Effect of aspect ratio on the elastic buckling of uniaxially loaded plates with eccentric holes

The Finite Element Method (FEM) has been employed to determine the elastic buckling load of uniaxially loaded rectangular perforated plates with length a and width b. Plates with simply supported edges in the out-of-plane direction and subjected to uniaxial end compression in their longitudinal direction are considered. Integer plate aspect ratios, a/b=1, 2, 3 and 4, have been chosen to assess the effect of aspect ratio on the plate buckling load. Two perforation shapes of different sizes are considered; circular, and rectangular with curved corners. The rectangular perforation is oriented such that either its long or its short side is parallel to the longitudinal direction of the plate. The center of perforation was chosen at different locations of the plate. The study shows that the buckling load of a rectangular perforated plate that could be divided into equal square panels is not the same as that of the square panel that contains the perforation when treated as a separate square plate. For rectangular plates, the study recommends not to have the center of a circular hole placed in a critical zone defined by the end half of the outer square panel, to try always to put the hole in an interior panel of the plate, and to have the distance between the edge of a circular hole and the nearest unloaded edge of the plate not less than 0.1b. The study concludes also that the use of a rectangular hole, with curved corners, with its short dimension positioned along the longitudinal direction of the plate is a better option than using a circular hole, from the plate stability point of view.     

Buckling of thin flat-walled structures by a spline finite strip method

A method of buckling analysis of thin flat-walled structures of finite length subjected to longitudinal compression and bending, transverse compression as well as shear is described. The analysis uses the spline finite strip method and allows for boundary conditions other than simply supported ends as required in the semi-analytical finite strip method of buckling analysis.Convergence studies with increasing numbers of section knots are described for plates in compression, bending and shear, and for long columns with different support conditions subjected to compression. A buckling analysis of a stiffened plate subjected to compression and shear is compared with results from a finite element analysis.     

Finite strip method for the free vibration and buckling analysis of plates with abrupt changes in thickness and complex support conditions

The free vibration problem of a stepped plate supported on non-homogeneous Winkler elastic foundation with elastically mounted masses is formulated based on Hamilton's principle. The stepped plate is modelled by finite strip method. To overcome the problem of excessive continuity of common beam vibration functions at the location of abrupt change of plate thickness, a set of C¹ continuous functions have been chosen as the longitudinal interpolation functions in the finite strip analysis. The C¹ continuous functions are obtained by augmenting the relevant beam vibration modes with piecewise cubic polynomials. As these displacement functions are built up from beam vibration modes with appropriate corrections, they possess both the advantages of fast convergence of harmonic functions as well as the appropriate order of continuity. The method is further extended to the buckling analysis of rectangular stepped plates. Numerical results also show that the method is versatile, efficient and accurate.     

Experimental verification of an inelastic plate theory based on plastic flow theory

The paper aims to verify a recently proposed inelastic plate theory by Becque (2010) [1]. In a first part, the theory is revisited and some necessary corrections to the derivation are presented.A total of 486 data points pertaining to the local buckling of aluminium and stainless steel plates and plate assemblies were collected from literature in order to be compared to the predictions of the theory. For plates with straightforward boundary conditions (either simply supported edges or free edges), the theoretical buckling load was obtained by solving Becque's differential equation analytically or numerically. For plate assemblies with less obvious boundary conditions, including plain channels, lipped cannels, Z- and H-sections, the buckling load was obtained by employing an inelastic semi-analytical finite strip method based on Becque's differential equation. Including all data points, the average ratio of the predicted buckling load to the experimentally measured buckling load was 0.98 with a standard deviation of 0.069, thus confirming the proposed theory.     

Elastic local buckling of perforated webs of steel cellular beam-column elements

In this study, the finite element method is employed to determine the critical in-plane longitudinal load at which elastic local buckling of the web of cellular beam-column elements occurs. To simplify the simulation of the problem, the interaction between the flanges and perforated web is approximated by modelling the web only as a long plate having aspect ratio (L/h_w≥10) with multiple circular perforations. The utilized model incorporates restrained out-of-plane displacements along the four edges of the plate. Analyzed plates are subjected to linearly varying in-plane loads to simulate various combinations of axial and flexural stresses. The effect of different geometrical parameters on the elastic buckling load of perforated web plate is investigated. These geometrical parameters include the plate’s length and width, and the perforations’ diameter and spacing. Comprehensive finite element analyses are conducted to identify the behaviour of wide spectrum of perforated web plates at buckling under various combinations of axial compressive load and bending moment. Outcomes of the study are expected to enhance the understanding of the elastic local buckling of web plates of cellular beam-column elements.     

Vibration and buckling of thin-Walled structures by a new finite strip

This paper presents the application of a new finite strip to the analysis of folded-plate structures. The displacement function of a flat shell finite strip is made up of two parts, namely, the two in-plane displacement interpolations and the out-of-plane displacement interpolation. Each of the three displacement components is interpolated by a set of computed shape functions in the longitudinal direction and, as usual, one-dimensional shape functions in the transverse direction. Only standard beam shape functions are involved in the longitudinal computed shape functions. When compared with other finite strips, the present finite strip is relatively simple in dealing with boundary and internal support conditions. In addition, the method can be easily implemented by incorporating a standard finite strip program with a continuous-beam program. The computation of the stiffness matrix involves no numerical integration. To verify the accuracy and efficiency of the new finite strip, a few numerical experiments are conducted in which the present finite strip results are compared with those using other finite strips and/or finite elements for the vibration and buckling of folded-plate structures with varying complexity.     

A finite strip method for the elastic—plastic large dlisplacement analysis of thin-walled and cold-formed steel sections

A large deflection elastic—plastic analysis has been developed using the finite strip method of structural analysis to determine the non-linear local buckling behaviour of thin-walled and cold-formed sections in compression. The analysis accounts for plate geometric imperfections, the variation of yield stress around a section, the stress—strain characteristics of the material forming the section and complex patterns of residual stress produced by the cold-forming process.

The analysis is verified against reliable solutions for the non-linear buckling behaviour of plates and plate assemblies in axial compression and the non-linear overall buckling behaviour of a strut. The analysis is further compared with the results of plates with a rounded stress—strain curve typical of cold-formed steel and aluminium.     

The stability of eccentrically compressed thin plates with a longitudinal free edge and with stress variation in the longitudinal direction

This paper presents results of the investigation of the stability of eccentrically compressed thin plates with one longitudinal free edge with the participation of loadings generating the variation of stresses in the direction of the length of a plate. A deflection function (5) is proposed to enable modelling of boundary conditions on the second longitudinal edge from a simply supported, through elastically restrained, to a built-in edge. Formulas are derived for work done by external force under loading generating the variation of stress according to a linear function and according to parabola 2°. Tables and plots of buckling coefficient (k) for variously supported and variously loaded plates with a free edge, which are not found in the literature, are determined. The effect of a longitudinal stress variation on the plate buckling modes is analyzed.     

Buckling optimization of variable thickness prismatic folded plates

This paper deals with the structural shape optimization of prismatic folded plates under buckling load consideration. Buckling loads are determined using linear, quadratic and cubic, variable thickness, C(0) continuity, Mindlin-Reissner finite strips. The whole structural optimization process is carried out by integrating finite strip analysis, cubic spline shape and thickness definition, semi analytical sensitivity analysis and mathematical programming algorithm. The objective is either the maximization of the critical buckling load or minimization of the cross-section of the prismatic folded plate with constraints on the volume and buckling loads. Several examples are included to illustrate various features of the optimization algorithm, including plates and stiffened panels.     

Finite strip elastic buckling solutions for thin-walled metal columns with perforation patterns

Approximate finite strip eigen-buckling solutions are introduced for local, distortional, flexural, and flexural-torsional elastic buckling of a thin-walled metal column with perforation patterns. These methods are developed to support a calculation-based strength prediction approach for steel pallet rack columns employing the American Iron and Steel Institute׳s Direct Strength Method, however they are generally posed and could also be useful in structural studies of thin-walled thermal or acoustical members made of steel, aluminum, or other metals. The critical elastic global buckling load including perforations is calculated by reducing the finite strip buckling load of the cross-section without perforations using the weighted average of the net and gross cross-sectional moment of inertia along the length of the member for flexural (Euler) buckling, and for flexural-torsional buckling, using the weighted average of both the torsional warping and St. Venant torsional constants. For local buckling, a Rayleigh–Ritz energy solution leads to a reduced thickness stiffened element equation that simulates the influence of decreased longitudinal and transverse plate bending stiffness caused by perforation patterns. The cross-section with these reduced thicknesses is input into a finite strip analysis program to calculate the critical elastic local buckling load. Local buckling at a perforation is also treated with a net section finite strip analysis. For distortional buckling, a reduced thickness equation is derived for the web of an open cross-section to simulate the reduction in its transverse bending stiffness caused by perforation patterns. The approximate elastic buckling methods are validated with a database of 1282 thin shell finite element eigen-buckling models considering five common pallet rack cross-sections featuring web perforations that include 36 perforation dimension combinations and twelve perforation spacing combinations.     

Post-local-buckling of fiber-reinforced plastic composite structural shapes using discrete plate analysis

In this paper, post-buckling of rectangular composite plates rotationally restrained at the longitudinal unloaded edges and subjected to end shortening strain at the simply-supported loaded edges is analyzed using the first-order shear deformation plate theory-based spline finite strip method, and its application to post-local-buckling of fiber-reinforced plastic (FRP) composite structural shapes is illustrated with discrete plate analysis. Two cases of elastically- and rotationally-restrained plates are analyzed using the spline finite strip method: rotationally-restrained along both the unloaded boundary edges (RR) and one rotationally-restrained and the other free along the unloaded edges (RF). The two cases of rotationally-restrained plates (i.e., the RR and RF plates) are further treated as the discrete plates of closed and open section FRP shapes, and by considering the effect of elastic restraints at the joint connections of flanges and webs, post-local-buckling of various FRP shapes under end shortening is studied. The numerical comparisons with the finite element modeling demonstrate that the proposed discrete plate analysis technique and spline finite strip method can be used as an efficient and valid tool for post-local-buckling analysis of FRP shapes.     

Ultimate strength of stiffened aluminium panels with predominantly torsional failure modes

The aim of this paper is to investigate the ultimate strength of aluminium plates with flatbar stiffeners with a torsional buckling or tripping failure mode. The formulations for torsional buckling of stiffeners in steel plating are still debated. Compared with steel structures, the ultimate strength of aluminium structures is sensitive not only to residual stresses and initial deformations, but also to the deterioration of mechanical strength in heat-affected zones (HAZ). In the present paper, the ultimate strength of stiffened aluminium panels with predominantly torsional failure modes is investigated by experimental and theoretical analysis. Stiffened panels made of the aluminium alloy AA5083-H116 and AA6082-T6 are considered. Various height of flatbar and various thickness of plate and stiffener were studied. The test results are compared with numerical predictions by using the finite element code ABAQUS (ABAQUS Version 5.7 (1997)), considering the influence of initial deflections, welding residual stresses and HAZ. The influence of HAZ and residual stresses on the ultimate strength of stiffened aluminium panels with the actual failure mode is discussed in detail. The numerical predictions are also compared with strength of material formulations used in DNV Rules for Classification of High Speed and Light Craft (Rules for classification of high speed and light craft, Hull structural design (1996)), NORSOK (Design of steel structures (1998)) all for steel, using the relevant values of the modulus elasticity and yield strength of aluminium, as well as EUROCODE 9 (Eurocode 9, Part 1-1: General rules (1998)).     

Buckling analysis of thin-walled cold-formed steel structural members using complex finite strip method

In this paper, a generalised complex finite strip method is proposed for buckling analysis of thin-walled cold-formed steel structures. The main advantage of this method over the ordinary finite strip method is that it can handle the shear effects due to the use of complex functions. In addition, distortional buckling as well as all other buckling modes of cold-formed steel sections like local and global modes can be investigated by the suggested complex finite strip method. A combination of general loading including bending, compression, shear and transverse compression forces is considered in the analytical model. For validation purposes, the results are compared with those obtained by the Generalized Beam Theory analysis. In order to illustrate the capabilities of complex finite strip method in modelling the buckling behavior of cold-formed steel structures, a number of case studies with different applications are presented. The studies are on both stiffened and unstiffened cold-formed steel members.     

Local buckling of steel plates in concrete-filled thin-walled steel tubular beam-columns

The availability of high strength steels and concrete leads to the use of thin steel plates in concrete-filled steel tubular beam-columns. However, the use of thin steel plates in composite beam-columns gives a rise to local buckling that would appreciably reduce the strength and ductility performance of the members. This paper studies the critical local and post-local buckling behavior of steel plates in concrete-filled thin-walled steel tubular beam-columns by using the finite element analysis method. Geometric and material nonlinear analyses are performed to investigate the critical local and post-local buckling strengths of steel plates under compression and in-plane bending. Initial geometric imperfections and residual stresses presented in steel plates, material yielding and strain hardening are taken into account in the nonlinear analysis. Based on the results obtained from the nonlinear finite element analyses, a set of design formulas are proposed for determining the critical local buckling and ultimate strengths of steel plates in concrete-filled steel tubular beam-columns. In addition, effective width formulas are developed for the ultimate strength design of clamped steel plates under non-uniform compression. The accuracy of the proposed design formulas is established by comparisons with available solutions. The proposed design formulas can be used directly in the design of composite beam-columns and adopted in the advanced analysis of concrete-filled thin-walled steel tubular beam-columns to account for local buckling effects.     

Strengthening of perforated plates under uniaxial compression: Buckling analysis

This paper focuses on the cutout-strengthening of perforated steel plates subjected to uniaxial compressive loads. The square plates considered each has a centrally placed circular hole and four simply supported edges in the out-of-plane direction. Four types of stiffeners named ringed stiffener (RS), flat stiffener (FS), longitudinal stiffener (LS) and transverse stiffener (TS) are mainly discussed. The finite element method (FEM) has been employed to analyse the elastic and elasto-plastic buckling behaviors of strengthened and unstrengthened perforated plates. The results show that the strengthened perforated plates have higher buckling strengths than those of the unstrengthened ones, while the elevations in elastic buckling stress and elasto-plastic ultimate strength are closely related to stiffener types (i.e., RS, FS, LS and TS) as well as plate geometric parameters (i.e., a plate slenderness ratio and a hole diameter ratio). Furthermore, comparisons of strengthening efficiency considering the variations of buckling stress with stiffener weight are carried out, and recommendations on the most efficient cutout-strengthening methods for the uniaxially compressed perforated square plates with centric circular holes are proposed.     

Local buckling of grouted and ungrouted internally stiffened double-plate HPS webs

High performance steel (HPS) is rapidly gaining attention as a desirable material for highway bridge girders largely due to its superior toughness properties and high strength. However, the benefits of using steels with nominal yield strengths of 485 MPa (70 ksi) or greater is restricted by factors such as web stability, deflection, and fatigue design limits, which may govern the design and prevent the effective utilization of the material strength. Therefore, new and innovative bridge design concepts are needed to take better advantage of the enhanced properties of HPS. One design innovation that provides a means of optimizing bridge girders for high strength material utilizes I-girders with double web plates. The web is composed of two steel face plates connected internally by continuous longitudinal stiffening elements. The voids between the face plates may be grouted or ungrouted. The stiffeners permit thin webs to be used, while still allowing the material to reach stresses as high as the yield strength without buckling. In the case of grouted webs, composite behavior increases the out-of-plane stiffness of the web, although bond between the two materials may be unreliable. Nevertheless, it is shown that even in a debonded state, the presence of the grout enhances the buckling capacity of the face plates significantly. Using classical plate buckling theory, design criteria are proposed for bend buckling of the web face plates, considering both the grouted and ungrouted cases. As a means of assessing the anticipated behavior of the plates, upper and lower bounds to the buckling strengths are established. In order to evaluate the ability of classical plate theory to predict the buckling of the face plates, tests were conducted on a series of web panels that simulate a portion of a girder web subjected to flexural compressive stresses. Two of the specimens were ungrouted, two were grouted with a cementitious material, and one was grouted with an epoxy grout. It was confirmed that the presence of grout increased the buckling capacity of the face plates and that the improved bond using epoxy grout served to delay buckling as well, although when the bond broke the failure was sudden. The experimentally determined buckling loads are used to validate the theory.     

Buckling analysis of thin-walled sections under general loading conditions

This paper presents an investigation of the buckling behaviour of thin-walled sections subjected to general loading conditions. The semi-analytical finite strip method is used. The existing results are only for sections subjected to a uniform loading, namely: uniform compression, uniform bending and uniform distributed loads, which are applied at the shear centre. For a general loading condition, we proposed the realizing linear analysis first to give longitudinal stresses. The stiffness matrix is provided in the standard manner. Each strip is divided into cells and longitudinal stresses are recorded in these cells. The integrations are performed on each cell domain and the sum of them provides the geometric matrix of the strip.     

基于扁壳理论的半解析有限条法用于冷弯构件的屈曲分析

提出了一种基于Marguerre的扁壳理论的有限条法。现有的大多数半解析有限条法都是基于基尔霍夫和Mindlin的板理论。在本文中,该板理论可以被视为扁壳理论的特殊情况。数值分析结果证实了基于扁壳理论的有限条法的有效性。     

Elastic/plastic buckling of a composite flat plate subjected to uniform edge compression

A theoretical approach, based on the plastic theory, has been developed for studying the elastic/plastic buckling behavior of a simply supported rectangular anisotropic plate subjected to edge compression. The said approach was employed to determine the critical buckling stresses of three different types of fiber reinforced composite plates, namely, carbon epoxy, glass epoxy and boron aluminum plates. Since the proposed approach is for a preliminary prediction of critical buckling loads of composite plates, the results obtained are deemed acceptable compared with those obtained by other authors experimentally.     

Buckling of cracked functionally graded plates under tension

Buckling of functionally graded cracked plates under tension has not been investigated so far. In this paper critical buckling load of functionally graded plates containing a crack has been obtained using classical plate theory through the finite element method. Displacement in vicinity of crack tips has been approximated using previous solutions related to bending of cracked plates. Effect on buckling of plate under uni-axial and bi-axial tension of different parameters, such as plate dimensions and material properties, are studied. Results show that the critical load decreases as material gradient index increases, while bi-axial loading leads to higher critical loads compared to uni-axial case.