Buckling and postbuckling investigations of imperfect curved stringer-stiffened composite shells. Part A: Experimental investigation and effective width evaluation

A box-like stringer-stiffened thin-walled CFRP structure was subjected to load cases well beyond the limits of local buckling. The development of the deflection pattern was recorded via optical means and analysed numerically. In addition, the structure was modelled and analysed using the MARC FE program in the nonlinear deflection range.

The geometrical imperfections of the test structure were recorded by mechanical scanning and optical methods and introduced into the mathematical model. For the perfect (‘ideal’) and the geometrically imperfect (‘real’) structural model, the results of the FE analyses were compared and used to judge the effect of geometric imperfections on the postbuckling behaviour of the structure.

The effective axial stiffness for the various postbuckling states was evaluated and related to analytical estimates of effective width values for orthotropic sheet-like panels.     

On the design of stiffeners in steel plate shear walls

This paper describes a numerical investigation to provide a practical design method for stiffening thin steel plate shear walls. The procedure considers one-sided transverse and longitudinal flat stiffeners located in various arrangements on shear plates which effectively divides the plate into subpanels and expands tension fields across the infill walls. The results obtained from several nonlinear static analyses are employed to draw applicable empirical relationships for evaluating optimal dimensions of stiffeners. The procedure also ascertains the effects of optimised stiffeners on the postbuckling behaviour and ultimate load bearing capacity of stiffened shear walls.     

Influence of longitudinal stiffeners on elastic stability of girder webs

In this paper, linear buckling analyses of plates with longitudinal stiffeners having various shapes and positions and subjected to axial force, in-plane bending and shear are developed. The aim is to give some new practical insights about the shape and optimum position of longitudinal stiffener in webs when axial force, bending moment and shear act. By means of a comprehensive numerical investigation, some practical issues for buckling phenomena in stiffened plates, taking into account (a) dimensions and shape (square and rectangular) of the plate, (b) dimensions and shape of the stiffener (with open and closed cross-sections), (c) location of the stiffener, and (d) load configuration (uniform compression, pure bending, combinations of axial compression and bending, and shear) are developed.     

The shear effective width of aluminium plates

The capacity of a plate to sustain shear load greater than the critical load is studied. The concept of effective width for square plates loaded in shear is developed to consider rectangular plates loaded in shear. The deformation of the plate at failure is formulated. Very good agreement is observed between the theoretical results and experimental measurements carried out on aluminium web plates.     

Moment–shear interaction of stiffened plate girders—Tests and numerical model verification

Results of four full-scale tests on plate girders stiffened with transverse and longitudinal stiffeners subjected to interaction of high bending moment and shear force are presented and discussed. In longitudinal direction the web was stiffened with open or closed stiffeners positioned in the compression zone. Detailed information on initial geometric imperfection and residual stresses is given. The experimental results were used to verify numerical model. The resistance is compared against reduced stress method and effective width method given in EN 1993-1-5.     

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.     

A simplified method for calculation of the natural frequencies of wall–frame buildings

A simple hand method is presented for the three-dimensional frequency analysis of buildings braced by frameworks, coupled shear walls, shear walls and cores. Lateral vibration is characterised by three types of deformation: the full-height ‘local’ bending of the individual columns/wall sections/shear walls/cores, the full-height ‘global’ bending of the frameworks/coupled shear walls, which is associated with the axial deformations of the columns/wall sections, and the shear deformation of the frameworks/coupled shear walls. Based on the stiffnesses associated with these three types of deformation, a closed formula is derived for calculation of the lateral frequencies. An analogy between bending and torsion is used to carry out the pure torsional frequency analysis. The coupling of the lateral and pure torsional modes is taken into account. The results of a comprehensive accuracy analysis covering 144 multi-storey structures demonstrate good agreement with the finite element solution, the maximum difference being 7%. A worked example with step-by-step instructions shows the easy use of the method.     

A generalized analytical approach to the buckling of simply-supported rectangular plates under arbitrary loads

An analytical approach for the elastic stability of simply-supported rectangular plates under arbitrary external loads is presented which, for the first time, may be described as ‘exact’. This is achieved through the use of exact solutions for the in-plane stresses and the adoption of double Fourier series for the buckled profiles which, together, ensure that accurate results are obtained in the Ritz energy technique. Several cases of plate buckling under direct, shear and bending loads (or their combinations) are studied to show the generality of the proposed approach, with the ensuing results compared with existing data (if available) and with numerical FE results.     

Theoretical and numerical analysis and design of stiffened steel plate shear walls having unequal subpanels

Following the adequate seismic behavior, the steel plate shear walls have gained popularity among the designers worldwide, and many buildings have been designed employing such walls as their lateral load‐carrying system. Despite the fact that the steel shear walls using unstiffened thin plates outnumber the ones benefitting from stiffeners, it should be kept in mind that application of stiffeners prevents the buckling induced by lateral loads such earthquake or wind. On the other hand, the stiffeners enhance the stiffness and shear strength of the plate. Accordingly, this paper deals with an analytical and numerical investigation into the steel plates stiffened by unequaled vertical and horizontal stiffeners. The results obtained for the analyses led to the development of the analytical relations to compute the shear strength, stiffness, and displacement. To validate the proposed relations, analytical outputs were compared with those obtained from the numerical investigation indicating great accuracy of the proposed analytical relations. Moreover, in this paper, the design process of the stiffened steel plate shear walls has been presented based on the interaction between the plate and frame. The results indicate the addition of stiffeners to the steel plate, improves its shear strength by 6 to 15%.     

Inelastic buckling behavior of stocky plates under interactive shear and in-plane bending

Inelastic buckling and postbuckling behavior of stocky plates under combined shear and in-plane bending stresses are investigated and compared to slender plates. Aluminum and steel plates having various slenderness ratios are modeled and analyzed by means of (i) numerical nonlinear finite element method and (ii) theoretical p-Ritz energy method; and both results are compared to the classic interaction equation. It is observed that whereas in slender plates, elastic buckling occurs prior to the material's proportional limit load, stocky plates buckle in an inelastic way within the post-yield stage. In contrast to slender plates with considerable postbuckling reserves, the buckling of stocky plates is immediately followed by softening. In addition, it is shown that the classic interaction equation overestimates buckling loads; and therefore, a modified equation that can safely be applied to both stocky and slender plates is proposed.     

Experiments of edge-stiffened plates in uniform compression

There is a large body of literature on the experimental behaviour of steel plates under a variety of loading and edge support conditions. Typically the longitudinal plates edges were either laterally restrained (stiffened) or laterally unrestrained (unstiffened). There are no such plate experiments reported in the literature of plates with a longitudinal edge stiffened with an edge-stiffener (partially stiffened). This paper presents 30 plate tests of plates simply supported on three sides, with the remaining (longitudinal) edge stiffened with an edge-stiffener. The edge-stiffener was increased in size from zero (unstiffened plate) to a size sufficient to create a stiffened edge (stiffened plate). Edge-stiffeners of an intermediate size resulted in partially stiffened plates. The behaviour of partially stiffened plates is fully documented with the aid of a 3D photogrammetry system, and the adequate size of edge-stiffeners required in order to create a stiffened plate edge are discussed. Edge-stiffener types investigated are simple lips, simple inclined lips and complex lips. Design procedures for partially stiffened elements previously inferred from numerical studies are shown to accurately and reliably determine the plate strengths, and further validate the procedure for use as a general strength approach in current international cold-formed steel specifications.     

边缘加劲板件有效宽厚比设计方法中的板组效应研究

本文根据能量原理,建立了边缘加劲板件的弹性屈曲理论和卷边槽形截面薄壁构件的板组相关屈曲理论。通过屈曲理论分析,得到了非均匀受压边缘加劲板件的屈曲系数及其反映板组效应的约束系数,并将其引入新修订的《冷弯薄壁型钢结构技术规范》(GB50018)的有效宽厚比设计方法中。本文介绍了50根冷弯薄壁型钢受压构件试验,并按修订后的规范方法进行了试件承载力计算,计算值与试验值比较,偏于安全。将考虑板组效应的有效宽厚比设计方法与国内外现行规范设计方法相比较,具有优越性。     

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.     

交叉加劲钢板剪力墙弹性屈曲研究

用有限元方法对交叉加劲钢板剪力墙的弹性屈曲性能进行了研究,重点研究了加劲肋与墙板的刚度比、墙板高厚比、边长比以及加劲肋宽厚比等对弹性屈曲系数k的影响,同时与十字加劲板的抗剪屈曲性能进行了对比。研究结果表明,设置交叉加劲肋能显著提高钢板剪力墙的弹性屈曲荷载;屈曲系数k随着墙板边长比、高厚比以及加劲肋宽厚比的增大而趋于减小;本文给出的交叉加劲板弹性屈曲系数k的计算公式与有限元法的结果较吻合。     

Accurate discrete modelling of stiffened isotropic and orthotropic rectangular plates

An analytical approach is presented here for simply supported blade-stiffened rectangular plates wherein important non-classical effects such as transverse shear deformation and rotary inertia are carefully accounted for. The analysis differs from full three-dimensional modelling of both the plate and the stiffener in that a plane stress idealization is used to model the kinematics of transverse bending of the stiffener while simple one-dimensional classical models are employed for lateral bending and torsion. Parametric studies are used to highlight the importance of non-classical effects in plate and stiffener kinematics and to finally lead to certain recommendations for accurate modelling of stiffened plates.     

荷载横向变位下箱梁顶板与底板的剪滞效应分析

不同于以往荷载作用于箱梁的肋板处时剪力滞效应的研究,考虑当荷载不作用在箱梁的肋板处时顶板与底板位移及剪力滞效应的差异,对顶板与底板分设不同的纵向位移差函数,采用二次抛物线作为箱梁翼缘板的纵向位移沿梁宽分布函数,通过能量变分法,研究荷载在顶板横向变位下箱梁顶板与底板的位移,应力及剪力滞效应的变化,根据对于简支箱梁受均布荷载作用下跨中剪力滞效应的计算结果,随着作用在顶板的对称荷载从对称中心向肋板处移动,顶板将经历一个产生负剪力滞效应到无剪力滞效应,再到产生正剪力滞效应的过程。而与此同时底板的剪力滞效应规律受荷载的移动的影响很小,始终保持正剪力滞效应。计算结果与有限元数值模拟结果吻合较好,验证了理论的可靠性,同时也说明当荷载不作用在箱梁的肋板处时,对顶板与底板分设不同的纵向位移差函数的考虑是正确的。     

Effects of anchoring tensile stresses in axially loaded plates and sections

Thin-walled compression members are commonly designed on the assumption that the loaded edges remain straight. Under this assumption, tensile stresses develop in the most flexible parts of the component plates at advanced local buckling deformation, and thus are assumed to be ‘anchored’ at the ends. However, current design rules for plate elements, such as the Winter formulae, are partly based on tests in which the load was applied by use of rigid platens that did not permit tensile stresses to develop. There exists an apparent inconsistency between the assumption of straight loaded edges and the use of a design curve calibrated from tests in which the loaded edges of component plates may not have remained straight.This paper addresses this apparent inconsistency by comparing finite element solutions for the conditions of straight loaded edges and loading by use of a contact surface between the plate edge and a non-deformable rigid body end platen, where there is no constraint for the plate edge to remain in contact with the rigid body. Solutions are provided for a single half-wavelength of unstiffened and stiffened plate elements simply supported along three and four edges, respectively. The effect of multiple half-wavelengths is also investigated, as is the effect of interaction between elements in practical sections comprising stiffened and unstiffened elements.     

Simulation of the behaviour of stiffened box girders with and without shear lag

A finite element program with large deformations and plasticity is used to analyse the behaviour of stiffened compression flanges of box girders. Simulations are performed for pure bending and for bending with shear in order to investigate the diminishing of the shear lag effect at collapse. It is concluded that at the actual ultimate load, shear lag has largely disappeared but that a design at ultimate load without consideration of the shear lag may entail an inelastic behaviour under service load. First yielding occurs in the centre of the stiffened panel for pure bending and in the middle of the unloaded edges when elastic shear lag becomes important. A substantial increase of the ultimate load can be expected from an increase of the stiffener rigidity.     

Interaction of breathing with the cumulation of damage in the webs of steel plate girders

The contribution describes the main results of two Prague experimental investigations into the ‘breathing’ of slender steel webs, the first of them focusing on web breathing under repeated partial edge loading and the other upon that under repeated shear. The effect of the initiation and propagation of cracks in breathing webs on the failure mechanisms and the ultimate load behaviour of steel plate girders is studied.     

边缘加劲板在均匀压力下的试验研究

目前,已有许多关于不同边界条件钢板在各种荷载下的试验研究。经典的类型是钢板纵向边缘受约束(加劲)或不受约束(不加劲)。但是现有文献没有关于钢板纵向边缘部分约束(部分加劲)的试验报告。对30块三边简支、单纵边加劲的钢板进行试验分析。试件的边缘加劲从零加劲(自由边)到刚性加劲变化,在此中间状态的边缘加劲就是部分加劲钢板。部分加劲钢板的性能由一套三维照相测量系统进行记录,并讨论了多大尺寸的边缘加劲肋才能提供足够刚性的约束。试验中的边缘加劲板的类型有简单边缘板、简单倾斜边缘板和复杂边缘板。根据数值研究得到关于部分加劲板强度计算的设计准则是准确可靠的,并且证明现有的国际冷轧钢结构规范所采用的一般强度方法有效。