Ultimate Capacity of Structural Steel Cross Section under Compression, Bending and Combined Loading

Call for Papers Engineering Journal, May 2019

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Ultimate Capacity of Structural Steel Cross Section under Compression, Bending and Combined Loading

Jyoti1, Rachna 2 , Satish Kumar 3, Jaidutt4

1,2,3,4Department of Civil Engineering, Applied college of Management & Engineering,

Palwal Haryana, India

Abstract- The Continuous Strength Method (CSM) is a strain based structural steel design approach which allows for the beneficial influence of strain hardening. The method has been previously developed for predicting compression and bending resistances in isolation. This paper describes extension of the method to enable the prediction of the ultimate cross-section resistance of I-sections and box sections under combined loading. At the core of the method is a base curve, which relates the deformation capacity of a cross-section to its cross- section slenderness. Deformation capacity is defined as the ratio of the maximum strain that a cross-section can endure relative to its yield strain. Knowing this limiting strain and assuming plane sections remain plane, the resistance of a cross-section to combinations of axial load and bending moments can be calculated, by integrating the stresses arising from a suitable strain hardening material model over the area of the cross-section. By considering a range of combinations of applied actions, analytical expressions and numerically derived interaction surfaces have been produced, which were then rationalised into simple expressions for use in design. The resulting CSM design predictions for box sections and I- sections have been compared with existing test data, and shown to give additional capacity over current design approaches and a reduction in scatter of the predictions.

Keywords- Compression; Bending; Combined loading; Steel structures; Strain hardening; Continuous Strength Method


    Design rules for structural steel cross-sections often include simplifications that allow quick and conservative estimates of capacity (e.g. the ability to withstand combinations of axial forces, shear forces and bending moments) to be obtained. Some of these simplifications are at the material level, where structural steel is typically assumed to have an elasticplastic or rigidplastic stress strain ( ) response, some are based on equations limited by elastic conditions, while other approximations involve the grouping of similar behaviour, as in the case of cross- section classification for the treatment of local buckling.

    At the ultimate limit state, a cross-section subjected to flexure is typically designed on the basis of its plastic (Mpl = Wplfy) or elastic moment capacity (Mel = Welfy), where Wpl and Wel are the plastic and elastic section moduli and fy is the material yield stress. The choice between the two is based on the susceptibility of the cross-section to local buckling, which is assessed by considering the width-

    to-thickness ratios of the elements that make up the cross- section through a process known as cross-section classification. For slender cross-sections, where local buckling occurs prior to the initiation of yielding, reduced moment capacities are assigned. This approach generally results in a step from Mel to Mpl at a particular slenderness limit.


    The stressstrain ( ) response of structural steel can differ depending on the material grade and how the material has been manufactured, subsequently mechanically worked, and ultimately tested. Hot-rolling ( Fig. 1a) or cold-forming ( Fig. 1b) can affect the material behaviour by altering the distinctiveness of the yield point, the length of the yield plateau, and the magnitude of the strain hardening slope. Variation in material properties around structural cross-sections is also possible, such as in the case of cold-formed sections, where higher strength but lower ductility are typically found in the corner regions..

    Fig. 1. Schematic stressstrain curves (a) for hot-rolled material, (b) cold-formed material and (c) the CSM material model.

    In Fig. 1, E is the Young's modulus, fy and fu are the yield and ultimate tensile stresses, y = fy/E and u are the strains at the yield and ultimate stress, 0.2 and t,0.2 are the 0.2% offset proof stress and corresponding strain, Esh is the strain hardening slope and fcsm and csm are the CSM limiting stress and strain. Traditionally a bi-linear, elasticperfectly plastic material model is used to model structural steel, with the key advantage of being very simple to analyse, but with the potential disadvantage of being overly conservative since no post-yield strain hardening is accounted for. Alternatives to the elasticperfectly plastic

    model include bi-linear (elasticlinear hardening), tri- linear, power, Ramberg and Osgood and other stress strain models. In principle, any material law can be used in conjunction with the deformation based CSM. The proposed material model ( Fig. 1c) is an elasticlinear hardening relationship, which consists of an initial linear region with Young's modulus E defining stresses up to the yield stress, followed by a strain hardening region, described by an appropriate strain hardening modulus Esh for the material. A maximum limiting strain is also set at

    15 times the yield strain (csm/y = 15), a value which corresponds to the material ductility requirements given in Clause 3.2.2(1) of EN 1993-1-1. This material model gives the following stressstrain relationship:



    The key characteristic to be defined in the adopted material model is the strain hardening modulus Esh, which should be representative of the whole cross-section.


    Local plate buckling may initiate before or after the onset of material yielding, with the key determining geometric factor being the relative width-to-thickness ratios of the plate elements that make up the cross-section. Plate slenderness is commonly defined in the non-dimensional form of equation(2)


    where cr is the elastic buckling stress, which is influenced by the boundary and loading conditions of the plate. The plate slenderness values of all the elements that make up the cross-section are evaluated, with the critical and governing element determined as that with the highest value of ¯p. Since basing the cross-section slenderness upon the most slender constituent plate element does not consider the connectivity between the plates.Cross-section slenderness reduces when element interaction is considered; this reduction is by a factor of between 0.85

    1.0 for axial loading, and approximately 0.701.0 for major or minor axis bending for typical hot-rolled structural profiles


    The CSM is a deformation based design approach, founded upon a derived relationship between the failure strain of a cross-section and its local slenderness. The results of both stub column and in-plane bending tests can be used in the derivation of this relationship. A stub column is defined herein as a column with a global non-dimensional slenderness ¯0.1, where ¯=Ny/Ncr, Ny being the yield load of the cross-section and Ncr the

    elastic buckling load of the member. While meeting the above requirement of ¯0.1 to avoid any significant influence from global buckling, the test lengths L of stub columns should also ideally be at least three times the larger cross-sectional dimension, in order to contain a representative distribution of geometric imperfections and residual stresses and to allow local failure modes to form without a strong influence fromend effects.A typical load end-shortening (N ) curve from a stub column test is shown in Fig. 2, where loads above the yield load will be reached if the cross-section slenderness is sufficiently low to allow stresses to enter the strain hardening regime. The end-shortening at the ultimate load Nlb (i.e. the peak load Nu achieved in the stub column test), is divided by the length of the specimen to obtain the average failure strain of the cross-section lb. The deformation capacity of the stub column is then defined as csm, which is taken directly as lb for materials that exhibit a distinct yield point and as lb 0.002 for materials with a rounded stressstrain curve. The subtraction of 0.2% strain in the case of rounded stressstrain curves is to ensure compatibility with the chosen material model of Fig. 1c and to avoid over- predictions of capacity. The CSM strain is normalised by the yield strain in Eqn. (3), and csm/y is referred to as the strain ratio.


    csmy=lbyforhotrolledmaterialswithdistinctyieldpointl b0.002yforcoldformedandnonlinearmaterials.

    Fig. 2. Stub column loaddeformation response.

    Cross-sections that fail at strain ratios greater than unity can exceed their yield loads and mobilise some of their strain hardening potential. Note that, in the present study, when considering test specimens that exhibit a variation in material properties around the cross-section (e.g. enhanced strength at corners due to cold-forming), area weighted average values have been used to establish Ny and y. For slender cross-sections (defined herein as cross-sections with ¯p>0.68), elastic local buckling is followed by stable post-buckling, which results in increased capacities but with reduced axial stiffness. The consequence of this is that a slender cross-section can have a high deformation capacity (i.e. strain at failure), greater than y, but a peak load still below the yield load. This would result in an over-prediction of capacity when using

    the CSM. To avoid this, the deformation capacity of slender cross-sections is defined by Eqn. (4).



    The value of ¯p=0.68 has been found by Afshan and Gardner to represent the transition point for which cross-sections behave as either slender (achieving peak loads below the yield values Ny and Mel), or as non-slender (achieving peak loads above the yield values). For cross- sections in bending, the strain distribution is assumed to be linearly-varying through the cross-section depth, and a relationship can be made between curvature and outer-fibre strains. With the height and width of a cross-section denoted as h and b, the yield curvatures y,y and y,z for the major and minor axes are defined as:



    These are the curvatures at which a cross-section will reach its major and minor axis elastic moments, Mel,y and Mel,z respectively. Assuming that plane sections remain plane during bending, there is a proportional relationship between the strains at ±h / 2 and ±b / 2 and curvature. This permits the equivalence of the strain ratio csm/y to the curvature ratio csm/y, where csm is the curvature when the strain csm is reached in the outer-most compressive outer- fibre of the cross-section.


    in which lb is the curvature at maximum moment Mlb (or Mu). Hence csm is related to lb in a similar manner to the relationship between csm and lb, as defined by Eqn. (6) for the case of the major axis. For slender cross-sections ¯p>0.68) in bending, the definition of curvature ratio, given by Eqn. (6) has been modified for similar reasons to those previously explained for compression:


    had been tested to failure,; the majority of the test cross- sections were non-slender and had therefore reached axial loads and strains above the yield values, Ny and y. In addition, test data on short or laterally restrained beams in four-point bending, and were also collated. These data are plotted in Fig. 3 on a graph of strain ratio versus cross- section slenderness, alongside equivalent stainless steel and and aluminium and test data.

    Fig. 3. CSM base curve with stub column and bending test data for structural steel, stainless steel and aluminium.

    The test data shows a clear trend of increasing deformation capacity with reducing cross-section slenderness (i.e. lower ¯p), with the strain at peak load sometimes exceeding 25y. For the slender cross-sections ¯p>0.68), the strain ratio drops below the elastic value of csm/y = 1. A non-linear least squares fit to the collected data set, excluding cross-sections where ¯p>0.68, is given by Eqn. (8), with an upper bound of 15 applied to the strain ratio to avoid excessive strains and to remain within the fracture ductility limits set out in EN 1993-1-1. Fig. 3 shows a good match between the collected test data and Eqn. (8). Similar agreement was observed between a slightly modified version of Eqn. (8) by [9] and the numerical results for a range of material models. Hence, Eqn. (8), referred to as the CSM base curve, may be used to predict the CSM failure strain csm from the cross-section slenderness ¯p.








      For a column unaffected by global buckling, the strains in

      This allows experimental bending data to be plotted on a common deformation capacityslenderness curve with axial test data;


The CSM base curve relates normalised cross-section failure strain csm/y to cross-section slenderness ¯p. Stub column test data were collected for specimens that

the cross-section are assumed to be uniform at A. When A < y, the cross-section is within its elastic material range. However, when A y the cross-section is deforming inelastically and, following the CSM strain hardening material model described in Section 2.1, will reach the CSM limiting stress fcsm. Thus, for csm/y > 1, the CSM axial resistance Ncsm given by Eqn. (9), will be greater than the yield load. In Eqn. (9), M0 is the partial factor for

cross-section resistance, with a recommended value of unity.


For box sections bending about the minor axis, the Ww

modulus in Eqn. (12) is similar to Eqn. (11): equation(12)




In bending, a cross-section that has a strain ratio equal to unity will have an outer-fibre stress equal to the yield stress, and will fail at the elastic moment capacity Mel. For higher strain ratios, the outer-fibre CSM stress fcsm will be greater than the yield stress fy, and the cross-section resistance will exceed the elastic moment capacity. This results from the fact that the integrals Mcsm,y = y dA and Mcsm,z = z dA, where stresses are defined from a linear strain distribution and the CSM bi-linear material model, and y and z are the distances from the neutral axes, will result in moments greater than Mel. Cross-sections with higher csm/y enter further into the strain hardening regime and can achieve Mcsm > Mpl.For the cross-section geometry shown in Fig. 4, an analytical moment resistance expression was presented by [9], which gave the normalised moment capacity of Eqn. (10), where the appropriate section moduli Wpl,y and Wel,y or Wpl,z and Wel,z are used for bending about the yy or zz axes.




Fig. 4. For box and I-sections bending about the major axis, the

Ww modulus is defined by Eqn. (11):


Wwtwp12csmy2for1+2tfhwcsmybp12csmy2 btw12hw232hwhcsmyfor1+2tfhw>csmy1.



I-sections perform notably different aout the minor axis and have modulus Ww as:


Ww=hb212csmy2forcsmybtwtw2hw212tw3bcsm y+tfb26csmy2forcsmy<btw.

The analytical equations are exact for the chosen bi-linear material model, but are lengthy for practical design due to the Ww term. However, accurate approximations to Eqns. (11), (12) and (13) are achieved using the simpler design expression of Eqn. (14):

Mcsm=WplfyM01+EshEWelWplcsmy11WelWpl/ csmy

where = 2 for I-sections bending about the major axis and box sections bending about either axis, and = 1.2 for I-sections bending about the minor axis. This CSM design bending equation requires the assignment of three ratios to the cross-section: the ratio of strain hardening modulus to Young's modulus Esh/E, the shape factor reciprocal Wel/Wpl, and the strain ratio csm/y.


    Combined loading differs from the basic cases of axial load or uni-axial bending in that stresses and strains depend on both the y and z coordinates (i.e. those in the plane of the cross-section). The proposed CSM model for doubly symmetric cross-sections under combined loading is based on orientations of a rotated (due to bending) planar strain surface shifted by a uniform compressive strain A and limited by the CSM strain csm. Examples of such strain planes are shown in Fig. 5, from which the stresses and cross-section resistances to axial forces and bending moments may be derived; this is undertaken by means of a numerical model, as described in Section 4.2.

    lower values of B and C). With the strain at each element known, the element stress fi normalised by the yield stress is calculated from Eqn. (16).


    Fig. 5. Strain distributions in cross-sections under combined



    axial load and bi-axial bending; compression is positive and NA is the neutral axis.

    For bending about both axes and with A = 0 (Fig. 5a), the area and distribution of stresses in compression equal those in tension and no net axial force is produced; when A is non-zero a net axial force exists. Failure of the cross- section is deemed to occur when the maximum compressive strain reaches the CSM limiting strain csm. With reference to Fig. 5, the maximum compressive strain will occur at y = h/2, z = b/2 for box and I-sections. The uniform compressive strain A has the effect of preventing the tensile corner strain (at y = h/2, z = b/2) from reaching csm and accelerating the compressive corner strain reaching csm ( Fig. 5b and Fig.5c).


    In this subsection, cross-section capacities under combined loading are determined by means of a numerical model. In the numerical model, the cross-section is divided into i elements each of area Ai, and with element centroids at distances yi and zi from the centroidal axes of the cross- section. There are three strain components that must be considered: A, B and C. The strain A is the uniform strain across the whole of the cross-section, associated with axial deformations. The strains B and C describe bending about the major and minor axes respectively. The linearly varying strains are associated with flexural deformations and are defined by 2yiB/h for the major axis and 2ziC/b for the minor axis. The numerical analysis procedure is initiated with a given value of A/csm from between 0 and 1, with A = 0 indicating no axial deformations and A = csm indicating only axial loading. The parameter B/csm is then varied from 0, for no major axis bending, to 1 A/csm for axial load plus major axis bending. This limits the maximum value of the minor axis bending parameter C/csm = 1 A/csm B/csm. The strain i at element i is then given by Eqn. (15), based on the addition of A, B and C, all normalised by csm. equation(15)

    icsm=Ac sm+Bcsm2yih+1AcsmBcsm2zib

    The failure criterion is based on 2y /h = 2z /b = 1, that is the

    The normalised cross-section resistances N (axial force), My (major axis bending) and Mz (minor axis bending) are then:


    NNy=1AififyAi equation(18)

    MyMpl,y=1Wpl,yififyyiAi equation(19)


    The output from the numerical model is a suite of interaction curves between My and Mz, as displayed in Fig. 6 for an I-section with a strain ratio of 15. This forms an interaction surface that can be used to create design interaction curves for different combinations of axial load and bending moments. The curves and surfaces can be plotted in two or three-dimensional space with respect to the plastic resistances Mpl,y, Mpl,z and Ny or, more suitably, with respect to the CSM resistances Mcsm,y, Mcsm,z and Ncsm.

    Fig. 6. Typical interaction surface generated from the numerical model for an I-section with a strain ratio csm/y = 15.

    The numerically generated surfaces can be used to create interaction curves between major axis and minor axis bending for a given axial load, and for a constant strain

    i i ratio. This gives contours of constant axial load by slicing

    CSM limiting strain csm being reached at the corner element where yi = h/2, zi = b/2, and A + B + C = csm. As A increases, the strain plane can rotate to a lesser extent before csm is reached and hence the lower the capacity of the cross-section to co-existing bending (corresponding to

    through the interaction surface with a series of parallel planes. Examples of these contours with curves normalised by the plastic moment capacities, are displayed in Fig. 7 and Fig. 8 for a typical I-section and box section respectively, both of which have a strain ratio of 15 and are

    plotted with and without strain hardening (Esh = E/100 and Esh = 0). curves provide a good representation of the numerically generated results for the case of no strain hardening, though some deviation may be seen in the middle region of the I-section curves. It may also been seen that as the axial load increases, the bi-axial bending interaction curves from both the numerical model and EN 1993-1-1 contract towards the origin and become more rectangular in shape. The influence of strain hardening, which is essentially to expand the interaction surfaces, may also be clearly seen from the numerical results. In the following subsection, design interaction expressions are developed on the basis of the numerical results.

    Fig. 7. Bi-axial bending interaction curves for a typical I- section at fixed axial load levels n = N/Ny.

    MR,csm,z, and map surfaces that conform well to the numerical model surfaces.equation(20)

    MyMR,csm,y+MzMR,csm,z1 equation(21)

    MR,csm,y=Mcsm,y1nay1byandMR,csm,z=Mcsm,z1naz 1bz

    The design equations simplify when the appropriate loading terms are taken as zero. For example, when there is no axial load (n = 0), the reduced moments in Eqn. (21) collapse to the CSM moments and convert Eqn. (20) into a bi-axial bending design equation, and when either My or Mz is zero, the equations collapse into the simple axial load and uni-axial bending forms of Mz MR,csm,z and My MR,csm,y respectively.The powers ay, az, by, bz, and are all defined in Table 1. The tabulated powers were found via a non-linear least squares fitting regime, and are based on the ratio of the cross-section web area to gross area a = Aw/A and the ratio of the major to minor axis plastic section moduli Wr = Wpl,y/Wpl,z. A strain ratio of 5 is required before the convergence of the powers for I- sections, compared to that of 3 needed for box sections. The powers ay, az, by, bz, and are all unity when csm/y < 3.


    Fig. 8.Bi-axial bending interaction curves for a typical box section at fixed axial load levels n = N/Ny.

    2 1.5n 1

    0.8 + 5n2.2


    2 + 0.15Wr 5n1.5


    0.8 + (15 Wr)n2.2


    1.75 + Wr(2n2 0.15) 1

    .7 + Wr

    1.6 + (3.5 1.5Wr)n2 3.

    7 Wr


    This subsection introduces the CSM design equation for combined loading, shows comparisons with the numerical model presented in Section 4.2, and defines a goodness of fit by a residual. The proposed design equations for the prediction of the combined axial load and bending capacity of a cross-section, are based on curve fits that rationalise the numerical model surface in the CSM domain. In CSM normalised space, the surfaces are bound to a unit box defined by Mcsm,y, Mcsm,z and Ncsm; this prevents the interaction surfaces from progressively expanding outwards away from the origin as the strain ratio (and hence cross-section capacity) increases.The proposed form of design equations, given by Eqns. (20) and (21), trace bi- axial bending interaction curves that are anchored by reduced moments MR,csm,y and MR,csm,z, which are functions of the axial load level n = N/Ncsm. Eqn. (20) contains two reduced moment normalised terms for major and minor axis bending, raised to powers and respectively.The equations provide smooth curves between MR,csm,y and

    The CSM design equation for a typical I-section (for strain ratios 515) and box section (for strain ratios 315) are plotted in Fig. 9 and Fig. 10 respectively, together with the corresponding numerical envelopes. For both cross- sections, the design equations fit well to the numerical model, providing interaction curves that pass through the middle of the envelopes, or giving predictions on the conservative side. For the I-section, the straight sides of the curves at high axial loads are produced by the larger powers.

    Fig. 9. I-section numerical model envelopes (strain ratios 5

    15) and the CSM design equation for combined loading.

    Fig. 10. Rectangular hollow section numerical model envelopes (strain ratios 315) and the CSM design equation for combined loading.

    In order to assess the conformity between the design interaction surfaces and the numerical results, a residual is defined. The selected residual definition is the distance between the design interaction surfaces produced by Eqns. (20) and (21) and an associated point produced by the numerical model, based on a projection from the origin to the surface. The line from the origin to the known numerical data point R, will intersect the trial surface at point S, with the distance from the origin to this intersection as S=cR. The scalar c for every interaction surface point is then found by solving implicitly by a numerical root finding procedure, and the residual is then expressed as 100(1 c) %. A positive value of the residual corresponds to the numerical point lying outside the design interaction surface and therefore a conservative prediction of capacity. For a negative value, the opposite is true.

    Fig. 11 and Fig. 12 show shaded surface maps of the residuals for a typical I-section and box section respectively. For the I-section, the magnitudes of the residuals are less than 3.5%, with a negative region appearing towards the centre of the surface, and two smaller concentrations at the Mcsm,y and Ncsm end points. The shaded residual plot is more uniform for the box section and gives residuals of lower magnitude, the highest being approximately 2.5%. A similar residual analysis based on the EN 1993-1-1 combined loading equations, but using the CSM axial and moment capacities in place of the yield load and plastic moment capacities, leads to residuals which are on average 12% greater in magnitude than the CSM design equations using the powers.

    Fig. 11. I-section residuals (%) between the CSM design equation and the numerical results (strain ratio of 15).

    Fig. 12. Rectangular hollow section residuals (%) between the CSM design equation and the numerical results (strain ratio of 15)


The Continuous Strength Method (CSM) is a strain based structural design approach that has been previously developed for cross-sections under either compression or bending. Herein, the method was extended to the case of combined loading. Analyses have been performed for structural steel I-sections and box sections via a strain based numerical model, and rationalised with simple equations suitable for use in design.

A bi-linear material model was represented that combined the stressstrain relationships of hot-rolled and cold- formed structural steel and allowed for the benefits of strain hardening. A maximum strain of 15 times the yield strain was allowed based upon ductility requirements, and a value of Esh = E/100 was selected to represent the strain hardening modulus. A strain ratio and curvature ratio were defined as the strain or curvature at the peak axial load or peak bending moment from tests, and then normalised by the yield values. Higher values of strain ratio indicated greater deformation capacity and hence increased resistance to local buckling. The relationship between the strain ratio and the cross-section slenderness was displayed on a base curve, for which a suitable expression was presented.

Using the two key components of the CSM the base curve and the material model, cross-section resistance expressions for axial compression and bending in isolation were presented. Comparisons with test data revealed that the CSM provides closer and more consistent predictions of capacity than existing design codes, through a rational allowance of strain hardening.

Extension of the strain based design from simple axial and bending resistances to the general case of combined loading was then presented. A numerical method was formulated to generate interaction surfaces for box sections and I-sections, by computing all permutations of uniform and linearly varying strains, which when combined did not exceed the CSM limiting strain in compression. From the numerical results, design expressions to describe the interaction surfaces, anchored to the CSM end-points, were developed, with the key parameters determined by means of non-linear least squares regression. The accuracy of the fits to the numerical model envelopes was assessed through a residual analysis, which showed excellent agreement with deviations of less than 3.5%. Use of the EN 1993-1-1 interaction curves, but with CSM end-points, also yielded accurate results. Comparisons of the resulting design expressions were then made with tests, which showed similarly good agreement to that observed under the individual load cases.


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Jyoti B.Tech., M. Tech. Scholar in Civil Engineering (Structural Design) from Ganga Institute of Technology and Management, Kablana, Jhajjar, Haryana (India) affiliated to Maharshi Dayanand University, Rohtak, Haryana (India).

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