Equivalent bow imperfections for use in design by second order inelastic analysis
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.
Section snippets
Eurocode 3 provisions for design by second order elastic analysis
In EN 1993-1-1 [3] and prEN 1993-1-1 [7], equivalent initial bow imperfection magnitudes e0 for design by second order elastic analysis may be determined in two ways; (1) by back-calculating the required imperfection from the buckling curves or (2) by the use of approximate tabulated values. The tabulated approach [15], [16] generally yields upper bound values of equivalent bow imperfections relative to the back-calculated values, resulting in safe-sided predictions of buckling resistance. The
Introduction and illustration of shortcomings in current provisions
The equivalent imperfections included in prEN 1993-1-1 [7] were derived on the basis of a second order elastic analysis and a linear M−N cross-section design check. Their use in design by second order elastic analysis gives the same result as the EN 1993-1-1 buckling curves if the back-calculated imperfection values are used, or a close, but safe-sided buckling resistance if the tabulated imperfection values are used. However, use of these imperfections, whether employing the back-calculated or
Assessment and illustration of design recommendations
The reliability of the developed design recommendations i.e. design by second order inelastic analysis with equivalent bow imperfections determined from Eq. (14) with β = 1/150 and cross-section checks (both strength and strain based) as specified in Section 3.4, is assessed in this section with respect to the benchmark FE ultimate loads determined using GMNIA of columns and beam-columns with geometric imperfection amplitudes of L/1000 and residual stresses. Comparisons are also made against
Conclusions
prEN 1993-1-1 provides equivalent bow imperfections that account for the combined effects of geometric imperfections and residual stresses for use in design by second order elastic analysis. If the back-calculated values are used, the resulting capacity is the same as that achieved using the buckling curves, while close, but safe-sided buckling resistances are obtained if the simplified tabulated values are used. While these imperfections are appropriate for use in design by second order
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
Funding for this investigation was received from the Imperial College PhD Scholarship scheme.
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