Buckling and postbuckling behavior of unstiffened slender curved plates under uniform shear

Buckling and postbuckling behavior of curved plates under in-plane shear are investigated. After revisiting classic elastic buckling results, the elastoplastic postbuckling behavior and the effects of curvature parameter and aspect ratio are simulated via geometrical and material nonlinear analyses. Imperfection sensitivity is studied for various imperfection shapes and magnitudes. An increase in curvature parameter raises the elastic buckling load, produces unstable buckling and reduces postbuckling reserves. The buckling load and shear capacity are higher in shorter plates. Small initial imperfections are found to have severe effects on the initial buckling load of plates with large curvature parameter, but little effect on ultimate postbuckling capacity.     

Imperfections in steel girder webs with and without perforations under patch loading

In this paper, the problem of stability of web plates with imperfections, subjected to patch load, is studied. The aim is to give some insights about the best way to take into account real imperfections in non-linear stability analyses of plates with and without perforations. In this context, the study is developed on the basis of the measured imperfections or numerically deriving the deformed model (theoretical imperfections). The influence of the patch load length, out-of-plane imperfection amplitude, dimension and position of the hole on stability behaviour and buckling strength are studied comparing some theoretical deformed configurations corresponding to different modal shapes. The results obtained with a three-dimensional model of the whole real beam with stiffeners, with experimentally measured imperfections, and each corresponding single web panel are compared and discussed obtaining some insights about the accuracy of the simplified (and conservative) model of the single panel.The main insights of this work are as follows. The deformed shape, corresponding to the first buckling mode, can be assumed as the initial configuration of the panels with and without holes to study post-critical behaviour until ultimate condition. The shape of the imperfection does not severely change the critical buckling stress. A longer patch load reduces the ultimate stress in the panel. An initial imperfection amplitude of less than 1% of the height of the panel does not reduce the ultimate load by more than about 5%.     

Shear resistance of longitudinally stiffened panels—Part 1: Tests and numerical analysis of imperfections

This paper deals with the results of four full-scale tests, numerical simulation of tests and initial geometric imperfection analysis for longitudinally stiffened panels in shear. The tests examine the influence of varying position and bending stiffness of one trapezoidal longitudinal stiffener on the panel shear resistance and its buckling behaviour. The stiffeners were designed such as to obtain both global and local buckling shapes. Numerical simulations (FEA), based on the test girder geometry, the measured initial geometric imperfections and elastic-plastic material characteristic from the tensile tests, demonstrate a very good agreement with the tests. The initial geometric imperfection study on different verified numerical models shows a limited sensitivity of the panel shear capacity to any kind of imperfection shape variation with amplitude at the allowable fabrication tolerances. Finally, the paper offers some ideas for modelling geometric imperfections with regard to the design or research demands.     

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.     

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.     

Degradation of the compression strength of square plates due to initial deflection

The influence of initial deflections on collapse strength of thin square plates in uniaxial in-plane compression is studied emphasising an integral energy measure of the imperfections. In calculations of the plate strength, measured data on distortions of stiffened plating in ships as well as buckling modes, compound and localised shapes are considered. The strength values are compared normalising the initial deflections by the energy measure or by the commonly employed amplitude to thickness ratio. The paper extends the previous study on strength of rectangular plates [Sadovský Z, Teixeira AP, Guedes Soares C. Degradation of the compression strength of rectangular plates due to initial deflection. Thin-Walled Structures 2005;43:65-82].     

Experimental and Numerical Study on Buckling Behavior of a Rigidly Stiffened Plate with Tee Ribs

In the bridge structures, stiffened plates are usually designed as rigidly stiffened when the orthotropic steel box girder is used as the main load-bearing structure. Therefore, the buckling mode of stiffened plates is plate buckling which occurs in subpanel supported by stiffeners. The orthotropic steel box girder is used as the main girder for Egongyan Rail Special Bridge, which is a self-anchored suspension bridge. Plates of the steel girder are rigidly stiffened with unequal spacing open ribs, and the most slender stiffened plate is the mid web stiffened with Tee ribs. In order to ensure the safety of the bridge, the buckling behavior of the web and orthotropic steel box girder under axial compression, including ultimate strength, post-buckling behavior and failure modes, should be clearly investigated by experimental and numerical methods. The design, loading and testing methods of the 1:4 scale model of the orthotropic steel box girder are introduced in detail firstly. The orthotropic steel box girder and the stiffened web finite element (FE) models are validated by the test results, and the effects of residual stress and the magnitude of geometric imperfections are discussed roughly. Based on the validated web FE model, a detailed parametric study is performed to systematically investigate the effects of residual stress and geometric imperfections on buckling behavior of the web. The effect of shapes of geometric imperfections discussed is highlighted. Through tracing stress states, the failure modes of stiffened plate are in agreement with the experimental phenomenon to some extent. Results show that shapes of geometric imperfections have significantly influenced post-buckling behavior and failure modes of the web, but slightly affected the ultimate strength. It is advised that residual stress and geometric imperfections should be controlled to make full use of excellent performance of steel materials.     

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.     

Dynamic buckling of fiber composite shells under impulsive axial compression

The paper deals with dynamic buckling due to impulsive loading of thin-walled carbon fiber reinforced plastics (CFRP) shell structures under axial compression. The approach adopted is based on the equations of motion, which are numerically solved using a finite element code (ABAQUS/Explicit) and using numerical models validated by experimental static buckling tests. To study the influence of the load duration, the time history of impulsive loading is varied and the corresponding dynamic buckling loads are related to the quasi-static buckling loads. To analyse the sensitivity to geometric imperfections, the initial geometric imperfections, measured experimentally on the internal surface of real shells, are introduced in the numerical models. It is shown numerically that the initial geometric imperfections as well as the duration of the loading period have a great influence on the dynamic buckling of the shells. For short time duration, the dynamic buckling loads are larger than the static ones. By increasing the load duration, the dynamic buckling loads decrease quickly and get significantly smaller than the static loads. Since the common practice is to assume that dynamic bucking loads are higher than the static ones, which means that static design is safe, careful design is recommended. Indeed, taking the static buckling load as the design point for dynamic problems might be misleading.     

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.     

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.     

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。     

Computational modelling of geometric imperfections and buckling strength of cold-formed steel

Computational modelling of the buckling strength of cold-formed steel members as influenced by initial geometric imperfections is studied. The geometric imperfections are represented by the member eigenmode shapes. Along with the classical measure — the amplitude of imperfections, an energy measure defined by the square root of the elastic strain energy hypothetically required to distort the originally perfect structural element into the considered imperfect shape is used. Based on the measures, two approaches for the choice of the most unfavourable imperfections are suggested. Normalising imperfections by the amplitude, the energy measure is calculated as indicative parameter of imperfection significance. Vice versa, when adopting normalisation by the energy measure, the amplitude is used as a supporting parameter. The suggestions are illustrated on calculating the strength of an axially compressed steel lipped channel column with eigenmodes exhibiting local-distortional interactions. For eigenvalue and geometrically and materially non-linear strength calculations, the FEM codes MSC.NASTRAN and COSMOS/M are employed.     

锈蚀箱梁极限强度试验中板变形量的分析

根据垂直弯曲荷载下3个锈蚀箱梁板的压应力试验结果,分析初始和倒塌后的板变形量。研究初始缺陷和腐蚀对倒塌后最终变形的影响,得出不同的荷载响应、初始缺陷和板高厚比之间的关系。分析初始缺陷、板高厚比和倒塌后最终变形,建立高厚比准则以计算倒塌后变形。     

Analysis of plate deflections during ultimate strength experiments of corroded box girders

The objective of this work is to analyze the initial and post-collapse plate deflections based on measurement records of the experiments of three corroded box girders subjected to pure vertical bending loading inducing a compressive stress on deck. The effect of initial imperfections and corrosion degradation on the final post-collapse deformation shape has been investigated and a relationship between different loading responses, shape of initial imperfections and plate slenderness has been derived. Analyzing initial imperfections, plate slenderness and final post-collapse deformations, a slenderness criterion has been established to predict the post-collapse deformation shape.     

Asymmetric interactive buckling of thin-walled columns with initial imperfections

In this paper the effect of the interaction between two or more simultaneous buckling modes on the postbuckling behaviour of uniformly compressed thin-walled members (TWM) is analysed by means of the general theory of elastic stability. The analysis is restricted to third-order terms of the energy expansion and therefore can be fruitfully applied to the investigation of structures with asymmetric postbuckling behaviour only. Initial imperfection effect is taken into account. A simplified procedure is suggested for solving the nonlinear equations relative to the evaluation of the bifurcated paths. By using the finite strip method an extensive parametric analysis is performed. It is found that when the flexural-torsional (FT) buckling interacts with a local symmetric and antisymmetric mode, sensitivity to initial inperfections is remarkable and is comparable to the one arising from the interaction between the Euler (E) and any local buckling.     

Dynamic buckling of thin-walled viscoplastic columns

The purpose of this paper is the analysis of strain-rate effect in dynamic stability of thin-walled orthotropic column of closed rectangular cross-section, subjected to in-plane pulse loading of finite duration. For the solution the first-order shear deformation theory displacement field is employed with the Green–Lagrange strain tensor application. The effect of strain rate sensitivity is included in the framework of the viscoplasticity constitutive Perzyna model for material behaviour under high strain rate loading. The numerical results are obtained with the finite element method application. In the performed analysis the strain-rate effect influence on the dynamic buckling load is examined as well as the initial imperfections of walls, pulse shape and the orthotropy ratio are considered. The results of dynamic criteria application are compared furthermore.     

Buckling of plates with a hole under tension

An FEM analysis is made on the elastic buckling of plates, each of which has a hole and is subjected to tensile loading.In a general way, no attention is paid to the buckling of plates if they are subjected to a tensile load. However, when a plate has a hole, compressive stresses appear near the hole under a tensile load, and the stress may cause local buckling of the plate.In this paper, stress distributions and buckling behaviours of such plates under tension are studied. Aspect ratios, shapes of holes and so on, are adopted as parameters.Through the analysis, variations of buckling coefficients and buckling modes against aspect ratios are obtained. The effects of the hole shapes on the buckling strength are also discussed.     

Buckling modes identification from FEA of thin-walled members using only GBT cross-sectional deformation modes

This paper presents the latest developments of an original method based on Generalized Beam Theory (GBT) capable to identify the fundamental deformation modes of global, distorsional or local nature, in general buckling modes provided by the shell finite element analysis (FEA) of isotropic thin-walled members. This method has the advantage of using only the GBT cross-sectional deformation modes instead of the member base mode shapes. The participation of each fundamental buckling mode can be calculated, allowing an accurate and quantitative evaluation of the coupled instability. There are no restrictions regarding the element cross-sectional shape, loading and quite recently discovered, boundary conditions.     

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

A new approach is extended to investigate the buckling and postbuckling behaviour of perfect and imperfect, stringer and ring stiffened cylindrical shells of finite length subject to combined loading of external pressure and axial compression. The formulations are based on a boundary layer theory which includes the edge effect in the postbuckling analysis of a thin shell. The analysis uses a singular perturbation technique to determine the buckling loads and the postbuckling equilibrium paths. Some interaction curves for perfect and imperfect stiffened cylindrical shells are given and compared well with experimental data. The effects of initial imperfection on the interactive buckling load and postbuckling behaviour of stiffened cylindrical shells have also been discussed.