Engineering Structures 25 (2003) 1033–1043www.elsevier.com/locate/engstruct

A second-order inelastic model for steel frames of taperedmembers with slender web

Jin-Jun Lia, Guo-Qiang Lib,∗, Siu-Lai Chanc

a R&D Department, Shanghai Maglev Transportation Development Company, 2520 Longyang Road, Shanghai, 201204, Chinab College of Civil Engineering, Tongji University, 1239 Siping Road, Shanghai, 200092, China

c Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

Received 22 June 2002; received in revised form 29 January 2003; accepted 29 January 2003

Abstract

A concentrated plasticity model is proposed for second-order inelastic analysis of the steel frames of tapered members with aslender web. Such significant effects as residual stresses, initial geometric imperfection, gradual section yielding at the elementends, distributed plasticity within the element and local web buckling are considered in this model. Numerical examples on taperedcompact columns, prismatic beam-columns with local buckling, a prismatic frame with local buckling and a tapered frame withlocal buckling are studied in this paper to verify the accuracy and efficiency of the proposed analytical model. As an application,the column curves of tapered steel columns are obtained with the proposed analytical model, both excluding and including the localbuckling effects of slender webs. Some meaningful conclusions are drawn in the end of this paper. 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Second-order inelastic analysis; Concentrated plasticity model; Steel frames; Tapered members; Slender web

1. Introduction

The present approaches for second-order inelasticanalyses of steel frame structures can be generally cate-gorized into two types: concentrated plasticity model(plastic hinge method) and distributed plasticity model(plastic zone method)[12].

Two different distributed plasticity models exist, 3Dshell FE and plastic zone method based on beam-columntheory. The 3D shell FE is the fundamental distributedplasticity model in common sense to predict the actualresponse of steel frame structures[3], where the webplate and flange plates are discreted with a large numberof 3D shell elements and non-linearities can be explicitlyand exactly reflected. Some widely applied software,such as ABAQUS and ANSYS are capable of beingtools for these analyses. However, the second-order plas-tic zone method based on beam-column theory, with

∗ Corresponding author: Tel:+86 21 6598 2975; Fax:+86 216598 3431.

E-mail address: gqli@mail.tongji.edu.cn (G.-Q. Li).

0141-0296/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0141-0296(03)00043-9

much less degrees of freedom and therefore much lessneed for computer effort, is more applicable in practicalanalysis of steel frame structures. The plastic zonemethod can consider the complex distribution of plas-ticity through the volume of a structure by discretizationof members along the length and over the cross-section[8,16,34], including the effects of residual stresses,initial geometrical imperfection, etc. However, it is stillvery time-consuming due to the necessary fine-meshdiscretizations, even based upon present powerful com-puters[12,19].

The simple plastic hinge method is a simple andefficient approach for representing the inelastic behaviorof steel frames. The material of a steel frame is assumedto be elastic-perfectly plastic and a second-order elasticanalysis of the structure is performed until the plasticmoment capacity is reached at the maximum momentsection. An imaginary hinge is then placed in the struc-ture at this location. This procedure is repeated until asufficient number of hinges have formed to produce amechanism[12]. However, it is found that the analyticalresults of the simple plastic hinge method are over-esti-mated in certain cases[12,19].

1034 J.-J. Li et al. / Engineering Structures 25 (2003) 1033–1043

Several novel plastic hinge methods, such as therefined plastic hinge method [29,30], modified plastichinge method [20,21], notional load plastic hingemethod [31], quasi plastic hinge method [2], and themodel used in the NIDA (Nonlinear Integrated Designand Analysis for steel frame structures) software [9,10]have been proposed in the past decade. These improvedplastic hinge methods seek to be as simple and efficientas the simple plastic hinge method while as accurate forthe assessment of load-bearing capacity of steel framesas the plastic zone method. Two modifications are madeto account for: (1) the secant stiffness degradation at theplastic hinge location; and (2) the member stiffnessdegradation between two plastic hinges.

The conventional plastic hinge analysis for steel framestructures assumes the section to be compact, and doesnot consider the degradation of the section capacitycaused by local buckling. Blandford and Glass [6] con-sidered the local web buckling of steel box sections inthe static and dynamic frame analysis with a simpleeffective breadth formula and therefore treated thebuckled prismatic members as non-prismatic members.Based on the refined plastic method, Kim and Lee [18]proposed the improved refined plastic-hinge method byintroducing the LRFD equations for local bucklingstrength, to account for local buckling effects in second-order inelastic analysis. With the similar form of therefined plastic hinge method, the pseudo plastic zonemethod can consider the local member buckling byimproving the stiffness parameters with accurate studbeam-column 3D FE analysis and defining some equa-tions such as the inelastic stability function and imper-fection reduction equations [5]. Of course, these investi-gations were initiated for the prismatic steel structures.

Steel frames comprising tapered beams and columnsnot only provide even distribution of structural strength,but also yield a design with less steel consumption. Sincethe structural strength along member length is non-uni-form, the plastic zone method is, strictly speaking,required to predict the second-order inelastic responseof tapered steel frames. Alternatively, step representationby dividing the tapered member into a large number ofprismatic elements can be used for the same purpose[32]. However, both the plastic-zone method and the sat-isfactory step representation need great computationalefforts and are not convenient for daily use in engineer-ing design of tapered and non-compact steel structures.

To the authors’ knowledge, few investigations werepreviously conducted on the second-order inelasticanalysis of tapered steel structures. Li [26] conducted aresearch study and this paper presents a simplified butaccurate concentrated plasticity model for second-orderinelastic analysis of tapered and non-compact steel framestructures, where a modified plastic-hinge method isadapted to tapered member based on the correspondingelastic stiffness matrix of tapered elements. Effects of

local web buckling on the structural capacity are con-sidered directly by re-calculating the elemental stiffnesswith the effective breadth formula under various stressstates. Numerical examples on a tapered and compactcolumn, prismatic and non-compact columns, a pris-matic and non-compact steel portal frame and a taperedand non-compact steel portal frame are employed toexamine the accuracy and efficiency of the analyticalmodel proposed in this paper. As an application, the col-umn curves of tapered steel columns are obtained withthe proposed analytical model, both excluding andincluding the effects of slender webs.

2. Basic assumption

1. The tapered steel member is shown in Fig. 1 andframes with such tapered members are studied in thispaper. The height of the web is linearly varied withthe flanges symmetric and uniform in width along themember length.

2. The material of steel members is assumed to be elas-tic-perfectly plastic.

3. All members are initially straight.4. The plane section remains plane after deformation.5. Shear deformation is considered in the elastic stiff-

ness matrix but its contribution to yielding is ignored.6. Only local web buckling is considered and local

flange buckling is ignored.7. The frame is braced in such a manner as to preclude

lateral torsional instability of any member.8. Elastic unloading at plastic hinge is not considered.

3. Proposed concentrated plasticity model

3.1. Elastic stiffness matrix

For the tapered member studied in this paper, theapplied forces and deformations can be modeled as in

Fig. 1. A steel tapered member.

1035J.-J. Li et al. / Engineering Structures 25 (2003) 1033–1043

Fig. 2. Following the same procedure for prismaticbeams [22,23], the flexual equilibrium differential equ-ation with non-dimensional form of the tapered Timosh-enko–Euler beam element simultaneously consideringeffects of axial forces and shear deformation can be writ-ten as [24,26,27],

a(z)·y��b(z)·N·y��N·y � b(z)·Q1�(M1�Q1·z) (1)

where

a(z) � E·I(z)·g(z), b(z) � E·I(z)·A�w(z)

G·A2w(z)

, g(z) � 1

�N

G·Aw(z),

and A, Aw and I are respectively the overall area, webarea and inertial moment of the cross-section at thelocation of distance from the original point of theelement; and E and G are respectively elastic andshear modulus.

By the Chebyshev Polynomial to represent functions,y, a, b can be expressed as,

y(x) � �Mn � 0

yn·xn (2a)

a(x) � �Mn � 0

an·xn (2b)

b(x) � �Mn � 0

bn·xn (2c)

for the solution of Eq. (1), the elastic stiffness equationof the tapered beam element could be obtained as,

[ke]·{d} � {f} (3)

where

{d} � [d1,q1,d2,q2]T,

{f} � [Q1,M1,Q2,M2]T

Fig. 2. Applied forces and deformation of an element.

[ke] � ��f1 f2 f1 f3

�f4 f5 f4 f6

f1 �f2 �f1 �f3

�f7 f8 f7 f9

�.

The expressions of fi(i = 1,2,...,9) are given in AppendixA and the detail of derivation of Eq. (2a–c) can be foundin the previous publications [24,26,27].

From Fig. 2 the axial force within the taperedelement is

N � EA(du / dx) (4)

and the axial equilibrium differential equation is

dNdx

� E�A(x)d2udx2 � �dA

dx�dudx� � 0 (5)

where u(x) is axial displacement. The element expressionof axial stiffness matrix is given by [17],

k11 � k22 � �k12 � �k21 �E

�L

0

1A(x)

dx

. (6)

Thus, the elastic stiffness equation of the tapered columnelement could be obtained as

[k]·{d} � {f} (7)

where

{d} � [u1,d1,q1,u2,d2,q2]T

{f} � [N,Q1,M1,N,Q2,M2]T

[k] � �k11 0 0 k12 0 0

0 �f1 f2 0 f1 f3

0 �f4 f5 0 f4 f6

k21 0 0 k22 0 0

0 f1 �f2 0 �f1 �f3

0 �f7 f8 0 f7 f9

�.

3.2. Initial and limit yielding surface equations

It is necessary to provide the initial and ultimate yield-ing surfaces for cross-sections for second-order inelasticanalysis using the concentrated plasticity model. A fullyyielding limit surface equation for the maximum strengthof I-sections was proposed [13]. This equation for thecase of uniaxial bending about the strong axis of thecross-section has the simple form as

�NNy�1.3

�MMp

� 1.0. (8)

1036 J.-J. Li et al. / Engineering Structures 25 (2003) 1033–1043

The initial yielding surface including the effect ofresidual stresses may be defined as

N0.8Ny

�cM

0.9Mp

� 1.0. (9)

In the above equations, Nand M are the axial load andmoment applied to the section respectively, Ny = fyA andMp are respectively the axial squash load and limit plas-tic moment of the section for axial load and momentapplied separately, fy is the yield stress of material andc is the shape factor of cross-section.

3.3. Elastic-plastic incremental stiffness matrix

Following the same procedure as the inelasticincremental stiffness equation for prismatic element[22,23], a similar equation for tapered element using theplastic-hinge model can also be obtained as [25],

[kp]·{�d} � {�f} (10)

where {�d} and {�f} are the vectors of incrementalnodal displacements and applied loads, [kp] is the elastic-plastic tangent stiffness matrix, which can be calcu-lated by

[kp] � [ke]�[ke][G][E][L][E]T[G]T[ke] (11)

where

[L] � ([E]T[G]T([ke] � [�][ke])[G][E])�1, [�]

� diag[a1,a1,a1,a2,a2,a2],

[E] � �1 1 1 0 0 0

0 0 0 1 1 1�T

, [G]

� diag�∂�1

∂N1,0,

∂�1

∂M1,∂�2

∂N21,0,

∂�2

∂M2�.

In matrix [G], �i (i = 1,2) is the yielding function atthe end i of the element, defined as,

� � �NNy�1.3

�MMp

. (12)

In matrix[�], ai (i = 1,2) is the plastic hinge parameterrepresenting the degree of gradual yielding at the end iof the element, defined by,

ai �Ri

1�Ri

(13)

where

Ri � 1 MMsN

1�M�MsN

MpN�MsNMsNMMpN

0 MMpN

M, MsN and MpN stand respectively for applied moment,initial yield moment and full plastic moment at the endi of the element in the presence of axial force.

3.4. Concept of tangent modulus for tapered elements

The concept of tangent modulus for prismaticelements was proposed to consider the plasticity spreadwithin the ends of elements [12,29,30], which may bethe results of residual stresses and large axial force tosquash load ratio. The tangent / elastic modulus ratiobased on the CRC column strength equations for pris-matic members can be expressed as [12,29,30],

Et

E� 1.0 if N0.5Ny (14a)

Et

E�

4NNy�1�

NNy� if N � 0.5Ny. (14b)

Since the ratios of axial force to squash load calcu-lated at the two ends of a tapered element are different,the tangent modulus for tapered elements can beapproximately re-written from Eq. (14a,b) as,

Et �A1

A1 � A2

Et1 �A2

A1 � A2

Et2 (15)

where A1 and A2 are respectively the sectional area ofthe two ends, Et1 and Et2 are representative of tangentmodulus at the two ends and can be determined byEq. (14a,b).

3.5. Effects of local web buckling

When a high and thin web plate is used for taperedmembers of steel frames, the structural economy maybe enhanced further. Under this circumstance, the flangeoutstand breadth-to-thickness ratio should be pro-portionally large in order that web and flange plates canprovide mutual support in work stresses and sectionshave good economy [11]. In the Chinese code [7] for thedesign of lightweight steel portal frames, the maximumpermitted web height-to-thickness ratio can be up to 250for steels with nominal yield strength of 235 MPa andabout 200 for steels with nominal yield strength of 345MPa. Elastic buckling for slender webs possibly occurseven when the section is fully elastic. The local webbuckling will generally lead to an evident reduction ofelement strength and therefore structural resistance butnot structural failure. Although local flange buckling canoccur after local web buckling and leads to further stiff-ness reduction [4], for the sake of simplicity only localweb buckling will be considered in this current research.

A series of sophisticated effective breadth formula forweb plates under compression and bending werereported [33] and are employed in this study. For simply

1037J.-J. Li et al. / Engineering Structures 25 (2003) 1033–1043

Fig. 3. Simply supported rectangular plate in compression andbending.

supported rectangular plate shown in Fig. 3, the effectivebreadth, defined in Fig. 4, can be determined by the fol-lowing formulae [33],

be1

b�a4l�b�b2�4l� (16)

b � 1 � C·(l�l0) � l (17)

l0 � A�B·ln��p

b �1.0 (18)

A � �0.05�0.542·exp��11.9src

sy� (19)

B � �0.09�0.107·exp��12.4src

sy� (20)

C � �157��p

b ��src

sy� � 43��p

b � � 1.2�src

sy� (21)

� 0.03

a �src

0.3sy�1 � 45

�p

bf� � �1�

src

0.3sy��1�

f2

16�. (22)

If both sides in compression (s1,s20),

be2

b� (1 � hf)

be1

b(23)

Fig. 4. Effective breadth of simply supported plates.

h �src

0.3sy�0.59�86

�p

b � � �1�src

0.3sy� (24)

�0.44 � 29�p

b �be1 � be2b. (25)

If one side in compression and the other in tension(s10,s2 � 0),

be2

b� (1 � h)

be1

b(26)

h �src

0.3sy��0.68�83

�p

b�

1.27f2 � (27)

� �1�src

0.3sy���0.53 � 29

�p

b�

0.97f3 �

be1 � be2bf

(28)

where

f �s1�s2

s1(29)

l � �s1

sE

(30)

sE �tb�

Ep2k12(1�n2)

(31)

k �8.4

(2.1�f)(0f1) (32a)

k � 10f2�13.37f � 11.36(1f2) (32b)

1500

�p

b

1150

(33)

0src

sy0.3. (34)

In above formulae, �p and srcrepresent respectively theeffects of initial imperfection and residual stresses onplate buckling, s1 and s2 are the maximum and mini-mum stresses along the plate sides (positive for tension),k and sE are the elastic buckling factor and the elasticbuckling stress for simply supported rectangular plates.

The above effective breadth formulae are used to re-calculate the sectional area and inertial moment once theweb boundary stresses are equal to the critical stresses.

1038 J.-J. Li et al. / Engineering Structures 25 (2003) 1033–1043

The reduction of structural stiffness is accomplishedthrough the re-calculated sectional area and inertialmoment in forming the elemental stiffness matrix foreach element with web buckled. The parameters �p /band src /sy in Eqs. (33) and (34) concerning initial geo-metric imperfection and residual stresses are assumed tobe constants and equal to 1/150 and 0.3 respectively inthe proposed model.

3.6. Numerical scheme for non-linear finite elementsolution

Second-order inelastic analysis of steel frames is atypical finite element non-linear analysis, and best solvedfor the incremental–iterative procedure. The modifiedNewton–Raphson method is employed to obtain the limitload of the structures in this paper.

4. Verification

4.1. Mesh convergence test

The taper ratio of tapered columns can be defined as,

r �d1

d2�1 (35)

where d1 and d2 are respectively the larger and smallersection height.

Defining the effective length factor K and non-dimen-sional slenderness parameter l̄ for tapered columns as,

K �pL�

EI2

Pcr

(36)

where Pcr is the elastic critical load of the tapered col-umn under axial compression, L is the member lengthand I2 is the inertial moment at the smaller end, and

l̄ �lp�sy

E�

KLprx2

�sy

E(37)

where rx2 is the gyrus radii at the smaller end, sy isyielding strength and slenderness parameter

l �KLrx2

. (38)

For a tapered compact column with r = 4, l̄ =1.265 and subjected to residual stresses, initial geometricimperfection with sinusoid pattern of L/1000 amplitudeand axial compression, the relationship between elementnumber and relative error of capacity results by the pro-posed method and step representation (ten prismaticelements) are shown in Fig. 5. It can be observed thateven for such sharply tapered columns, i.e. r = 4, four

Fig. 5. Relative error vs number of elements for a sharply taperedmember.

proposed elements are enough to obtain the strengthcapacity with sufficient accuracy.

4.2. Prismatic non-compact beam-columns

To examine section capacity of thin-walled I-sectionsin combined compression and major axis bending, Testseries I and II were performed at the University of Syd-ney [15]. The section capacities of eight prismatic beam-columns with local web buckling were obtained in TestSeries I. These tested steel beam-columns are analyzedhere, both ignoring and considering the effects of localweb buckling. The ratios of analysis-to-test results areillustrated in Fig. 6.

It can be observed that a great deviation exist in allcases but the last one with pure bending (moment-to-thrust ratio being 1) when ignoring local web buckling,nevertheless good coincidence occurs when consideringthose effects.

Fig. 6. Prismatic non-compact steel beam-colimns.

1039J.-J. Li et al. / Engineering Structures 25 (2003) 1033–1043

4.3. A prismatic non-compact frame

A series of large-scale tests of prismatic steel frameswith non-compact sections were conducted at theQueensland University of Technology [4] to validate thedistributed plasticity analysis and pseudo plastic zoneanalysis. Test Frame 4 is used here to verify the pro-posed method to prismatic steel frames with local buck-ling. Fig. 7 gives the horizontal load–displacement (forthe top of right column) curves of the test and analyticalresults, both ignoring and considering the effects of localweb buckling.

The ratios of analysis-to-test limit loads are respect-ively 1.20 when ignoring local buckling, and 1.06 whenconsidering local web buckling. As shown in Fig. 7 theanalytical curve with local web buckling provides a bet-ter prediction to the test result.

4.4. A tapered non-compact frame

A series of large-scale tests of tapered steel portalframes with non-compact sections have been conductedat Tongji University [26,28], where the Test Frame 1 ispresented here to verify the proposed method of theanalysis of tapered steel frames with member local buck-ling. Test Frame 1 is a full-scale, single-bay and taperedsteel portal frame with sufficient out-of-plane restraints,rigid knee and ridge joints and pinned column bases, asshown in Fig. 8. The member sections are non-compact.The web height-to-thickness ratio and flange outstandbreadth-to-thickness ratio of the members for the frameare 127 and 10.42 respectively. Such a frame may occurat local web buckling even in fully elastic state and localflange buckling in the elasto-plastic state [11].

Vertical loads were only applied to Test Frame 1,exerted by jacks incrementally till the structural failure.The general arrangement of the test set-up for TestFrame 1 is illustrated in Fig. 9. The on-site overview ofthe test of Test Frame 1 is shown in Fig. 10. The elasticmodulus and yielding strength of the steel used for Test

Fig. 7. A prismatic non-compact steel frame.

Frame 1 are respectively 197 GPa and 394 MPa, whichwere obtained by standard material tension tests.

At a load of 32 kN exerted by each jack, Test Frame1 began to unload, indicating that the maximum capacityof the frame had been achieved. Test Frame 1 failed inthe mode of in-plane instability due to the yielding andspread of plasticity over the frame members caused bythe combination of axial compression force, bendingmoment, residual stress and local buckling. Plastic localbuckling deformations were observed at the regionadjacent to the knee beam-to-column connection and theridge beam-to-beam connection, as shown in Fig. 11.

Curves of the experimental and the analytical load vsvertical displacement at the ridge joint are plotted in Fig.12. A good correlation between the two sets of resultswas observed.

5. Application

5.1. Column curves of tapered compact I-columns

Tapered columns are widely used in steel portal framestructures. For safety, the strength and stability of tap-ered columns are specified in the codes for steel structuredesign. For example, the equation for prismatic columnchecks is adapted to tapered columns after introducingthe special effective length factors Kr in the AmericanCode [1]. The Chinese Code [7] adopts a similar treat-ment for the safety check of tapered columns. But theeffective length factors are generally calculated from theelastic stability analysis, and therefore it is necessary toexactly predict the actual column capacities of taperedcolumns and examine the adequacy of abovementionedcode method.

Adopting the elastic analysis with the obtained elasticstiffness equation in this paper, the effective length fac-tor K of simply supported tapered columns defined inEq. (36) can be expressed as a function of tapered ratio[26] as,

K � 0.3155 � (1�0.3155)exp(�r / 1.56). (39)

Adopting the second-order inelastic analysis for tap-ered compact columns with various tapered ratios [seeEq. (35)] and various slenderness [see Eqs. (37) and(38)], column curves are obtained and shown in Fig. 13.All calculated tapered columns are subjected to residualstresses, initial geometric imperfection with sinusoid pat-tern of L/1000 amplitude and axial compression, withoutconsideration of local web buckling. The coded columncurves respectively from LRFD [1] and GBJ 17-88 [14]for prismatic compact columns are also given in Fig. 13,which correspond with the curve of r = 0 obtained bythe proposed model.

The approach treating tapered steel columns as pris-matic columns with specified effective length factor K

1040 J.-J. Li et al. / Engineering Structures 25 (2003) 1033–1043

Fig. 8. The dimensionsof test frame 1 (mm).

Fig. 9. Test set-up fpr test frame 1.

is generally conservative for compact sections in design,since from Fig. 13 the real column strengths increasewith the taper ratio under the same slenderness factors.

5.2. Column curves of tapered I-columns with slenderweb

As previously mentioned, tapered I-columns arewidely used in lightweight steel portal frame structuresand generally comprise a slender web. So, it is moremeaningful to predict the actual column capacities fortapered columns with slender web than those of taperedcompact ones. To achieve this, a series of tapered col-umns manufactured with Chinese Q345 steel material,of maximum web height-to-thickness ratio of 200 andconstant flange outstand breadth-to-thickness ratio of 10are selected, and the predicted column curves are plottedin Fig. 14.

Comparing Fig. 14 with Fig. 13 we can see local webbuckling reduces the capacities of tapered non-compactcolumns. The maximum possible degradation can bemore than 5% in the cases considered in this paper.Further, the coded approaches for tapered and non-com-pact steel columns such as CECS and LRFD are uncon-

servative when the ratio of compression-to-squash loadis larger than 0.8

6. Conclusions

To develop a theoretical approach for second-orderinelastic analysis of steel frames of tapered memberswith slender web, a concentrated plasticity model is pro-posed in this paper. The following concluding remarkscan be drawn.

1. In the proposed analytical model, the effects ofresidual stresses, initial geometric imperfection, grad-ual section yielding at the element ends, distributedplasticity within the element and local web bucklingcan be taken into account.

2. Numerical examples show that the proposed methodhas a fast rate of mesh convergence since only fourtapered elements are sufficient even for the analysisof sharply tapered (r = 4) steel members, while 20elements are needed for the similar accuracy whenstep representation is adopted.

3. Validation of the proposed model is conducted by

1041J.-J. Li et al. / Engineering Structures 25 (2003) 1033–1043

Fig. 10. On-site overview of test frame 1.

comparing the analytical results with those of experi-mental dates for prismatic non-compact I-sections, aprismatic non-compact steel portal frame and a tap-ered non-compact steel portal frame. Good accuracyof the proposed model is confirmed.

4. Column curves of tapered compact column are pre-dicted by the proposed analytical model and com-pared with the coded prismatic column curves ofLRFD and GBJ 17-88. It can be found from the col-

Fig. 11. Plastic local buckling deformations at the failure of test frame 1.

Fig. 12. A tapered non-compact steel frame.

Fig. 13. Column curves for tapered compact columns.

umn curves that treating tapered columns as prismaticones through an appropriate effective length factor isgenerally conservative for capacity checks.

5. Local web buckling evidently reduces the capacitiesof tapered columns with slender web. The maximumpossible degradation can be more than 5% in the casesconsidered in this paper. Although the axial forces intapered columns of the steel portal frames are small,it should be noted that the coded formula for tapered

1042 J.-J. Li et al. / Engineering Structures 25 (2003) 1033–1043

Fig. 14. Column curves for tapered non-compact columns.

and non-compact columns such as CECS and LRFDare unconservative when the ratio of compression-to-squash load is larger than 0.8.

Acknowledgements

The authors acknowledge the financial support of theproject ‘ Advanced Analysis & Design of Steel PortalFrames Considering Integrated Limit States’ , in theScheme of University Principle Professor Support, bythe Education Ministry of PR China.

Appendix A

f1 �y15

y11y15�y14y12, f2 �

y13y15�y16y12

y11y15�y14y12, f3 �

�y12y14

y11y15�y14y12

, f4 �1�y11f1

y12

,

f5 �y13�y11f2

y12, f6 � �

y11f3

y12, f7 � f1L � N�f4, f8

� f2L�f5, f9 � f3L�f6,

y1 � �L

G·Aw(0)·g(0), y2 �

Lg(0)

, y3 �

�L

G·Aw(1)·g(1), y4 �

Lg(1)

,

y5 �L·b0(N·y1 � L)

2a0

, y6 � �L2

2a0

, y7 �L·N·b0·y2

2a0

,

y8 �2(L·N·b0�a1)y5 � L·(b1 � L)(N·y1 � L)

6a0, y9

�2(L·N·b0�a1)y6

6a0

y10 �2(L·N·b0�a1)y7 � L·N(b1 � L)y2

6a0

, y11 � c1y1

� c2y5 � c3y8 � c4, y12 � c2y6 � c3y9,

y13 � �(c1y2 � c2y7 � c3y10), y14 � c5y1 � c6y5

� c7y8 � c8�y3, y15 � c6y6 � c7y9,

y16 � �(c5y2 � c6y7 � c7y10)

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