Experimental study of inelastic buckling strength and stiffness requirements for longitudinally stiffened panels

An experimental study was conducted in order to verify a proposed equation for the minimum required stiffness for longitudinal stiffeners attached on compression plates. Nine test specimens of stiffened plates were fabricated and tested to their ultimate strengths. Test results are compared with the maximum strengths predicted for stiffened plates by finite element analysis and by the proposed SSRC type critical stress curve. Fairly good correlations were observed, thereby confirming experimentally the adequacy of the proposed equation for the minimum required stiffness of the longitudinal stiffener attached to the compression panel and a modified SSRC type critical stress curve encompassing both the buckling ranges of plastic yield and the transition zones. A set of conclusions is drawn from the experimental studies as to the buckling behaviors of stiffened plates.     

Compression tests of longitudinally stiffened plates undergoing distortional buckling

This paper describes a series of compression tests performed on longitudinally stiffened plates fabricated from a mild steel plate of thickness of 4.0 mm with nominal yield stress of 235.0 MPa. The stiffened plates with longitudinal stiffeners of a range of rigidities were tested to failure. The ultimate strengths and performances of the longitudinally stiffened plates in compression undergoing distortional buckling or interaction between local and distortional buckling were investigated experimentally and theoretically. The compression tests indicated that the critical buckling mode was dependent mainly on the rigidity of the longitudinal stiffeners and the width-to-thickness ratio of the sub-panels. A noticeable interaction between local and distortional buckling was also observed for some stiffened plates. A significant post-buckling strength reserve was shown for those sections with distortional buckling and for those sections showing interaction between local and distortional buckling. A limiting strength curve for distortional buckling of longitudinally stiffened plates was studied. Simple design strength formulas in the direct strength method are proposed to account for the distortional buckling and the interaction between local and distortional buckling of longitudinally stiffened plates. The strength curves were compared with the test and FE results conducted. The adequacy of the strength curve was confirmed. A set of conclusions on the buckling behavior of longitudinally stiffened plates was drawn from the experimental studies.     

Torsional buckling behaviour of stiffened cylinders under combined loading

In this study cylindrical boundary conditions for finite element analysis are formulated that allow torsional displacement and buckling of a sector of a cylinder of half axial height, and of a circumferential arc angle that will divide into 360°. Finite element tests are carried out on un-stiffened elastic cylinders to verify the method of analysis against classical elastic torsional buckling theory.Elastic–plastic limit point finite element tests are carried out on ring and stringer stiffened and stringer stiffened cylinders to investigate the effects of stiffeners on post-buckling behaviour in torsion.A stringer stiffened cylinder is subjected to many combinations of axial force and surface pressure in the elastic range of response and then tested to failure in torsion to investigate the effects of axial and surface pressure loads on the resistance to plastic collapse in torsion.     

On the buckling strength of stiffened elliptic paraboloidal steel panels

This paper reports the results of a numerical study undertaken on the buckling behaviour of lightly stiffened elliptic paraboloidal steel panels intended for use as long-span shuttering for lightweight concrete bridge decks, walkways and floors. Steel panel shutters of double curvature may allow the casting of concrete over relatively large spans while avoiding the use of supporting scaffolding and other intermediate props. Such long-span shutters are desirable when the ground below the deck cannot adequately support scaffolding, or the space below the deck carries traffic carriageways which should not be obstructed by scaffolding. In this study, the effect of the wet concrete is simulated as a uniform pressure normal to the shell surface (limiting case), while the dead weight of the hardening concrete is simulated as a uniformly distributed loading on the horizontal projection of the shell surface. The results show that panel rise and aspect ratio have a considerable influence on the buckling strength of the shutter. Tentative viability limits are established.     

Improved prediction of simultaneous local and overall buckling of stiffened panels

In this paper, improved expressions for elastic local plate buckling and overall panel buckling of uniaxially compressed T-stiffened panels are developed and validated with 55 ABAQUS eigenvalue buckling analyses of a wide range of typical panel geometries. These two expressions are equated to derive a new expression for the rigidity ratio (EI_x/Db)_CO that uniquely identifies “crossover” panels—those for which local and overall buckling stresses are the same. The new expression for (EI_x/Db)_CO is also validated using the 55 FE models. Earlier work by Chen (Ultimate strength analysis of stiffened panels using a beam-column method. PhD Dissertation, Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, 2003) had produced a new step-by-step beam-column method for predicting stiffener-induced compressive collapse of stiffened panels. An alternative approach is to use orthotropic plate theory. As part of the validation of the new beam-column method, ABAQUS elasto-plastic Riks ultimate strength analyses were made for 107 stiffened panels—the 55 crossover panels and 52 others. The beam-column and orthotropic approaches were also used. A surprising result was that the orthotropic approach has a large error for crossover panels whereas the beam-column method does not. Some possible reasons for this are suggested.     

The computational post buckling analysis of fuselage stiffened panels loaded in compression

Fuselage panels are commonly fabricated as skin–stringer constructions, which are permitted to locally buckle under normal flight loads. The current analysis methodologies used to determine the post buckling response behaviour of stiffened panels relies on applying simplifying assumptions with semi-empirical/empirical data. Using the finite element method and employing non-linear material and geometric analysis procedures, it is possible to model the post buckling behaviour of stiffened panels without having to place the same emphases on simplifying assumptions or empirical data. Investigation of element, mesh, idealisation, imperfection and solution procedure selection has been undertaken, with results validated against mechanical tests. The research undertaken has demonstrated that using a commercial implicit code, the finite element method can be used successfully to model the post buckling behaviour of flat riveted panels. The work has generated a series of guidelines for the non-linear computational analysis of flat riveted panels subjected to uniform axial compression.     

The buckling characteristics of some integrally formed bead stiffened composite panels

Composite panel stability can easily be improved by using vertical male beads. In this paper, new methods of stabilizing techniques used for the panels, webs and ribs of composite structures are studied. A parametric study is performed to assess the effects of important design considerations such as, bead length, number of beads, bead radius, bead depth and bead spacing on the initial buckling load of the panels. The results show that, there is an optimum bead spacing for each panel containing more than one bead which can be estimated using a simple equation. Integration of vertical beads with a length of less than 0.5 times the panel's length has no significant effect on the buckling load. There are no significant changes on the buckling loads of the beaded panels with bead depths greater than 0.6 times the bead radius. In this investigation, the instability of the nose and main ribs of a light airplane wing structure made of woven E-glass material and stiffened by P.V.C foam core and vertical male beads are also studied using experimental methods. The experimental results show that we can easily improve the buckling capability of the panels and webs by using vertical male beads instead of sandwiched construction. It is estimated that this would cause a weight reduction of about 50% and a manufacturing time reduction of about 50%.     

Shear resistance of longitudinally stiffened panels — Part 2: Numerical parametric study

This paper presents the FE analysis of the influence of different parameters on the shear resistance of panels with different arrangements of longitudinal stiffeners. The studied parameters were the stiffener bending stiffness, the panel aspect ratio, the stiffener position, the web slenderness and the flange rigidity. Longitudinal stiffeners of trapezoidal shape were compared to open T-stiffeners. The former proved to be more efficient, since a larger panel resistance is achieved, for which in addition a smaller stiffness of trapezoidal stiffeners is needed. Different features of the new Eurocode rules were verified against the FEA results as well. Three different procedures for the determination of panel slenderness were tested and the reduction of stiffener bending stiffness, prescribed due to a better correlation with tests on open stiffeners, was verified for both closed and open stiffeners. The influence of bending moment was also considered and the verification of shear and bending interaction was discussed. Finally, the flange contribution to shear resistance was critically analysed.     

A semi-analytical model for global buckling and postbuckling analysis of stiffened panels

A computational model for global buckling and postbuckling analysis of stiffened panels is derived. The loads considered are biaxial in-plane compression or tension, shear, and lateral pressure. Deflections are assumed in the form of trigonometric function series, and the principle of stationary potential energy is used for deriving the equilibrium equations. Lateral pressure is accounted for by taking the deflection as a combination of a clamped and a simply supported deflection mode. The global buckling model is based on Marguerre’s nonlinear plate theory, by deriving a set of anisotropic stiffness coefficients to account for the plate stiffening. Local buckling is treated in a separate local model developed previously. The anisotropic stiffness coefficients used in the global model are derived from the local analysis. Together, the two models provide a tool for buckling assessment of stiffened panels. Implemented in the computer code PULS, developed at Det Norske Veritas, local and global stresses are combined in an incremental procedure. Ultimate limit state estimates for design are obtained by calculating the stresses at certain critical points, and using the onset of yielding due to membrane stress as the limiting criterion.     

Analytical calculation of local buckling and post-buckling behavior of isotropic and orthotropic stiffened panels

A methodology for the analytical assessment of local buckling and post-buckling behavior of isotropic and orthotropic stiffened plates is presented. The approach considers the stiffened panel segment located between two stiffeners, while the remaining panel is replaced by equivalent transverse and rotational springs of varying stiffness, which act as elastic edge supports. A two-dimensional Ritz displacement function (pb-2 Ritz) is utilized in the solution of the local buckling problem of isotropic and laminated symmetric composite panels with arbitrary edge boundary conditions. The buckling analysis of the segment provides an accurate and conservative prediction of the panel local buckling behavior. Consequently, the developed methodology is extended in the prediction of the post-buckling response of stiffened panels of which the skin has undergone local buckling. Of high importance for the calculation of the post-buckling behavior is the selection of appropriate boundary conditions for the structural members analyzed. A comparison of the present methodology results to respective finite element (FE) results has shown a satisfactory agreement.     

Local buckling and slenderness limits for flange outstands at elevated temperatures

This paper presents a study of the inelastic local buckling of flange outstands in thin-walled steel beams at elevated temperatures. A spline finite strip method of buckling analysis is modified in order to model the properties of the steel section at elevated temperatures, so that slenderness limits can be proposed. These limits are needed in describing the behaviour of members under fire loading, since the development of benign catenary action in a restrained beam that has received widespread research attention in fire engineering design is reliant on the formation of plastic hinges, and of the influence of local buckling on the formation of these hinges. A sensitivity analysis is undertaken in order to propose plasticity and yield slenderness limits at elevated temperature, and it is shown that these limits depend on the parameters describing the uniaxial stress-strain relationship at the elevated temperature. The local buckling temperature is predicted for two different boundary conditions for a flange outstand, and it is shown that at high temperatures the local buckling of the beam is highly significant in accelerating the favourable development of catenary action.     

A simplified method for elastic large deflection analysis of plates and stiffened panels due to local buckling

A computational model for analysis of local buckling and postbuckling of stiffened panels is derived. The model provides a tool that is more accurate than existing design codes, and more efficient than nonlinear finite element methods. Any combination of biaxial in-plane compression or tension, shear, and lateral pressure may be analysed. Deflections are assumed in the form of trigonometric function series. The deformations are coupled such that continuity of rotation between the plate and the stiffener web is ensured, as well as longitudinal continuity of displacement. The response history is traced using energy principles and perturbation theory. The procedure is semi-analytical in the sense that all energy formulations are derived analytically, while a numerical method is used for solving the resulting set of equations, and for incrementation of the solution. The stress in certain critical points are checked using the von Mises yield criterion, and the onset of yielding is taken as an estimate of ultimate strength for design purposes.     

Numerical investigation of longitudinally stiffened web channels predominantly in shear

In practice, there is a wide variety of commercially available channel sections with complex shapes where the web is stiffened by adding longitudinal intermediate stiffeners. These stiffeners may improve the shear capacity of the channels. Recently, the Direct Strength Method (DSM) of design of cold-formed sections has been extended in the North American Specification for Cold-Formed Steel Structural Members (NAS S100:2012) to include shear based on research by the authors. The prequalified sections include flat webs and webs with small intermediate longitudinal stiffeners. To extend the range to larger intermediate stiffeners as occurs in practice, a series of predominantly shear tests of lipped channel sections with one web stiffener of different shapes and various sizes has been performed at the University of Sydney. Six different types of stiffened web channel sections were tested along with an additional reference plain section. All tests were conducted with straps screwed on the top flanges adjacent to the loading points. These straps provide torsion/distortion restraints which may enhance the shear capacities of the sections. The test failures were observed mainly in the combined bending and shear modes. Numerical simulations based on the Finite Element Method (FEM) using the software package ABAQUS/Standard are also performed to compare with and calibrate against the tests. The accurate results from the FEM models allow extension of the test data. Based on the reliable FEM models, a series of FEM modelling of predominantly shear tests for stiffened web channels has been performed without straps attached to the top flanges adjacent to loading points. The test and FEM results are subsequently plotted against the DSM interaction curves between bending and shear where the interaction is found to be significant. An extended range of DSM prequalified sections with longitudinally stiffened web channels in shear is proposed in this paper.     

Elastic web bend-buckling analysis of longitudinally stiffened I-section girders

The linear elastic web bend-buckling behavior of I-section girders with a longitudinal stiffener is investigated by a threedimensional finite element analysis in which the web, top and bottom flanges, and transverse and longitudinal stiffeners are modeled as thin shell elements. After obtaining the bend-buckling moment for an I-section girder by finite element analysis, the buckling stress and subsequently the buckling coefficient of a rectangular web panel with a longitudinal stiffener are calculated and compared with those obtained by AASHTO LRFD and Eurocode 3. To observe the effect of various parameters on the buckling behavior of an I-section girder under pure bending, the following aspects are considered: the non-symmetry of the cross-section, the location of the longitudinal stiffener, the boundary conditions along the transverse and longitudinal stiffeners and flanges, the flexural rigidity of the longitudinal stiffener, the spacing of the transverse web stiffeners, and the slenderness of the web. Based on the numerical results, new design equations for the bend-buckling coefficient of a longitudinally stiffened I-girder are proposed.     

Elasto-plastic buckling of integrally stiffened panels (ISP): An optimization approach for the design of cross-section profiles

In this work, three optimization algorithms—Levenberg–Marquardt, simulated annealing and a newly developed and proposed hybrid differential evolution particle swarm optimization—are considered in the shape optimizations of cross-sections of integrally stiffened panels (ISP) for aeronautical applications, when subjected to buckling deformation modes within the elasto-plastic range. The proposed algorithm is shown to be robust and more effective, in the tested examples, than conventional optimization algorithms, leading to optimum designs of ISP cross-sections for a pre-defined buckling load-carrying capacity, combined with elasto-plastic nonlinear behaviors.     

The behavior and design of longitudinally stiffened thin-walled compression elements

The behavior and design of cold-formed steel sections with single and multiple intermediate stiffeners in the compression, flange is investigated. Existing experimental data are used to evaluate critically the AISI specification and Eurocode. For bending strength prediction of sections with multiple intermediate stiffeners, the AISI specification can be quite unconservative and Eurocode often Yields overly conservative results. Existing experimental data are used to calibrate a finite element model. An extensive parametric study is conducted using the finite element model. The results of the study are used to gain a better understanding of the behavior of these elements, and to help evaluate alternatives to the existing design procedures. Based on the existing data and the finite element stud two alternatives to the existing strength prediction procedures are advanced.     

Ultimate resistance of longitudinally stiffened webs subjected to patch loading

Longitudinal stiffening increases the ultimate resistance of slender girder webs subjected to patch loading when the stiffener is placed rather close to the loaded flange. In the past, some attempts have been made to quantify this influence, however none of these formulae have been successfully implemented due to poor correlation between predictions and experimental results. This paper describes a methodology to determine the ultimate resistance of longitudinally stiffened girder webs subjected to patch loading. The methodology accounts for the influence of: (a) the relative position of the stiffener, (b) the ratio of flange-to-web thickness, and (c) the ratio between the yield strength of the flange and the web. Finally, theoretical predictions show good agreement with experimental results.     

Compressive tests on stiffened panels of intermediate slenderness

Test results are presented of eight stiffened panels subjected to axial compression until collapse and beyond. The specimens are three-bay stiffened panels with associated plate made of very high tensile steel S690. The use of this very high strength steel led to the unconventional solution of using U stiffeners and this paper aims at understanding the difference of performance of this stiffener type as compared with the conventional ones. Four different configurations are considered for the stiffeners, which are made of mild or high tensile steel for bar stiffeners and mild steel for ‘L’ and ‘U’ stiffeners. The influence of the stiffener's geometry on the ultimate strength of the stiffened panels under compression is analyzed.     

Multi-criteria optimal design of stiffened plates-II. Mathematical modelling of the optimal design of longitudinally stiffened plates

In the second part of these series of papers we present the mathematical modelling of the multi-criteria optimization problem of longitudinally stiffened plates. This leads to a non-linear multi-criteria optimization problem with a finite number of side-conditions. As an application we consider the optimal design of such plates where the weight should be as small as possible and the ultimate buckling load as high as possible. The chosen design variables are the number, the thickness and the height of the stiffeners for a specific plate thickness.For the computation of the optimal points the multi-criteria optimization problem is transformed to a family of ordinary optimization problems using a method developed in Ref.¹ (Segundas Jornadas Latino Americanas de Matematica Aplicada, 1983, 2, 483–495). An investigation of the special side-conditions of the considered optimal design problem leads to a simplification of the method of Ref.¹ The optimal design problem for weight minimization and for maximization of the ultimate buckling load is considered explicitly. Some numerical examples conclude the second part of the paper.     

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.