Design of laterally restrained web-tapered steel structures through

a stiffness reduction method

Merih Kucuklera,b,∗, Leroy Gardnerb

aDepartment of Civil Engineering, Bilecik Seyh Edebali University, Bilecik, 11210, TurkeybDepartment of Civil and Environmental Engineering, South Kensington Campus, Imperial College

London, London, SW7 2AZ, UK

Abstract

A stiffness reduction method for the design of laterally restrained web-tapered steel struc-tures fabricated through the welding of individual steel plates is presented in this paper.Stiffness reduction functions for welded members, accounting fully for the deleterious influ-ence of the spread of plasticity and imperfections on the structural resistance, are developed.The method is implemented through (i) dividing tapered members into prismatic segmentsalong their lengths, (ii) reducing the flexural stiffness of each segment by means of thedeveloped stiffness reduction functions considering the first-order forces and cross-sectionproperties of each segment, (iii) performing Geometrically Nonlinear Analysis and (iv) mak-ing cross-section strength checks. Essentially, it is proposed to replace the current typicalapproach to structural design of conducting a simple elastic (with nominal stiffness) struc-tural analysis followed by elaborate member checks with an integrated process utilising moresophisticated second-order analysis (with stiffness reduction) but very simple design checks.The distribution of internal forces within the structure is captured more accurately due tothe allowance for imperfections, residual stresses and plasticity through stiffness reductionand the allowance for frame and member instability effects through the use of second-orderanalysis. The need for determining effective lengths and for conducting member bucklingchecks is also eliminated. Verification of the proposed approach against the results obtainedfrom nonlinear shell finite element modelling is presented for various tapering geometries,slenderness values and loading conditions. Assessment of the proposed method against theEuropean and North American steel design codes for tapered steel structures is also provided.

Keywords: Stiffness reduction; tapered steel members ; inelastic buckling; imperfections;residual stresses; finite element modelling; tapered steel frames

∗Corresponding authorEmail addresses: merihkucukler10@imperial.ac.uk (Merih Kucukler),

leroy.gardner@imperial.ac.uk (Leroy Gardner)

Preprint submitted to Journal of Constructional Steel Research October 27, 2017

1. Introduction

To achieve greater economy in material use, web-tapered steel members, where deepercross-sections are used within the regions subjected to larger internal forces, are commonlyemployed within steel structures. However, the methods provided in current structuralsteel design specifications [1, 2] for the instability assessment of web-tapered steel membersare based largely on those developed for prismatic steel members, often leading to overly-conservative estimations of their ultimate strengths and thus limiting the efficiency gainsachieved through their use.

For the purpose of establishing accurate methods for the design of tapered steel mem-bers, various design approaches have been proposed in the literature. Shiomi and Kurata[3] recommended empirical formulae for the transformation of tapered members into pris-matic members with equivalent lengths. This method is straightforward to apply but leadsto rather conservative results, particularly for tapered beam-columns due to its adoptionof a linear interaction equation. Raftoyiannis and Ermopoulos [4] proposed the use of theequivalent geometrical imperfections provided in [1] for the design of eccentrically loadedtapered columns. Salem et al. [5] put forward an empirical equation for the flexural buck-ling assessment of tapered columns with slender cross-sections. Braham and Hanikenne [6]adopted the Merchant-Rankine equation for the lateral-torsional buckling assessment of ta-pered beams. Though these approaches led to strength predictions close to those obtainedthrough nonlinear finite element modelling, they were only investigated for rather few cases.Marques et al. [7] put forward design equations with a clear mechanical background, whichwere calibrated to the results of a large number of finite element analyses. This method [7]is straightforward to apply for the design of tapered columns, but becomes more complexand less accurate in the case of tapered beam-columns.

The primary objective of this paper is to develop a both practical and accurate methodfor the in-plane design of web-tapered steel structures fabricated through the welding of in-dividual plates, extending the stiffness reduction approach proposed by Kucukler et al. [8, 9]for hot-rolled members to welded tapered structures. Stiffness reduction functions for weldedmembers able to fully consider the detrimental influence of plasticity and imperfections arederived. The proposed approach is applied, using beam elements, through (i) the divisionof tapered members into prismatic segments, (ii) the reduction of flexural stiffness of eachprismatic segment through the developed stiffness reduction functions considering the first-order forces at the middle of the segment, (iii) performing Geometrically Nonlinear Analysis(GNA-SR) and (iv) making cross-section strength checks. Owing to the full consideration ofthe deleterious influence of the spread of plasticity, geometrical imperfections and residualstresses on the structural resistance through the developed stiffness reduction functions, theproposed method only requires cross-section checks after structural analysis with stiffnessreduction. Hence the conventional analysis and design stages are merged, obviating the needto determine effective lengths or conduct elaborate member design checks, which can be par-ticularly complicated for tapered structures [10, 11]. The practicality of the use of stiffnessreduction in the design of both irregular and regular steel members has also been recognisedby [12–14]. Since the proposed approach is based on the division of tapered members into

2

prismatic segments, it can be applied through any conventional structural analysis softwarecapable of performing elastic Geometrically Nonlinear Analysis on beam element models.The approach proposed in this paper constitutes part of a design framework based on theseparation of in-plane (i.e. in the plane of bending) and out-of-plane instability assessmentsof tapered steel structures. Since the latter is not required for laterally restrained struc-tures, the method developed in this study can be applied in their design without the needfor any further global instability assessment. Similar to [7], the global instability responseof tapered steel structures is considered in this study assuming the cross-sections are ableto reach their full plastic strengths; the proposed approach is therefore currently applicableto Class 1 and 2 cross-sections only, but will be extended to consider the interaction of localand global instability effects in future work.

In the following sections, shell finite element models of tapered members and frames withdifferent geometries are created and then utilised to assess the accuracy of the proposed ap-proach. The development of the stiffness reduction functions is illustrated and comparisonsare also made against the design methods provided in EN 1993-1-1 [1] and AISC DesignGuide 25 [15] for both tapered members and frames. A range of slendernesses, loading casesand tapering geometries are considered for thorough verification of the proposed approach.

2. Finite element modelling

The results of shell finite element modelling are used to verify the stiffness reductiondesign approach proposed in this study. Details of the adopted finite element modellingapproach are described in this section. The finite element models were created throughthe finite element analysis software Abaqus [16]. A four-node reduced integration shellelement, which is designated as S4R in the Abaqus [16] element library and takes intoaccount transverse shear deformations and finite membrane strains, was used in all thenumerical simulations. 16 elements were employed for each flange and web plate so that thespread of plasticity through the depth of the cross-sections could be accurately considered.To avoid overlapping of the flange and web plates, the web plates were offset consideringthe thickness of the flanges. The default Simpson integration method was used, with fiveintegration points through the thickness of each element. The Poisson’s ratio was taken as0.3 in the elastic range and 0.5 in the plastic range by defining the effective Poisson’s ratioas 0.5 to allow for the change of cross-sectional area under load. A tri-linear stress-strainrelationship was used in the adopted material model, which is shown in Fig. 1, where E isthe Young’s modulus, Esh is the strain hardening modulus, fy and �y are the yield stressand strain respectively and �sh is the strain value at which strain hardening commences.The parameters fu and �u correspond to the ultimate stress and strain values respectively.Esh was assumed to be 2 % of E and �sh was taken as 10�y, conforming to the ECCSrecommendations [17], though more representative values for these two parameters haverecently been proposed [18], which will be considered in future studies. In the finite elementmodels, isotropic strain hardening and the Von Mises yield criterion with the associatedflow rule were employed. Since the constitutive formulations of Abaqus [16] adopt theCauchy (true) stress-strain assumption for shell finite elements, the engineering stress-strain

3

Esh

Stress, f

Strain, ε

fy

fu

εy εsh εu

E

Figure 1: Material stress-strain curve used in finite element models

relationship shown in Fig. 1 was transformed into a true stress-strain relationship. Unlessotherwise indicated, S235 steel was considered in all the simulations. Using the defaultconvergence criteria recommended by Abaqus [16], the load-displacement response of thefinite element models was determined by means of GMNIA (Geometrically and MateriallyNonlinear Analysis with Imperfections) using the modified Riks method [19, 20].

The ECCS residual stress pattern [17], illustrated in Fig. 2 ,which is recommended for

fy

0.25fy

0.075b

0.125b

+

_

0.075(h-2tf)

b

h

tf

0.125(h-2tf)

+_

fy

0.25fy

0.25fy

fy

Figure 2: Residual stress pattern applied to finite element models (+ve tension, -ve compression)

cross-sections fabricated through the welding of plates was applied to the finite elementmodels. It should be noted that the welds were conservatively not modelled in the finiteelement simulations, only the residual stresses resulting from the welding were accountedfor. Geometrical imperfections were assigned to the models in the shape of the lowest globalbuckling mode, with the maximum imperfection scaled to 1 / 1000 of the member length.Since the present study is limited to the global in-plane instability response of tapered

4

structures, local imperfections were not considered and the finite element models were fullyrestrained in the out-of-plane direction; a similar approach was followed in [7]. Note thoughthat for tapered members with slender cross-sections, local buckling can control the designand will be considered in future work. Comprehensive validation of the adopted finiteelement modelling approach against results from a range of experimental studies reportedin the literature has been provided by Kucukler [21], for both individual steel members andframes containing nonprismatic segments.

3. Stiffness reduction functions and ultimate cross-section strength equation

In this section, stiffness reduction functions required for the implementation of the pro-posed stiffness reduction approach are developed and the adopted ultimate cross-sectionstrength equation is described. The proposed stiffness reduction functions are able to con-sider fully the detrimental influence of the spread of plasticity, geometrical imperfectionsand residual stresses, thus enabling the establishment of a very practical stiffness reductionmethod only requiring cross-sections strength checks without the need to model geometricalimperfections or use member design equations.

3.1. Stiffness reduction function for axial loading

The stiffness reduction function for welded steel members under axial loading τN isexpressed as:

τN =4κnψ

2

α2NEd/Npl

[1 +√

1 − 4ψβnNEd/Npl−1α2NEd/Npl]2 but τN ≤ 1.0

where ψ = 1 + 0.2αNEdNpl

− NEdNpl

and κn = 0.88 if My,Ed = 0,

κn = 1.00 if My,Ed > 0, (1)

in which NEd is the applied axial load, Npl is plastic axial force resistance equal to the cross-sectional area A times the yield stress fy (i.e. Npl = Afy). In eq. (1), the imperfection factorα is equal to 0.49 (i.e. α = 0.49), the auxiliary coefficient βn is taken as 0.6 (i.e. βn = 0.6)and the auxiliary coefficient κn is taken as 0.88 in the case of a member subjected to purecompression and taken as 1.00 in the case of a member subjected to bending moment My,Ed;the latter case requires the use of the stiffness reduction function for combined bending andcompression τMN which utilises τN , as will be illustrated in the subsection 3.3. The proposedfunction τN is in the same form as that recommended in [8] for hot-rolled steel members withthe exception of the use of the auxiliary coefficients βn and κn. In this study, the proposedstiffness reduction function τN is derived using a Perry-Robertson equation calibrated to theultimate strengths obtained from GMNIA results of a large number of laterally-restrainedwelded columns failing about the major axis following the approach described by Kucukleret al. [8, 22]. The calibrated Perry-Robertson equation used for the derivation of τN is inthe same form as that recommended for the lateral-torsional buckling assessment of steel

5

beams in Clause 6.3.2.3 of EN 1993-1-1 [1] with the imperfection factor α and auxiliarycoefficients βn equal to 0.49 and 0.6 respectively. Note that the values of the imperfectionfactor α and the auxiliary coefficient βn recommended in Eurocode 3 [1], which are equal to0.34 and 1.0 respectively, could also be used in the proposed stiffness reduction function τN ,with κn = 1.0, resulting in the same, rather inaccurate, strength predictions obtained from[1] for welded columns failing about the major axis. The inaccuracy of the current designrules in Eurocode 3 [1] for the ultimate strength prediction of welded columns failing aboutthe major axis is shown in [23]. Inclusion of the κn factor, equal to 0.88, in the applicationof the proposed stiffness reduction method (SRM) to members under pure axial loadingreflects the immediate reduction in major axis elastic flexural stiffness of a welded I-sectionarising from the applied residual stress pattern shown in Fig. 2, which features materialat yield at the web-to-flange junctions. In the case of combined bending and compression,the initial reduction of the flexural stiffness is considered through the stiffness reductionfunction for bending τM as described in the following subsection, thus κn is taken as 1.0 inτN for members subjected to combined bending and compression, to avoid double-counting.It is worth noting that the initial reduction of the flexural stiffness is not recognised inthe Perry-Robertson concept, which, dubiously, assumes that the strengths of very slenderwelded columns are very close to their elastic flexural buckling loads Ncr, and only considersexplicitly the influence of geometrical imperfections.

3.2. Stiffness reduction function for bending

In this subsection, a stiffness reduction function for welded sections under bending τMis derived through calibration to the moment-curvature (MEd − ϕ) response obtained fromGMNIA of shell finite element models of laterally-restrained beams with 20 different non-slender welded cross-sections subjected to uniform major-axis bending. The consideredwelded members were assigned cross-sections properties the same as those of 20 differentEuropean IPE and HE cross-sections with the exception of the presence of the fillets. Therange of the cross-sections examined are presented in Table 1, where h is the cross-sectiondepth, b is the flange width, and tw and tf are the web and flange thicknesses respectively.For the determination of the stiffness reduction functions from GMNIA, initially the secant

Table 1: Range of the considered cross-section properties for the derivation of τM

h/b b/tf h/twMaximum 3.34 25.0 60.6Minimum 0.91 5.3 10.0

stiffnesses EIs were obtained by dividing the applied bending moments by the correspondingcurvatures of the beams ϕ, for bending moment values ranging from zero to the plasticbending moment resistance of the cross-section My,pl (i.e. EIs,i = My,Ed,i/ϕi). Divisionof the secant stiffnesses by the initial flexural stiffness EI furnished the stiffness reduction

6

function for bending τM (i.e. τM = EIs,i/EI). The stiffness reduction function τM calibratedto the generated GMNIA results is as follows:

τM = τM0 if My,Ed/My,pl ≤ φy,

τM = τMl + (τM0 − τMl)

[1 −

(My,Ed/My,pl − φy

ξm − φy

)βm]1/βmif φy ≤My,Ed/My,pl ≤ ξm,

τM = τMl

[1 −

(My,Ed/My,pl − ξm

1 − ξm

)]1/δmif ξm ≤My,Ed/My,pl, (2)

where My,Ed is the applied major axis bending moment, My,pl is the major axis plastic bend-ing moment resistance equal to the plastic section modulus Wpl,y multiplied by the yieldstress fy (i.e. My,pl = Wpl,yfy), and ξm, βm, δm, φy, τM0 and τMl are the auxiliary coefficientsprovided in Table 2. In the table, Wely,y is the elastic section modulus of the cross-section,

Table 2: Parameters for the stiffness reduction function for welded I-sections in major axis bending τM

τM0 τMl φy ξm βm δm0.88 0.70 0.75Wel,y/Wpl,y 0.88(Aw/Af )

−0.05 1.10 3.0

and Aw and Af are the web area and the area of the one flange of the cross-section respec-tively. The stiffness reduction function for bending τM is a piecewise function representingthree stiffness reduction stages observed in the stiffness reduction patterns obtained throughGMNIA. As can be seen from Table 2, the developed stiffness reduction function τM is equalto 0.88 even before the cross-section is loaded (i.e. for My,Ed = 0), which results from theyielded regions around the web-flange junction in the adopted residual stress pattern forwelded cross-sections shown in Fig. 2. The auxiliary coefficients of τM were calibrated tothe GMNIA results; calibration of the auxiliary coefficient ξm following regression analysisis illustrated in Fig. 3. The proposed stiffness reduction function is compared against thatobtained from GMNIA in Fig. 3 for a welded-cross-section with the geometric properties ofan IPE 600 section. As can be seen from the figure, the proposed stiffness reduction functionprovides a very accurate prediction of the actual stiffness reduction pattern obtained fromGMNIA; the same high accuracy was also observed for the other considered cross-sectionshapes.

3.3. Stiffness reduction function under combined axial loading and bending

For the derivation of the stiffness reduction function under combined axial loading andbending τMN , the stiffness reduction functions developed for pure axial loading τN andpure bending τM were utilised. The proposed stiffness reduction function for combinedcompression and bending τMN is given by the following expression:

τMN = τMτN

{1 −

[(NEdNpl

)(My,EdMy,pl

)]0.5}. (3)

7

0.88(Aw/Af)-0.05

(a) Calibration of ξm

y

z

My,Ed

IPE 600

(b) Comparison between τM obtained throughGMNIA and that proposed for a welded sectionequivalent to an IPE 600

Figure 3: Calibration and accuracy of the stiffness reduction function for bending τM proposed in this study

The proposed stiffness reduction function τMN was developed through calibration to the ul-timate strength predictions obtained through GMNIA of prismatic and web-tapered weldedbeam-columns in this paper. The stiffness reduction function for combined axial loadingand bending τMN for a welded section with the cross-section properties the same as thoseof an HEB 200 cross-section is illustrated in Fig. 4, showing decreasing τMN values withincreasing axial loading and bending moments.

3.4. Ultimate cross-section strength equations

Owing to its high accuracy and simple form, the ultimate cross-section strength equationproposed by Duan and Chen [24] given by

1

αult,c=

(NEdNpl

)1.3+My,EdMy,pl

≤ 1.0 (4)

is adopted in this paper, where 1/αult,c represents the cross-section utilisation factor.

4. Implementation of the stiffness reduction method for web-tapered structures

Application of the proposed SRM to web-tapered steel members is illustrated, through afour-step process, in Fig. 5. Unless otherwise indicated, the tapered members were dividedinto 10 prismatic segments in all the considered cases in this paper and the use of this numberis recommended in the implementation of the proposed approach. In lieu of prismatic beam

8

0

My,Ed

/ My,pl

0.20.4

0.60.8

1.01.00.8

0.6N

Ed / N

pl

0.40.2

0

0.4

0.8

1.0

0

0.2

0.6

τM

N

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 4: Stiffness reduction function for bending and axial loading τMN

elements, which are able to accurately represent the in-plane behaviour of tapered membersprovided a sufficient number of elements are used in their modelling [15], beam elementsspecifically developed to represent the response of tapered steel members [25] could also beused in the implementation of the SRM. The use of both element types in conjunction withthe SRM is expected to lead to very close strength predictions for tapered members, providedthe same and sufficiently high number of elements is adopted in their modelling. Owing totheir wide availability in conventional structural analysis packages used in practice, prismaticbeam elements are utilised in this study. In the case of tapered members subjected to purecompression, Linear Buckling Analysis of the segmented model with reduced stiffness shouldbe performed in the Step 3 where the lowest buckling load factor αcr,i should be greater thanor equal to 1.0 (i.e. αcr,i ≥ 1.0) for a satisfactory design. Note that the same steps describedin Fig. 5 should be followed in the application of the proposed approach to tapered frames,where the initial out-of-plumbness of the frame should be taken as equal to the maximumvalue of 0.002 rad permitted in EN 1090-2 [26]; this imperfection can be directly modelledor its effect can be considered by applying notional horizontal forces to each storey of theperfect structure in accordance with the rules described in [1, 2]. Application of the proposedSRM to web-tapered frames is illustrated within Section 7. Finally, in accordance with [15],the application of the proposed SRM to members with web-tapering angles smaller than orequal to 15o is recommended, excluding the cases where conventional beam theory may beinsufficient to represent the behaviour.

9

NEd

My,Ed

1

2

3

4

NEd

My,Ed

1

2

3

4

A1, Iy,1

A2, Iy,2

A3, Iy,3

A4, Iy,4

My,Ed

My,Ed,4

My,Ed,3

My,Ed,2

My,Ed,1

NEdMy,Ed

1

2

3

4 tMN(My,Ed,4) *EIy,4

tMN(My,Ed,3) *EIy,3

tMN(My,Ed,2) *EIy,2

tMN(My,Ed,1) *EIy,1

Divide member intoprismatic segments

Determine stiffness reduction factors tMN for eachsegment and reduce their flexural stiffnesses EI yi

A1, Iy,1

A2, Iy,2

A3, Iy,3

A4, Iy,4

Step 1 Step 2

Perform Geometrically NonlinearAnalysis of the segmented member

with reduced stiffness

Step 3

My,Ed

My,Ed,4

My,Ed,3

My,Ed,2

My,Ed,1

Perform cross-section strength checks and ensure that1 /ault,c ≤ 1.0 along the length

Step 4

NEd

NEd

NEd

NEd

NEd

NEd

My,Ed

1

2

3

4

A1, Wpl,y,1

A2, Wpl,y,2

A3, Wpl,y,3

A4, Wpl,y,4 1 /ault,c,4 ≤ 1.0

1 /ault,c,end ≤ 1.0

1 /ault,c,3 ≤ 1.0

1 /ault,c,2 ≤ 1.0

1 /ault,c,1 ≤ 1.0

1 /ault,c,start ≤ 1.0

Figure 5: Application of the proposed stiffness reduction method (SRM)

5. Application of the stiffness reduction method to web-tapered columns

In this section, application of the proposed stiffness reduction approach to web-taperedsteel columns subjected to either uniform or non-uniform axial loading is addressed, pro-viding extensive verification against the results obtained through GMNIA. The proposedapproach is also compared against the alternative design methods for tapered membersprovided in Eurocode 3 [1] and AISC Metal Building Manufacturers Association (MBMA)Design Guide 25 [15].

5.1. Application of the stiffness reduction method to web-tapered columns under uniformaxial loading

The proposed stiffness reduction method (SRM) is applied to a series of web-taperedcolumns with various slenderness values and different tapering ratios ζh, defined as the ratio

10

of the cross-section depth at the deep end hi to that at the shallow end h (i.e. ζh = hi/h),in this subsection. A comparison between the different design approaches and the ultimatestrengths obtained through GMNIA of the finite element models is shown in Fig. 6, where

h=2.0

NEd

hi

h

h=5.0

h=1.0

h=3.0

h = hi / h

z

y

HEB 200

(a) HEB 200

h=2.0

NEd

hi

h

h=5.0

h=1.0

h=3.0

h = hi / h

y

z

HEM 100

(b) HEM 100

h=3.0

h=2.0

NEd

hi

h

h=5.0

h=1.0

h = hi / h

z

y

IPE 240

(c) IPE 240

NEd

hi

h = hi / h

h

h=3.0

h=2.0

h=5.0

h=1.0 z

y

IPE 240

2500

Le

L

(d) IPE 240 EC3

Figure 6: Comparison of the proposed stiffness reduction method (SRM) against GMNIA, AISC MBMAand Eurocode 3 for web-tapered columns with various tapering ratios and non-dimensional slendernesses

Npl is the axial yield load of the cross-section at the shallow end and λ is the non-dimensionalslenderness equal to the square root of the axial yield load divided by the elastic critical

11

load Ncr (i.e. λ =√Npl/Ncr). The elastic critical buckling loads of the tapered columns

Ncr were determined through elastic linear buckling analyses of shell finite element models.Note that the cross-section properties at the shallow end of the investigated members werethe same as those of HEB 200, HEM 100 and IPE 240 hot-rolled sections, with the exceptionof the presence of the fillets; tapering was only applied to the cross-section depth. As canbe seen from Fig. 6, the proposed SRM provides very accurate strength predictions for allthe considered tapering ratios, cross-section shapes and slenderness values.

To assess the accuracy of the proposed approach against the design methods recom-mended in the European and North American structural steel design codes, the ultimatestrength predictions obtained through the equivalent imperfection approach of Eurocode 3[1] and the method recommended in the AISC MBMA design guide [15] for tapered mem-bers are also compared against those of GMNIA in Fig. 6. In the implementation of theequivalent imperfection approach of Eurocode 3 [1], (i) a parabolic imperfection scaled to 1/ 250 of the member length L (i.e. e0 = L/250) was applied to the tapered member, (ii) Ge-ometrically Nonlinear Analysis of the imperfect tapered member divided into 10 prismaticsegments was performed and (iii) cross-section strength checks were made using a linearinteraction equation with plastic (Npl and My,pl) end points in accordance with Eurocode 3[1]. Fig. 6 (d) shows that the equivalent imperfection approach of Eurocode 3 [1] leads tooverly-conservative strength predictions in all the considered cases and the proposed SRMprovides considerably more accurate results. In Fig. 6, the ultimate strength predictionsobtained from AISC MBMA [15] are also compared against those of GMNIA. In the appli-cation of this approach, the safety factor was taken as equal to 1.0 instead of 0.9 in thispaper for comparison purposes, since a safety factor was also not applied to the yield stressor stiffness in the shell GMNIA. Note that AISC MBMA recommends the use of a singlebuckling curve regardless of the tapering ratio. As can be seen from Fig. 6, the method leadsto less accurate results in comparison to the SRM, overestimating the ultimate strengths forlow tapering ratios. This may be related to the adoption of a single buckling curve for bothhot-rolled and welded sections by this method.

A comparison of the ultimate strength predictions for 150 web-tapered columns obtainedthrough the SRM, AISC MBMA and Eurocode 3 equivalent imperfection approach againstthose of GMNIA is provided in Table 3, where N is the number of considered columns, Sav,SCOV , Smax, Smin are the average, coefficient of variation, maximum and minimum of theratios of the strength predictions obtained through the SRM, AISC MBMA or Eurocode 3equivalent imperfection approach to those of the benchmark shell GMNIA respectively. Theconsidered tapered columns have shallow sections with cross-section properties the same asthose of IPE 240, HEB 200 and HEM 100 sections and non-dimensional slendernesses λyranging between 0.2 and 2.0. As can be seen from Table 3, the proposed SRM provides veryaccurate ultimate strength predictions for all the considered tapered columns. Table 3 alsoillustrates that the Eurocode 3 [1] equivalent imperfection approach provides less accurateand overly conservative results in comparison to the SRM, while the AISC MBMA [15] isless accurate than the SRM for all the considered cases.

12

Table 3: Accuracy of the proposed stiffness reduction method in comparison to Eurocode 3 and MBMA forweb-tapered welded columns with three different cross-section shapes (i.e. HEB 200, HEM 100, IPE 240) atthe shallow end

Loading conditions ζh N Sav SCOV Smax SminSRM

NEd 1.0 30

1.00 0.018 1.03 0.97EC3 equiv. imp. 0.91 0.041 0.96 0.86

AISC MBMA 1.05 0.040 1.13 1.00

SRM

NEd 2.0 30

1.00 0.013 1.03 0.97EC3 equiv. imp. 0.92 0.044 1.00 0.86

AISC MBMA 1.01 0.028 1.06 0.94

SRM

NEd 3.0 30

1.00 0.014 1.04 0.98EC3 equiv. imp. 0.92 0.043 1.00 0.87

AISC MBMA 1.00 0.035 1.05 0.92

SRM

NEd 4.0 30

1.00 0.015 1.05 0.99EC3 equiv. imp. 0.93 0.042 1.00 0.87

AISC MBMA 1.00 0.041 1.05 0.91

SRM

NEd

5.0 301.01 0.017 1.04 0.98

EC3 equiv. imp. 0.93 0.041 1.00 0.88AISC MBMA 1.00 0.050 1.06 0.88

5.2. Application of the stiffness reduction method to web-tapered columns under non-uniformaxial loading

In this subsection, the proposed SRM is applied to a series of welded web-tapered columnssubjected to varying axial loading along their lengths, with the shallow section having thecross-section properties the same as those of an HEM 100 section. A comparison of theultimate strength predictions for two different tapering ratios ζh, equal to 2.0 and 4.6, isillustrated in Fig. 7, where Npl,2 is the axial resistance of the cross-section at the deep end.As can be seen from Fig. 7, half of the total axial load NEd was applied at the shallow end andthe remaining half was applied as a uniformly-distributed axial load n = 0.5NEd/L along thelength of the tapered column. The non-dimensional slenderness of the columns for flexuralbuckling about the major axis λy was determined by dividing the load factor resulting inyielding at the deep end αp (which was the critical cross-section under the considered loadingscenario), which is equal to the ratio between the axial yield load at this location Npl,2 andthe corresponding total applied axial load NEd, to the elastic flexural buckling load amplifierαcr of the tapered column (i.e. λy =

√αp/αcr =

√(Npl,2/NEd)/αcr). The elastic buckling

load amplifiers αcr were determined through the linear buckling analyses of the shell finiteelement models of the considered tapered columns. It should be emphasised that for theinvestigated loading condition, the linearly web-tapered columns with the tapering ratio ζhequal to 4.6 (i.e. ζh = 4.6) are under uniform axial stresses along their lengths, while thecolumns with the tapering ratio equal to 2.0 (i.e. ζh = 2.0) have non-uniform axial stressesalong their lengths. Thus, for low to moderate slendernesses λy, the former columns (i.e.

13

h2

h1

h = 2.0

y

z

HEM 100 L

0.5NEd

L

NEd5.0

NEd 0.5NEd

(a) ζh = h2/h1 = 2.0

h2

h1

h = 4.6

y

z

HEM 100 L

0.5NEd

L

NEd5.0

NEd 0.5NEd

(b) ζh = h2/h1 = 4.6

Figure 7: Comparison of the proposed stiffness reduction method against GMNIA, Eurocode 3, AISC MBMAGuide and Marques et al. [7] for web-tapered columns under non-uniform axial loading

those with ζh = 4.6) are expected to experience more severe levels of plasticity along theirlengths in comparison to the latter columns (i.e. those with ζh = 2.0) at failure, which is ofvital importance for their ultimate load carrying capacities NEd,max relative to their axialyield loads at the deeper ends Npl,2 (i.e. NEd,max/Npl,2).

Fig. 7 shows that the proposed SRM leads to very accurate strength predictions fortapered columns under non-uniform axial loading, when the level of axial stresses is eitheruniform or non-uniform along the member length, owing to the capability of the SRM toconsider the different extents of plasticity experienced by the different cross-sections throughits stiffness reduction functions. The ultimate strengths obtained through the Eurocode 3[1] equivalent imperfection approach and AISC MBMA [15] guide are also provided in Fig.7, showing that the proposed SRM leads to more accurate results (i.e. closer to GMNIA) incomparison to both of these approaches as neither of them takes into account the differentlevels of plasticity experienced along the lengths of the tapered columns.

The ultimate strength predictions obtained through the method recommended by Mar-ques et al. [7] are also illustrated in Fig. 7, where it can be seen that unlike the proposedSRM, the method of [7] leads to some over-predictions of GMNIA results for columns withthe tapering ratio equal to 4.6, overestimating the ultimate strengths by up to 16 %. Thisresults from the fact that the design equations of [7] were calibrated considering GMNIA re-sults of tapered columns under uniform axial loading only, while the case of tapered columnssubjected to non-uniform axial loading along their lengths was not considered. A commonexample of this loading condition is that of the rafters in pitched-roof portal frames (seeSection 7 of this paper), where the level of axial force increases towards the eaves. On the

14

other hand, for the tapering ratio equal to 2.0, the method of [7] underestimates the ulti-mate strengths of tapered columns with low to moderate slendernesses since these columnsundergo less extensive plasticity relative to those under constant axial loading with the sametapering ratio which was the only case considered in the calibration of their design equations[7].

6. Application of the stiffness reduction method to web-tapered beam-columns

The proposed SRM is applied to web-tapered beam-columns in this section, where itsverification against GMNIA is provided for all the considered cases. To compare the SRMagainst the alternative design approaches for tapered members, ultimate strength predictionsobtained through the Eurocode 3 equivalent imperfection approach [1] and AISC MBMA [15]are also presented. In the application of the SRM, EC3 [1] equivalent imperfection approachand AISC MBMA [15], Geometrically Nonlinear Analyses of the segmented models of web-tapered members are performed through a Matlab [27] code developed on the basis of thestability function approach recommended by Chen and Toma [28].

6.1. Application of the stiffness reduction method to web-tapered beam-columns under uni-form bending plus axial compression

In this subsection, the application of the SRM to simply-supported web-tapered beam-columns under uniform bending plus axial compression is addressed. Three non-dimensionalslenderness values λy equal to 0.4, 1.0 and 1.5 were considered so as to investigate theaccuracy of the SRM for tapered beam-columns of low, moderate and high slenderness. Inthe considered tapered beam-columns, the shallow sections have the dimensions of HEB 200,HEM 100 and IPE 240 sections. Four tapering ratios ζh equal to 2.0, 3.0, 4.0 and 5.0 wereinvestigated, along with prismatic beam-columns where ζh is equal to 1.0.

A comparison of the ultimate strength predictions obtained through the SRM againstthose of GMNIA is provided in Fig. 8. Note that Npl and My,pl are the axial yield loadand plastic bending moment resistance at the shallow end of the tapered members. In thefigure, the ultimate bending moment predictions of GMNIA that exceed the plastic bendingmoment resistance owing to strain hardening are limited to My,pl. As can be seen from Fig.8, the proposed SRM leads to very accurate results for beam-columns with different taperingratios and slendernesses, and various levels of axial loading and bending. The method onlyleads to some underpredictions of the ultimate strengths for tapered members subjectedto predominantly bending with small slendernesses and large tapering ratios, which resultsfrom the neglect of the beneficial influence of strain hardening by the proposed SRM.

A comparison of the results of the proposed SRM and GMNIA for 305 web-taperedbeam-columns subjected to uniform bending plus axial compression with three differentnon-dimensional slendernesses λy equal to 0.4, 1.0 and 1.5 and with the shallow sectionshaving properties the same as those of HEB 200, HEM 100 and IPE 240 sections is providedin Table 4. The accuracy of the SRM is assessed using the � parameter determined through

15

5.1y

NEd

HEB 200

z

y

My,Ed My,Ed

0.1y

4.0y

(a) ζh = 1.0, HEB 200

NEd

HEB 200 z

y

My,Ed My,Ed

4.0y

0.1y

5.1y

(b) ζh = 5.0, HEB 200

NEd

My,Ed My,Ed

4.0y

0.1y

5.1y

z

y

IPE 240

(c) ζh = 3.0, IPE 240

z

NEd

My,Ed My,Ed

4.0y

0.1y

5.1y y

HEM 100

(d) ζh = 2.0, HEM 100

Figure 8: Comparison of the proposed stiffness reduction method against GMNIA for web-tapered beam-columns under uniform bending plus axial compression

the following equation:

� =RSRMRGMNIA

=

√(NEd,max,SRM/Npl)2 + (My,Ed,max,SRM/My,pl)2√

(NEd,max,GMNIA/Npl)2 + (My,Ed,max,GMNIA/My,pl)2(5)

in which, RSRM and RGMNIA are the radial distances measured from the origin to theultimate strength on the My,Ed − NEd interaction curves determined through the SRMand GMNIA respectively. In eq. (5), NEd,max,SRM and My,Ed,max,SRM are the ultimate

16

axial compression and bending strengths obtained through the SRM and NEd,max,GMNIAand My,Ed,max,GMNIA are the ultimate axial compression and bending strengths obtainedthrough GMNIA respectively, while βR is the angle between the radial distances (i.e. RSRMand RGMNIA) and the horizontal axis, as illustrated in Fig. 9. The � values were determined

GMNIA

SRM

My,Ed / My,pl

NEd / Npl

RSRM

RGMNIA

My,Ed,max,SRM

βR

My,Ed,max,GMNIA

NEd,max,SRM NEd,max,GMNIA

ϵ = RSRM / RGMNIA

βR =11.25o to 90o

with ∆βR =11.25o

Figure 9: Determination of radial errors for the proposed stiffness reduction method (SRM)

considering the same angles βR for the SRM and GMNIA radial distances (i.e. RSRM andRGMNIA). As can be interpreted from eq. (5), � values larger than 1.0 correspond to uncon-servative strength predictions. In Table 4, �av and �COV are the average and coefficient ofvariation of the � values, while �max and �min are the maximum and minimum of the � valuesrespectively. For all the considered beam-columns, � values were determined considering8 different angles βR varying between 11.25

◦ and 90◦ with an increment of 11.25◦. In thecases where the strengths of beam-columns obtained from GMNIA exceeded the ultimatecross-section strength surface due to strain hardening, the � values were not calculated. Ascan be seen from Table 4, the proposed SRM provides very accurate strength predictionsfor tapered beam-columns under uniform bending moments for the wide range of consideredcases, with �av very close to 1.0 and very small �COV values. Minor overestimations wereobserved only for very slender (i.e. λy = 1.5) tapered beam-columns with tapering ratiosequal to 5.0 (i.e. ζh = 5.0), which are generally outside of the range used in practice, asdescribed by [15]. The ultimate strength predictions obtained through the Eurocode 3 [1]equivalent imperfection approach and AISC MBMA [15] are also compared against those ofGMNIA in Table 4, where the � values are the ratios of the radial distances to the interactioncurves from these methods to those obtained through GMNIA. It can be seen from the tablethat the proposed SRM leads to more accurate results in comparison to both of these designapproaches.

17

Table 4: Accuracy of the proposed stiffness reduction method in comparison to Eurocode 3 and MBMA forweb-tapered welded beam-columns under uniform bending and axial compression

Loading conditions ζh N �av �COV �max �min

SRM

NEd

My,Ed My,Ed

1.0 81

1.00 0.015 1.02 0.96

EC3 equiv. imp. 0.97 0.051 1.06 0.86

AISC MBMA 1.02 0.030 1.09 0.87

SRM

NEd

My,Ed My,Ed

2.0 62

1.00 0.014 1.04 0.97

EC3 equiv. imp. 0.97 0.049 1.06 0.87

AISC MBMA 1.00 0.032 1.06 0.93

SRM

NEd

My,Ed My,Ed

3.0 56

1.00 0.014 1.05 0.96

EC3 equiv. imp. 0.97 0.046 1.05 0.88

AISC MBMA 0.99 0.035 1.05 0.91

SRM

NEd

My,Ed My,Ed

4.0 54

1.00 0.015 1.05 0.97

EC3 equiv. imp. 0.97 0.045 1.05 0.88

AISC MBMA 0.98 0.042 1.05 0.89

SRM

NEd

My,Ed My,Ed

5.0 52

1.01 0.015 1.05 0.98

EC3 equiv. imp. 0.97 0.043 1.04 0.89

AISC MBMA 0.97 0.051 1.07 0.87

6.2. Application of the stiffness reduction method to web-tapered beam-columns subjected tovarying end-moments plus axial compression

The accuracy of the proposed SRM for tapered beam-columns under varying bendingmoments along their lengths is investigated in this subsection. The ultimate strengthsobtained through the SRM are compared against those of GMNIA in Fig. 10 for differentbending moment values, non-dimensional slendernesses, cross-section shapes and taperingratios. In the figure, My,pl,2 and My,pl,1 are the major axis bending moment resistance ofthe cross-sections at the deep and shallow ends of the member respectively, and Npl,1 is theaxial resistance of the cross-section at the shallow end. As can be seen from the figure, theproposed SRM leads to very accurate results for all the considered tapered beam-columnsunder various levels of single and double-curvature bending, only underpredicting strengthsin the case of the occurrence of strain hardening. The accuracy of the AISC MBMA [15] fortapered beam-columns subjected to varying bending moments is also investigated in Fig. 10,which shows that the AISC MBMA [15] provides overly conservative strength estimationsfor tapered beam-columns under double curvature bending, while it is accurate for membersunder single-curvature bending.

Verification of the proposed SRM against GMNIA is provided for 1111 web-tapered steel

18

HEB 200 z

y

NEd My,Ed

My,Ed

4.0y

0.1y

5.1y

My,Ed

My,Ed

(a) ζh = 5.0, HEB 200

HEB 200 z

y

2My,Ed NEd My,Ed

4.0y

5.1y

0.1y

My,Ed

2My,Ed

(b) ζh = 4.0, HEB 200

0.1y

2My,Ed

4.0y

5.1y

z

y

IPE 240

2My,Ed

NEd

My,Ed

My,Ed

(c) ζh = 2.0, IPE 240

NEd

My,Ed

4.0y

0.1y

5.1y

z

y

HEM 100

My,Ed

(d) ζh = 3.0, HEM 100

Figure 10: Comparison of the accuracy of proposed stiffness reduction method against GMNIA and AISCMBMA for web-tapered beam-columns under non-uniform bending plus axial compression

beam-columns under different bending moment gradients, tapering ratios, slendernesses andcross-section shapes at the shallow end in Table 5. As can be seen from Table 5, the proposedSRM provides very accurate ultimate strength predictions for the various considered cases.Only some minor overpredictions were observed for very slender members (λy = 1.5) withthe tapering ratio ζh equal to 5.0, which do not generally fall into the range used in practice.

19

Table 5: Accuracy of the proposed stiffness reduction method for web-tapered welded beam-columnsunder different end moments and axial compression with three different non-dimensional slendernessesλy = 0.4, 1.0, 1.5 and HEB 200, HEM 100 and IPE 240 section shapes at the shallow end

Loading conditions ζh N �av �COV �max �min

SRM

2My,Ed

NEd

My,Ed

1.0 65 1.00 0.018 1.05 0.96

2.0 68 1.00 0.017 1.04 0.97

3.0 64 1.00 0.016 1.05 0.97

4.0 58 1.00 0.016 1.05 0.96

5.0 55 1.01 0.017 1.05 0.97

SRM

My,Ed

NEd

1.0 59 1.00 0.016 1.05 0.97

2.0 69 0.99 0.024 1.04 0.93

3.0 71 0.99 0.026 1.05 0.93

4.0 71 1.00 0.027 1.05 0.93

5.0 71 1.00 0.029 1.05 0.92

SRM

2My,Ed

NEd

My,Ed

1.0 53 1.00 0.017 1.05 0.97

2.0 43 0.99 0.023 1.04 0.95

3.0 39 0.99 0.023 1.05 0.94

4.0 42 1.00 0.019 1.05 0.96

5.0 42 1.01 0.019 1.05 0.96

SRM

My,Ed

NEd

My,Ed

1.0 51 0.98 0.027 1.04 0.93

2.0 49 1.00 0.017 1.04 0.97

3.0 47 1.00 0.015 1.05 0.98

4.0 47 1.00 0.017 1.05 0.97

5.0 47 1.01 0.019 1.05 0.96

7. Application of the stiffness reduction method to the Watwood [29] frame

The proposed SRM is applied to the Watwood [29] web-tapered gable frame in thissection. The geometrical properties and loading conditions of the Watwood [29] frame,which is similar to an actual failed gable frame investigated by the author of [29], areillustrated in Fig. 11. The frame has short stiff columns, which leads to the developmentof large axial loads in the rafters, which are inclined at a large angle to the horizontal,resulting in an instability sensitive frame. This feature of the Watwood frame [29] makesit an excellent benchmark example to be used in the validation of the proposed SRM. Theframe was fabricated from steel with a yield stress of 344 MPa used for both flange and webplates, had a bay width equal to 6.1 m and had Z-section roof purlins spaced at 1.61 m, as

20

3150304.8

1422.4

A

B

C

D

E

F

3962

203.2

412

1219.2

11 spaces @

1607

1346

260

9728

17932

C

Section Web Flange Depth7.95 254 × 9.525 304.8 - 1219.2

B 7.95 _ KneeC 7.95 254 × 12.7 1219.2 - 878.69D 5.38 177.8 × 9.525 878.69 - 609.6E 4.57 177.8 × 7.137 609.6F 4.57 177.8 × 7.950 609.6

Bay width : 6.1 mUniform load : 2492 Pa

609.2

A

Figure 11: Geometrical properties of the Watwood [29] frame (all dimensions are in mm)

illustrated in Fig. 11.A beam finite element model of the Watwood [29] frame was created through the MAS-

TAN2 [30] structural analysis software considering the centreline dimensions and with thetapered members divided into prismatic segments using the actual cross-section propertiesat the middle of each segment, as shown in Fig. 12. This beam element model was used inthe implementation of the SRM, AISC MBMA [15] and Eurocode 3 [1] equivalent imper-fection approach. The same cross-section properties for each segment (i.e. element), loadingvalues and nodal geometry used in the beam element model of Watwood [29] were adoptedfor the beam element model developed in this study. Detailed information with respect toall these properties of the beam element model is provided in Watwood [29]. The proposedSRM was applied through (i) performing Linear Elastic Analysis of the segmented beamelement model of the frame considering an out-of-plumbness equal to 0.002 rad, which wassimulated by applying a notional horizontal force equal to 0.002 times the total gravityload at node 6, (ii) determining stiffness reduction factors τMN through eq. (3) for eachsegment by considering the cross-section properties and first-order forces at the middle ofeach segment and reducing their flexural stiffnesses, (iii) performing Geometrically Nonlin-ear Analysis of the frame with reduced stiffness (and the notional horizontal force), and (iv)making cross-section strength checks using eq. (4). The stiffness reduction factors τMN foreach segment calculated using the section forces obtained from a Linear Elastic Analysis atthe load factor equal to 1.0 are provided in Table 6. Note that the stiffness reduction factorsfor the segments in the other half of the frame are almost exactly the same as those of the

21

1

C

2345

6 78

910

11 1213

141516

1718

19

1234

56

78

910

11

1213

14

15

16 1718

P1P2

P2P2

P2

P2P2

P2P2

P2

P2

P3

MP1 = 30.95 kNP2 = 23.13 kNP3 = 30.63 kN

M = 12.38 kNm

873 mm

17932 mm

3726 mm

9728 mm

Element number

Node numberNode point

PR

OD

UC

ED

B

Y A

N A

UT

OD

ES

K E

DU

CA

TIO

NA

L P

RO

DU

CT

PRODUCED BY AN AUTODESK EDUCATIONAL PRODUCT

PR

OD

UC

ED

B

Y A

N A

UT

OD

ES

K E

DU

CA

TIO

NA

L P

RO

DU

CT

PRODUCED BY AN AUTODESK EDUCATIONAL PRODUCT

Figure 12: Beam element model of the Watwood [29] frame using the segmental approach

corresponding symmetric segments, and that for higher load factors, the stiffness reductionfactors would be calculated under the proportionally higher section forces.

Table 6: Ratios between axial forces and axial yield loads NEd/Npl, ratios between first order bendingmoments and plastic bending moment resistances My,Ed/My,pl and stiffness reduction factors τMN for eachsegment of the Watwood [29] frame at a load factor equal to 1.0

Segment no. NEd/Npl My,Ed/My,pl τMN Segment no. NEd/Npl My,Ed/My,pl τMN1 0.11 0.18 0.75 10 0.12 0.04 0.822 0.09 0.32 0.73 11 0.13 0.04 0.813 0.08 0.35 0.73 12 0.16 0.20 0.724 0.07 0.36 0.74 13 0.16 0.27 0.705 0.07 0.33 0.74 14 0.15 0.27 0.706 0.06 0.30 0.76 15 0.15 0.25 0.717 0.07 0.23 0.77 16 0.14 0.17 0.748 0.09 0.21 0.76 17 0.14 0.05 0.819 0.12 0.16 0.76 18 0.13 0.19 0.74

For the purpose of validating the proposed SRM for the design of tapered frames, a shellfinite element model of the Watwood frame was also created, which is shown in Fig. 13 with

22

the Von Mises stress values at the ultimate load factor obtained through GMNIA. In the

(Avg: 75%)fraction = −0.774597S, Mises

2 31 59 88116145173202230259287316344

X

Y Z

Figure 13: Shell finite element model of the Watwood [29] frame (stresses in N/mm2)

shell finite element simulation, an elastic perfectly-plastic material model was used with ayield stress equal to 344 MPa, the shape of the geometrical imperfections was assumed asthe global sway buckling mode of the frame and the ECCS residual stress [17] pattern shownin Fig. 2 was adopted.

The Eurocode 3 [1] equivalent imperfection approach and AISC MBMA [15] were alsoapplied to the frame so as to compare the qualities and limitations of the SRM againstthese design methods. In the EC3 equivalent imperfection approach, the out-of-plumbnessof the frame was assumed as 0.005 rad, the member out-of-straightnesses were modelled asa bow shape with the largest imperfection value equal to the member length divided by250 in the most detrimental direction and the ultimate cross-section strength checks wereperformed through a linear interaction equation. In the application of the AISC MBMA[15], two alternative approaches were used: (i) the direct analysis method (AISC MBMADAM) where a stiffness reduction factor equal to 0.9 was applied in the structural analysisusing the elastic critical stresses calculated considering the flexural buckling of the individualmembers of the frame in the design equations and (ii) the effective length method (AISCMBMA eff. length) approach where the nominal stiffness of the frame is considered in thestructural analysis while the elastic critical stresses resulting in the global sway buckling ofthe frame are used in the design equations. The stiffness reduction factor τDAM of the AISCMBMA DAM equal to 0.9 was determined by dividing the recommended stiffness reductionfactor of 0.8 by its safety factor equal to 0.9 in accordance with the recommendations of[31], since a safety factor was not used in the corresponding GMNIA. It should be notedthat local buckling effects were not considered in either the investigated design methods orthe GMNIA.

23

The ultimate load factors αult of the Watwood [29] frame determined through GMNIA,SRM, AISC MBMA DAM, AISC MBMA eff. length and the Eurocode 3 equivalent imper-fection approach are provided in Table 7. Note that since all the methods are based on

Table 7: Ultimate load factors αult obtained through GMNIA, SRM, AISC MBMA and Eurocode 3 equiv-alent geometrical imperfections for the Watwood frame

AISC MBMA AISC MBMA Eurocode 3GMNIA SRM DAM eff. length equiv. imp.

αult 2.35 2.14 1.97 1.90 1.90

Geometrically Nonlinear Analysis, these ultimate load factors αult were determined throughiteration until the utilisation ratio of the most critical cross-section, which was located at theapex of the frame, became equal to 1.0; the cross-section resistances were reduced throughbuckling reduction factors considering the corresponding elastic critical stresses in the AISCMBMA [15] and the stiffness reduction factors of the segments provided in Table 6 wererecalculated considering the corresponding first-order section forces which were readily de-termined through the principle of superposition in the SRM. As can be seen from Table7, the proposed SRM leads to considerably more accurate ultimate strength prediction incomparison to all the other design methods, with an ultimate load factor αult closest to thatdetermined through GMNIA of the frame.

The load-lateral deformation paths obtained through GMNIA, SRM, AISC MBMA [15]and the Eurocode 3 equivalent geometrical approach are compared in Fig. 14. Note that theSRM load-lateral deformation path was obtained using the lateral deformation values forten different load factors between 0 and 2.14, calculating stiffness reduction factors of eachsegment by considering the first-order forces for each corresponding load factor. As can beseen from the figure, both the AISC MBMA [15] and Eurocode 3 [1] equivalent imperfectionapproach underestimate the actual deformations at high load factors, providing a frameresponse which is stiffer than that obtained through GMNIA. Owing to stiffness reduction,the AISC MBMA DAM provides a load-lateral deflection response closer to that of GMNIAin comparison to the AISC MBMA effective length method where the nominal elastic stiffnesswas considered, but still underestimates the stiffness degradation. In contrast, the proposedSRM takes into account stiffness degradation at collapse more realistically owing to itsadvanced stiffness reduction functions, providing an accurate and conservative estimation ofthe lateral deflection.

8. Conclusions

A stiffness reduction method (SRM) for the in-plane design of web-tapered steel struc-tures fabricated through the welding of individual plates is developed in this paper. Theproposed approach is performed by (i) dividing tapered members into prismatic segmentsalong their lengths, (ii) reducing the flexural stiffness of each segment through the stiff-ness reduction functions on the basis of the first-order forces, (iii) performing Geometrically

24

P3

P1

P2 P2

P1

M M

Figure 14: Comparison of the load-displacement paths obtained through GMNIA, SRM, AISC MBMA –DAM, AISC MBMA – Effective length method and Eurocode 3 equivalent imperfection approach for theWatwood [29] frame

Nonlinear Analysis of the structure using beam elements with reduced stiffness (GNA-SR)and (iv) making cross-section strength checks. Stiffness reduction functions capable of con-sidering fully the deleterious influence of the spread of plasticity, geometrical imperfectionsand residual stresses are developed. Owing to the use of these advanced stiffness reductionfunctions, the proposed approach obviates the need of using member design equations ormodelling member out-of-straightnesses, but only requires cross-section strength checks, soleading to a very practical and direct approach for the design of tapered steel structures.

Shell finite element models of welded web-tapered steel members and frames were createdfor the verification of the proposed SRM. Initially, the SRM was verified for 150 taperedcolumns under uniform axial compression with different slenderness, tapering ratios andcross-section shapes, where it was observed that the proposed method leads to very accu-rate and consistent ultimate strength predictions. To compare the SRM against alternativedesign approaches for tapered members, the ultimate strengths obtained through the Eu-rocode 3 [1] equivalent imperfection approach and AISC MBMA [15] design guide were alsodetermined. It was observed that the proposed approach leads to considerably more accu-rate results in comparison to these approaches. The SRM was also verified against GMNIAfor tapered columns under non-uniform axial loading, where it was shown that it is muchmore accurate than the Eurocode 3 equivalent imperfection approach, AISC MBMA [15]and the method of [7] which provided some unconservative results for tapered memberssubjected to uniform axial stresses along their lengths. The proposed SRM was thoroughly

25

verified against GMNIA for 305 tapered beam-columns under uniform bending plus axialcompression and for 1111 tapered beam-columns subjected to different bending momentgradients and axial compression, with various levels of bending moments and axial loads,different slendernesses and different cross-section shapes. It was observed the proposed SRMis generally more accurate than Eurocode 3 [1] and AISC MBMA [15] for the design of web-tapered beam-columns. After the verification of the proposed SRM for individual members,it was assessed for a web-tapered frame using a benchmark example from the literature. Itwas observed that the proposed SRM leads to a very accurate ultimate strength predictionfor the considered tapered frame, providing a safe estimation of the actual stiffness degra-dation. The comparison of the SRM against AISC MBMA and the Eurocode 3 equivalentimperfection approach illustrated that the proposed approach is more accurate in predictingthe ultimate load factor and displacement of the frame.

The proposed stiffness reduction approach provides an accurate, direct and practicalmeans of designing web-tapered steel structures, realistically accounting for the stiffnessdegradation undergone by this type of structure. Future research will be directed towardsthe extension of the method to the consideration of the interaction of local and globalbuckling effects, and the flexural-torsional buckling assessments of tapered steel structures,following the approach adopted by Kucukler et al. [22, 32].

References

[1] EN 1993-1-1, Eurocode 3 Design of steel structures-Part 1-1: General rules and rules for buildings.European Committee for Standardization (CEN), Brussels; 2005.

[2] AISC 360-10, Specifications for structural steel buildings. American Institute of Steel Construction(AISC), Chicago; 2010.

[3] Shiomi, H., Kurata, M.. Strength formula for tapered beam-columns. Journal of Structural Engineer-ing, ASCE 1984;110(7):1630–1643.

[4] Raftoyiannis, I.G., Ermopoulos, J.C.. Stability of tapered and stepped steel columns with initialimperfections. Engineering Structures 2005;27(8):1248–1257.

[5] Salem, A., El Aghoury, M., Fayed, M., El Aghoury, I.. Ultimate capacity of axially loaded thin-walledtapered columns with doubly symmetric sections. Thin-Walled Structures 2009;47(8):931–941.

[6] Braham, M., Hanikenne, D.. Lateral buckling of web tapered beams: An original design methodconfronted with a computer simulation. Journal of Constructional Steel Research 1993;27(1):23–36.

[7] Marques, L., da Silva, L.S., Rebelo, C., Santiago, A.. Extension of EC3-1-1 interaction formu-lae for the stability verification of tapered beam-columns. Journal of Constructional Steel Research2014;100:122–135.

[8] Kucukler, M., Gardner, L., Macorini, L.. A stiffness reduction method for the in-plane design ofstructural steel elements. Engineering Structures 2014;73:72–84.

[9] Kucukler, M., Gardner, L., Macorini, L.. Development and assessment of a practical stiffnessreduction method for the in-plane design of steel frames. Journal of Constructional Steel Research2016;126:187–200.

[10] Lee, G.C., Morrell, M., Ketter, R.L.. Design of tapered members. Welding Research Council Bulletin1972;173:1–32.

[11] Fraser, D.. Design of tapered member portal frames. Journal of Constructional Steel Research1983;3(3):20–26.

[12] White, D.W., Jeong, W.Y., Toğay, O.. Comprehensive stability design of planar steel mem-bers and framing systems via inelastic buckling analysis. International Journal of Steel Structures2016;16(4):1029–1042.

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