Bow Imperfections Research Articles

A consistent approach to the definition of initial geometric imperfections for in-plane stability design of steel moment frames

Geometric imperfections can have a significant influence on the behaviour and ultimate resistance of steel structures. For the purposes of structural design, particularly design by geometrically and materially nonlinear analysis with imperfections (GMNIA), a suitable choice of imperfections, in terms of both amplitude and shape, is therefore required; an inappropriate choice of imperfections can lead to misleading results and unsafe resistance predictions. The development of a practical strategy for defining imperfection shapes in the in-plane GMNIA-based design of steel moment frames is the focus of the present study. The proposals are also suitable for GNIA-based design. Both overall sway imperfections (i.e. frame out-of-plumbness) and individual member bow imperfections, together with combinations thereof (i.e. sway-member imperfection combinations), are considered. Several methods for introducing imperfections are assessed. Each method is categorised depending upon whether the imperfection is established through: (1) direct definition or (2) the scaling of eigenmodes. The recommended direct definition-based method introduces a sway imperfection in sympathy with the applied lateral loading and member imperfections with alternating directions between storeys, as dictated by the column base boundary conditions. The recommended eigenmode-based method uses the first sway mode and determines the number of non-sway modes according to the eigenvalues under the design loading, with all non-sway modes for which the eigenvalues αcr,ns < 25 also contributing to the imperfection. The eigenmodes are scaled to ensure the bounds of the prescribed imperfection amplitudes are suitably maintained. The consistency and accuracy of the proposed methods are demonstrated for 21 different frame configurations.

Equivalent sway imperfections and sway-member imperfection combinations for GMNIA-based steel design

Two types of geometric imperfection typically need to be considered in the analysis and design of structural systems: global frame out-of-plumbness, also referred to as sway imperfections, and member out-of-straightness, also known as bow imperfections; the effect of local cross-section imperfections is generally accounted for through cross-section resistance checks. For modelling convenience, equivalent geometric imperfections are often employed to allow for the combined influence of initial geometric imperfections and residual stresses. For design by geometrically and materially nonlinear analysis with imperfections (GMNIA), amplitudes for equivalent bow imperfections for members are provided in EN 1993–1-14; however, no provisions exist for global sway imperfections. Therefore, equivalent sway imperfections suitable for use in the design of steel structures by GMNIA ϕ0,GMNIA are established herein. Two proposals for calculating ϕ0,GMNIA are presented: (1) ϕ0,GMNIA=ϕ0,geom+α/100, with ϕ0,geom representing the pure geometric sway imperfection, and (2) ϕ0,GMNIA=α/50. In both proposals, the varying degree and influence of residual stresses for different cross-section geometries and axes of buckling is captured by the imperfection factor α, as defined in EN 1993–1-1 for carbon steel and EN 1993–1-4 for stainless steel, respectively. A flow chart that provides guidance on the appropriate selection of frame and member imperfections to avoid duplication of the effects of residual stresses in design by GMNIA is also presented.

Equivalent global sway imperfections for use in design by GMNIA

AbstractThree types of geometric imperfection need to be considered in the design of structural systems: global frame out‐of‐plumbness, also referred to as sway imperfections, member out‐of‐straightness, also known as bow imperfections, and local cross‐section imperfections. For modelling convenience, equivalent geometric imperfections are often employed to account for the combined influence of initial geometric imperfections and residual stresses. For design by geometrically and materially nonlinear analysis with imperfections (GMNIA), amplitudes for equivalent bow imperfections for members are provided in prEN 1993‐1‐14; however, no provisions exist for global sway imperfections. Therefore, equivalent sway imperfections suitable for use in the design of steel structures by GMNIA, expressed by the out‐of‐plumbness Φ0,GMNIA are established herein. Two proposals for calculating Φ0,GMNIA are presented: (1) Φ0,GMNIA = Φ0,geom + α/100 with Φ0,geom representing the amplitude of the pure geometric sway imperfection, and (2) Φ0,GMNIA = α/50. In both proposals, the varying influence of residual stresses from different cross‐section geometries and axes of bending is captured by the imperfection factor α, as defined by the buckling curves in EN 1993‐1‐1 for carbon steel and EN 1993‐1‐4 for stainless steel, respectively.

Influence of Imperfections in the Design of Crane Runway I‐Beams

AbstractCrane runway beams are generally loaded by vertical and horizontal forces as well as not‐negligible torsional actions. Horizontal and torsional deformations originate already from the 1st order analysis. The effects of imperfections of crane runway beams could, at least partially, be simulated by these deformations. While for the purely vertically loaded beams it is necessary to consider the horizontal bow imperfection, for the generally loaded beams with horizontal forces as well, the influence of horizontal bow imperfection may not be as important, due to the occurrence of horizontal deflection from primary actions. Eurocode EN 1993‐6 allows to consider either eccentricities of vertical forces or introduction of horizontal bow imperfections in the design of crane runway beams. In this article, using a self‐developed FEM‐program, the effect of horizontal bow imperfection for the generally loaded crane runway beams is investigated. The two options of EN 1993‐6 are compared in the FEM analyses with the simplified design methods like the Annex A method of the same standard and the Naumes method.

Applicable equivalent bow imperfections in GMNIA for Eurocode buckling curves – SHS, RHS and welded box sections

The direct resistance check using advanced numerical models is spreading among designers nowadays. Therefore, the development of accurate and up-to-date design rules supporting the FEM-based design of slender steel structures is an important task and prospective research field. Using the direct resistance check, the load carrying capacity of the steel structure under investigation is obtained as a result of geometrical and material nonlinear analysis using imperfections (GMNIA). The present paper investigates the imperfection magnitudes of slender steel columns and specifies the accurate value of the equivalent geometric imperfection to be applied for welded square, cold-formed and hot-rolled square (with rounded corners), and rectangular sections. Using the proposed imperfection magnitude, the load carrying capacities can be determined by GMNI analysis and the obtained results will be identical or very close to the buckling resistances calculated by the buckling curves of the Eurocode. The developed imperfection magnitude is given in closed form as a function of relative slenderness, steel strength and buckling curve parameter (imperfection factor α). Details on the determination of the necessary equivalent geometric imperfection and the obtained trends on the flexural buckling resistance are presented in the current paper.

COMPARISON OF GEOMETRICAL IMPERFECTION DEFINITIONS ON ENCASED COMPOSITE COLUMNS

AbstractThe imperfections stemming from manufacturing or construction practices have effects on capacities of structural columns. Bow imperfections or residual stresses are commonly investigated and well known examples of these imperfections. Since the composite construction generally has different stages than reinforced concrete or steel construction, there are other sources of imperfections for this field. Different approaches to consider geometrical imperfections are studied for this paper. These are namely the approach of EN‐1992‐1‐1 [1],EN 1994‐1‐1 [2] and EN 13670 [3]. Eurocode 2 and Eurocode 4 specify geometric imperfection for columns differently. Former introduces them as load eccentricity and latter introduces them as bow imperfections. EN 13670 giving the construction tolerances for reinforced concrete structures, has different methods to limit geometrical imperfections. For the purpose of comparing different approaches and their effect on capacity calculations, several composite columns were simulated using physical and geometrical non‐linear analysis, their load‐deflection relationships were monitored and capacities compared. An outlook of the ongoing investigation for the adaptation of geometrical imperfection for composite columns is further discussed.

ASSESSMENT OF GENERAL METHOD FOR COMPOSITE COLUMN DESIGN IN EN 1994‐1‐1 AND COMPARISON WITH SIMPLIFIED METHOD

AbstractComposite column design in accordance with EN 1994‐1‐1 can be computed by the general method or by simplified methods. For the general method, EN 1994‐1‐1 refers to a comprehensive FE‐Analysis. However, the recent Eurocode 4 does not provide sufficient guidance for the application of the general method and there is not a harmony in the literature on how to do it. Additional information may be found as non‐contradictory complementary information (NCI) in National Annexes (NA). This paper points out the procedure for the general method and refers to the changes and additional information provided by the draft of prEN 1994‐1‐1:2021. Furthermore, the modelling and computation by FE‐Analysis for several software applications are presented by implementing specified initial bow imperfections and predefined residual stresses for the steel section. The results obtained with the general method's approach are compared with that calculated by simplified method by the European buckling curves. In addition, the impact of different stress‐strain curves for concrete and strain limits introduced by Codes are investigated.

Equivalent geometric imperfections for the LTB‐design of members with I‐sections

AbstractFor the stability assessment of members and structures according to EN 1993‐1‐1, the equivalent member method, the geometrical non‐linear calculation with equivalent geometrical imperfections or the GMNIA analysis with geometrical imperfections and residual stresses can be used alternatively. The second possibility requires a corresponding model for the cross‐section resistance. For the verification of lateral torsional buckling, bow imperfections e0 out‐of‐plane are defined, which lead in combination with the given loading in‐plane and the geometrical non‐linear analysis to bending moments Mz and torsion of the members. The amplitudes of the imperfections are highly dependent on the nature of the approach (e.g., scaling of the buckling shape, assumption of bow imperfections) and the verification method for the members. Within the framework of the scientific work supervised by the TU Dresden and TU Darmstadt, extensive parametric studies were conducted to calibrate imperfections for lateral torsional buckling based on the GMNIA. After determining the nature of imperfections and the design models for section resistance, this article presents results of these parametric studies and shows the calibration of imperfections for a standardisation proposal based on EN1993‐1‐1. The evaluation of the data in combination with the necessary simplifications for the design practice leads to appropriate definitions of imperfection values e0,LT and the necessary differentiations.

Imperfections for the LTB‐design of Members with I‐sections

AbstractFor the stability assessment of members and structures according to EN 1993‐1‐1 the equivalent member method, the geometrical nonlinear calculation with equivalent geometrical imperfections or the GMNIA analysis with geometrical imperfections and residual stresses can be used alternatively. The second possibility requires a corresponding model for the cross‐section resistance. For the verification of lateral torsional buckling, bow imperfections e0 out of plane are defined, which lead in combination with the given loading in plane and the geometrical non‐linear analysis to bending Mz and torsion of the members. The amplitudes of the imperfections are highly dependent on the nature of the approach (e.g., scaling of the buckling shape, assumption of bow imperfections) and the resistance model for the members. Within the framework of the scientific work supervised by the TU Dresden and the TU Darmstadt [1], extensive parameter studies were carried out to calibrate imperfections for lateral torsional buckling based on the GMNIA. After determining the nature of imperfections and the design models for section resistance, this paper presents results of these parameter studies and shows the calibration of imperfections for a standardisation proposal based on EN1993‐1‐1. The evaluation of the data in combination with the necessary simplifications for the design practice leads to appropriate definitions of imperfection values e0,LT and the necessary differentiations.

Applicable Equivalent Bow Imperfections in GMNIA for Eurocode Buckling Curves – in Case of Box Sections

AbstractThe flexural buckling resistance of compressed columns is typically determined using the flexural buckling curves of EN1993‐1‐1. With advances in computing capabilities, it has become possible to obtain the flexural buckling resistance as a result of a geometrically and materially nonlinear analysis (GMNIA) using imperfections. In design practice, equivalent geometric imperfections are usually used, but their values are calibrated against GNIA analysis. In the present research work, welded box‐section columns are investigated and a proposal for relative slenderness and yield strength dependent equivalent geometric imperfection magnitudes is developed. The required equivalent bow imperfection magnitude is determined to achieve the same resistance level as the calculation according to Eurocode‐based buckling curves. GMNIA analysis are conducted on simply supported columns subjected to concentrated force. The steel grade varied between S235 and S960. Several different section geometries are studied, the relative slenderness ratio is varied between 0.3 and 2.2. Design curve is fitted to the results and recommendation is given for applicable equivalent geometric imperfection magnitudes for steel welded box‐section columns.

The Beneficial Influence of the Roller‐straightening Process on the Bearing Capacity of Steel Columns

AbstractThe roller‐straightening process is a steel cold production technique which consists of passing a steel element through a series of rolls that bend the member, what leads progressively to a reduction of the initial geometrical imperfections along its weak axis. In addition, this post‐treatment process induces a continuous yielding of the steel member and thus may present a second interest which is the reduction of the compression residual stresses at flange tips. This modification of the residual stress pattern could lead to an increase of the carrying capacity of steel columns.Indeed, the capacity of columns in compression is evaluated through the European buckling curves reported in EN1993‐1‐1 in which the compression resistances are highly affected by the initial bow imperfections and residual stresses resulting from the rolling process.However, whilst all long products seem to be systematically straightened nowadays, EN1993‐1‐1 does not consider the potential beneficial effect of the roller‐straightening process. In other words, it means that the current recommendations of EN1993‐1‐1 neglect the potential benefit of this process on the buckling capacity.In this context, a collaboration for a research study has been initiated between ArcelorMittal and the University of Liège to numerically estimate the residual stress patterns after the roller‐straightening process of three relevant heavy H‐shape profiles (HE 600 B, HE 500 M and HD 400 x 262) in HISTAR®460 and evaluate whether these new distributions could lead to a significant increase of the column bearing capacity.

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.

Equivalent bow imperfections for design by second order inelastic analysis

AbstractThe stability of compression members is typically assessed through buckling curves, which include the influence of initial geometric imperfections and residual stresses. Alternatively, the capacity can be obtained more directly by carrying out a second order analysis using equivalent bow imperfections that account for both geometric imperfections and residual stresses. For design by second order elastic analysis, following EN 1993‐1‐1 recommendations, the magnitudes of the equivalent bow imperfections can either be back‐calculated for a given member to provide the same result as would be obtained from the member buckling curves or taken more simply as a fixed proportion of the member length. In both cases, a subsequent linear axial + bending cross‐section check is required. For design by second order inelastic analysis, there are currently no suitable recommendations for the magnitudes of equivalent bow imperfections and, as demonstrated herein, it is not generally appropriate to use equivalent bow imperfections developed on the basis of elastic analysis. Equivalent bow imperfections suitable for use in design by second order inelastic analysis are therefore established in the present paper and shown to yield reliable results when compared to benchmark results and current design guidance.

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.

Equivalent bow imperfections for use in design by second order inelastic analysis

The stability of compression members is typically assessed through buckling curves, which include the influence of initial geometric imperfections and residual stresses. Alternatively, the capacity may be obtained more directly by carrying out either an elastic or an inelastic second order analysis using equivalent bow imperfections that account for both geometric imperfections and residual stresses. For design by second order elastic analysis, following the recommendations of EN 1993-1-1, the magnitudes of the equivalent bow imperfections can either be back-calculated for a given member to provide the same result as would be obtained from the member buckling curves or can be taken more simply as a fixed proportion of the member length. In both cases, a subsequent M–N (bending + axial) cross-section check is also required, which can be either linear elastic or linear plastic. For design by second order inelastic analysis, also referred to as design by geometrically and materially nonlinear analysis with imperfections (GMNIA) there are currently no suitable recommendations for the magnitudes of equivalent bow imperfections and, as demonstrated herein, it is not generally appropriate to use equivalent bow imperfections developed on the basis of elastic analysis. Equivalent bow imperfections suitable for use in design by second order inelastic analysis are therefore established in the present paper. The equivalent bow imperfections are calibrated against benchmark FE results, generated using geometrically and materially nonlinear analysis with geometric imperfections of L/1000 (L being the member length) and residual stresses. Based on the results obtained, an equivalent bow imperfection amplitude e0 = αL/150 (α being the traditional imperfection factor set out in EC3), is proposed for both steel and stainless steel elements and shown to yield accurate results. The reliability of the proposed approach is assessed, using the first order reliability method set out in EN 1990, against the benchmark FE ultimate loads, where it is shown that partial safety factors of 1.0 for steel and 1.1 for stainless steel can be adopted.

Parameter studies on imperfections for the LTB‐design of members based on EN 1993‐1‐1

AbstractSelected, extended paper from the SDSS 2019 special session ECCS/TC8 – Structural StabilityThe stability design of members and structures based on a geometrically nonlinear analysis with equivalent initial geometric imperfections (GNIA) is a commonly used method. In the case of lateral torsional buckling (LTB) of members, the regulations in EN 1993‐1‐1 are unsatisfactory. With the current code, but also with the new draft prEN 1993‐1‐1 results are achieved, some of which are significantly too conservative as well as lacking in terms of safety. The reasons for this are the specified shape of imperfection as bow imperfections e0 out of plane, an inappropriate differentiation related to the cross‐section shape and yield strength, as well as neglecting the influence of the moment distribution over the member lengths. Parameter studies have shown that a distinction is required due to the different structural behaviours of members loaded with pure bending, pure compression, or a combination of bending and compression. This article presents current research results that consider the case of pure bending of I‐ and H‐profiles. The dependencies of the profile shape, the cross‐sectional resistance model and the influence of steel grades are analysed with respect to the height of the equivalent geometric imperfection to be applied.

Stability analysis of steel frames according to Eurocode 3

In the paper, the qualitative and quantitative comparison of EC3 methods to verify the global stability of the structure is presented. The steel portal frame subjected to varied loads is considered. The initial global sway imperfection and the initial local bow imperfections of member frame are taken into account. The sensitivity of a structure to the 2nd order effects is assessed indirectly using the elastic critical load. The elastic critical load of a frame is calculated according to the buckling mode. The 2nd order effects are taken into account using the finite element method. Keywords: second order effects, steel frame, global stability, critical load.

Numerical equilibrium analysis of a stack of steel post pallets

A method for analyzing the equilibrium of a stack of loaded post pallets is presented. The finite element method was used to investigate the behavior of the bottom pallet in the stack during the addition of successive pallets. The stack was regarded as a self-stable multi-storey structure without bracings which is subjected to the weight of loaded pallets, horizontal forces resulting from sway and bow imperfections, and the impact of a forklift truck. The definite quadratic form of the tangent stiffness matrix after every increment in load was determined by nonlinear analysis to indicate the loss of post stability. An analysis of the stacking process of the evaluated pallets did not reveal a buckling trend in the posts of the bottommost pallet and demonstrated that the loss of equilibrium can lead to the collapse of the entire stack when a critical number of pallets is reached.

Discussion on the applicability of the M-N interaction curve for the fire resistance design of CFT members

When subjected to combined axial force and bending moments at ambient temperature, concrete filled tubes (CFTs) show a marginal increase in bending resistance at low level of axial compression, but the bending resistance reduces when the compression force is high. To evaluate the buckling resistance of CFT member subject to axial compression (N) and moment (M), a second order analysis approach is proposed by EC 4: EN 1994-1-1 in which the cross section check can be carried out using the N-M interaction curve which is derived based on the plastic design principle. When the CFTs are further subjected to high temperature due to fire, the design method is however not available in EC 4: EN 1994-1-2. This paper proposes a modified M-N interaction curve based on plastic stress distribution as in EC 4: 1994-1-1 for fire resistance design of CFT members. The superiority of the M-N interaction curve is the universality to both axially loaded members and members subjected to combined axial force and bending moments, regardless of the moments being induced by load eccentricities or initial bow imperfections. Backgrounds of the existing simple calculation model in EC 4 using buckling curves and the proposed M-N interaction curve were first discussed. Then the validity of the said M-N interaction curve was established by comparisons with test results. The proposed M-N interaction curve was found to provide better predictions of the fire resistance than the simple calculation model. Hence, it could be safely extended to EC 4: 1994-1-2 for the design of CFT members under fire.

Initial bow imperfections e0 for the verification of Flexural Buckling According to Eurocode 3 Part 1‐1 – additional considerations

AbstractAccording to Eurocode 3 Part 1‐1, the resistance of compression members may be calculated in two different ways: through the design buckling resistance based on the reduction factor χ or through the cross section resistance based on internal forces according to a second order analysis, also taking into account equivalent initial bow imperfections e0. The second way is especially advantageous in the case of axial forces and bending. Values for equivalent initial bow imperfections e0 are given in codes like Eurocode 3 [4] or DIN 18800 [3]. The presently used values from Table 5.1 of Eurocode 3 will have to be changed in future, as was demonstrated recently by intensive investigations, especially for cases of axial compression and bending moments My or Mz respectively. Additional investigations concerning high‐strength steel S700, welded sections and circular hollow sections are presented. The proposed changes included in the newest draft of prEN1993‐1‐1 are explained through the evaluation of several ultimate load calculations. It is also shown that for the case that the initial bow imperfection e0 is calculated dependent on the non‐dimensional slenderness λ̅, restrictions regarding interaction and the ratio of Mpl/Mel are also necessary.