Engineering Structures 31 (2009) 1734–1744

Contents lists available at ScienceDirect

Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Second order approximate analysis of unbraced multistorey frames with singlecurvature regionsJostein Hellesland ∗Mechanics Division, Department of Mathematics, University of Oslo, P.O. Box 1053 – Blindern, NO-0316 Oslo, Norway

a r t i c l e i n f o

Article history:Received 6 June 2008Received in revised form13 October 2008Accepted 11 February 2009Available online 26 March 2009

Keywords:Multistorey sway framesColumnsSecond order analysisStorey magnifier methodGlobal second order effectsLocal second order effectsStorey interactionSway magnifiersEffective lengthsNegative restraints

a b s t r a c t

Present approximate second order methods for the analysis of unbraced multistorey frames maysignificantly underestimate moments in single curvature regions. To clarify reasons for this that may notbe well understood, the mechanics of column interaction in single curvature regions are studied. Suitabletools for sidesway description are derived, including a nearly exact, explicit free sway effective lengthexpression, that, when high accuracy is required, eliminates the need in many cases for cumbersome,iterative solutions of exact effective lengths from the transcendental instability equation. Two reasons forthe underestimation are identified. One is related to the local second order Nδ effects, and the other tosecond order effects causing changes in rotational restraint stiffness at column ends due to vertical, inter-storey column interaction. A modified approximate storey magnifier approach is proposed that accountsfor these local second order effects through two separate ‘‘flexibility factors’’. The approach is sufficientlysimple to be viable in practical analyses, and predictions are found to compare well with more accurateresults.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In frame and member analysis, it is often necessary to considersecond order load effects on sway, moments and stability causedby axial loads acting on the displacements of the frame and framemembers. In frames with sidesway, second order effects affectinterconnected members in three ways: (1) in an overall, globalsense, due vertical loads acting on the sidesway of the framesystem as such (‘‘N∆’’ effects), (2) in an individual, local sense,due to axial member loads acting on the deflections away from thechord between member ends and thus causing nonlinear (curved)moment distributions along the members (‘‘Nδ’’ effects), and(3) in a local sense in multi-level columns and frames by changingrotational restraint stiffness at member ends due to vertical, inter-level (inter-storey) column interaction.Subdivision into global and local second order effects is very

common in approximate analyses, such as in the so-called N −∆ type methods, that consider second order effects separatelyfollowing a conventional first order analysis. A valuable asset ofsuch methods is their transparency with respect to the importantvariables, and they have been dealt with in a number of studies

∗ Tel.: +47 22855950; fax: +47 22854349.E-mail address: josteinh@math.uio.no.

0141-0296/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2009.02.015

over the last 40 years or so. Some of these are reviewed inHellesland [1].Global effects are well taken care of in such methods, and to

some extent also local Nδ effects, through a factor often labelled‘‘flexibility factor’’ [2], ‘‘bending shape factor’’ [3], or ‘‘stiffnessreduction factor’’ [4]. In these and other relevant papers, e.g, [5–7], and textbooks, e.g., [8], it is stated or implied incorrectly thatthe increased column flexibility may be 1 to 1.22 (1.2) times thefirst order flexibility. This range is acceptable for many practicalframes, but may not be adequate for unbraced or partially bracedmultibay frames that include columns subjected to axial loads farin excess of the free sway critical load. For such columns, theflexibility factor may be considerably greater than indicated bythis range. Extensions to include such cases, which require a loaddependent flexibility factor that reflects the transition from swayto braced column response, have been presented and discussedelsewhere [1].Even for some frames with reasonably low axial column loads,

below the free sway critical loads of the columns consideredunbraced, local second order effects may be considerably greaterthan reflected by a flexibility factor range above. This is typicallythe case in single curvature bending regions of multistorey,unbraced frames. In such regions, with no inflection pointsbetween column ends, which imply negative rotational restraints

J. Hellesland / Engineering Structures 31 (2009) 1734–1744 1735

Notation

Bs Sway magnification factor;EI, EIb Cross-sectional stiffness of column and beam;Gj Relative rotational restraint flexibility at member

end j;H Applied lateral storey load (sum of column shears

and bracing force);L, Lb Lengths of considered column and of restraining

beam(s);L0 Distance from the base of a column row of a frame

to the first order inflection point along the row;N Axial (normal) force;Ncr Critical load in general (=π2EI/(βL)2);Ncb,Ncs Critical load of columns considered fully braced, and

free to-sway;NE The Euler buckling load of a pinned-end column

(= π2EI/L2);Rj Rotational degree of fixity at member end j;Rm Mean rotational degree of fixity of the two member

ends;S0 First-order lateral ‘‘storey’’ stiffness;SB Lateral bracing stiffness;V0, V First-order and total (first+second order) shear

force in a column;kj Rotational restraint stiffness (spring stiffness) at end

j;xts Distance from the base of a column row of a frame

to the top of the considered storey;αcr Member (system) stability index (=N/Ncr );αb, αs Load index of column considered fully braced, and

free to-sway;αss Storey (system) stability index;αE Nominal load index of a column (=N/NE);β Effective length factor (from system instability);βb, βs Effective (buckling) length factor corresponding to

Ncb and Ncs;γ , γn Flexibility factor in general, and load (N-) dependent

flexibility factor;γs, γ0 Flexibility factor at free sway, and at zero axial load;γk Flexibility factor for restraint stiffness correction;∆0,∆ First-order and total lateral displacement;κj Relative rotational restraint stiffness at end j

(= kj/(EI/L)).

at upper column ends, the conventional N −∆ type methods maygrossly underestimate moments [9].In single curvature regions there are two effects that will

increase the flexibility beyond what is presently accounted for:(1) Negative restraints will be inflicted at one column end, whichin itself may result in flexibility factors outside the mentionedrange. This is seemingly not well-known. (2) In addition, thereis a strong inter-storey interaction, beyond that reflected in afirst order analysis, between columns framing into the same joint.These interaction effects, resulting from changes in the columns’rotational restraint stiffnesses due to the axial column loads (thirdtype second order effects mentioned above), have received littleattention.The emphasis of this paper is on such second order effects in

linear elastic two dimensional, unbraced multistorey frames withsingle curvature regions (typical for ‘‘stiff column-flexible beam’’frames). Each column has uniform sectional stiffness and axial loadalong the length. These properties may vary from storey to storey.The main objective of the study is directed towards deriving

a modified approximate storey magnifier approach that may

Fig. 1. (a) Laterally loaded unbraced column, (b) moments and displacementshapes for given positive/positive end restraints, and (c) for negative/positive endrestraints.

account for the second order effects reviewed above. Towardsthis goal, (1) the basics of the storey magnifier approach arereviewed, (2) suitable tools for sidesway description are derived(sway magnifier, critical load, and first order storey stiffnessexpressions), (3) available approximate methods for computinglocal second order (Nδ) effects (flexibility factors) are reviewedand their accuracy evaluated for a variety of positive and negativeend restraint combinations, and (4) themechanics of restraints andvertical column interaction in single curvature regions are studiedto clarify aspects that are not well understood.

2. Second-order column analysis

2.1. Positive and negative end restraints

End restraint types may be briefly discussed with referenceto the isolated, laterally loaded unbraced column in Fig. 1. Ithas rotational end restraints, defined by springs with rotationalstiffness k1 and k2 at the member ends. When these representthe interaction with a larger structure, both positive and negativevalues are possible.At the ends with positive restraints, end moments act in

the opposite direction to that of the end rotation and thereby‘‘strengthens’’ the member by restraining the end rotation. At endswith negative restraints, rotational disturbances are inflicted asreflected by end moments acting in the same direction as the endrotations. Normally, this will ‘‘weaken’’ the member.In the general case, several bending modes may be possible.

The two most common are shown in Fig. 1(b,c). The first, withone inflection point between ends, will result provided the endrestraints have positive values. This is typically the case forcolumns in unbraced frames with stiff (‘‘strong’’) beams. Thesecond, with a negative rotational restraint inflicted at the upperend, and with no inflection point between ends, is typical forcolumns in lower stories of unbraced frames with very flexible(‘‘weak’’) beams, such as in so-called ‘‘stiff column-flexible beam’’structures. Such columns contribute to the rotational restraint ofthe columns on the storey above. As a consequence, they havenegative restraints inflicted upon themselves at their upper ends.

2.2. Unbraced column

In second order analysis, three quantities are normally ofinterest: the critical load (effective lengths), the sway and themoment magnifier. Reasonably simple, approximate expressionsfor these can be derived using principles of the so-called N − ∆analysis approach. The approach is well-known, and only a briefreview is given below for the purpose of deriving and defining thequantities of interest for this study.

1736 J. Hellesland / Engineering Structures 31 (2009) 1734–1744

2.2.1. Sway magnifierIf the column is subjected to a vertical (axial) load, the relative

end displacement will increase from the first order value ∆0, dueto the lateral load V0, to

∆ = Bs∆0 (1)

where Bs is the sway magnification factor that reflects the secondorder (global) overturning moment effect, and also the secondorder (local) member effects through a separate factor, herelabelled γn. By replacing the overturning moment effect (N∆) byan equivalent horizontal load (γnN∆/L) and scaling up the firstorder displacement, the final displacement can be written, ∆ =∆0(1+ (γnN∆/L)/V0). Solving for∆, it can be given in the form ofEq. (1), with the sway magnifier given by

Bs =∆

∆0=

11− αs

(2)

where

αs =γnN/LS0

and (3a)

S0 =V0∆0

(3b)

are the sidesway ‘‘stability index’’ and the first order lateralstiffness, respectively. This sway magnifier can also be establishedin other ways, including in an iterative, Vianello type manner(leading to a geometric series). The γn factor reflects an increasedflexibility caused by local (member) second order effects. It isstrictly axial load dependent, but is most often taken as loadindependent. It is discussed in more detail below (Section 4).

2.2.2. Critical load and effective lengthThe critical load factor, causing infinite displacements, is equal

to the inverse of the ‘‘stability index’’ (λcr = 1/αs). The unbraced,or free sway, critical load can therefore be expressed by

Ncs =Nαs

or Ncs =V0Lγs∆0

(4)

where γn is denoted γs at the free sway condition. For a singlecolumn, this is the true critical load (Ncr = Ncs). For a framedcolumn considered in isolation, Ncs is a pseudo-critical free swayload.Alternatively, the critical compression load of an elastic

member of length L and both uniform cross-sectional bendingstiffness EI and axial force along the member, may be written inthe conventional form as

Ncr =π2EI(Le)2

with Le = βL (5)

Le is the effective length (or buckling length), and β is the effective(buckling) length factor of the member. Physically, the bucklinglength is as known equal to the distance between inflection points(points of contraflexure) located on the buckled shape, or on themathematical continuation of the buckled shape.From Eqs. (4) and (5), the free sway effective length (β = βs)

can then be expressed by

βs =

√NENαs with NE =

π2EIL2

. (6)

2.2.3. Moment magnifierEndmoments due to lateral and axial loads can be expressed by

M = BMM0 ≈ BsM0 (7)

where BM is a moment magnification factor that is commonlyapproximated by the sway magnifier Bs. This is a reasonableapproximation for unbraced columns, and for framed columns thatcontribute to the lateral resistance of the frame, i.e., columns withaxial load levels below the free sway critical loads. It is such casesthat are of main interest here.

2.3. Interacting columns

2.3.1. Laterally interacting columnsFor frames with laterally interacting columns, the appropriate

‘‘storey sway magnifier’’ and critical load can be obtained byreplacing αs above by a ‘‘storey (system) stability index’’ αss,obtained from Eq. (3) by replacing the numerator and denominatorby the corresponding sums over all the interacting columns and apossible bracing. Thus, for the storey sway magnifier,

Bs =∆

∆0=

11− αss

(8)

where

αss =

∑(γnN/L)S0

with (9a)

S0 =H∆0=

∑V0

∆0+ SB (9b)

S0 is now the first order lateral storey stiffness, H is the totallateral storey load, and SB is the lateral stiffness of bracings,if present. The shear stiffness may be included in S0, but it isnormally neglected. Above, it is tacitly assumed that axial beamdeformations are negligible such that the sidesway displacementis the same in all column axes included in the summation. Detailsof the derivation of such sway magnifier formulations, recentadvances, simplifications and corresponding limitations, and codeadaptations, are available elsewhere [1].

2.3.2. Vertically interacting columnsThe storey magnifier approach above, based on the first order

lateral storey stiffness, is commonly applied also to individualstoreys in multistorey frames. In particular for frames withrestraining beams that are sufficiently stiff to cause inflectionpoints (zero moment) within the column lengths, the agreementwithmore accurate methods is good, both for such structures withsingle and multiple bays [9]. However, for storeys with columnsin single curvature bending (with negative rotational restraintsat an end), the approach may significantly underestimate thestorey stability indices and sway magnifiers. This has been clearlydemonstrated by Lai and MacGregor [9], and others.An effort is made in this study to clarify the reason for this, and

to propose a modified approach that may improve predictions insingle curvature regions and that is sufficiently simple to be viablein practical analyses.

3. First order properties

The first order lateral column stiffness V0/∆0 in Eqs. (3) and(4) can readily be established by the differential equation, themoment-area theorem, or by other methods, and can be found inthe literature. Here, first order properties are derived in forms thatare suitable for use in applications later in the paper.

J. Hellesland / Engineering Structures 31 (2009) 1734–1744 1737

With reference to the laterally loaded column in Fig. 1, endmoments and end rotations (θ0) are taken as positive when theyact in the clockwise direction, and segment lengths are defined aspositive when they are oriented from the respective column endstoward the other end, as shown in Fig. 1(b). With this definition,a negative segment length implies an inflection point outside themember length such as in Fig. 1(c).

3.1. Inflection points, moments, restraints

The location of the inflection point, and a very suitable restraintfixity parameter, can be determined from the slope continuitycondition at the common inflection point C of each of the twocantilever segments in the figure. Equal slope can be expressed (forinstance using the moment-area theorem) by

θ0C =(−M01)L12EI

+ θ01 =(−M02)L22EI

+ θ02. (10)

Substituting −M0j = V0Lj, which follows from momentequilibrium, and noting that L1+ L2 = L, Eq. (10) can be expressedin terms of one of the unknown member segment lengths as

LjL=−M0jV0L=

RjR1 + R2

j = 1, 2 (11)

where

Rj =kj

kj + cEI/L=

11+ c/κj

with c = 2 (12)

and

κj =kj

(EI/L)(13)

Here, R is a first order ‘‘rotational degree of fixity factor’’, and κis the non-dimensional rotational restraint stiffness.The first order fixity factors R are seen to be directly

proportional to the end moment, and to the first order inflectionpoint distance from the end at which the factor is computed. Itis, consequently, a very useful parameter. Its definition evolvesnaturally from the mathematics of the problem and is closelyrelated to the physics of the column response. At a rotationallyfixed end,R = 1, and at a pinned end,R = 0 (zero fixity). A negativeR at an end implies an inflection point located away from the end,outside the column length.Similar factors defining the approximate inflection point

locations of a buckled column, obtained by replacing c = 2 inEq. (12) by c = 2.4, is given and discussed in Hellesland [10] forboth positive and negative restraints.

3.2. Lateral displacement and stiffness

The total relative lateral displacement can be given by the sumof that of each cantilever segment (∆0 = a01 + a02, Fig. 1):

∆0 =(−M01)L213EI

+ θ01L+(−M02)L223EI

+ θ02L. (14)

Expressing Lj and −M0j = V0Lj in terms of the R-factors (Eq. (11))and substituting into Eq. (14), the first order relative displacementcan be written as

∆0 =

(6

R1 + R2− 2

)V0L3

EI(15)

and the corresponding first order lateral stiffness as

V0∆0= cv

EIL3

where (16a)

cv =12Rm3− 2Rm

(16b)

Here, Rm = 0.5(R1 + R2) is the mean first order fixity factor. Thelateral stiffness is consequently a function of the sum of the fixityfactors, and not of the individual components.Restated in terms of the restraint flexibility parameter G,

Eq. (16) becomes

cv =12(G1 + G2 + 6)

2G1G2 + 4(G1 + G2)+ 6(17)

where

Gj = bo(EI/L)ki=boκi

j = 1, 2 (18)

which, in this general form, allows for both positive or negativerestraint values [10,11]. The coefficient bo is a reference restraintstiffness coefficient, normally taken equal to bo = 6 correspondingto that of a beam bent in antisymmetrical curvature. This referencevalue is also adopted here. Apart from the generalised G factordefinition, the lateral stiffness coefficient in Eq. (17) is on a similarform previously derived along different lines by others, e.g., [4].

4. Flexibility factors for Nδ effects

4.1. Background

An axial force gives rise to a nonlinear moment distributionalong a column. These local second order (Nδ) effects lead inturn to a reduction in lateral column stiffness and to a sidewaysdisplacement that is greater than that due to a linear momentdistribution. This effect can be accounted for through a factoroften denoted γ , and previously (1976) labelled flexibility factorby the author [2]. It reflects, in other words, the increased lateralflexibility of an axially loaded column as compared to that of acolumn with a (first order) linear moment distribution.This flexibility factor is one of two local second order aspects of

importance for correct predictions of column and frame responseto lateral loads. It will be considered in some detail, both withregard to alternatives available and accuracy.The flexibility factor is a function of the column axial load (the

normal load). The subscript ‘‘n’’ in γn is added in order to indicatethis dependence. A general, load dependent version of γn is givenin Hellesland [1]. Normally, like in this study, simplifications arewarranted. For axial loads approaching zero and the critical freesway load, γn takes on values that for convenience will be labelledγ0 and γs, respectively.Between the zero and free sway axial load levels, the variation

in γn is verymodest. In this range, which is the one ofmain interestin this paper, the flexibility factor may, with good accuracy, betaken independent of axial loads, and for instance equal to γs orγ0. In multibay frames, where some columns may have axial loadsfar in excess of the free sway critical load, γn for a column maystill be approximated by γs. However, the accuracy decreases withincreasing axial load, in particular as the axial load approaches thecritical load of the column considered braced [1].Common for earlier studies of the flexibility factor [2–7,12] is

that they deal with positive end restraints only, and that they, withone exception [12], statewithout any reservations that γs is limitedto values between 1 and 1.22, or 1 and 1.2 for γ0. This is a commonmisconception. As shall be seen below, it is correct for the specialcase of isolated free sway columns, for which end restraints willalways be positive, but not necessarily correct for other cases, suchas framed columns for which also negative end restraints may beinflicted through the interactionwith other members in the frame.

1738 J. Hellesland / Engineering Structures 31 (2009) 1734–1744

Fig. 2. The flexibility factor γ = γs (at the free sway condition) in terms of the larger (RMAX) and smaller (RMIN) of the first order rotational restraint fixity factors at memberends (labelled A and B in the figure).

4.2. Exact γs variation

Evaluated at the free sway (zero shear) condition, the resultingγs can be determined by equating Ncs in Eq. (4) to the exact freesway critical compression load Ncr . It can given in either of the twoforms

γs =(V0L/∆0)Ncr

or γs =cvβ2sπ2

(19)

In the second form, V0/∆0 has been replaced by Eq. (16a), andNcr by Eq. (5) with β = βs. The exact βs can be found from thetranscendental instability equation (defined later by Eq. (24)).Results obtained in this manner are shown in Fig. 2 for various

positive and negative end restraint combinations. The restraintsare conveniently defined in terms the smaller (RMIN) and the larger(RMAX) of the rotational fixity factors R (Eq. (12)) at the twomemberends labelled A and B, respectively, in the figure. For instance, in acase with RA = 0.5 and RB = −0.6, RMAX = 0.5 and RMIN = −0.6.For positive restraint stiffness values κ , the R values will always

be positive and take on values between 0 (pinned end) and 1.0(fully fixed end). For large negative κ values (strong negativerestraints), the R values will become greater than 1.0, and forsmall negative κ values (weak negative restraints), the R valueswill become negative. Results are symmetrical about the the +45degree diagonal RA = RB. Therefore, only those above the lineare shown.End restraint combinations giving infinite effective lengths

represent outer limits and can be given by [10]

1κ1+1κ2= −1 or R1 + R2 = 0 (20)

in terms of κ and R factors (and G1 + G2 = −6 in terms of Gfactors with bo = 6). This restraint combination is shown by the−45 degree diagonal in Fig. 2. In the figure, γ = γs values havearbitrarily been terminated at 1.44. This is probably beyond therange of practical interest.It may be useful for the understanding of these results, to relate

them to selected column bending (buckling) shapes presented anddiscussed in [10], and shown in Fig. 3. Results in the lower rightquadrant, for 0 < RMAX < 1 and 0 < RMIN < 1, correspond tobending shapes such as illustrated in Fig. 3(c,d), and those in thelower left quadrant, for 0 < RMAX < 1 and −1 < RMIN < 0,to bending shapes of the type in Fig. 3(e). Results in these two

Fig. 3. Selected buckling modes of an unbraced compression member.

Fig. 4. A flexibility factor definition (Nδ effects).

quadrants represent the most common bending shapes. Results inupper right quadrant for RMAX > 1 and 0 < RMIN < 1, correspondto buckling shapes of the type in Fig. 3(b), and those in the upperleft quadrant for RMAX > 1 and−1 < RMIN < 0 to bending shapesof the type in Fig. 3(f). The label ‘‘limk’’ in the figure refers to thelimits given by Eq. (20).

4.3. Approximate γs factors

Flexibility factor expressions can be derived in variousways. Forinstance, and the simplest, is to combine Eq. (19)with approximatefree sway effective length expressions. Alternatively, they canbe established directly by other means. The latter approach isconsidered below.Based on results obtained using the principle of minimum

potential energy, Rubin [3,13] defined the factor, as illustratedin Fig. 4, as the ratio of the vertical displacements at the top ofa flexurally deformed column and a column rotating as a rigid

J. Hellesland / Engineering Structures 31 (2009) 1734–1744 1739

pendulum, γ = u/up. For a column with given end moments andlateral displacement ∆, a deflected shape approximated by thatcorresponding to a linear moment variation (i.e., a third degreeparabola), and neglecting axial deformations, he derived a factor,here denoted γ0 and expressed by

γ0 = 1+L4

180(EI∆)2[M1M2 + 4(M1 −M2)2

]. (21)

Also based on energy considerations, Girgin et al. [7] recentlyderived along similar lines a factor that may be expressed in theexact same form as that above. In the elastic case with positiveend restraints, Eq. (21) gives values between 1 and 1.2, which arecorrect for columns with negligible axial loads for which the thirddegree parabola assumption is correct. At the free sway criticalload, which is the state of interest here, the correct range forpositive restraints is 1 to 1.216 (1.22), as seen in Fig. 2. This minordifference, which is of no practical importance, is adjusted forbelow by replacing the numeral 180 in Eq. (21) by 167.In the elastic case, the first order values of ∆, end moments

and restraints can be adopted. In this case, the expression can alsobe rewritten in terms of first order fixity factors R (Eq. (12)) or Gfactors (Eq. (18)) using the the first order properties, Eq. (11) andEq. (16), or these equations in terms of G factors (with bo = 6).The three following alternative forms of the modified factor thenresult:

γs = 1+L4

167(EI∆0)2[M01M02 + 4(M01 −M02)2

](22a)

γs = 1+ 0.216R1R2 + 4(R1 − R2)2

(R1 + R2 − 3)2(22b)

γs = 1+ 0.216(G1 + 3)(G2 + 3)+ 4(G1 − G2)2

[(G1 + 2)(G2 + 2)− 1]2. (22c)

Values at the zero axial load limit, γ0, can be obtained by replacing0.216 above by 0.2. The latter expression with 0.216 replaced by0.2, has been presented before in [9], also there based on Eq. (21).Eq. (22) will be used later in this study.Other flexibility factor expressions at the free sway (γs) or

zero axial load (γ0), or equivalent expressions, are available inthe literature. Prior to learning of the work by Rubin, the authorderived a general, but cumbersome expression [2]. A simplerexpression [14], given in terms of the larger and the smaller ofthe G factors at the column ends (Gmax and Gmin), was proposed in1981 (during a stay at the University of Alberta, Edmonton). Otherflexibility expressions can be extracted from an effective lengthfactor or critical load expression given by Lui [5] in 1992 (for γ0),and by Aristizabal-Ochoa [6] in 1994 (for γs). A load dependentfactor can be obtained from an approximate elastic lateral columnstiffness expression obtained in 2002 by Xu and Liu [12] from asecond order Taylor series expansion of the exact lateral stiffness.The resulting expression at zero axial load (γ0) is almost identicalto the expression that can be extracted from Aristizabal-Ochoa’swork.A detailed overview of the different factors is given in [15].

For practical analysis and design, all of the cited flexibility factorsprovide acceptable results for given rotational end restraints. It isa question of preference as to which of them to use.

5. Effective length factor expressions

Approximate effective length factors for unbraced columns,β = βs, can now be obtained from Eq. (6) with the γs factorsabove andwith the first order lateral stiffness given by V0/∆0 from

Eq. (16) or (17). They may be given in any of the three following,convenient alternative forms:

βs =

[π2EIL2·γs∆0

V0L

]1/2(23a)

βs =

[γsπ

2

12

(3Rm− 2

)]1/2(23b)

βs =

[γsπ

2

12·2G1G2 + 4(G1 + G2)+ 6

G1 + G2 + 6

]1/2. (23c)

The accuracy of the γs expressions is investigated implicitlyby comparing predictions by Eq. (23) with exact results for anunbraced member with given rotational end restraints. Exacteffective length results of such a column with uniform sectionstiffness and axial force along the member, can be obtainedin standard fashion from the zero determinant condition ofthe stiffness matrix of the column. This leads to a well-knowninstability condition (transcendental equation) that for instancemay be given in the form of

(π/β)2 − κ1κ2

κ1 + κ2=

(π/βs)

tan(π/βs)(24)

where the nondimensional rotational end restraints κ are definedby Eq. (13). Selected comparisons with exact results given in [10],are presented in Table 1 for combinations of positive/positive,positive/negative and negative/negative rotational restraints. Themost relevant results in the table are those for positive/positive endrestraint combinations in the lower left quadrant, corresponding tothe buckling shapes in Fig. 3(c, d), and thosewith positive/negativecombinations in the lower right quadrant, corresponding to thebuckling shapes in Fig. 3(e).For combinations of positive restraints, which are most

common in practical cases, predictions by Eq. (23), with γs givenby Eq. (22), are generally within 0.1% of exact results. This is anextremely good accuracy, and may for most practical cases beconsidered ‘‘exact’’. Thus, in cases when exact or nearly exactresults are required, the tedious iterations required to obtainsolutions from the transcendental equation (Eq. (24)), can beavoided. In the lower right quadrant (positive/negative), theaccuracy is not quite as good, but still very good and generally exactto two decimals.This effective length factor will be used later in this study.

However, when the high accuracy of this factor is not required,a number of other, simpler approximate effective length factors[10] that are valid for both positive and negative restraints, maybe used.

6. Restraint mechanics—Vertical interaction

Another local second order aspect that is of great importancefor correct predictions of column and frame response to lateralloads, in particular in single curvature situations, is connected tothe vertical inter-column (inter-storey) interaction. A brief reviewrelevant to the present work is considered useful.The rotational end restraint stiffness of a column at a joint ‘‘j’’

may be defined accurately by

kj = −Mjθj

(25)

or

kj = fj kbj with (26a)

fj =Mj

(∑Mcol)j

. (26b)

1740 J. Hellesland / Engineering Structures 31 (2009) 1734–1744

Table 1Evaluation of free sway effective length factor formula (Eq. (23)): Ratios of βAPPROX,Eq. (23)/βEXACT .κ2 (R2) κ1 (R1)

∞ 24 6 1.5 0 −0.3 −0.4 −0.6 −0.75(0 0.92 0.75 0.43 1 −.18 −.25 −.43 −.60)

−12 (1.20) 1.016 1.015 1.013 1.009 1.008 1.008 1.008 1.008 1.009−24 (1.09) 1.004 1.003 1.003 1.002 1.002 1.003 1.004 1.005 1.007∞ (1.00) 1.000 0.999 1.000 0.999 1.000 1.001 1.002 1.004 1.00624 (0.92) 0.999 0.999 0.998 0.999 1.001 1.001 1.003 1.0056 (0.75) 1.000 0.999 0.999 1.000 1.001 1.002 1.0041.5 (0.43) 1.000 1.000 1.001 1.001 a

0.75 (0.27) 1.000 1.001 1.001Results that can be obtained by reversing κ1 and κ2 are not shown. Rj = 1/(1+ (2/κj)); Gj = 6/κj .a Exact eff. length is infinite.

Here, Mj is the end moment and θj the rotation, both includingsecond order effects, caused at the considered column end by theother members framing into it. For later discussions in this study,this definition is most useful. Moments and rotations are definedpositive when they act in the same direction (here taken as theclockwise direction). The stiffness kj becomes positive when theendmoment (Mj = −kjθj) acts in the opposite direction to the endrotation, and negative otherwise.In the second definition, kbj is the rotational stiffness (equal

to the moment giving a unit rotation) of the restraining beams,tension members, etc., at joint j, and f is the fraction (or multiple)of kb that is provided to, or ‘‘demanded’’ by, the considered columnend. This factor, given by Eq. (26b), is determined from momentequilibrium and rotation compatibility requirements at joint j.The summation in the denominator is the sum over all column(compressionmember) moments at joint j. The rotational restraintoffered by beams etc. at the joint is in other words shared between(or distributed to) the columns meeting at the joint in proportionto their end moments at the joint.Normally, of course, moments and rotations that include the

second order effects are not known, and they are therefore oftenreplaced, such as in storey magnifier approaches, by first ordervalues. Inherent in first order analyses are the restraint stiffnessesobtained from Eq. (25) or (26b) when the total moments androtations are replaced by the respective first order values (M0,j,M0,col, θ0j). Eq. (26b) has been given with first order values beforefor braced frames [16], but it is valid at any frame joint.It may be recalled that the vertical restraint assessment

implied by the ‘‘conventional’’ G factor, as defined by Gj =∑(EI/L)j/

∑(EI/L)b,j, is given by

fj =EI/L

(∑EI/L)j

(27)

where EI/L (numerator) is for the considered column and the sum(denominator) is over all columns framing into the consideredcolumn end j. This interaction result, which is discussed in detailelsewhere [11], can be derived for the case of buckling of a 3-column frame assembly with given boundary conditions at the farends of members framing into the ends of the considered middlecolumn, equal load levels N/NE , etc. [8]. Apart from being veryapproximate in practical cases, it allows for positive restraints only,and is therefore, unlike the G factor in Eq. (18), not very suitable instudies such as the present one with negative restraints.The stiffness definitions of Eq. (25) or (26), are not much

affected by second order axial load effects in ‘‘flexible column-stiff beam’’ frames, in which the beams (kb) are sufficientlystiff to ensure that the storeys act reasonably independent ofneighbouring storeys. The same is not the case for ‘‘stiff column-flexible beam’’ frames, in which the beams are not stiff enoughto provide double curvature bending. Such a case is illustrated bythe four storey frame in Fig. 5, where the columns of the threelower storeys are bent in single curvature. The first order stiffnessapproximations are studied in more detail for such cases below.

Fig. 5. Multistorey ‘‘stiff column-flexible beam’’ frame.

7. Frames in single curvature bending

7.1. 4-storey unbraced frame

The extreme case of a frame with very flexible beams isobtained by neglecting the beams altogether. The restraint at thebottom of a column in one storey is then required to be providedentirely by the column in the storey below. Such a case is illustratedby the cantilever column shown by the insert in Fig. 6. The columnis fixed at the base and can be divided into an arbitrary numberof ‘‘storeys’’, or segments. First-order and second order analysissolutions for the cantilever column can easily be obtained fromrather straightforward hand calculations. Details of the analyseswill therefore not be given.Resulting rotational stiffnesses k(x) at various heights (x) and

axial load levels, given as fractions of the exact critical load of thecolumn (αcr = N/Ncr,exact; Ncr,exact = π2EI/(4h2), are shown inthe figure. They are computed from the corresponding momentsand rotations (Eq. (25)), and are given in the figure in terms of thecorresponding first order values k0(x).Once rotational stiffnesses are known, all quantities of interest

can be computed using the previously developed expressions(Section 2). They will become approximate values if the correctstiffnesses, corresponding to the appropriate load level of thesituation studied, are not used. This will be illustrated below.

7.2. Effective lengths

The effective length will be computed for the lowest Column(segment) S1 of the 4 ‘‘storey’’ frame in Fig. 7(a). At the fixed base,κ2 = ∞ and R2 = 1. The rotational restraint stiffness at theupper end (κ1) is obtained from Fig. 6 and given in Table 2 forthe three different axial load levels, as defined by αcr = 1, 0.5and 0. The first of these corresponds to buckling, the last to thefirst order case, and the middle one to a level at which it might be

J. Hellesland / Engineering Structures 31 (2009) 1734–1744 1741

Fig. 6. Rotational stiffness of cantilever column at various sections versus axial loadlevel.

Fig. 7. (a) 4-‘‘storey’’ column row; (b) sway magnifiers; (c) moments based on 1storder stiffness; and (d) moments based on exact stiffness. (Exact and 2nd orderapproximate effects for N = 0.5Ncr ).

Table 2Effective length computations of Column S1 (Fig. 7) for different end restraints (loadlevels).

Col.1 αcr = N/Ncr,exact1 0.5 0

κ1 −0.948 −0.899 −0.857

R1 −0.901 −0.816 −0.750γs 1.348 1.337 1.329βs 8.066 5.804 4.905Le/h 2.017 1.451 1.226

Le/Le,exact 1.008 0.725 0.613

R2 = 1; First-order results for αcr = 0; Results at buckling for αcr = 1.

of interest to compute the sway magnifier. For corresponding R1(Eq. (12)) and γs values (Eq. (22b)), also given in the table, theunbraced effective length factors can for instance be computeddirectly from Eq. (23b).Although the restraint stiffness at the upper end 1 does not

seem to be too different for the different load levels, it is obviousfrom the effective length results (βs = Le/L; Le/h) that theyhave significant effects. With restraints pertaining to the truebuckling condition (αcr = 1), the predicted effective length is verygood (0.8% above exact result) even though the fixity factor R1 is

strongly negative, and significantly outside the range for whichcomparisons were made in Table 1. On the other hand, the useof restraint stiffnesses obtained for the load levels αcr = 0.5 andαcr = 0 (first order), is seen to give poor predictions that are about30% to 40% below exact results. This illustrates the importance ofusing restraints relevant for the actual load state.Otherwise itmay be noted that the flexibility factor γs (between

1.33 and 1.35) is very little affected by the different load levels. So,the example serves to demonstrate that it is the restraint stiffnessthat is themost important parameter in isolated analysis of columnsegments in single curvature as here. The same will be the case forframes with stiff columns-flexible beams where there will still bestrong interaction between columns in adjacent stories.When, unlike here, restraint stiffnesses are not known,methods

that include the stiffness as an unknown could have beenemployed (e.g., Hellesland [10], Tong et al. [17]). However,except for smaller systems, these are generally cumbersome. Analternative approach will be pursued here.

7.3. Moments for first order restraint stiffness

In order to study the accuracy of using first order restraints inmoment predictions, the same four ‘‘storey’’ column in Fig. 7(a) isconsidered, but now subjected at the top to a lateral load H andan axial load N = 0.5Ncr,exact (αcr = 0.5). Magnified moments arefirst computed using sway magnifiers obtained in a conventionalstorey analysis based on first order analysis.A sample calculation of the lowest segment S1 is first

demonstrated. Based on the first order rotational stiffnesses κ02 =∞ at the base, κ01 = k01/(EI/L) = −3.429/4 = −0.857 at the topof S1 (from Fig. 6), and γn taken equal to γs, the following quantitiesare computed:(1) Rotational fixity factors, Eq. (12): R02 = 1.0, R01 = −0.750;(2) First-order relative displacement between the ends of S1,Eq. (15): ∆0 = 1.837HL3/EI; (3) Flexibility factor, Eq. (22b):γs = 1.329; (4) Sway stability index, Eq. (3): αs = 0.188; (5) Swaymagnifier, Eq. (2): Bs = 1/(1− 0.188) = 1.231.Sway magnifiers obtained in this manner, Bs = 1.23, 1.62, 2.11

and 2.53 for S1 to S4, respectively, are compared to correspondingexact moment magnifiers, BM = 1.82, 2.00, 2.14 and 2.22, inFig. 7(b). Approximate total moments computed by

M = BsM0are shown by the stepped line in Fig. 7(c). Compared to the exactmoments in the figure (dashed line), the correspondence is good inthe upper half (S3 and S4), but considerably below exact momentsin the lower half (−32% and−19% in S1 and S2, respectively).This example confirms results of previous studies that the

conventional storeymagnifier approach is not applicable, or at bestvery approximate, in single curvature bending regions, or in thisparticular case, in the lower half of the region.

7.4. Moments for exact restraint stiffness

The inaccuracy in the flexibility factor due to the approximationγn ≈ γs at αcr = 0.5, is quite small (about 1%–1.5% below the exactvalue). The reason for the large discrepancy in moments abovemust therefore be due to the first order stiffness properties impliedby the first order analysis. To verify this, magnified momentcomputations are repeated using sway magnifiers obtained basedon the exact restraint stiffnesses at the considered axial load level.Modified first order quantities are now computed in exactly the

same manner as above, but for the different restraint stiffnesses.A prime is added to distinguish modified quantities from theconventional first order quantities above. A sample computationis here given for S1. Nondimensional rotational stiffnesses (from

1742 J. Hellesland / Engineering Structures 31 (2009) 1734–1744

Table 3Moment computations of Columns S1–S4 (Fig. 7(a)) for αcr = 0.5 and corresponding exact restraint stiffnesses.

x/h κj Col. R′2/R′

1 ∆′0 γ ′s B′s M ′02 M2,exactB′sM′02

M2,exact(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

1.0 0 S4 0.069/0 7.136 1.002 2.237 −0.250 −0.556 1.0060.75 ±0.147 S3 0.153/−0.079 6.630 1.005 2.057 −0.520 −1.069 1.0000.5 ±0.361 S2 0.301/−0.220 5.402 1.027 1.747 −0.863 −1.502 1.0040.25 ±0.899 S1 1/−0.816 2.554 1.337 1.358 −1.359 −1.817 1.0160 ∞

κj = kj/(EI/L);∆′ = ∆′/(HL3/EI);M = M/Hh;M ′01 = −M ′02 − 0.25Hh.

Fig. 6) at the actual axial load level of αcr = 0.5 are κ02 = ∞, κ01 =k01/(EI/L) = −1.048·3.429/4 = −0.899. These give themodifiedquantities: R′2 = 1.0, R

′

1 = −0.816;∆′

0 = 2.554HL3/EI; γ ′s =

1.337;α′s = 0.263, and, finally, B′s = 1/(1− 0.263) = 1.358.

The modified magnifier (1.358) is only about 10% greater thanthat based on the first order stiffness (1.231 in Section 7.3).However, in order to obtain the appropriate total momentpredictions, the modified magnifier must be applied to first ordermoments that also is modified to comply with the exact restraints.The ‘‘modified first order moments’’ can be obtained from Eq. (11)with the fixity factors R ′1(2) above. Expressed in terms of the totalheight h = 4L, the modified first order moments become M ′01 =−(−0.816/(1− 0.816))0.25Hh = 1.109Hh, andM ′02 = (−1/(1−0.816))0.25Hh = −1.359Hh. The corresponding approximate,sway magnifier based, total momentsM = B′sM

′

0

becomeM2 = −1.846Hh andM1 = 1.506Hh.These, and similar results for the other column segments, are

summarised and compared to exactmoment results in Table 3. Theaccuracy is seen to be very good (within 0% to +1.6%). Similarly,total relative displacements, ∆ = B′s∆

′

0, have been found to bewithin−0.1% and+0.7% of exact displacements.Modified first order moments and total moment predictions

are also shown (by filled dots) in Fig. 7(d). The stepped first ordermoment distribution in the figure was to be expected as the slopeof the individual portions must be the same since the shear isconstant (= H) along the segments.This last exercise clearly demonstrates, again as expected, that

the storey magnifier approach as such is applicable provided thecorrect stiffness values are used both in the computation of the firstorder (‘‘modified’’) moment and of the magnifier.

8. Modified sway magnifier method (MSM)

8.1. General remarks

In order to extend the use of the conventional storey magnifierapproach to single curvature regions, means of estimatingstiffnesses reasonably accurate at column ends are required, ormore approximate procedures must be used in single curvatureregions. In this study, the latter alternative is pursued.In the lower half of the structure in Fig. 7(c), where the

conventional storey magnifier approach severely underestimatesthe moments, the change in restraint stiffness caused by the axialloading cause an increase of approximately 10% (1.358/1.231 =1.10) in the sway magnifier B′s, and 39% (2.554/1.8837 = 1.39)in the first order displacement ∆′0. In the second storey, theincreases are smaller (8% and 12%, respectively), but togetherquite significant. The stiffness change manifests itself in increasedcolumn flexibility.Based on these observations, a simple approach has been

studied whereby a ‘‘restraint correction flexibility factor’’ isintroduced to account for the local effects of axial forces onthe rotational restraint stiffnesses k in single curvature bendingregions. Indications are that it might be a viable approach, and aproposal on this basis is presented below.

8.2. Modified sway magnifier proposal

Consideringpreviously presented results (primarily Section7.3),it is clear that the restraint correction flexibility factor, here la-belled γk, must increase with increasing distance from the inflec-tion point towards the base (i.e., the distance from the top inFig. 7(c)) in order to compensate for the lack of agreement be-tween accurate moments and predicted moments based on firstorder stiffness values. From a study of typical storey swaymagnifi-cation factors in the single curvature regions ofmultistorey frames,it was found, after several trials, that reasonably simple expres-sions might be adequate. A nonlinear variation (defined below) interms of a ratio defining the location of the considered storey inthe single curvature moment region, was chosen due to its relativesimplicity.It is now proposed to compute the storey sway magnifier from

Eq. (8),

Bs =1

1− αssbut with γn in the sway stability index αss (Eq. (9)) replaced by acombined flexibility factor γc such that

αss =

∑(γcN/L)S0

(28)

Here,

γc = γn γk (29)

and

γk =

(c1L0xts+ c2

)g≥ 1 (30)

The flexibility factors in Eq. (29) account for second order effectsof axial forces acting on the bent shape (Nδ effects) and on therotational restraints (vertical interaction effects), respectively.L0 is the distance along the structure from the base (of the

column row (vertical axis) on which the considered column islocated) to the first order inflection point (at the top of thesingle curvature region in the considered column row), and xts isthe distance from the base to the top of the storey considered.Finally, c1, c2 and the exponent g are constants. They will bedependent on several factors, including axial load levels, and theyshould be chosen such as to provide acceptable predictions. Severalcombinations of the constants have been considered. Tentatively,the values

c1 = 0.11, c2 = 0.89, g = 3 (31)

are suggested for computation of sway magnifiers for commonframes in the practical magnifier range of about Bs = 1.3–1.5.These constants are also used in computations below.The definition above is applicable to frames both with, and

without, single curvature regions. However, for frames withcolumns in double curvature, and for columns of storeys abovesingle curvature regions, predictions are not affected by stiffness

J. Hellesland / Engineering Structures 31 (2009) 1734–1744 1743

Table 4Moment computations by the modified sway magnifier method for Columns S1–S4 (Fig. 7(a)).

αcr = 0.5 αcr = 0.3Col. ∆0 γs γk Bs BM,exact (5)/(6) Bs BM,exact (8)/(9)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

S4 7.834 1.000 1.0 2.525 2.223 1.136 1.568 1.522 1.031S3 6.829 1.004 1.114 2.436 2.138 1.139 1.547 1.486 1.041S2 4.832 1.024 1.368 2.091 2.001 1.045 1.456 1.429 1.019S1 1.837 1.329 2.353 1.789 1.817 0.984 1.360 1.351 1.007

∆ = ∆/(HL3/EI); αcr = N/Ncr,exact (Ncr,exact = π2EI/(2h)2).

changes since xts > L0 and γk = 1 in such cases. The γn factor maybe approximated by γs (or γ0) in most practical applications This isin particular the case for columns in single curvature regions.The various columns in a multibay storey may have different γs

and γk factors (through different L0/xts ratios) due to differencesin restraints at the various axes (column rows). The sway isdependent on the combined effect from all axes, as reflectedthrough the summation in Eq. (28). Therefore, in cases withdifferent L0 values in the various vertical axes, it will normally beacceptable to adopt an average value to be used for all axes.In common frames with reasonably regular distribution of

horizontal and gravity loads on the stories, stiffness distributionsacross the bays and along the height in the various column rows,etc., it is believed that the suggested constants (c1, c2, g) are nottoo sensitive to these frame parameters. However, in futurework itwould be of interest to investigate the applicability of the proposaland constants for a wider range of frames and frame parameters.As shown in Section 7.2, predictions of effective lengths in single

curvature regions were quite sensitive to restraint assessments(Table 2). The constants (c1, c2, g) suggested above, determinedby comparison with frame results obtained for axial load levelssignificant below the critical loads, may consequently not beadequate for effective length predictions. Suitable constants forsuch predictions are not pursued further here.

8.3. Typical calculation procedure

In summary, a typical sway magnifier calculation for a generalframe will include the following elements:1. Carry out a conventional first order analysis to provide first ordersidesway, moments and shears in the columns on each floor, andcalculate the first order lateral stiffness of the various stories (S0,Eq. (9b))2. Calculate the flexibility factors γs (γn ≈ γs) from Eq. (22), orequivalent expressions, for the various columns on each floor.3. Calculate the lengths of the single curvature regions (L0) in eachcolumn row, and calculate the flexibility factors γk from Eq. (30)for the various columns on each floor.4. Calculate the storey swaymagnifiers Bs with the stability indicesαss, Eq. (28), and approximate total moments at column ends byEq. (7).

9. Applications to single curvature regions

9.1. 4-‘‘storey’’ column in single curvature

First, the proposal is applied to the 4-‘‘storey’’ continuouscolumn in Fig. 7(a), with L0 = h = 4L, and xts = L, 2L, 3L and4L for stories S1 to S4, respectively, and γc = γsγk.The calculations now follow the same routine described before

(Section 7.3), and the values of∆0 and γs are the same used there.Results of the calculations are summarised in Table 4 for the twoaxial load levels of αcr = 0.3 and 0.5. These give sway magnifierpredictions in the ranges of 1.4–1.6 and 1.8–2.5, respectively.

Fig. 8. Magnifiers for 24-storey ‘‘stiff column-flexible beam’’ frame.

Compared to the exact moment magnifiers in the table, thepredictions are seen to be good, and best for the lower load level(αcr = 0.3), which is most realistic in practical cases. It should benoted that the predictions for S4 is not affected by the γk factor.

9.2. 24-storey frame

The applicability of the method is now demonstrated for asymmetrical 24-storey, 1-bay frame with equal storey heightspreviously analysed by Lai andMacGregor [9]. The lower 17 storeysare shown in Fig. 8(a). The column stiffnesses are identical withinthree sets of eight storeys, but significantly different for the threesets. The columns of the lower eight storeys are considerably stifferthan the beams (EI/L of columns are about 15 times EIb/Lb ofbeams). Thus, the frame represents an example of a ‘‘stiff column-flexible beam’’ frame. The same vertical loading is applied at allstorey levels, and it seems that only a single lateral load is appliedat the top of the frame. The first order bending moment diagramshows single curvature bending in the bottom four storeys.Exact moment magnification factors (BM = M2/M02) at

the column end with maximum moment, and correspondingsway magnification factors based on conventional storey stabilityindices (here denoted Bs,lai), were computed and presented indiagram form by Lai and MacGregor. Such magnification values,read from an enlarged version of this diagram (consideredsufficiently accurate for the present purpose), are replotted inFig. 8(b) for the lower seven storeys. The estimated inflection pointlocation from the base is L0 = 4.7L, where L is the storey height.Compared to the exact moment magnifiers (solid, stepped

lines), the conventional storey magnifiers (filled circles) are seento significantly underestimate the moments in the lower stories.This is typical, as alsomentioned previously, for regionswith singlecurvature bending.

1744 J. Hellesland / Engineering Structures 31 (2009) 1734–1744

Modified stability indices are calculated from the results in Laiand MacGregor (‘‘lai’’) using αss,lai = 1 − (1/Bs,lai) and αss,mod =αss,lai · γsγk/γss,lai ≈ αss,lai · γk. The present γs values are herefor simplicity approximated by the γs,lai values (1.2 in the bottomstorey and 1.05 in the others [9]), although a more correct γs valuefor the lower storey probably is somewhat greater than 1.3. ForL0 = 4.7L and L0/xts = 4.7, 2.35, 1.57, 1.18 and 0.94 for storiesS1 to S5, respectively, the corresponding γk values become 2.79,1.52, 1.20, 1.06 and 1.0.Modified storey swaymagnifier results, corresponding to Bs (Eq.

(2)) defined with αss,mod, are also shown in the figure (dashed,stepped lines). They are in good agreementwith the exactmomentmagnifiers (between 5% above and 2% below).

10. Summary and conclusions

Present approximate storey magnifier methods for the analysisof unbraced multistorey frames may significantly underestimatesecond order effects in single curvature regions. To provide abetter understanding of the reasons for this, the mechanics ofcolumn interaction in single curvature regions have been studied.For this purpose, suitable tools for sidesway description werederived, including a nearly exact, explicit free sway effective lengthexpression, that, when high accuracy is required, eliminates theneed for cumbersome, iterative solutions of exact effective lengthsfrom the transcendental instability equation.Two reasons for the underestimation are identified, and their

relative importance have been clarified. One is related to the localsecond order Nδ effects, which may be greater than commonlyassumed, and the other, and most important one, is relatedto second order effects causing changes in rotational restraintstiffness distributions at column ends due to vertical, inter-storeycolumn interaction.Amodified approximate storeymagnifier approach is proposed

that accounts for these local second order effects through twoseparate ‘‘flexibility factors’’, γn (γs) and γk. Both factors increasethe flexibility of the columns of the frame beyond their first ordervalues. The first factor (reflecting Nδ effects (or bending shape))may typically vary between 1 and about 1.35. The second factor(‘‘restraint correction flexibility factor’’) may vary between muchwider limits, such as 1 and 2.8 in the examples considered.

The proposed approach has been found to provide predictionsthat compare well with more accurate results. It is appealing inthat it is sufficiently simple to be viable in practical analyses.Comparisons with accurate analysis results for a wider range offrame parameters are recommended prior to possible inclusion inrelevant codes and standards.

References

[1] Hellesland J. Extended second order approximate analysis of frames withsway-braced column interaction. J Construct Steel Res 2009;65(5):1075–86.doi:10.1016/j.jcsr.2008.08.008.

[2] Hellesland J. Approximate second order analysis of unbraced frames. Technicalrep. Oslo (Norway): Dr. Ing. A. Aas-Jakobsen Ltd. 1976. 43 pp.

[3] Rubin H. Das Q∆-Verfahren zur vereinfachten Berechnung verschieblicherRahmensysteme nach dem Traglastverfahren der Theorie II. Ordnung DerBauingenieur 1973;48(8):275–85.

[4] LeMessurierWM.Apracticalmethod of secondorder analysis. Eng J AISC 1977;14(2):49–67.

[5] Lui EM. A novel approach for K factor determination. Eng J AISC 1992;29(4):150–9.

[6] Aristizabal-Ochoa JD. K-factor for columns in any type of construction:Nonparadoxical approach. J Struct Eng ASCE 1994;120(4):1272–90.

[7] Girgin K, Ozmen G, Orakdogen E. Buckling lengths of irregular frame columns.J Construct Eng 2006;62(6):605–13.

[8] Chen WF, Lui EM. Stability design of steel frames. Boca Raton, Ann Arbor,Boston, London: CRC Press; 1991.

[9] Lai S-MA, MacGregor JG. Geometric nonlinearities in unbraced frames. J StructEng ASCE 1983;109(11):2528–45.

[10] Hellesland J. Mechanics and effective lengths of columns with positive andnegative end restraints. Eng Struct 2007;29(12):3464–74.

[11] Hellesland J, Bjorhovde R. Restraint demand factors and effective lengths ofbraced columns. J Struct Eng ASCE 1996;122(10):1216–24.

[12] Xu L, Liu Y. Story stability of semi-braced steel frames. J Construct Steel Res2002;58(4):467–91.

[13] Massonet CE. European approaches to P−∆method of design. J Struct DivisionASCE 1978;104(ST1):193–8.

[14] Hellesland J. Discussion of an improved K-factor formula by Cheong-Siat-Moy.F J Struct Eng ASCE 2000;126(5):633–5.

[15] Hellesland J. Approximate second order analysis of unbraced framesreflecting inter-storey interaction in single curvature regions. Research rep.in mechanics. No. 08–2. Oslo (Norway): Mechanics Division, Dept. of Math.,Univ. of Oslo; 2008. 26 pp.

[16] Lai S-MA, MacGregor JG, Hellesland J. Geometric non-linearities in non-swayframes. J Struct Eng ASCE 1983;109(12):2770–85.

[17] Tong GS, Wang JP. Column effective lengths considering inter-columninteractions in sway-permitted frames. J Construct Steel Res 2006;62(5):413–23.