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 and postbuckling behaviors of imperfect cylindrical shells subjected to torsion

Buckling and postbuckling behaviors of imperfect cylindrical shell subjected to torsion are investigated. The governing equations are based on the Karman–Donnell-type nonlinear differential equations. A boundary layer theory of shell buckling is applied to obtain the analytic solutions that meet the boundary conditions strictly. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. Numerical results reveal that the current theory gives quite good estimates of the postbuckling paths of cylindrical shells. The effects of the geometric parameters on the buckling and postbuckling behaviors of the cylindrical shells are analyzed. It is confirmed that the postbuckling equilibrium paths of cylindrical shells subjected to torsion are unstable and the relatively shorter shells have higher postbuckling equilibrium paths. Finally, the effects of the initial imperfections on the buckling and postbuckling behaviors of the cylindrical shells are clarified. The illustrated results of the imperfect shells with different initial transverse deflections show that extremely small imperfections do indeed reduce the buckling loads and make the postbuckling equilibrium paths be lower. The buckling and postbuckling of cylindrical shells under torsion exhibit obvious imperfect sensitivity. Furthermore, the effects become greater following with the larger imperfections.     

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.     

Imperfection sensitivity of internally-pressurized torispherical shells

The imperfection sensitivity of torispherical shells subjected to internal pressure was investigated numerically, using the finite element displacement method. Imperfection patterns affine to the first buckling mode at the bifurcation points of the perfect structure were considered. Load-normal displacement curves were obtained for the pre- and postbuckling regions of the perfect computation model and also for imperfect computation models with different imperfection amplitudes. The dependence of the critical pressure and the postbuckling minimum on the imperfection amplitudes was presented graphically. Torispherical shells subjected to internal pressure failed to demonstrate any significant imperfection sensitivity for the imperfection patterns analysed.     

基于改进一致缺陷模态法的轴压圆柱薄壳极限承载力分析

对结构进行缺陷稳定分析的主要方法是一致缺陷模态法和随机缺陷模态法,一致缺陷模态法对薄壁圆柱壳结构进行非线性分析得到的极限承载力与其实际承载能力有一定差距,随机缺陷模态法则工作量很大。基于圆柱薄壳轴压失稳呈现出多模态屈曲的特点,本文提出改进一致缺陷模态法,通过对圆柱壳分别施加不同屈曲模态找到最不利缺陷分布形式。文中通过有限元法验证了改进一致缺陷模态法的可靠性,同时指出按照某一类高阶屈曲模态施加初始缺陷能得到薄壳的最不利极限承载力。     

Recent advances on postbuckling analyses of thin-walled structures: Beams, frames and cylindrical shells

The lateral postbuckling response of thin-walled structures such as bars and frames with members having steel rolled shapes as well as circular cylindrical shells under axial compression is thoroughly reconsidered. More specifically via a simple and very efficient technique it is found that beams with rolled shapes (symmetric or non symmetric) under uniform bending and axial compression exhibit a stable lateral-torsional secondary path with limited margins of postbuckling strength. New findings for the static and dynamic stability of frames with crooked steel members-due to the presence of residual stresses-are also reported. It is comprehensively established that the coupling effect due to initial crookedness and loading eccentricity may have a beneficial effect on the load-carrying capacity of the frames. Moreover, simple mechanical models are proposed for simulating the buckling mechanism of axially compressed circular cylindrical shells. Very recently Bodner and Rubin proposed an 1-DOF mechanical model whose buckling parameters correlated to those of the shells by using an empirical formula based on experimentally observed shell buckling loads. In the present analysis a new 2-DOF model for the static and dynamic buckling of axially compressed circular cylindrical shells, which can include mode coupling, is presented.     

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.     

Postbuckling analysis and imperfection sensitivity of viscoplastic plates and cylindrical panels

The postbuckling behaviour of elastic-viscoplastic rectangular plates and cylindrical panels is analysed. The rate-dependent inelastic material behaviour is modelled by a unified theory of viscoplasticity. Initial geometrical imperfections of the inelastic structure are included and their effect on the postbuckling behaviour is investigated. The analysis relies on an incremental approach in which at each loading increment the Galerkin method is employed. Results are presented for a plate made of an elastic-viscoplastic material and subjected to a uniaxial compression, applied at a constant strain rate on both edges. The results display the applied loading against the out-of-plane displacement, and the effects of loading rate and imperfection sensitivity. It turns out that there is a significant difference between the postbuckling behaviour of a plate in the two cases: (1) when the bifurcation buckling of a geometrically perfect plate occurs in the elastic region, and (2) when it occurs after a plastic flow takes place.     

扭转作用下有缺陷柱形壳的屈曲和后屈曲性能

分析扭转作用下有缺陷柱形壳的屈曲和后屈曲性能。基于Karman-Donnell-Type非线性微分方程建立计算公式,采用壳屈曲的边界层理论进行分析,以获得能严格满足边界条件的解决方案。采用奇摄动技术,以确定屈曲载荷和后屈曲平衡路径。数值结果显示,目前的理论能对柱形壳的后屈曲性能进行较好评估。同时分析了几何参数对柱形壳的屈曲和后屈曲性能的影响。证实了扭转作用下柱形壳的后屈曲平衡路径并不稳定,并且相对更短的壳体具有更高的后屈曲平衡能力。最后,指出初始缺陷对柱形壳屈曲和后屈曲性能的影响。对具有初始横向挠曲的有缺陷壳体的分析结果显示：即便是非常小的缺陷,也确实会减少屈曲承载力,并使得后屈曲稳定性变差。扭转作用下柱形壳的屈曲和后屈曲性能显示出明显的缺陷敏感性。此外,如果缺陷更大,那么带来的影响也随之会变得更大。     

Buckling strength of the cylindrical shell and tank subjected to axially compressive loads

This paper aims to develop practical design equations and charts estimating the buckling strength of the cylindrical shell and tank subjected to axially compressive loads. Both geometrically perfect and imperfect shells and tanks are studied. Numerical analysis is used to evaluate buckling strength. The modeling method, appropriate element type and necessary number of elements to use in numerical analysis are recommended. According to the results of the parametric study of the perfect shell, the buckling strength decreases significantly as the diameter-to-thickness ratio increases, while it decreases slightly as the height-to-diameter ratio increases. These results are different from those in the case of columns. The buckling strength of the perfect tank placed on an extremely soft foundation and a stiff foundation increases by up to 1.6% and 5.6%, respectively, compared with that of the perfect shell. The buckling strength of the shell and tank decreases significantly as the amplitude of initial geometric imperfection increases. Convenient and sufficiently accurate design equations and charts used for estimating buckling strength are provided.     

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.     

Tailoring the elastic postbuckling response of thin-walled cylindrical composite shells under axial compression

The buckling of cylindrical shells has long been regarded as an undesirable phenomenon, but increasing interests on the development of active and controllable structures open new opportunities to utilize such unstable behavior. In this paper, approaches for modifying and controlling the elastic response of axially compressed laminated composite cylindrical shells in the far postbuckling regime are presented and evaluated. Three methods are explored (1) varying ply orientation and laminate stacking sequence; (2) introducing patterned material stiffness distributions; and (3) providing internal lateral constraints. Experimental data and numerical results show that the static and kinematic response of unstable mode branch switching during postbuckling response can be modified and potentially tailored.     

Simple solution for buckling of orthotropic circular cylindrical shells

Most papers dealing with the analysis of the buckling behaviour of orthotropic circular cylindrical shells provide solutions which are complex in nature and difficult to use. In this paper the theory has been developed to the point that relatively simple solutions of a general nature have been formulated. Based on Flugge's linear theory for isotropic cylindrical shells, a general buckling solution under combined axial compression and external pressure was derived. For moderate-length orthotropic cylindrical shells loaded by either external pressure or axial compression, buckling loads are formulated in a simple form.     

Numerical and experimental investigations of the response of aluminum cylinders with a cutout subject to axial compression

Understanding how a cutout influences the load bearing capacity and buckling behavior of a cylindrical shell is critical in the design of structural components used in automobiles, aircrafts, and marine applications. Numerical simulation and analysis of moderately thick and thin unstiffened aluminum cylindrical shells (D/t=45, 450 and L/D=2, 5, 10), having a square cutout, subjected to axial compression were systematically carried out in this paper. The investigation examined the influence of the cutout size, cutout location, and the shell aspect ratio (L/D) on the prebuckling, buckling, and postbuckling responses of the cylindrical shells.An experimental investigation on the moderately thick-walled shells was also carried out. A good correlation was observed between the results obtained from the finite element simulation and the experiments. Furthermore, empirical equations, in the form of a ‘buckling load reduction factor’ were developed using the least square regression method. These simple equations could be used to predict the buckling capacities of several specific types of cylindrical shells with a cutout.     

真空引起的薄壁圆筒屈曲压力预测

研究均匀外压力下初始缺陷对圆柱形薄壳结构屈曲性能的影响。对缩尺薄壁圆筒的外形进行分析以测量壳体表面的几何缺陷。有限元分析时将这些初始缺陷考虑在内,并进行静态几何非线性分析。在实验室进行圆筒的倒塌试验,并将试验结果和有限元分析结果进行比较。结果表明,有限元分析能够准确预测圆筒的破坏坍塌压力和后屈曲模态。     

Parametric study on buckling behaviour of dented short carbon steel cylindrical shell subjected to uniform axial compression

Generally, thin cylindrical shells are susceptible for geometrical imperfections like non-circularity, non-cylindricity, dents, swellings, etc. All these geometrical imperfections decrease the static buckling strength of thin cylindrical shells, but in this paper only effect of a dent on strength of a short (Lc/Rc∼1, Rc/t=117, 175, 280) cylindrical shell is considered for analysis. The dent is modeled on the FE surface of perfect cylindrical shell for different angles of inclination and sizes at half the height of cylindrical shell. The cylindrical shells with a dent are analyzed using non-linear static buckling analysis. From the results it is found that in case of shorter dents, size and angle of inclination of dents do not have much effect on static buckling strength of thin cylindrical shells, whereas in the case of long dents, size and angle of inclination of dents have significant effect. But both short and long dents reduce the static buckling strength drastically. It is also found that the reduction in buckling strength of thin cylindrical shell with a dent of same size and orientation increases with increase in shell thickness.     

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.     

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.     

Buckling and postbuckling investigations of imperfect curved stringer-stiffened composite shells. Part B: Computational investigations

Computational investigations of the buckling and postbuckling behaviour of a stringer-stiffened composite wing torsion box employing the finite element method are presented. Perfect and imperfect configurations — considering geometrical imperfections as well as initial stresses - are discussed. Two different loadcases are investigated: pure axial loading and axial loading with a superimposed constant torsion moment. The buckling behaviour is determined by tracing the load-deflection curves using Riks' path following method. Additional investigations, such as accompanying eigenvalue analyses and first-ply failure calculations, are performed for the first loadcase. Special attention is put on the modelling of the stiffened regions, where eccentrically placed layers have to be taken into account due to the bonding of the stiffeners to the inner surface of the box. The results show interesting phenomena, such as additional buckling points in the postbuckling region, which, however, can hardly be detected by simply considering the load-axial displacement path.     

Delamination buckling for composite laminated cylindrical shells in Hamilton system

In this article, a semi-analytical three-dimensional model based on the modified Hellinger–Reissner (H–R) variational principle and a nonlinear spring-layer model are presented for the buckling analysis of composite laminated cylindrical shells with a delamination. The method allows the effect of transverse shear deformation in the control equations of the composite laminated structures. In addition, it uses a two-dimensional mesh and can ensure that the number of variables is independent of the layer number. The nonlinear spring-layer model between the exterior and interior sub-laminates ensures the continuity of transverse stresses and displacements in the undelaminated region by specifying infinite values of springs and therefore avoids the possibility of material penetration phenomenon in the delaminated region. As an application of the present method, the influence of the delamination length on the critical buckling loads of delaminated composite laminated stiffened cylindrical shells is investigated.