Postbuckling analysis of cylindrical shells under combined external liquid pressure and axial compression

A postbuckling analysis is presented for a circular cylindrical shell of finite length which is subjected to combined loading of external liquid pressure and axial compression. The formulations are based on a boundary layer theory which includes effects of the nonlinear prebuckling deformation, the nonlinear large deflection in the postbuckling range and the initial geometrical imperfection of shells. The analysis uses a singular perturbation technique to determine interactive buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect cylindrical shells. Typical results are presented in dimensionless graphical form.     

Postbuckling analysis of circular cylindrical shells under external pressure

The postbuckling behavior of circular cylindrical shells of finite length under uniform external pressure is analysed using the spline finite strip method. A Total Lagrangian formulation on the displacement dependent pressure load in the orthogonal curvilinear reference frame is derived. An improvement for the arc-length iteration method is presented. The postbuckling equilibrium path and the contour map of equal radial deflection computed are in good agreement with the experimental and analytical results reported in Esslinger, M. and Geier, B., Postbuckling Behaviour of Structures, Springer-Verlag, Wien, New York, 1975.     

Buckling and postbuckling behaviour of cylindrical shells under combined external pressure and axial compression

Buckling and postbuckling behaviour of perfect and imperfect cylindrical shells of finite length subject to combined loading of external pressure and axial compression are considered. Based on the boundary layer theory which includes the edge effect in the buckling of shells, a theoretical analysis for the buckling and postbuckling of circular cylindrical shells under combined loading is presented using a singular perturbation technique. Some interaction curves for perfect and imperfect cylindrical shells are given. The analytical results obtained are compared with some experimental data in detail, and it is shown that both agree well. The effects of initial imperfection on the interactive buckling load and postbuckling behaviour of cylindrical shells have also been discussed.     

Buckling of cracked cylindrical thin shells under combined internal pressure and axial compression

Linear eigenvalue analysis of cracked cylindrical shells under combined internal pressure and axial compression is carried out to study the effect of crack type, size and orientation on the buckling behavior of cylindrical thin shells. Two types of crack are considered; through crack and thumbnail crack. Our calculations indicate that depending on the crack type, length, orientation and the internal pressure, local buckling may precede the global buckling of the cylindrical shell. The internal pressure, in general, increases the buckling load associated with the global buckling mode of the cylindrical shells. In contrast, the effect of internal pressure on buckling loads associated with the local buckling modes of the cylindrical shell depends mainly on the crack orientation. For cylindrical shells with relatively long axial crack, buckling loads associated with local buckling modes of the cylindrical shell reduce drastically on increasing the shell internal pressure. In contrast, the internal pressure has the stabilizing effect against the local buckling for circumferentially cracked cylindrical shells. A critical crack length for each crack orientation and loading condition is defined as the shortest crack causing the local buckling to precede the global buckling of the cylindrical shell. Some insight into the effect of internal pressure on this critical crack length is provided.     

Buckling of intermediate ring supported cylindrical shells under axial compression

This paper is concerned with the elastic buckling of axially compressed, circular cylindrical shells with intermediate ring supports. The simple Timoshenko thin shell theory and the more sophisticated Flügge thin shell theory have been adopted in the modeling of the cylindrical shells. We used these two representative theories to examine the sensitivity of the buckling solutions to the different degree of approximations made in shell theories. By dividing the shell into segments at the locations of the ring supports, the state-space technique is employed to derive the solutions for each shell segment and the domain decomposition method utilized to impose the equilibrium and compatibility conditions at the interfaces of the shell segments. First-known exact buckling factors are obtained for cylindrical shells of one and multiple intermediate ring supports and various combinations of boundary conditions. Comparison studies are carried out against benchmark solutions and independent numerical results from ANSYS and p-Ritz analyses. The influence of the locations of the ring supports on the buckling behaviour of the shells is examined.     

Dynamic and static axial crushing of axially stiffened cylindrical shells

The axial impact of cylindrical tubes, which incorporate axial stiffeners, is examined in this paper. For comparison purposes, the effect of static loading is also studied. An examination is made into the influence of stiffener depth (T), number of stiffeners (N) and the effect of placing the stiffeners externally or internally.

The experimental results on mild steel specimens show that there are considerable differences between the static and dynamic modes of failure, and that an optimum T/D ratio may exist for a given value of N.     

Experimental studying of buckling of stringer cylindrical shells under axial compression

Results of tests on axial compression of small-sized quality steel cylinder shells strengthened by 24 and 36 longitudinal thin-walled stiffeners are presented. The shell length was varied. Shells both with inside and outside stiffening were tested at simply supported and clamped edges. The shell carrying capacity that was governed in the tests by overall buckling in the elastic range was compared with the estimated critical loads based on structural-orthotropic theory. The satisfactory quantitative correlation has been received only for the long simply supported shells with 36 inner stiffeners, which demonstrated insignificant effect of local undulation that preceded overall deflections. The experimental and the theoretical results differed significantly (twice as much) when the actual mechanism of lateral deflection caused by the intensive local undulation differed from the adopted model.     

Creep buckling of cylindrical shells under external lateral pressure

In this paper, the authors present the results of an analysis of creep buckling of cylindrical shells under external lateral pressure. The nature of the material, the dimensions of the specimens and the type of boundary permit an elastic analysis of this phenomenon. The results of tests and calculations show on the one hand that this phenomenon is very sensitive to the presence of initial geometric imperfections; and on the other hand that it is possible to establish a monograph giving the critical time of an electroplated nickel shell as a function of its initial imperfections and of the nominal applied load. In this manner, the same reasoning can be applied to other shells made up of other materials that are sensitive to creep (resins, concretes, wood, metals at high temperatures, etc.).     

Behaviour of FRP-jacketed circular steel tubes and cylindrical shells under axial compression

Fibre-reinforced polymer (FRP) jackets have been widely used to confine reinforced concrete (RC) columns for enhancement in both strength and ductility. This paper presents the results of a recent study in which the benefit of FRP confinement of hollow steel tubes was explored. Axial compression tests on FRP-confined steel tubes are first described. Finite element modelling of these tests is next discussed. Both the test and the numerical results show that FRP jacketing is a very promising technique for the retrofit and strengthening of circular hollow steel tubes. In addition, finite element results for FRP-jacketed thin cylindrical shells under combined axial compression and internal pressure are presented to show that FRP jacketing is also an effective strengthening method for such shells failing by elephant’s foot collapse near the base.     

Buckling and postbuckling of imperfect cylindrical shells under external pressure

A comprehensive nonlinear finite element elastic stability analysis was performed with the aim of explaining the effect of a deep single longitudinal initial dent on the load-carrying behaviour of an externally-pressurized cylindrical shell. The numerical results are compared with available test results and agree well, at least qualitatively. Imperfection sensitivity studies were carried out comparing the effect of the single dent with the effect of evenly distributed periodic initial imperfections. Imperfection sensitivity tends to vanish for increasing amplitudes of the idealized case of a single dent.     

Buckling of cylindrical shells with longitudinal joints under external pressure

In this article, the bucking of cylindrical shells with longitudinal joint has been investigated through the experimental and numerical analysis. It was clarified that the buckling behavior of cylindrical shells with longitudinal joints under lateral external pressure is not only related to its dimension, but also longitudinal joint and an imperfection. The buckling of cylindrical shells with rigid joint buckles only once and in multi-lobe buckling, whereas one with flexible joints buckles twice and firstly in single-lobe buckling in the vicinity of the joint, secondly in multi-lobe buckling in remaining un-deformed area. And the more flexible the longitudinal joint, the lower the critical pressure, with respect to the same dimension of jointed cylindrical shells and imperfection condition. Moreover the numerical analysis approaches were also presented and verified, by which the imperfection can greatly enlarge the effect of joint on buckling has been demonstrated.     

Reduced stiffness buckling of sandwich cylindrical shells under uniform external pressure

A reduced stiffness lower bound method for the buckling of laterally pressure loaded sandwich cylindrical shell is proposed. Also, an attempt is made to assess the validity of the proposed reduced stiffness lower bound with FEM numerical examples. In addition, the proposed method is compared with classical and Plantema's approaches of the buckling of the laterally pressure loaded sandwich cylindrical shell. Comparison of the proposed reduced stiffness lower bound with that obtained from non-linear FEM analysis verifies that it indeed provides a safe lower bound to the buckling of laterally pressure loaded sandwich cylindrical shells. The attractive feature of the proposed reduced stiffness method is that it can be readily used in designing laterally pressure loaded sandwich cylindrical shells without being concerned about geometrical imperfections.     

Dynamic buckling of stiffened cylindrical shells of revolution under a transient lateral pressure shock wave

A modal method of analysis is used to determine the response of an infinitely long stiffened cylindrical shell of revolution to a transient lateral pressure produced by an underwater explosion and propagating in an acoustic fluid. The shell is initially immersed, hence prestressed by the external hydrostatic pressure. A theory of dynamic buckling is then developed for cylindrical shells subjected to transverse pressure pulses of different durations.     

Robust design of composite cylindrical shells under axial compression — Simulation and validation

Thin-walled shell structures like circular cylindrical shells are prone to buckling. Imperfections, which are defined as deviations from perfect shape and perfect loading distributions, can reduce the buckling load drastically compared to that of the perfect shell. Design criteria monographs like NASA-SP 8007 recommend that the buckling load of the perfect shell shall be reduced by using a knock-down factor. The existing knock-down factors are very conservative and do not account for the structural behaviour of composite shells. To determine an improved knock-down factor, several authors consider realistic shapes of shells in numerical simulations using probabilistic methods. Each manufacturing process causes a specific imperfection pattern; hence for this probabilistic approach a large number of test data is needed, which is often not available. Motivated by this lack of data, a new deterministic approach is presented for determining the lower bound of the buckling load of thin-walled cylindrical composite shells, which is derived from phenomenological test data. For the present test series, a single pre-buckle is induced by a radial perturbation load, before the axial displacement controlled loading starts. The deformations are measured using the prototype of a high-speed optical measurement system with a frequency up to 3680 Hz. The observed structural behaviour leads to a new reasonable lower bound of the buckling load. Based on test results, the numerical model is validated and the shell design is optimized by virtual testing. The results of test and numerical analysis indicate that this new approach has the potential to provide an improved and less conservative shell design in order to reduce weight and cost of thin-walled shell structures made from composite material.     

Buckling of cylindrical shells with stepwise variable wall thickness under uniform external pressure

Cylindrical shells of stepwise variable wall thickness are widely used for cylindrical containment structures, such as vertical-axis tanks and silos. The thickness is changed because the stress resultants are much larger at lower levels. The increase of internal pressure and axial compression in the shell is addressed by increasing the wall thickness. Each shell is built up from a number of individual strakes of constant thickness. The thickness of the wall increases progressively from top to bottom.Whilst the buckling behaviour of a uniform thickness cylinder under external pressure is well defined, that of a stepped wall cylinder is difficult to determine. In the European standard EN 1993-1-6 (2007) and Recommendations ECCS EDR5 (2008), stepped wall cylinders under circumferential compression are transformed, first into a three-stage cylinder and thence into an equivalent uniform thickness cylinder. This two-stage process leads to a complicated calculation that depends on a chart that requires interpolation and is not easy to use, where the mechanics is somewhat hidden, which cannot be programmed into a spreadsheet leading to difficulties in the practical design of silos and tanks.This paper introduces a new “weighted smeared wall method”, which is proposed as a simpler method to deal with stepped-wall cylinders of short or medium length with any thickness variation. Buckling predictions are made for a wide range of geometries of silos and tanks (unanchored and anchored) using the new hand calculation method and compared both with accurate predictions from finite element calculations using ABAQUS and with the current Eurocode rules. The comparison shows that the weighted smeared wall method provides a close approximation to the external buckling strength of stepped wall cylinders for a wide range of short and medium-length shells, is easily programmed into a spreadsheet and is informative to the designer.     

Multiobjective optimization of orthogonally stiffened cylindrical shells for minimum weight and maximum axial buckling load

In the present research, the weight and axial buckling optimization of orthogonally stiffened cylindrical shells is carried out by the Genetic Algorithm. Constraints include two nondimensional functions of weight and buckling load in such a way that the stiffened shell has no increase in the weight and no decrease in the buckling load with respect to the initial unstiffened shell. In analytical solution, the Rayleigh–Ritz energy procedure is applied and the stiffeners are treated as discrete members. The optimization is implemented for shells with simply supported end conditions stiffened by four shapes of stiffeners including rectangular-, cee-, I-, and hat-shaped ones. The results show that the I-section and rectangular-section stiffeners are, respectively, the most and the least efficient in designing stiffened cylindrical shells for minimum weight and maximum critical axial buckling load.     

Experimental study on damaged cylindrical shells under compression

Sensitivity to initial imperfections under compressive loading has been extensively studied in shell structures. However, due to the existence of a wide range of imperfections with various shapes and amplitudes, the real behavior of such structures needs to be further investigated when they face with a damaged area. This study presents an experimental program in which buckling and failure response of damaged shell specimens are analyzed. The results of this study can be generalized for many kinds of cylindrical shells to full scale of applications with similar D/t ratios.     

Buckling and ultimate strength criteria of stiffened shells under combined loading for reliability analysis

The work presented in this paper forms part of a broader task in establishing a guide to serve as technical documentation for buckling and ultimate strength assessment of various types of marine structural components using the best state-of-the-art knowledge for extreme environmental loading. This paper concentrates on buckling and ultimate strength assessment of ring stiffened shells and ring and stringer stiffened shells involving various modes of buckling and under various loading like axial compression, radial pressure and combined loading. Comparisons are made with screened test data, which have realistic imperfections and various radius to thickness ratio values in the range generally used in offshore structures. The statistical data of model uncertainty factors in terms of bias and coefficient of variation (COV) are calculated and may be used in a further reliability study. Comparisons are also made with the codified rules, API BUL 2U and DNV buckling strength of shells.     

Dynamic local and global buckling of cylindrical shells under axial impact

In this paper, the local and global buckling of cylindrical shells under axial, compressive impact loads is studied. A Hamiltonian system is introduced in the problem. The fundamental problem in the system can be described mathematically by the Hamiltonian dual equations, which are expressed in four pairs of dual variables. The problem is reduced to a problem of eigenvalues and eigensolutions for critical loads and buckling modes, respectively. The buckling modes can be described by their respective orders and they are grouped into two classes, the short-wave or local buckling and the long-wave or global buckling. The solutions are obtained analytically and numerically, and some rules observed are indicated.     

Theoretical elastoplastic buckling of stringer stiffened cylindrical shells

A linear analysis method is offered to predict the theoretical elastoplastic buckling of stringer stiffened cylindrical shells subjected to longitudinal loading. Welding residual stresses are taken into account in the calculation, but effects of geometrical imperfections and pre-buckling displacements are ignored.The examples analysed show a good correlation between the analytical results and those obtained experimentally with stocky models of moderate geometrical imperfections.