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.     

Static buckling analysis of thin cylindrical shell with centrally located dent under uniform lateral pressure

One of the common failure modes of thin cylindrical shell subjected to external pressure is buckling. The buckling pressure of these shell structures are dominantly affected by the geometrical imperfections present in the cylindrical shell which are very difficult to alleviate during manufacturing process. Dent is one of the common geometrical imperfections present in thin shell structures which may be formed due to mechanical damage caused by accidental loading or impact. In this work, influence of various dent parameters (dent length, dent width, dent depth and angle of orientation of the dent) on the critical buckling pressure of thin cylindrical shells with a centrally located dent is studied using non-linear static finite-element analysis of ANSYS under external pressure with simply supported boundary conditions at the top and bottom edges of the thin cylindrical shell.     

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%.     

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.     

A semi-analytical approach for linear and non-linear analysis of unstiffened laminated composite cylinders and cones under axial,torsion and pressure loads

A semi-analytical model for the non-linear analysis of simply supported, unstiffened laminated composite cylinders and cones using the Ritz method and the Classical Laminated Plate Theory is proposed. A matrix notation is used to formulate the problem using Donnell׳s and Sanders׳ non-linear equations. The approximation functions proposed are capable to simulate the elephant׳s foot effect, a common phenomenon and a common failure mode for cylindrical and conical structures under axial compression. Axial, torsion and pressure loads can be applied individually or combined, and solutions for linear static, linear buckling and non-linear buckling analyses are presented and verified using a commercial finite element software. The presented non-linear buckling analyses used perturbation loads to create the initial geometric imperfections, showing the capability of the method for arbitrary imperfection patterns. The linear stiffness matrices are integrated analytically and for the conical structures an approximation is proposed to overcome the non-integrable expressions.     

变截面H型钢柱相关屈曲性能研究

采用板壳单元离散变截面H型钢柱,并同时考虑构件整体与局部初始缺陷、残余应力、材料屈服、几何非线性等因素的影响。根据数值分析的结果,重点研究了腹板宽厚比、腹板楔率、翼缘宽厚比、长细比等参数对板件局部屈曲与构件整体稳定的影响,总结了压力作用下变截面H型钢柱局部屈曲、整体屈曲和相关屈曲的发展过程以及影响因素。     

用于评估外力作用下圆环体和圆柱壳承载力的改进方程(1):分析推导

在两部分系列文章中,分析了外力作用下圆环体的承载力,这作为经典Levy-Timoshenko方法假设的出发点。该方法也是大多数圆柱壳体的设计准则。本文提出了一个这类问题的更加完善的方程式。该方程式结合各种几何缺陷效应,可以准确地分析塑性性能。先提出一个严格的非线性方程,后建立一个代数表达式,从而避免数值解求解过程。     

Simplified procedure for design of liquid-storage combined conical tanks

Steel vessels in the form of combined conical-cylindrical shells are commonly used for liquid storage in elevated water tanks. A number of such tanks collapsed in different places around the globe due to instability of the steel shells. An essential cause of those collapses is the lack of adequate design procedures for such structures. In this study, a simplified design approach is developed to ensure safety of hydrostatically loaded combined steel conical tanks against buckling. The study is conducted numerically using a non-linear finite element model that accounts for the effects of large deformations and geometric imperfections on the stability of combined conical tanks. The finite element results together with a non-linear regression analysis are used to develop magnification functions that relate the overall shell stresses to the membrane stresses which can be evaluated analytically. Numerical examples are presented to explain application of the suggested design approach.     

Numerical study of built-up double-Z members in bending and compression

The use of the finite element method (FEM) for the design of composed, thin-walled, structural steel members is considered. The bolted double-Z frame member is an interesting and economical engineering solution, already used in practice [1]. However, the European recommendations for the design of steel structures do not consider built-up members from cold-formed steel profiles. Finite element analysis is used to capture the various buckling effects that shape the response of slender thin-walled members. From the finite element model, the importance of initial imperfections and stiffness of connections is identified. The experimentally validated model predictions show that a non-linear finite element analysis can predict the member behaviour, in terms of failure mode and ultimate load, yield line pattern, overall stiffness and local strain in the cold-formed profiles. To obtain a good prediction, overall and localised initial imperfections should be considered and included in the analysis.     

Ultimate compressive strength of plate elements in aluminium: Correlation of finite element analyses and tests

A study with the objective of assessing the reliability of non-linear finite element analyses in predictions of the ultimate strength of aluminium plates subjected to in-plane compression is presented. Outstand elements of alloy AA6082 in tempers T4 and T6 and internal elements of alloys 5083 M and 6082 TF were analysed for a range of b/t-ratios. For the latter class both non-welded and welded plates were studied, the non-welded plates having two levels of geometric imperfections. The accuracy of the predictions was evaluated by comparison with existing experimental results. It was found that the overall correlation between the experimental and predicted ultimate compressive strengths was good. The finite element analyses reproduced the main effects of slenderness, stress-strain curve (i.e. alloy and temper), geometric imperfections and welding (i.e. residual stresses and heat-affected zones) that were observed in the experiments     

Numerical Modelling of Thin-Walled Stainless Steel Structural Elements in Case of Fire

In this paper, the structural response of stainless steel thin-walled elements submitted to fire is analysed numerically by means of the geometrically and materially non-linear Finite Element program SAFIR, including imperfections. In order to make these simulations, two main changes in the program were made: (i) the code was changed in order to deal with the stainless steel 2D material constitutive law to be used with shell elements and (ii) the possibility of the program to take into account residual stresses with shell finite elements was introduced. The stainless steel stress–strain relationship at high temperatures was based on the one presented in part 1.2 of Eurocode 3. To model the strain hardening exhibited by the stainless steels, using the shell element formulation, an approximation to the Eurocode 3 constitutive law was needed. Local and global geometrical imperfections were considered in the simulations. The paper shows the influence of the residual stresses on the ultimate load-carrying capacity of thin-walled stainless steel structural elements in case of fire.     

Numerical investigation of channel columns with complex stiffeners—part I: test verification

A numerical investigation on fixed-ended cold-formed steel channel columns with complex stiffeners subjected to axial load is described in this paper. The complex stiffeners of the channel sections consist of simple lips with return lips. The specimens were brake-pressed from high strength zinc-coated structural steel sheets having a nominal yield stress of 450 MPa. A non-linear finite element model is developed and verified against experimental results of fixed-ended channel columns with complex stiffeners. Initial geometric imperfections and material non-linearity are included in the model. The calibrated model is shown to provide accurate predictions of the experimental ultimate loads and failure modes of the tested columns.     

An improved formulation for the assessment of the capacity load of circular rings and cylindrical shells under external pressure. Part 1. Analytical derivation

In this two-part set of articles the capacity load of circular rings under external pressure is investigated assuming as a starting point the classic Levy–Timoshenko approach, which is still at the bases of most design codes for cylindrical shells. This first paper presents a perfected analytical formulation of the problem, which accurately accounts for the onset of plasticity and incorporates the effect of various geometrical imperfections. A rigorous non-linear formulation is first derived and, subsequently, an algebraic expression which avoids the recourse to numerical solution strategies is established.     

带垂直加劲肋的波纹板组成的圆柱形金属筒的三维屈曲分析

介绍了采用垂直加劲肋加强的水平波纹板组成的漏斗式圆柱形金属筒的准静态屈曲性能的三维分析结果。考虑了欧洲规范1规定的散粒体对筒壁产生的对称与不对称荷载、不同的初始几何缺陷和荷载分布的不均匀性,给出了无缺陷和有缺陷的筒仓壳体的线性屈曲、几何和材料的非线性的分析结果。将计算得到的临界荷载值与欧洲规范3给出的允许值进行对比。     

单层穹顶网壳的荷载-位移全过程及缺陷分析

本文以一个50m跨度的单层穹顶网壳为例进行了荷载-位移全过程分析。研究了该结构作为理想壳的非线性稳定性能,以及加荷过程中若干次屈曲后的结构位移形态。单层穹顶网壳对初始缺陷极为敏感,而实际结构不可避免存在初始安装偏差,但国内针对有缺陷网壳的稳定问题研究很少。本文在这方面做了探索,提出两种缺陷分析方法,并论证了这些方法用于实际工程的可行性。     

Experimental investigation for the behaviour of battened beam-columns composed of four equal slender angles

This study presents series of compression tests on battened columns that are composed of four equal slender angles. The angles are formed by bending thin steel sheets, such that the legs outstand width–thickness ratio is slender. Twenty specimens varied in their plate element width–thickness ratio as well as covered short and medium member slenderness were tested. The angles were assembled by batten plates by means of bolts. Measurements of residual stresses and geometrical imperfections were carried out. Moreover the specimens were simulated by a finite element model using shell element that accounts for both geometric and material non-linearities. The measured geometric imperfections and residual stresses were included in the numerical model. Finally, the test results have been compared with those of non-linear finite element model, and also with the predicted ultimate strengths determined by the American and European specifications. Results show that the interaction between slender outstanding width–thickness ratios, overall angle slenderness and overall column slenderness decrease the strength of battened columns. Also, the results of bolted finite element model were in reasonably good agreement with test results that neglect the effect of bolt holes.     

On the incorporation of equivalent member imperfections in the in-plane design of steel frames

This work deals with the incorporation of equivalent member imperfections in the global analysis of steel frames and, in particular, is intended to clarify the Eurocode 3 (EC3) provisions involved in such procedure. In fact, these provisions stem from the well-known “European column buckling curves”, which means that they are based on the behaviour of simply supported isolated members under uniform compression (columns). First, one addresses the geometrically non-linear behaviour of isolated columns displaying arbitrary support conditions and different initial geometrical configurations. Then, the results obtained are used to propose a systematic and rational method to evaluate the appropriate “equivalent initial imperfections” that need to be incorporated in the second-order global elastic analysis of a frame or isolated compressed member. This method (i) is fully consistent with the EC3 column buckling curves and (ii) adopts critical buckling mode shape initial imperfections with amplitudes determined by means of closed-form analytical expressions. In order to enable a better grasp of the concepts involved and also to illustrate the capabilities of the proposed methodology, several numerical examples are presented and discussed throughout the paper.     

基于能量法的钢框架结构屈曲分析方法

钢框架结构的屈曲分析通常分两个步骤。首先对结构进行线弹性分析得到内力和弯矩,然后对各个杆单元进行屈曲分析和设计,并且考虑初始缺陷产生的影响。目前,许多国家和地区的钢结构设计规范都是依靠杆单元计算长度系数μ来进行屈曲分析,该系数随杆单元的屈曲形式而定,而实际上它的值是近似的。为此,基于能量法从杆单元的初始缺陷引起的初始变形开始对结构进行非线性分析,从而避免了使用计算长度系数μ。     

Residual stresses in cold-rolled stainless steel hollow sections

Stainless steel exhibits a pronounced response to cold-work and heat input. As a result, the behaviour of structural stainless steel sections, as influenced by strength, ductility and residual stress presence, is sensitive to the precise means by which the sections are produced. This paper explores the presence and influence of residual stresses in cold-rolled stainless steel box sections using experimental and numerical techniques. In previous studies, residual stress magnitudes have been inferred from surface strain measurements and an assumed through-thickness stress distribution. In the present study, through-thickness residual stresses in cold-rolled stainless steel box sections have been measured directly by means of X-ray diffraction and their effect on structural behaviour has been carefully assessed through detailed non-linear numerical modelling. Geometric imperfections, flat and corner material properties and the average compressive response of stainless steel box sections were also examined experimentally and the results have been fully reported. From the X-ray diffraction measurements, it was concluded that the influence of through-thickness (bending) residual stresses in cold-rolled stainless steel box sections could be effectively represented by a rectangular stress block distribution. The developed ABAQUS numerical models included features such as non-linear material stress-strain characteristics, initial geometric imperfections, residual stresses (membrane and bending) and enhanced strength corner properties. The residual stresses, together with the corresponding plastic strains, were included in the FE models by means of the SIGINI and HARDINI Fortran subroutines. Of the two residual stress components, the bending residual stresses were found to be larger in magnitude and of greater (often positive) influence on the structural behaviour of thin-walled cold-formed stainless steel sections.     

3D buckling analysis of a cylindrical metal bin composed of corrugated sheets strengthened by vertical stiffeners

The paper presents 3D results of a quasi-static buckling analysis of a funnel-flow cylindrical metal bin composed of horizontally corrugated sheets strengthened by vertical columns. A linear buckling and a non-linear analysis with geometric and material non-linearity were carried out with a perfect and an imperfect real silo shell by taking into account axisymmetric and non-axisymmetric loads imposed by a bulk solid following Eurocode 1 and different initial geometric imperfections and load non-uniformities around the circumference. The calculated buckling forces were compared with the permissible one given by Eurocode 3.