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    Moment-Curvature-Thrust Relationships for Beam-Columns

    Andrew Liewa,*, Leroy Gardnerb, Philippe Blocka

    a Institute of Technology in Architecture, ETH Zurich, Switzerlandb Department of Civil and Environmental Engineering, Imperial College London, United Kingdom

    A R T I C L E I N F O

    Keywords:Moment curvature thrustCurvesCross-sectionMetalBeamColumnPlatedHollow sections

    A B S T R A C T

    Momentcurvaturethrust relationships (MN) are a useful resource for the solution of a variety of inelasticand geometrically non-linear structural problems involving elements under combined axial load and bending. Anumerical discretised cross-section method is used in this research to generate such relationships for I-sections,rectangular box-sections and circular or elliptical hollow sections. The method is strain driven, with themaximum strain limited by an a priori defined local buckling strain, which can occur above or below the yieldstrain depending on the local slenderness of the cross-section. The relationship between the limiting strain andthe local slenderness has been given for aluminium, mild steel and stainless steel cross-sections through the basecurve of the Continuous Strength Method. Moment-curvature-thrust curves are derived from axial force andbending moment interaction curves by pairing the curvatures and moments for a given axial load level. Thesemomentcurvaturethrust curves can be transformed into various formats to solve a variety of structuralproblems. The gradient of the curves is used to find the materially and geometrically non-linear solution of anexample beam-column, by solving numerically the momentcurvature ordinary differential equations. Theresults capture the importance of the second order effects, particularly with regard to the plastic hinge formationat mid-height and the post-peak unloading response.

    1. Introduction

    For a given cross-section, such as an open or closed metal I-sectionor tubular section, momentcurvature (M) curves can be created.Such curves can be used to describe the behaviour of each cross-sectionand subsequently the entire length of a structural member, subjected toa given applied load. Generating the M curve is straightforward whenthere is no applied axial load, since the strains throughout the cross-section are exclusive to flexure, which can be described as linearlyvarying with depth, with the highest strains at the outer fibres. This isbased on the assumption that plane sections remain plane duringbending, which has been shown to be valid for practical structuralsteel cross-sections in bending, as determined from strain gauge read-ings on I-sections up to and beyond the plastic moment [1]. This alsostems from the fact that cross-section dimensions are generally con-siderably smaller than beam lengths, permitting the neglect of sheardeformations [2]. Combining this assumed strain profile with aparticular material model, M curves can be generated analytically.

    The determination of M curves in the presence of a given axialload is more challenging, due to the interaction between the axial andbending strains and material non-linearity. Expressions for solidrectangular sections with an elastic-perfectly plastic material model

    can be found in [2], which also describes other approximations fordifferent cross-section shapes. Finding accurate analytical curves forcross-section shapes typically used in structural applications and withmore realistic material stressstrain curves is significantly more chal-lenging as a continuous function is needed in the entire MN domain,that is initially straight in the elastic region and then transitionsthrough to a curved shape in the inelastic regime. The calculationand application of momentcurvaturethrust relationships in the lit-erature include: steel reinforced rectangular masonry sections usingnon-linear constitutive models [3]; momentcurvature relationships forvarious tubular cross-sections with residual stresses, geometric imper-fections and hydrostatic pressures via the tangent stiffness Newmarkmethod in [4]; using MN curves to analyse the ultimate strength ofdented tubular members by [5]; creating curves by results from finiteelement analyses as in [6]; non-linear analyses of reinforced concretebeams considering tension softening and bond slip using momentcur-vature curves from a section analysis in [7]; momentcurvature curvesand comparisons with experimental results of CFRP-strengthened steelcircular hollow section beams by [8] and with concrete-filled hollowsection tubes [9]. Fibre based models, where the cross-section isdiscretised into a finite number of thin strips, have been used to modelconcrete-filled steel tubes [10], as well as the static and dynamic 21 February 2017; Received in revised form 22 May 2017; Accepted 23 May 2017

    * Corresponding author at: Institute of Technology in Architecture, Zurich 8093, Switzerland.E-mail address: (A. Liew).

    Structures 11 (2017) 146154

    Available online 24 May 20172352-0124/ Crown Copyright 2017 Published by Elsevier Ltd on behalf of Institution of Structural Engineers. All rights reserved.


  • response of reinforced concrete columns and beam-columns inframes [11,12].

    In the present study, a numerical procedure is employed in whichcross-section interaction curves are first formed and then used to findall possible strain distributions that correspond to the same axial loadlevel, leading to the required MN curves. The approach is demon-strated for cross-sections with at least one line of symmetry, such as I-sections and tubular sections and for a bi-linear material model, but themethod can be used for any cross-section geometry. Key features of themethod are that: 1) it is not specific to a particular material model, asany stressstrain curve can be used with the strain driven approach, 2)local buckling of thin-walled elements can be explicitly includedthrough a limiting strain ratio due to the integration with theContinuous Strength Method, 3) any cross-section shape that can bediscretised into discrete elements may be analysed, and 4) theprocedure allows for efficient calculation of the inelastic flexuralstiffness and plastic hinge regions of a member, subjected to givenaxial load levels.

    2. Cross-section model

    The central aspects of the cross-section model are described in thissection. The cross-section strain and stress distributions are presented inSection 2.1 and a limiting strain is set-out in Section 2.2 to define cross-section failure through local buckling.

    2.1. Strain and stress distributions

    The strain and stress distributions for a cross-section under axialcompression, uni-axial bending and combined axial compression anduni-axial bending are shown in Fig. 1 for the case of a material with arounded stressstrain relationship.

    For the pure axial load state (Fig. 1a), the strains are uniformthroughout the cross-section at A. When the uniform strain is less thanthe material yield strain, A< y, the cross-section stresses are below fy,and when A y the cross-section is deforming inelastically accordingto the chosen material stressstrain curve, up to limiting ultimate valuesfor the cross-section of u and fu.

    For the case of simple bending (Fig. 1b), there is no uniform strainpresent, only linearly-varying flexural strains with a maximum value ofB. The strain and stress profiles are antisymmetric about the zero strainneutral axis (which is located at mid-depth for symmetric sections), andthe upper and lower outer-fibres reach u and fu.

    The combination of an axial load and bending moment is illustratedin Fig. 1c, where the compressive outer-fibre strain is limited to u. Theinteraction of axial strains and bending strains is taken as the summa-tion of the uniform strains A and the linearly varying strains withmaximum magnitude B. This combination leads to strain and stressprofiles between the axial and flexural states. Although strains A and Bare linearly superimposed, the stresses are based on the given materialmodel. Therefore A is not solely responsible for defining the axial forceand neither is B exclusive to bending; it is the combination of A and Bthat defines the axial and bending capacity.

    The approach adopted herein follows the principles laid-out inresearch by the first two authors on the deformation-based ContinuousStrength Method (CSM) [13,14]. In the CSM, the limiting straindistribution is first established and the stresses and hence capacityfollow; this is in contrast to traditional design [15]. The approachdescribed has been used previously to analyse the combined loading (N,My, Mz) of cross-sections to produce interaction surfaces and designcurves [16].

    Since a strain driven approach makes no assumptions on thematerial model, the same method can be used for a variety of structuralmaterials. This includes metallic construction materials such as struc-tural steel, stainless steel and aluminium, as well as compositeconstruction materials such as reinforced concrete or fibre-reinforcedpolymers. For cross-sections consisting of two or more materials, thekey condition to satisfy for the proposed method, is that there is acompatibility of strains at the interfaces between materials (such asproviding full bond to rebar for reinforced concrete cross-sections).

    In this resea


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