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

Abstract

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.

Introduction

Imperfections inevitably occur in practice in the manufacturing and fabrication of steel members and in the construction of structural systems. The load-carrying capacity of structural elements in compression is sensitive to imperfections; it is therefore essential that their deleterious influence be accounted for in design. Member imperfections include both initial geometric out-of-straightness and residual stresses. Depending on the adopted method of analysis and design, different magnitudes of initial imperfection are required in order to achieve a given load-carrying capacity. The appropriate choice of imperfection magnitude depends on: (i) the type of imperfection considered i.e. geometric imperfections only or equivalent imperfections accounting for both geometric out-of-straightness and residual stresses, (ii) the analysis type, (iii) the considered cross-section failure criterion and (iv) the benchmark resistance against which the choice of imperfection is assessed. EN 1090-2 [1] specifies manufacturing tolerances on member out-of-straightness and EN 1993-1-5 [2] recommends that 80% of the geometric manufacturing tolerance be applied as the geometric imperfection.

The stability of compression members is typically assessed through buckling curves, in which the effects of imperfections are incorporated. The European buckling curves [3], originally developed in [4], [5], [6], are based on the Perry–Robertson formulation and are underpinned by extensive test and numerical data; member resistances determined using the EN 1993-1-1 [3] buckling curves are therefore presupposed to be “correct” and have been taken as the target results in the development of imperfection rules for design by second order elastic analysis set out in EN 1993-1-1 and its upcoming revision prEN 1993-1-1 [7].

The use of buckling curves and member buckling checks can be avoided if member buckling can be directly captured in the structural analysis. To achieve this, a second order – also referred to as a geometrically nonlinear or advanced – analysis with appropriate member imperfections, is required. Member imperfections can be accounted for either through modelling the imperfect geometry or by applying a set of equivalent horizontal forces. Direct modelling of residual stresses in an analysis can present challenges to the designer; EN 1993-1-1 [3] therefore provides ‘equivalent’ bow imperfections that implicitly account for the combined effects of geometric and material (i.e. residual stresses) imperfections. These equivalent bow imperfections are for use with second order elastic analysis i.e. geometrically nonlinear analysis with imperfections (GNIA). There are no equivalent provisions for use with second order inelastic, or geometrically and materially nonlinear, analysis; this limitation is addressed herein.

Geometrically and materially nonlinear analysis with imperfections (GMNIA), incorporates material nonlinearity (plastic zone/fibre element model) as well as geometric nonlinearity in the analysis, and allows for accurate predictions of the full load–deformation response of a structure. Since the effects of loss of stiffness due to buckling and plasticity of the individual members within the structure are captured, a more accurate representation of the actual distribution of forces and moments, as compared to first order elastic analysis, is achieved. Hence, both safer, more consistent and more efficient design can result from design by advanced inelastic analysis with imperfections or GMNIA. With improvements in computational power and software, design by GMNIA is becoming more widespread in practice and is receiving a growing level of attention in research [8], [9], [10], [11], [12], [13], [14]. Suitable equivalent bow imperfections for use in design by advanced inelastic analysis are therefore urgently needed. While significant capacity benefits are not expected from the use of second order inelastic analysis at the individual member level, the increased accuracy in capturing stiffness reductions and resulting deformations can lead to considerable benefits at the system level, both in terms of a more streamlined design process and structural safety and efficiency. Appropriate recommendations for both steel and stainless steel elements are made in the present paper.