [5, 6], the theory of buckling of rods and plates is developed in the framework of the dominating bending model [7], where the forces in the midplane of the plate were determined independently of the solution of the bending problem. In what follows, we use the Karman scheme [8] to derive two basic differential equations of the coupled theory of the plane stress state and bending. We solve the problem of buckling of a rectilinear plate for linearly viscous (Newtonian) medium and show that the Karman scheme gives an essential correction to the solution of the bending problem for initial deflections comparable with the plate thickness.     

Nonlinear deformation and buckling of curvilinear pipes loaded by external pressure

The problem of loading of a thin-walled elastic pipe (a toroidal shell) by external pressure is examined in a geometrically nonlinear formulation. A numerical algorithm is used to study the nonlinear deformation of the shell and the stability of its equilibrium states when its cross section has undergone a significant change in shape. Results are presented from a determination of the critical stresses of curvilinear pipes with allowance for moments in the subcritical state. These results are compared with the approximate solution. Chaplygin Siberian Aviation Institute, Novosibirsk 630051. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 162–166, July–August, 1998.     

On buckling of ellipsoidal cups under internal pressure

Summary A method for the calculation of the critical load for a semi-ellipsoidal shell with a stiffening rib at the edge is presented. The external and internal buckling energy of the shell is described. The Rayleigh quotient is used as the static buckling criterion, assuming the deflection function to depend on four shape parameters. Some numerical examples are presented showing the influence of the rib stiffness and the shell dimensions on the critical pressure value.

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Stabilitätsproblem von Ellipsoidschalen unter Innendruck

Übersicht Eine Methode zur Berechnung der kritischen Belastung von Halbellipsoidschalen mit versteifendem Kragen am Rande wird vorgestellt. Die äußere und innere Energie der Schale wird berechnet und der Rayleigh-Quotient als statisches Kriterium der Stabilität benutzt. Ihm liegt eine Funktion der Biegung mit vier Gestaltparametern zugrunde. Einige numerische Beispiele werden vorgestellt, welche den Einfluß der Kragensteifigkeit und der Schalenabmessungen auf den kritischen Druck illustrieren.

     

Elastic buckling of honeycomb structure under out-of-plane pressure

A theoretical model and an analytical method, which are suitable for initial elastic buckling analysis of two-dimensional honeycomb structures including hexagonal honeycombs with walls of equal or unequal thickness, rectangular and triangular honeycombs etc., are developed in this paper. The results given in present paper agree well with the experimental data of hexagonal honeycombs with walls of unequal thickness.The project was supported by National Natural Science Foundation of China.