Equivalent Imperfection Research Articles

Imperfections for the calculation of steel structures by the finite element method. Part 1

It is well known that imperfections are always present in structural elements. Imperfections can significantly affect the behavior and bearing capacity of steel structures, especially in the case of stability-related tasks. Therefore, inconsistencies must be taken into account in the loadbearing capacity model and their correct application (setting the shape and value) is a key point in the numerical analysis process. In recent decades, much attention has been paid in the domestic scientific space to updating imperfection models for use in numerical models, including taking into account modern more accurate manufacturing and installation technologies for steel structures. The purpose of this study is an analytical review and analysis of scientific research and technical literature, followed by synthesis and elaboration of recommendations on imperfections in relation to the calculation of steel structures using computer modeling technology, including the finite element method. The results of the study contain instructions on how to set the shapes and values of imperfections for different groups of imperfections. The article consists of two parts. The first part is devoted to the study of geometric imperfections, residual stresses and rules for the combination of imperfections, the second part of the article is devoted to equivalent imperfections.

Equivalent geometrical imperfections for local and global interactive buckling of welded rectangular box-section columns

In recent years, the construction industry has witnessed a growing preference for welded box-sections due to their ease of fabrication and improved stability characteristics. Nonetheless, welded box-sections are prone to various instabilities, which can be classified into three categories: global buckling, local buckling, and interactive buckling. amongst these, interactive buckling is of primary concern as research in this area is still limited. Currently, there is a lack of viable options to effectively address the nonlinear interaction of global and local buckling. The equivalent imperfections provided by the Eurocode were developed based on an elastic design approach and are not suitable for geometrically and materially nonlinear analysis (GMNIA). Therefore, the applicable equivalent geometric imperfections should be revised and improved to make them suitable for plastic analysis. The authors previously investigated the interactive buckling resistance of welded square box-sections and developed suitable imperfections that can be used for GMNI analysis. However, the study was focused only on square box-sections. In the current research, the interactive buckling capacities of square and rectangular welded box-sections are investigated. A general rule for imperfection combination is proposed, which can be applied to all welded box-section columns. This makes the application of numerical model-based design more accurate for box-section columns, as well as significantly improves the general applicability of GMNI analysis-based resistance calculations.

Imperfections for Local and Global Interaction Buckling of Welded Square High-strength Steel Box-section Columns

The interaction behavior between local and global buckling modes in high-strength steel box-section columns has received limited research attention. Currently, there is a lack of a validated equivalent geometric imperfection that can be effectively employed in nonlinear plastic analysis to estimate the interaction buckling resistance. This research aims to find equivalent geometric imperfections that can be used in geometrical and materially nonlinear analysis using imperfections (GMNIA) to estimate the interaction buckling resistance of square welded box-section columns made of high-strength steel. It extends prior investigations by the authors on equivalent imperfections for normal-strength steel welded box-sections. A developed and validated numerical model is used to perform parametric studies to estimate the accurate buckling capacity using previously developed and verified combinations of imperfections and residual stresses. The accurate buckling capacities are used to calibrate equivalent local and global imperfection combinations that can be used in FEM-based design. A reliability assessment study is also performed to check the safety level of the proposed imperfection combinations.

A consistent approach for global buckling of steel structures

AbstractThe ”General Method“ according to chapter 6.3.4 of EN1993‐1‐1 or 8.3.4 of prEN1993‐1‐1 deals with the verification of stability problems of frame and beam structures with loadings in‐plane. The ”General method“ according to Eurocode is intended for the general case of a combined moment and normal force. For the choice of the reduction factor χop, there are two options for applying the reduction curve, either the minimum of lateral torsional buckling (LTB) and flexural buckling (FB) or a weighted combination of both. With the use of analytical derivations, it can be shown that this approach leads to inconsistencies which can possibly lead to incorrect results. In this paper an alternative approach is presented which leads to unification and fully mechanical justification in verification. The approach allows e.g. the verification of stability of an entire frame system directly and without interaction factors (as in the equivalent member method). In addition, the issue of a consistent equivalent imperfection is addressed. With the use of this method, geometrical non‐linear analyses of structures showing out‐of‐plane instability can be omitted; only an in‐plane calculation and a determination of a linear buckling analysis is necessary. Finally, examples are presented in this paper that demonstrate the application of the approach shown.

Decision tree for local + global imperfection combinations in double‐symmetric prismatic members – Practical recommendations in the framework of advanced analysis

AbstractBetter and simpler possibilities of structural optimization due to increasing computational power but also for reasons of environmental sustainability, the use of materials and their reusability lead to greater acceptance towards more advanced numerically intensive, so‐called ‘design by analysis' methods like geometrically and materially non‐linear analyses with imperfections (GMNIA). The general choice of imperfections and their combination in such models, especially for slender cross sections of intermediate length prone to an interaction between a global and local plate buckling, is crucial in terms of the reached load‐bearing capacity. Annex C of EN 1993‐1‐5:2010 makes use of the ‘70 %‐rule' for the combination of imperfection modes and amplitudes. This rule postulates that two GMNIA calculations should be conducted when local and global interactive buckling may be dominant; one with 100 % + 70 % of the maximum specified amplitude in either case. In addition, extended information is provided on the choice and combination of imperfections in the newly introduced and currently available draft of the prEN 1993‐1‐14:2020 (design assisted by finite element analysis). Although information is provided on how the local and global imperfections should be combined, it is not stated when it is relevant to consider those. Based on conducted GMNIA simulations on SHS/RHS (square and rectangular hollow sections) and I‐shaped sections, this article presents general decision support on the choice of equivalent imperfections. On the basis of numerical analysis, the developed flow chart and design routine allow for the decision whether the consideration of the interaction of local and global imperfections is required or not.

Out‐of‐plane Stability Design by GMNIA – Equivalent Imperfections and Strain Limits

AbstractAn accurate and consistent approach to the out‐of‐plane stability design of steel beams and structures utilising geometrically and materially nonlinear analysis with imperfections (GMNIA) and strain limits is proposed. The method is implemented using computationally efficient beam elements, with the ultimate structural resistance defined either by (i) the ultimate load factor or (ii) the load factor at which a strain limit, determined on the basis of the continuous strength method (CSM), is attained, whichever occurs first. In the present paper, the scope of this method is extended from the in‐plane design of steel structures and structural components to scenarios in which out‐of‐plane stability effects, with a focus on lateral‐torsional buckling (LTB), govern. First, two shapes and corresponding amplitudes of equivalent imperfections, which consider the combined influence of both geometric imperfections and residual stresses, for the out‐of‐stability design of steel and stainless steel members by GMNIA are established. The existing CSM strain limits for in‐plane design are then extended to allow for the influence of shear, torsion and warping, to enable their general applicability to three‐dimensional buckling problems. It is shown that employing the proposed design method together with the proposed equivalent imperfections generally provides more accurate and consistent ultimate strength predictions than traditional Eurocode design calculations and also streamlines the design process by eliminating the need for cross‐section classification and individual member design checks.

On the Choice of Eigenmode Shapes for Interactive Cases of Local and Global Buckling of RHS/SHS and I‐Shaped Sections

AbstractBetter and simpler possibilities of structural optimization due to increasing computational power but also for reasons of environmental sustainability, the use of materials and their reusability lead to greater acceptance towards more advanced numerically intensive, so‐called “design by analysis” methods like geometrically and materially nonlinear analyses with imperfections (GMNIA). The general choice of imperfections and their combination in such models, especially for slender cross‐sections of intermediate length prone to an interaction between a global and local plate‐buckling, is crucial in terms of the reached load‐bearing capacity. Annex C of EN 1993‐1‐5 make use of the “70%‐rule” for the combination of imperfection modes and amplitudes. This rule postulates that two GMNIA calculations should be carried out when local and global interactive buckling may be dominant; one with 100% + 70% of the maximum specified amplitude in either case. In addition, extended information is provided on the choice and combination of imperfections in the newly introduced and currently available draft of the prEN1993‐1‐14:2020 (Design assisted by finite element analysis). Although information is provided how the local and global imperfections should be combined, it is not stated when it is relevant to consider those. Based on carried out GMNIA simulations on SHS/RHS and I‐shaped sections, this paper presents a general decision support on the choice of equivalent imperfections. Based on numerical analysis, the developed flowchart and design routine allows for the decision whether the consideration of the interaction of local and global imperfections is required or not.

A Consistent Approach for Stability Verification of Steel Structures

AbstractThe “General Method” according to chapter 6.3.4 of EN1993‐1‐1 [1] or 8.3.4 of prEN1993‐1‐1 [2] deals with the verification of stability problems of frame and beam structures with loadings in‐plane. The “General method” according to Eurocode is intended for the general case of a combined moment and normal force. For the choice of the reduction factor χop, there are two options for applying the reduction curve, either the minimum of that of lateral torsional buckling (LTB) and flexural buckling (FB) or a weighted combination of both. With the use of analytical derivations, it can be shown that this approach leads to inconsistencies which can possibly lead to incorrect results. The intended generality of the EC3 formulation therefore should be put under scrutinizing. The approach presented in this paper can solve this problem and leads to unification, which allows the verification of stability of e.g. an entire frame system directly and without interaction factors (as in the equivalent member method). In addition, the issue of a consistent equivalent imperfection is addressed. With the use of the General Method non‐linear analyses of structures showing out‐of‐plane instability can be omitted; only an in‐plane calculation and a determination of the critical load is necessary. Finally, examples are presented in this paper that demonstrate the application of the approach shown.

Equivalent imperfections for the out-of-plane stability design of steel beams by second-order inelastic analysis

In current structural design specifications, such as EN 1993-1-1 for steel and EN 1993-1-4 for stainless steel, the stability of members is typically assessed through the use of buckling curves, which consider the influence of initial geometric imperfections and residual stresses. An alternative, more direct, approach is to perform either an elastic or inelastic second-order analysis of the member or structure with imperfections. For modelling convenience, so-called ‘equivalent’ imperfections are typically utilised, which consider the combined influence of both geometric imperfections and residual stresses. Equivalent imperfections for the design of columns and beams by second-order elastic analysis, also referred to as geometrically nonlinear analysis with imperfection (GNIA), are provided in the current design specifications. For columns, equivalent imperfections for design by second-order inelastic analysis, also referred to as geometrically and materially nonlinear analysis with imperfections (GMNIA), were recently developed, but for beams that are susceptible to lateral-torsional buckling (LTB), there are currently no appropriate provisions. The aim of this study is therefore to develop equivalent imperfections for use in out-of-plane stability design of steel and stainless steel members by GMNIA. The proposals are calibrated against the results of benchmark finite element (FE) simulations performed on a large number of steel and stainless steel members with geometric imperfections and residual stresses subjected to major axis bending. Two proposals for equivalent imperfection amplitudes are developed: (1) e0,mod, for use with eigenmode-affine imperfections and (2) e0,bow, for use with sinusoidal bow imperfections. The latter is applied solely in the lateral direction and as a summation of a half-sine wave and a full-sine wave. Relative to the traditional Eurocode design calculations, employing the proposed LTB imperfections in GMNIA provides generally more accurate member resistance predictions, while remaining safe-sided relative to the benchmark FE results. The reliability of the design provisions is demonstrated through statistical analysis, where it is shown that partial safety factors of 1.0 for steel and 1.1 for stainless steel can be safely adopted.

Stability of Single-Bolted Thin-Walled Steel Angle Members with Stochastic Imperfections

The focus of this paper is on the probabilistic estimation of the buckling capacity of single-bolted members from plain or lipped angle sections with stochastic geometric imperfections. A joint experimental-stochastic mechanics approach is adopted, employing in-house imperfection measurements of angle members in combination with detailed numerical models. Different slenderness values, plain and lipped angle sections, as well as single-brace or X-bracing diagonals are investigated in all cases accounting for the single-bolt connection. The member buckling loads are obtained via numerical analyses considering geometric and material nonlinearities with initial imperfections. The random variables are the geometric imperfections, the material properties, and the lateral load (wind pressure) on the member. Quatro-variate single-dimensional stochastic processes (the spectral representation method in connection with the method of separation) are used for the modeling of the geometric imperfections. It is shown that the influence of imperfections for typical lattice tower angle members is 3–6 times lower than the influence of material properties and lateral loading. All in all, current EU and US design provisions are found to be mostly conservative for the design of such members, while the use of equivalent imperfections in a numerical model can provide a more rational safety margin.

Assessment of in-Plane Behavior of Metal Compressed Members with Equivalent Geometrical Imperfection

A new procedure was presented with the objective of proving that it is the generalization of current attempts in designing compressed members and structures which is able to solve cases where other authors have problems. It is the further development of the former methods published by Chladný, Baláž, Agüero et al., which are based on the shape of the elastic critical buckling mode of the structure. Chladný’s method was accepted by CEN/TC 250 working groups creating Eurocodes. Both current Eurocodes EN 1993-1-1:2005 and EN 1999-1-1:2007 in their clauses 5.3.2(11) enable applying the geometrical equivalent unique global and local initial (UGLI) imperfection. The imperfection has the shape of the elastic critical buckling mode with amplitude defined in 5.3.2(11). UGLI imperfection is an alternative to the global sway and local bow initial imperfections defined in 5.3.2(3) and to the imperfections described in the clause 5.3.2(6). The determination of the location and value of UGLI imperfection proved to be onerous by some authors, especially in cases of members with variable cross-sections or/and axial forces. The paper also provides for special cases a procedure to detect the critical cross-section along the member which is defined as the one in which the utilization factor obtains maximum values. The new approach is validated by the investigation of five complex structures made of steel and one made of aluminum alloy solved by other authors. Comparisons of the results with those of other authors and with the Geometrically and Materially Nonlinear Analysis with Imperfections (GMNIA) results showed very good agreements with negligible differences. The information concerning the differences between current Eurocodes EN 1993-1-1:2005 and EN 1999-1-1:2007 is provided. Working drafts of Eurocodes of new generation prEN 1993-1-1:2020 and prEN 1999-1-1:2020 are also commented on.

An equivalent imperfection-based FE simulation of the stability of dented cylindrical shells accounting for unintended imperfections

Metal cylindrical shells subjected to axial compressions are prone to buckling. The buckling strength is highly sensitive to geometric imperfections. Experimentally tested cylinders with intended imperfection have already, in most cases, inevitable (unintended) forms of imperfections, such as weld depressions at the weld joints, irregularity in the shape of the cylinder and imperfections due to loading/supporting conditions. The interaction between the effects of the intended and the unintended imperfections in the experimental investigations makes the determination of the sole effect of either one hard to be identified. To evaluate the buckling strength reduction due a certain intended form of imperfection such as a localized dent, the effect of the other hidden (unintended) imperfections should be accounted for, otherwise a misleading strength reduction value will be obtained. In this study, a simple approximate procedure accounting for the effect of the unintended imperfections is proposed and explained though a finite element (FE) analysis of experimentally tested specimens with intended dent imperfections. Geometric imperfections equivalent in their strength reduction to that of the unintended imperfections are introduced to the FE analysis and referred to as “equivalent imperfection”. The experimental value of the buckling strength for the intact specimen (without a dent but having unintended imperfections) along with the FE model are used to calibrate the equivalent imperfection that represents all possible forms of unintended imperfections. On the light of this study, general recommendations for conducting combined experimental-numerical parametric studies are proposed.

The Validity of the Universal Transformation Method in Global Buckling Design

ABSTRACTIn this paper, the limit and validity of the universal transformation method in global buckling design are examined. To calculate the stability resistance of compressed steel members an alternative approach, so‐called equivalent imperfection method, was applied into Eurocode 1993‐1‐1:2005 (1). Later, Papp showed that the equivalent unique global and local imperfection concept is valid for lateral‐torsional buckling and coupled buckling too (2). This method is based on the equivalent reference member method, where the buckling resistance of the irregular member should be equal to the buckling resistance of the reference member. The reference member has pinned ends, its cross‐section, axial force and bending moments are the same as in the equivalent cross‐section of the original member. Between the real member and the reference member forward and backward transformation are needed. In order to investigate the consistency of the above‐mentioned transformation technique on prismatic and tapered beam‐columns several geometrically and materially nonlinear analysis with imperfection were executed. The (semi‐) probabilistic safety level assessment of the universal transformation method performed according to (3), where the difference between the theoretical and experimental resistance is being used to measure the accuracy.

Buckling resistance of steel H-section beam–columns: In-plane buckling resistance

Buckling resistance predictions resulting from flexural and flexural–torsional buckling of double tee section members subjected to compression and bending are considered. A novel analytical model is developed for establishing design criteria based on decomposition of the member buckling behaviour into in-plane and out-of-plane resistance. The former is based on second-order bending relationships of load effects of structural members with in-plane equivalent imperfections, while the latter is based on the stability theory of thin-walled open sections. First part of this study presents an analytical formulation of the in-plane buckling resistance of beam–columns. In this regard, further decomposition is postulated for the in-plane first-order bending moment diagram that results in the loading state to be the superposition of two components. The first component is related to symmetrical loading and the second to antisymmetric loading. To consider the second-order effects, prebuckling displacements generated by the abovementioned loading components are amplified with regard to the inversion of the residuum of the buckling force utilisation ratio in order to obtain approximate values of second-order displacements and internal moments. As a result, the in-plane interaction curve, expressed in dimensionless coordinates, that describes the beam–column in-plane flexural buckling resistance without considering lateral–torsional buckling effects, is obtained. The results of nonlinear finite element simulations are used for the verification of the developed analytical formulation. It is concluded that this proposal yields less conservative predictions than those based on the interaction relationships of clause 6.3.3 of Eurocode 3, Part 1–1.

Buckling behaviour of high strength concrete encased steel composite columns

This paper provides an insight into the buckling behavior of high strength concrete encased steel (CES) columns through a comprehensive investigation including experimental, numerical and analytical analyses. Three long CES column specimens made of high strength concrete C100 and S355 H-section were tested under axial compression. The maximum test loads obtained from these tests were compared with the buckling resistance predicted by EN 1994-1-1, AISC 360-10 and ACI 318-08. Nonlinear finite element analyses were performed to predict the buckling resistance and trace the load displacement behaviour of these columns. In the finite element model, the column initial imperfections were carefully chosen to predict the maximum resistance and the load-displacement response, and compared with the equivalent imperfection values stipulated in the modern design codes. In order to examine the validity and limitation of the current design approaches in predicting the buckling resistance of CES columns made of high strength concrete, a statistical study was undertaken based on the established database covering a wider range of material strength and geometric configurations. In the statistical study, the buckling resistance and effective flexural stiffness obtained from the tests were compared with the codes' predictions from EN 1994-1-1, AISC 360-10 and ACI 318-08. Finally, the reliability of the current design methods is assessed by correlating their accuracy with respect to the variation of material strength, column slenderness ratio, load eccentricity and steel contribution ratio.

04.07: On the axial buckling of very thin‐walled cylindrical shells

ABSTRACTMajor advances in the buckling analysis of cylindrical shells have been achieved in the last two decades. Eurocode nowadays provides rules for tanks up to r/t = 5000. However, since only few literature has become available in the past, dealing specifically with very thin walled shells in the range between r/t = 1500 and r/t = 10000, code provisions either are quite conservative or, in case of r/t > 5000, do barely exist. Employing modern construction methods, close‐to perfect, very thin‐walled shells can be built that are out of the scope of the design codes. Therefore, to achieve more economical designs and a better sustainability, sophisticated design tools have to be used.In this paper, modifications and extensions of the current design procedure are proposed to overcome current limitations and drawbacks. A review of experiments with a special focus on cylinders with high r/t‐ratios was conducted and modified lower bound approximation curves for the knock‐down factor α deduced. They were checked against representative experiments from literature. Test results recalculated using an axisymmetric imperfection to explore their equivalent imperfection depth, proved that even small imperfections drastically reduce the buckling strength. Using a parametric study, the result range of α was extended up to r/t = 10000. The design proposal could be approved and an equation for the determination of an equivalent imperfection depth that relates the dependence of α to the imperfection depth and the r/t‐ratio is proposed.

13.09: Yield stress independent column buckling curves

ABSTRACTUsing GMNIA and shell finite element modelling of steel columns it is ascertained that the buckling curves for given imperfections and residual stresses are not only dependent on the relative slenderness ratio and the cross section shape but also on the magnitude of the yield stress. The influence of the yield stress is to some inadequate degree taken into account in the Eurocode by specifying that steel grades of S460 and higher all belong to a common set of “raised” buckling curves. This is not satisfying as it can be shown theoretically that the current Eurocode formulation misses an epsilon factor in the definition of the normalised imperfection magnitudes. By introducing this factor it seems that the GMNIA analysis and knowledge of the independency of residual stress levels on the yield stress can be brought together and give results showing consistency between numerical modelling and a simple modified Ayrton‐Perry formulation. In this paper magnitudes of imperfections and residual stresses in relation to the Eurocode will be discussed. It will be shown that the use of equivalent imperfections may be very conservative if considered by finite element analysis as described in the current Eurocode code. A suggestion is given for a slight modification of the Eurocode formulations of imperfections leading to adequate inclusion of modern high grade steels within the four bucking curves.

Sensitivity of the mode locking phenomenon to geometric imperfections during wrinkling of supported thin films

Although geometric imperfections have a detrimental effect on buckling, imperfection sensitivity has not been well studied in the past during design of sinusoidal micro and nano-scale structures via wrinkling of supported thin films. This is likely because one is more interested in predicting the shape/size of the resultant patterns than the buckling bifurcation onset strain during fabrication of such wrinkled structures. Herein, I have demonstrated that even modest geometric imperfections alter the final wrinkled mode shapes via the mode locking phenomenon wherein the imperfection mode grows in exclusion to the natural mode of the system. To study the effect of imperfections on mode locking, I have (i) developed a finite element mesh perturbation scheme to generate arbitrary geometric imperfections in the system and (ii) performed a parametric study via finite element methods to link the amplitude and period of the sinusoidal imperfections to the observed wrinkle mode shape and size. Based on this, a non-dimensional geometric parameter has been identified that characterizes the effect of imperfection on the mode locking phenomenon – the equivalent imperfection size. An upper limit for this equivalent imperfection size has been identified via a combination of analytical and finite element modeling. During compression of supported thin films, the system gets “locked” into the imperfection mode if its equivalent imperfection size is above this critical limit. For the polydimethylsiloxane/glass bilayer with a wrinkle period of 2µm, this mode lock-in limit corresponds to an imperfection amplitude of 32nm for an imperfection period of 5µm and 8nm for an imperfection period of 0.8µm. Interestingly, when the non-dimensional critical imperfection size is scaled by the bifurcation onset strain, the scaled critical size depends solely on the ratio of the imperfection to natural periods. Thus, the computational data generated here can be generalized beyond the specific natural periods and bilayer systems studied to enable deterministic design of a variety of wrinkled micro and nano-scale structures.

Numerical analysis of flexural buckling resistance of non-uniform compression members

This paper presents parametric linear-elastic analysis of flexural buckling of idealised non-uniform member and nonlinear analysis of flexural buckling with equivalent imperfections, using software package Abaqus. The analysis includes hinged and cantileverstepped members, where the stiffness ratio of the upper and lower segments and the ratio of the values of axial forces at the top and at the change in the cross-section are varied. The aim of this paper is to define graphic and table models for determining effective lengths coefficients of non-uniform members' segments based on a relevant and reliable database that was obtained using the Finite element method. In addition, the reliability of the method for calculating the flexural stability of the compressed columns according to EC3 was evaluated in which the critical load value was determined in the previous step using the Elastic buckling analysis of idealised elements.

Beam-Column In-Plane Resistance Based on the Concept of Equivalent Geometric Imperfections

AbstractAssessment of the flexural buckling resistance of bisymmetrical I-section beam-columns using FEM is widely discussed in the paper with regard to their imperfect model. The concept of equivalent geometric imperfections is applied in compliance with the so-called Eurocode’s general method. Various imperfection profiles are considered. The global effect of imperfections on the real compression members behaviour is illustrated by the comparison of imperfect beam-columns resistance and the resistance of their perfect counterparts. Numerous FEM simulations with regard to the stability behaviour of laterally and torsionally restrained steel structural elements of hot-rolled wide flange HEB section subjected to both compression and bending about the major or minor principal axes were performed. Geometrically and materially nonlinear analyses, GMNA for perfect structural elements and GMNIA for imperfect ones, preceded by LBA for the initial curvature evaluation of imperfect member configuration prior to loading were carried out. Numerical modelling and simulations were conducted with use of ABAQUS/Standard program. FEM results are compared with those obtained using the Eurocode’s interaction criteria of Method 1 and 2. Concluding remarks with regard to a necessity of equivalent imperfection profiles inclusion in modelling of the in-plane resistance of compression members are presented.