Guidance for the Design of Spliced Columns

Guidance for the Design of Spliced ColumnsAna M. Girão Coelho, Ph.D.1; Pedro D. Simão, Ph.D.2; and Frans S. K. Bijlaard3

Abstract: Steel columns in sway and nonsway frames that are spliced along their length generally have a lower strength capacity in compres-sion. This load capacity can be further reduced owing to the inevitable presence of small geometrical imperfections in the form of the out-of-straightness of the column and column segment misalignment. The current work examines the buckling behavior of a framed spliced columnwith initial imperfections and the possibility of a nonuniform cross section. A geometrically nonlinear model accounting for imperfect elasticbuckling behavior is formulated using the differential equations of equilibrium. This is followed by a study of the imperfection sensitivity to thelinearly evaluated critical load. A discussion on the variation of the load-carrying capacity with the level of imperfections on a practical splicedcolumn is also presented. The findings suggest that a spliced column can be considered equivalent to a prismatic Euler column, with an appro-priate effective length, for design purposes. In this context, some implications for design are deduced from the presented analysis. DOI:10.1061/(ASCE)ST.1943-541X.0000546. © 2012 American Society of Civil Engineers.

CE Database subject headings: Connections; Imperfection; Standards and codes; Steel columns; Structural stability.

Author keywords: Analytical modeling; Buckling; Columns; Geometrical Imperfections; Splices.

Introduction

Research on the stability of steel columns is longstanding. Euler’sclassic work on the stability of an elastic pin-ended column wascompleted in the eighteenth century. Since then, many studies oncolumn buckling have been carried out to address problems asso-ciated with the influence of (1) the degree of column end restraints,(2) material nonlinearity, (3) residual stresses, (4) initial imper-fections, and (5) load eccentricity on the load capacity (Johnston1983; Bjorhovde 2010). Some of these studies have been reflected inthe development of models and engineering-oriented formulas forinclusion inmodern design codes, such as the AISC specification forstructural steel buildings (AISC 2010) and the European code ofpractice for the design of steel structures, EN 1993 (CEN 2005).

The current design approach adopted in both codes is based onthe behavior of individual members. Column design accounts for theeffect of the interaction among neighboring framing members. Inmost structural configurations, the columns are supported by othermembers (beams and columns) that provide some kind of restraintat the column ends. Consider Column c of length L of the

subassemblage model of a frame illustrated in Fig. 1. The basicmechanical model is also shown in Fig. 1. Ends a and b are restrainedby beam-to-column connections that usually exhibit an elastic initialresponse (partially restrained connections). Here, Ku is the corre-sponding rotational spring constant and is defined as the moment thespring can sustain for a unit rotation. The beams framing into thecolumn also provide some kind of restraint against lateral deflection.The degree of fixity depends on the beam flexibility. Here, KD rep-resents the spring extensional stiffness and is defined as the transverseforce the spring sustains for a unit displacement. This type of modeloffers a clear physical illustration and solid grounds in themechanicsof the problem. However, it completely disregards the constructiondetails such as column splices.

In recent papers (Girão Coelho et al. 2010; Simão et al. 2010) theeffect of splices on column behavior has been examined and amodelthat combined the flexible column end support conditions with thepossibility of change in the column section serial size and splicedalong its length has been proposed. This model is significantly morecomplex but also more accurate in terms of the evaluation of thestability of columns as part of the framework. The model was an-alyzed using an energy-based approach. An important conclusionwas that of the possible detrimental effect of the splice on the load-carrying capacity of the column, depending on the location and thestiffness. The key development was the inclusion of the concept ofend fixity factor C in simplifying the analysis of the results. Thisfactor is equivalent to the so-called effective length Leq of a columnwith arbitrary end conditions, which represents the length of theequivalent Euler column (Timoshenko and Gere 1961). Simplerelationships between C and the relevant characteristics of thecolumn and splice have been derived to a point where the criticalload can be readily determined by hand or with a computer.

This study extends the earlier work by introducing geometricimperfections to the model. Imperfections can change the response ofthe system. An imperfect column does not exhibit bifurcation of equi-librium.The stability problembecomes that of an in-plane bending type.The formulation of the model is based on the exact solution of thegoverning equations for buckling.The following assumptions aremade:• The analysis is purely elastic (the stress-strain relationship is

completely linear).

1Researcher, Dept. of Structural and Building Engineering, Steel andTimber Structures, Faculty of Civil Engineering and Geosciences, DelftUniv. of Technology, P.O. Box 5048, 2600 GA, Delft, Netherlands; andInstitute of Computers and Systems Engineering of Coimbra (INESC-Coimbra), Coimbra 3000-033, Portugal (corresponding author). E-mail:[email protected]

2Assistant Professor, Dept. of Civil Engineering, Univ. of Coimbra,Coimbra 3030-788, Portugal; and Institute of Computers and SystemsEngineering of Coimbra (INESC-Coimbra), Coimbra 3000-033, Portugal.E-mail: [email protected]

3Professor, Dept. of Structural and Building Engineering, Steel andTimber Structures, Faculty of Civil Engineering and Geosciences, DelftUniv. of Technology, P.O. Box 5048, 2600 GA, Delft, Netherlands. E-mail:[email protected]

Note. This manuscript was submitted on May 27, 2011; approved onNovember 30, 2011; published online on December 2, 2011. Discussionperiod open until February 1, 2013; separate discussions must be submittedfor individual papers. This paper is part of the Journal of StructuralEngineering, Vol. 138, No. 9, September 1, 2012. ©ASCE, ISSN 0733-9445/2012/9-1079–1088/$25.00.

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• Residual stresses are ignored.• The column is centrally loaded.• The column is considered transversely supported; thus, the

possibility of buckling about the weak axis is precluded.• The cut surfaces at the column splice are in perfect contact in the

case of bearing splices.• The initial out-of-straightness of the column is taken into account

in the analysis.• The possibility of column segment misalignment is also taken

into account.The current paper starts with a stability analysis of a framed

spliced column using the classical equilibrium approach. The pos-sibility of columns with a stepped cross section is also included inthe analysis. This is followed by a study of the imperfection sensi-tivity to the linearly evaluated critical load. Geometric imperfectionsin the form of the out-of-straightness of the column and columnsegment misalignment are considered. The initial curvature is ap-plied by using the critical mode shapes from linear buckling anal-ysis. A discussion on the variation of the load-carrying capacity withthe imperfection magnitude on a practical spliced column is alsopresented. It is shown that the geometric imperfections can producehighly unstable behavior. These findings are important for designersaiming to achieve safer and more efficient and economical designsfor frame stepped columns. Some guidelines to represent this in-formation in a suitable form for subsequent inclusion in a maximumstrength column analysis are also proposed.

Elastic Buckling of Spliced Columns

Differential Equation of Equilibrium of an Ideal Column

The fourth-order equilibrium equations for an initially straightspliced column loaded axially by a compressive loadNEd are derivedusing the equilibrium method. The force NEd retains its direction asthe column deflects. In this classical approach, the problem is re-duced to an eigenboundary-value problem and the critical conditionsare the eigenvalues.

The column in Fig. 2(a) consists of two independent members, Iand II, of lengths LI5aL andLII5 (12a)L, 0#a# 1, respectively,connected by a spring at point S. Here, Kuc is the tangent elasticstiffness coefficient for this spring. Each member has a constantbending stiffness EII and EIII. The mathematical formulation of thisproblem is given subsequently. Fig. 2(b) shows the free-body diagram

of an infinitesimal segment of this column. The moment equilibriumequation is written in the following form (Chen and Lui 1987):

d2M

dx22NEd

d2w

dx2¼ 0 ð1Þ

where w 5 lateral displacement. For linearly elastic materials, thebending moment and the column curvature are related as follows:

M ¼ 2EId2w

dx2ð2Þ

From Eqs. (1) and (2) the general fourth-order differentialequilibrium equation is obtained as follows:

d2

dx2

�2EI

d2w

dx2

�2NEd

d2w

dx2¼ 0 or wIV þ m2w0 ¼ 0

ð3Þwhere m is given by

m2 [ m2I ¼ NEd

EIIor m2 [ m2

II ¼ NEd

EIIIð4Þ

The general solution of this equation isMember I:

wI ¼ A1sinmIx þ A2cosmIx þ A3x þ A4 ð5aÞ

Member II:

wII ¼ B1sinmIIx þ B2cosmIIx þ B3x þ B4 ð5bÞ

whereAi,Bi5 constants (i5 1, 2, 3, 4). This solutionmust satisfy theboundary conditions that are given by the following:

At xI 5 0:

wI ¼ 0

2EIIw0I ¼ 2Kuaw9Ið6aÞ

Fig. 1. Structural framed member

Fig. 2. Ideal column model: (a) frame spliced column system; (b)column segment

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At the splice location:

2EIIw0IjxI5LI¼ 2Kucðw9IIjxII ¼ 02w9IjxI5LI

Þ

2EIIw0IjxI5LI¼ 2EIIIw0IIjxII50

wIjxI5LI¼ wIIjxII50�

2EIIw09I 2NEdw9I���

xI5LI¼ �

2EII�Iw09II 2NEdw9II���

xII50

(6b)

At xII 5 LII:

2EIIIw0II ¼ Kubw9II2EIIIw09II 2NEdw9II ¼ 2KDbw

ð6cÞ

From Eqs. (5) and (6), the following eight linear homoge-neous algebraic equations in the eight constants Ai and Bi areobtained

266666666664

0 1 0 1 0 0 0 0KuamI NEd Kua 0 0 0 0 0c3;1 c3;2 2Kuc 0 KucmII 0 Kuc 0c4;1 c4;2 0 0 0 2NEd 0 0

sinðmILIÞ cosðmILIÞ LI 1 0 21 0 210 0 2NEd 0 0 0 NEd 00 0 0 0 c7;5 c7;6 2Kub 00 0 0 0 c8;5 c8;6 c8;7 2KDb

377777777775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}C

266666666664

A1

A2

A3

A4

B1

B2

B3

B4

377777777775

¼

266666666664

00000000

377777777775

|fflffl{zfflffl}b

ð7Þ

where ci,j are coefficients that are defined as follows:

c3;1 ¼ NEd sinðmILIÞ2mIKuc cosðmILIÞc3;2 ¼ NEd cosðmILIÞ þ mIKuc sinðmILIÞc4;1 ¼ NEd sinðmILIÞc4;2 ¼ NEd cosðmILIÞc7;5 ¼ NEd sinðmIILIIÞ2mIIKub cosðmIILIIÞc7;6 ¼ NEd cosðmIILIIÞ þ mIIKub sinðmIILIIÞc8;5 ¼ 2KDb sinðmIILIIÞc8;6 ¼ 2KDb cosðmIILIIÞc8;7 ¼ NEd 2KDbLII

ð8Þ

Anontrivialsolutionexistsifanyoftheeightconstantsisnotequaltozero. This happens if the determinant ofmatrixC vanishes. The ex-pansion of this determinant leads to the characteristic equation. Thesmallest positive route yields the buckling load Ncr and the shape ofthe deflection curve—Eqs. (5a) and (5b) (first eigenvector).

Imperfect Columns

Initial CurvatureConsider the initial imperfect spliced column represented in Fig. 3(a).Before the load is applied, the column is assumed to be slightly bowedand to be stress free. The initial curvature is introduced by using thefirst critical buckling mode shape. This is a standard procedure inbuckling analysis, although this is not necessarily the worst casescenario in the sense that it may not be associated with the largestsecond-order moments, as highlighted by Gonçalves and Camotim(2005). The shape functions for the geometric imperfections as-sociated with the out-of-straightness of the column are thus given byMember I:

w0;I ¼ d0

�A0;1 sinmcr;Ix þ A0;2 cosmcr;Ix þ A0;3x þ A0;4

ð9aÞ

Member II:

w0;II ¼ d0

�B0;1 sinmcr;IIx þ B0;2 cosmcr;IIx þ B0;3x þ B0;4

ð9bÞ

where d0 5 amplitude of the shape function; A0,i, B0,i 5 first ei-genvector coefficients (i5 1, 2, 3, 4); andmcr,I andmcr,II are obtainedfrom Eq. (4) at critical buckling.

The total deflection wT (x) is obtained by superimposing thelateral deflection w(x) to the imperfection w0(x), for each member

wTðxÞ ¼ w0ðxÞ þ wðxÞ ð10Þ

Because the bending strains are caused by the change in curvaturew0and not by the total curvaturew0T , the internal bendingmoment is still

Fig. 3. Initially curved column model: (a) geometric imperfections;(b) column segment

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given by Eq. (2). By equating this moment to the external appliedmoment, NEdwT [Fig. 3(b)], the following equilibrium equation isobtained:

wIV þ m2ðw0 þ w00Þ ¼ 0 ð11Þ

where m is given by Eq. (4). The complementary solution wc hasthe form of Eqs. (5a) and 5(b). The particular solution wp isobtained by using the method of undetermined coefficients. Thisleads toMember I:

wp;I ¼ C1 sinmcr;Ix þ C2 cosmcr;Ix ð12aÞ

Member II:

wp;II ¼ D1 sinmcr;IIx þ D2 cosmcr;IIx ð12bÞ

where Ci and Di are the following constants (i 5 1, 2):

C1 ¼ A0;1d0NEd

Ncr 2NEd

C2 ¼ A0;2d0NEd

Ncr 2NEd

D1 ¼ B0;1d0NEd

Ncr 2NEd

D2 ¼ B0;2d0NEd

Ncr 2NEd

ð13Þ

The general solution of Eq. (11) is thusMember I:

wI ¼ A1 sinmIx þ A2 cosmIx þ A3x þ A4

þ d0NEd

Ncr 2NEd

�A0;1 sinmcr;Ix þ A0;2 cosmcr;Ix

ð14aÞ

Member II:

wII ¼ B1 sinmIIx þ B2 cosmIIx þ B3x þ B4

þ d0NEd

Ncr 2NEd

�B0;1 sinmcr;IIx þ B0;2 cosmcr;IIx

ð14bÞ

Use of the boundary conditions given by Eq. (6) and theprocedure described in the preceding section gives the in-tegration constants Ai and Bi and total deflection for the imper-fect column. Note that Conditions 6 and 8 in Eq. (6) are nowdefined as follows:�2EIIw09I 2NEdw9T;I

���xI5LI

¼�2EII�Iw09II 2NEdw9T;II

���xII50

2EIIIw09II 2NEdw9T ;II ¼ 2KDbwII (15)

The characteristic equation is obtained from Eqs. (7) and (8). Theindependent term becomes b � 0 and is a function of d0, A0,i, B0,i,Ncr, and NEd and the ratio NEd/(Ncr 2 NEd).

Column Segment MisalignmentA frame stepped column with an eccentricity e0 between the twocentroid segment axes is considered [Fig. 4(a)]. This eccentricity canbe a mere manufacture additional imperfection or a deliberate

erection misalignment. In many practical applications, the twocolumn segments are aligned vertically to the flanges, as shownin Fig. 4(b). The governing differential equation for this case isstill given by Eq. (1). The boundary conditions for this problemare given by Eq. (6), except for Conditions 3 and 4 that nowchange to

2EIIw0IjxI5LI¼ 2Kuc

�w9IIjxII ¼ 02w9IjxI5LI

�2NEde0

2EIIw0IjxI5LI¼ 2EIIIw0IIjxII502NEde0 ð16Þ

The characteristic equation is obtained from Eqs. (7) and (8) withb � 0. The independent term is defined as follows:

b ¼ ½ 0 0 2NEde0 2NEde0 0 0 0 0 �T ð17Þ

Imperfection Sensitivity Study

Examples

The variation of the relative basic amplitudes of the imperfection d0and the column segments eccentricity e0 are considered to investigatethe imperfection sensitivity of a nonuniform spliced column. Thedimensions and properties of the column used is this study were asfollows:• Column length: L 5 4 m;• Member I cross section: HE240B (II 5 11,260 3 104 mm4);• Member II cross section: HE200B (III 5 5,696 3 104 mm4);• Splice location: a5 0.5 (LI5 2 m) and a5 0.125 (LI5 0.5 m);• Splice stiffness: Kuc 5 EIII/L and Kuc 5 10EIII/L;• Young’s modulus: E 5 210 kN/mm2; and• End conditions:

• Sway prevented case: clamped ends;• Sway prevented case: pinned ends;• Sway case: fixed-guided ends; and• Sway case: pinned-guided ends.

First, a linear elastic eigenvalue analysis was conducted to obtainthe buckling loadNcr for each column. The equations were solved byusing the algebraic manipulator Mathematica (Wolfram Research2007). The results are given in Table 1 and agree well with the

Fig. 4.Column segmentmisalignment: (a) structuralmodel (e0. 0); (b)typical example

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analytical results obtained bymeans of an energy-based formulationof the problem (Girão Coelho et al. 2010; Simão et al. 2010). Theinfluence of the degree of end fixity on the stabilizing effect on thecolumns was well demonstrated: the stiffer the beam-to-columnconnection, the larger is the corresponding increase in the loadcapacity. The results in Table 1 also highlight the strength-reducingeffect owing to the column sway. The subsequent section also showsthat the presence of initial geometric imperfections reduces thestiffening effect of the end restraint.

The shape coefficients of the relevant column buckling mode arepresented in Table 2. Fig. 5 shows the buckling modes. For im-perfection sensitivity studies, the selected amplitudes for the initialout-of-straightness were chosen as a percentage of the columnlength: 2, 1, 0.5, 0.25, and 0.1%. For the eccentricity, two valueswere considered: e0 5 5 and 20 mm. The first value is a limit valueproposed by Lindner (2008). The second value corresponds toa column arrangement similar to Fig. 4(b).

Influence of Imperfections on the ColumnLoad-Carrying Capacity

The effect of the amplitude of the initial curvature d0 on the maximumload of the column is depicted in Fig. 6 for the cases of sway preventedcolumns and sway columns, respectively. The curves in Fig. 6 pertainto both splice locations, a 5 0.125 and 0.5.The diagrams are rep-resented by the axial loadNEd (#0.99Ncr) versus themaximum lateraldisplacement wmax with a variation of d0. The maximum deflectionwmax is obtained from the requirement dw=dx5 0 for sway preventedcolumns and corresponds to the absolute lateral displacement at theend that is able to sway in the case of sway columns.

The two nondimensional graphs are identical. The response wasinitially linear. As the load increased, the increase in displacement be-came disproportionately larger. The increase in the amplitude of theimperfection reduced the prebuckling stiffness and the carrying capacityof the column. This type of behavior is independent of the columnboundary conditions and the splice stiffness because the imperfectionsare always related to the critical mode shape (see also Fig. 5).

This study also shows that

• The magnitude of the column initial out-of-straightness has anunfavorable effect that may result in a very significant deterio-ration of the load bearing capacity.

• As d0→ 0, the plot represents the behavior of the perfect system.• The maximum lateral displacement becomes unbounded when

the applied load NEd approaches the critical load.• Theequilibriumcurves seem toconvergeon to thepathNEd /Ncr51.• The maximum lateral displacement is proportional to a factor

d0NEd/(Ncr –NEd), irrespective of the column end restraints (Fig. 7).As a result, the analysis of a spliced column can be reduced to the

basic pin-ended column element that has been traditionally used asa reference when real columns are compared and designed, providedthat the critical conditions are properly characterized (Fig. 8).

Fig. 9 shows the representative equilibrium paths for sway pre-vented columns and sway permitted columns with centroid mis-alignment. The curves are represented in terms of normalized axialload NEd /Ncr (NEd # 0.99Ncr) and normalized maximum displace-ment wmax/L. The graphs are qualitatively identical to those repre-sented inFig. 6. Thismeans that the effect of the columnmisalignmentcan be easily compared with an initially curved column. The repre-sented equilibrium paths now depend on the splice stiffness andlocation, as well as on the boundary conditions because the eccen-tricity is taken as an absolute value. In the case of an initially curvedcolumn, the out-of-straightness effects in all examples are pro-portional to the first buckling mode and, thus, the final curves assumeidentical forms. The following observations are also made:• The maximum load is a function of the eccentricity between the

two segment centroids.• As e0→ 0, the plot represents the behavior of the perfect system.• The magnitude of the lateral displacement can be bounded by

adequating the type of column end restraint and the splicestiffness and location.

Design Implications

In the current AISC specifications (AISC 2010) and EN 1993 (CEN2005), the column design procedure involves column curves based

Table 1. Buckling Load

Case a KucL=EIII

Ncr (kN)

Equilibriumsolution

Energy-basedformulation

Sway prevented columns

Clamped ends 1 0.5 1 16,161 16,1612 10 31,500 31,5013 0.125 1 22,044 22,0484 10 29,111 29,123

Pinned ends 5 0.5 1 2,368 2,3866 10 7,547 7,5747 0.125 1 4,608 4,6598 10 7,199 7,230

Sway columns

Fixed-guided ends 9 0.5 1 9,136 9,13610 10 10,442 10,44211 0.125 1 3,777 3,77812 10 6,986 6,987

Pinned-guided ends 13 0.5 1 910 91214 10 1,822 1,82415 0.125 1 1,684 1,68716 10 1,833 1,835

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on the maximum strength concept and incorporate residual stressesand geometric imperfections. TheAISC specification adopts a singlecolumn curve, whereas the EN 1993 represents the strength ofcolumns by multiple column curves. The basis of the curve for-mulation is briefly reviewed in the following section.

Maximum Strength Criteria

The maximum stress in an initially curved steel column is expressedas the sum of axial and bending stress

smax ¼ NEd

Aþ Mmax

Welð18Þ

where A 5 area of cross section; Wel 5 section modulus corre-sponding to the fiber with maximum elastic stress; and Mmax 5maximum moment in the column. It has been shown that the be-havior of a framed column can be considered equivalent to that ofprismatic pin-ended columns with effective lengths Leq. Themaximum bending moment can thus be computed as follows (Chenand Lui 1987):

Mmax ¼ Ncr

Ncr 2NEdNEddi ð19Þ

where di 5 equivalent out-of-straightness at midheight of theprismatic pin-ended column. The classical elastic limit analysisrequires that the most compressed fiber reaches the yield stress fy.Eq. (18) can then be written as

NEd

Ny

1 þ 1

12NEd=Nyl2h

!¼ 1 ð20Þ

whereNy5 column squash load; h5 imperfection parameter; andl 5 column slenderness parameter defined as follows:

l ¼ffiffiffiffiffiffiffify A

Ncr

rð21Þ

Eq. (20) is the Ayrton-Perry equation (Ayrton and Perry 1886).The global imperfection parameter h integrates geometricalimperfections and residual stresses. Here, h can be calculated fromthe Rondal-Maquoi equation (Rondal and Maquoi 1979) and isdefined by

h ¼ aðl2 l0Þ ð22Þ

Table 2. First Eigenmode Coefficients

Case A0,1 A0,2 A0,3 A0,4 B0,1 B0,2 B0,3 B0,4

1 0.095 20.593 20.078 0.593 20.251 0.143 20.078 0.4372 0.034 20.553 20.039 0.553 0.017 0.397 20.039 0.4753 20.206 20.097 0.199 0.097 0.888 20.182 0.199 0.1964 20.026 20.464 0.029 0.464 0.413 20.408 0.029 0.4795 0.796 0.000 0.000 0.000 20.381 0.471 0.000 0.0006 0.742 0.000 0.000 0.000 0.012 0.671 0.000 0.0007 0.966 0.000 0.000 0.000 0.145 0.212 0.000 0.0008 0.835 0.000 0.000 0.000 0.501 0.227 0.000 0.0009 0.000 20.397 0.000 0.397 0.714 20.128 0.000 0.39710 0.000 20.522 0.000 0.522 0.407 20.125 0.000 0.52211 0.000 20.325 0.000 0.325 0.762 20.319 0.000 0.32512 0.000 20.490 0.000 0.490 0.238 20.472 0.000 0.49013 0.912 0.000 0.000 0.000 0.215 0.349 0.000 0.00014 0.803 0.000 0.000 0.000 0.419 0.423 0.000 0.00015 0.886 0.000 0.000 0.000 0.448 0.118 0.000 0.00016 0.821 0.000 0.000 0.000 0.560 0.114 0.000 0.000

Fig. 5. Buckling modes: (a) sway prevented columns; (b) sway columns

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The column curve adopted in the AISC specifications (AISC 2010)is based on the Structural Stability Research Council (SSRC)Curve 2 (Ziemian 2010) that adopts l0 5 0.15 and a 5 0.293. InEN 1993 (CEN 2005), l0 5 0.2 and a attains different valuesdepending on the cross-section shape, fabrication process, and axisof bending.

Recommendations

The theoretical developments that have been presented suggest thatthe current column design provisions can easily be extended to thecase of spliced stepped columns. The effects of the splice on thecolumn behavior should be accommodated within the framework ofexisting design procedures with a minimum of disturbance. In thiscontext, the design of these members in sway and nonsway framesbased on the column curve concept still relies on the conversion ofthe real column problem into equivalent pin-ended columns (Fig. 8)because the column curves relate to pin-ended members. This

implies the use of an effective slenderness entry into a columncurve. In the case of the spliced column, the column slenderness hasto be defined separately for each segment as follows:

li ¼ffiffiffiffiffiffiffiffiffify Ai

Ncr

ri ¼ I; II ð23Þ

The column curve defined by Eq. (20) should be adopted asa strength curve for design consistency. The values of the imper-fection parameter, Eq. (22), have to be developed for this type ofproblem. In practice, the global imperfection parameter h is ob-tained by curve fitting to available test data. Experimental andnumerical tests on spliced columns with variations in column sizemust be carried out as a follow up to this study. This parametershould also account for possible column misalignment because thisis also an important strength parameter in spliced columns. It may beexpected to exceed the usual tolerances on straightness so that theproposed strength curves are designed to err on the safe side.

Fig. 6. Equilibrium paths with initial curvature: (a) sway prevented columns; (b) sway columns

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Concluding Remarks and Future Work

The work reported in this paper has focused on the elastic stabilityanalysis of nonuniform stepped columns with initial geometricimperfections in the form of the out-of-straightness of the column andcolumn segment misalignment. The formulation of the problem isbased on the exact solution of the governing equations for buckling.

The initial curvature is applied by using the first mode shape fromlinear buckling analysis and its influence on the load-carrying ca-pacity is studied compared with the perfect column. The introductionof the imperfection in the form of the first eigenmode will not nec-essarily provide a lower limit to the prebuckling stiffness and collapseload. Analyses would need to be carried out based on imperfectionsin the form of a linear combination of the column buckling modes toensure that the results are conservative and can safely be used indesign calculations (Gonçalves and Camotim 2005).

The effect of the magnitude of geometric imperfections is ex-amined in the case study of a practical framed spliced column and

Fig. 7. Representative relationship d0NEd/(Ncr – NEd) versus wmax/L: (a) sway prevented columns; (b) sway columns

Fig. 8. Prismatic equivalent column having the same smaller crosssection and critical load

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by adopting various relative basic values of the imperfection. Theresults indicate that spliced columns can be highly sensitive to theinitial imperfections in the geometry. However, small imperfectionsin these structures are inevitable and may result in a very significantdeterioration of their load bearing capacity. The margin between themaximum strength and the design load should be decided by com-plementary elastic-plastic imperfection sensitivity studies that thewriters are undertaking. The investigated configuration was ratherlimited to a particular case. Investigation with other configurationcases would be necessary to obtain more general conclusions. Also,additional imperfections such as imperfect contact between the cutsurfaces at the column splice deserve further investigation. Exceptfor thework by Lindner (2008) and Popov and Stephen (1977), whoinvestigated contact splices, data on the behavior of such structuralmembers are scarce.

This work affords some basis to produce design guidance onframed spliced columns. Designers and steel fabricators would po-tentially be interested in this issue and the writers are furtherextending this topic to set up sound design criteria regarding (1) therequirements for stiffness and strength of column splices and (2) thegeneralization of the column buckling curves adopted in design

codes for spliced columns by means of appropriate forms of thegeneralized imperfection factors.

References

AISC. (2010). “Specification for structural steel buildings.” ANSI/AISCStandard No. 360-10, Chicago.

Ayrton, W. E., and Perry, J. (1886). “On struts.” Engineer (London), 62,464–465.

Bjorhovde, R. (2010). “Some issues for column stability criteria.” Proc.,Int. Colloquium on Stability and Ductility of Steel Structures(SDSS’Rio 2010), E. Batista, P. Vellasco, and L. de Lima, eds., FederalUniversity of Rio de Janeiro, Rio de Janeiro, 9–20.

Chen, W. F., and Lui, E. M. (1987). Structural stability: Theory andimplementation, Prentice Hall, New York.

European Committee for Standardization (CEN). (2005). “Eurocode 3:Design of steel structures—Part 1-1: General rules and rules forbuildings.” EN 1993-1-1, Brussels.

Girão Coelho, A. M., Simão, P. D., and Bijlaard, F. S. K. (2010). “Stabilitydesign criteria for steel column splices.” J. Constr. Steel Res., 66(10),1261–1277.

Fig. 9. Equilibrium paths with column segment misalignment: (a) sway prevented columns (pinned ends); (b) sway columns (fixed-guided ends)

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n 09

/14/

12. F

or p

erso

nal u

se o

nly.

No

othe

r us

es w

ithou

t per

mis

sion

. Cop

yrig

ht (

c) 2

012.

Am

eric

an S

ocie

ty o

f C

ivil

Eng

inee

rs. A

ll ri

ghts

res

erve

d.

Gonçalves, R., andCamotim, D. (2005). “On the incorporation of equivalentmember imperfections in the in-plane design of steel frames.” J. Constr.Steel Res., 61(9), 1226–1240.

Johnston, B. G. (1983). “Column buckling theory: Historic highlights.”J. Struct. Eng., 109(9), 2086–2096.

Lindner, J. (2008). “Old and new solutions for contact splices in columns.”J. Constr. Steel Res., 64(7-8), 833–844.

Popov, E. P., and Stephen R. M. (1977). “Capacity of columns with spliceimperfections.” Eng. J., 14(1), 16–23.

Rondal, J., and Maquoi, R. (1979). “Single equation for SSRC columnstrength curves.” J. Struct. Div., 105(1), 247–250.

Simão, P. D., Girão Coelho, A. M., and Bijlaard, F. S. K. (2010). “Influenceof splices on the stability behaviour of columns and frames.” Proc., Int.Colloquium on Stability and Ductility of Steel Structures (SDSS’Rio2010), E. Batista P. Vellasco, and L. de Lima, eds., FederalUniversity ofRio de Janeiro, Rio de Janeiro, 619–626.

Timoshenko, S. P., and Gere, J. M. (1961). Theory of elastic stability,McGraw-Hill, Singapore.

Wolfram Research. (2007). Mathematica version 6, Wolfram Research,Champaign, IL.

Ziemen, R. D., ed. (2010). Guide to stability design criteria for metalstructures, Wiley, New York.

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