# Buckling strength of multi-story sway, non-sway and partially-sway frames with semi-rigid connections

**ABSTRACT** The objective of this paper is to propose a simplified approach to the evaluation of the critical buckling load of multi-story frames with semi-rigid connections. To that effect, analytical expressions and corresponding graphs accounting for the boundary conditions of the column under investigation are proposed for the calculation of the effective buckling length coefficient for different levels of frame sway ability. In addition, a complete set of rotational stiffness coefficients is derived, which is then used for the replacement of members converging at the bottom and top ends of the column in question by equivalent springs. All possible rotational and translational boundary conditions at the far end of these members, featuring semi-rigid connection at their near end as well as the eventual presence of axial force, are considered. Examples of sway, non-sway and partially-sway frames with semi-rigid connections are presented, where the proposed approach is found to be in excellent agreement with the finite element results, while the application of codes such as Eurocode 3 and LRFD leads to significant inaccuracies.

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Page 1

Journal of Constructional Steel Research 62 (2006) 893–905

www.elsevier.com/locate/jcsr

Buckling strength of multi-story sway, non-sway and partially-sway frames

with semi-rigid connections

Georgios E. Mageirou∗, Charis J. Gantes1

Laboratory of Metal Structures, Department of Structural Engineering, National Technical University of Athens, 9 Heroon Polytechniou, GR-15780, Zografou,

Athens, Greece

Received 2 August 2005; accepted 30 November 2005

Abstract

The objective of this paper is to propose a simplified approach to the evaluation of the critical buckling load of multi-story frames with semi-

rigid connections. To that effect, analytical expressions and corresponding graphs accounting for the boundary conditions of the column under

investigation are proposed for the calculation of the effective buckling length coefficient for different levels of frame sway ability. In addition, a

complete set of rotational stiffness coefficients is derived, which is then used for the replacement of members converging at the bottom and top

ends of the column in question by equivalent springs. All possible rotational and translational boundary conditions at the far end of these members,

featuring semi-rigid connection at their near end as well as the eventual presence of axial force, are considered. Examples of sway, non-sway and

partially-sway frames with semi-rigid connections are presented, where the proposed approach is found to be in excellent agreement with the

finite element results, while the application of codes such as Eurocode 3 and LRFD leads to significant inaccuracies.

c ? 2005 Elsevier Ltd. All rights reserved.

Keywords: Buckling; Effective length; Stiffness coefficients; Multi-story sway; Non-sway and partially-sway frames; Semi-rigid connections

1. Introduction

Nowadays, the buckling strength of a member can be

evaluated using engineering software based on linear or also

non-linear (in terms of large displacements and/or material

yielding)procedureswithanalyticalornumericalmethods[15].

Nonetheless, the large majority of structural engineers still

prefer analytical techniques such as the effective length and

notional load methods [26]. These two methodologies are

included in most modern structural design codes (for example,

Eurocode 3 [9], LRFD [23]).

The objective of this work is to propose a simplified

approach for the evaluation of critical buckling loads of multi-

story frames with semi-rigid connections, for different levels

of frame sway ability. To that effect, a model of a column in a

multi-story frame is considered as individual. The contribution

of members converging at the bottom and top ends of the

∗Corresponding author. Tel.: +30 210 9707444; fax: +30 210 9707444.

E-mail addresses: mageirou@central.ntua.gr (G.E. Mageirou),

chgantes@central.ntua.gr (C.J. Gantes).

1Tel.: +30 210 7723440; fax: +30 210 7723442.

column is taken into account by equivalent springs. Namely,

the restriction provided by the other members of the frame

to the rotations of the bottom and top nodes is modeled

via rotational springs with constants cb and ct, respectively,

while the resistance provided by the bracing system to the

relative transverse translation of the end nodes is modeled

via a translational spring with constant cbr. This is shown

schematically in Fig. 1. The rotational stiffness of the springs

must be evaluated considering the influence of the connection

non-linearity.This modelhas beenusedby severalinvestigators

(for example, Wood [27], Aristizabal-Ochoa [1], and Cheong-

Siat-Moy [6]) for the evaluation of the critical buckling load of

the member, and is adopted by most codes.

The stiffness of the bottom and top rotational springs

is estimated by summing up the contributions of members

convergingat the bottom and top ends, respectively:

?

A frame is characterized as non-sway if the stiffness cbr of

the bracing system is very large, as sway if this stiffness is

negligible, and as partially-sway for intermediate values of this

cb=

i

cb,i,

ct=

?

j

ct,j.

(1)

0143-974X/$ - see front matter c ? 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jcsr.2005.11.019

Page 2

894

G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905

Notations

A, B,C, D integration constants

E

modulus of elasticity

G

distributionfactor at the end nodesof the column,

according to LRFD

I

moment of inertia

K

effective buckling length coefficient

L

span length of adjoining members

M

bending moment

N

axial force of adjoining members

P

compressive load

Φ

factor of slope deflection method

a

factor of slope deflection method, factor for the

effect of the boundary condition at the far end

nodes of the member

c

stiffness coefficient

c

ratio of flexural stiffness to span

c#

dimensionless rotational stiffness

d

factor for the effect of the axial force

h

column height

k

non-dimensional compressive load

?

effectivebucklinglengthcoefficient,accordingto

EC3

n

ratio of member’s compressive force to Euler’s

buckling load

x

longitudinal coordinate

z

dimensionless distribution factor at the end nodes

of the column

w

transverse deflection

δ

relative transverse deflection between the end

nodes of the member

η

distributionfactor at the end nodesof the column,

according to EC3

θ

rotation at the end nodes of the member

Subscripts:

A

B

E

EC3

FEM

LRFD

c

cr

b

bm

br

i

n

r

t

bottom end node of the column

top end node of the column

Euler

Eurocode 3

Finite Element Method

Load Resistance Factor Design

column

critical

bottom

beam

bracing system

member i

node

rigid connection

top

stiffness. Eurocode 3 and LRFD provide the effective length

Kh of columns in sway and non-sway frames via graphs or

Fig. 1. (a) Multi-story steel frame; (b) proposed model of column under

investigation.

analytical relations as functions of the rotational boundary

conditions without considering the connection non-linearity

and the partially-sway behaviour of the frame. The critical

buckling load is then defined as:

Pcr=π2EIc

(Kh)2

(2)

where EIcis the flexural resistance.

The main source of inaccuracy in the above process lies

in the estimation of the rotational boundary conditions. LRFD

makes no mention to the dependence of the rotational stiffness

of members converging at the ends of the column under

consideration on their boundary conditions at their far end or

their axial load. Annex E of EC3 is more detailed in accounting

for the contribution of converging beams and lower/upper

columns, but ignores several cases that are encountered in

practice, and are often decisive for the buckling strength. Both

codes ignore the partially-sway behaviour of the frames as well

as the connection non-linearity.

This problem has been investigated by several researchers.

The work of Wood [27] constituted the theoretical basis of

EC3. Cheong-Siat-Moy [5] examined the k-factor paradox

for leaning columns and drew attention to the dependence

of buckling strength not only on the rotational boundary

conditions of the member in question but also on the overall

structural system behavior. Bridge and Fraser [4] proposed

an iterative procedure for the evaluation of the effective

length, which accounts for the presence of axial forces in

the restraining members and thus also considers the negative

values of rotational stiffness. Hellesland and Bjorhovde [11]

proposeda new restraint demand factor consideringthe vertical

and horizontal interaction in member stability terms. Kishi

et al. [14] proposed an analytical relation for the evaluation of

the effective length of columns with semi-rigid joints in sway

frames. Essa [8] proposed a design method for the evaluation

of the effective length for columns in unbraced multi-story

frames considering different story drift angles. Aristizabal-

Ochoa examined the influence of uniformly distributed axial

load on the evaluation of the effective length of columns in

sway and partially-sway frames [2]. He then examined the

behavior of columns with semi-rigid connections under loads

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G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905

895

Fig. 2. Model of column in (a) non-sway frame, (b) sway frame, and (c) partially-sway frame, and (d) the sign convention used.

such as those produced by tension cables that always pass

through fixed points or loads applied by rigid links [3]. What

is more, Kounadis [16] investigated the inelastic buckling of

rigid-jointed frames.

Christopher and Bjorhovde [7] conducted analyses of a

series of semi-rigid frames, each with the same dimensions,

applied loads and member sizes, but with different connection

properties, explaining how connection properties affect

member forces, frame stability, and inter-story drift. Jaspart

and Maquoi [12] described the mode of application of the

elastic and plastic design philosophies to braced frames

with semi-rigid connections. The buckling collapse of steel

reticulated domes with semi-rigid joints was investigated by

Kato et al. [13] on the basis of a nonlinear elastic–plastic hinge

analysis formulated for three-dimensional beam–columns with

elastic, perfectly plastic hinges located at both ends and

mid-span for each member. Lau et al. [17] performed an

analytical study to investigate the behavior of subassemblages

with a range of semi-rigid connections under different test

conditions and loading arrangements. They showed that

significant variations in the M–ϕ response had a negligible

effect on the load carrying capacity of the column and the

behavior of the subassemblage. A method for column design

in non-sway bare steel structures which takes into account

the semi-rigid action of the beam to column connections

was proposed by Lau et al. [18]. In [19], closed-form

solutions of the second order differential equation of non-

uniform bars with rotational and translational springs were

derived for eleven important cases. A simplified method

for estimating the maximum load of semi-rigid frames was

proposed by Li and Mativo [20]. The method was in the

form of a multiple linear regression relationship between the

maximum load and various parameters (frame and section

properties), obtained from numerous analyses of frames. Liew

et al. [21] proposed a comprehensive set of moment-rotation

data, in terms of stiffness and moment capacity, so that a

comparative assessment of the frame performance due to

different connection types could be undertaken. Reyes-Salazar

and Haldar [24], using a nonlinear time domain seismic

analysis algorithm developed by themselves, excited three steel

frames with semi-rigid connections by thirteen earthquake time

histories. They proposed a parameter called the T ratio in

order to define the rigidity of the connections. This parameter

is the ratio of the moment the connection would have to

carry according to the beam line theory and the fixed end

moment of the girder. In [25], the equilibrium path was traced

for braced and unbraced steel plane frames with semi-rigid

connections with the aid of a hybrid algorithm that combines

the convergence properties of the iterative-incremental tangent

method, calculating the unbalancing forces by considering the

elementrigidbodymotion.Yuetal.[28]describedthedetailsof

atest programofthreetest specimensloadedtocollapseandthe

test observations for sway frames under the combined actions

of gravity and lateral loads.

However, all these studies mention nothing about the

dependence of the rotational stiffness of the members

converging on the column under consideration, from the

boundaryconditions at their far ends and from their axial loads.

This dependence is investigated in the present work for multi-

story frames with semi-rigid connections for different levels

of lateral stiffness cbr. Easy to use analytical relations and

corresponding graphs are proposed for the estimation of the

columns’ effective length for sway, non-sway and partially-

sway frame behaviour. Furthermore, analytical expressions are

derived for the evaluation of the rotational springs’ stiffness

coefficients for different member boundary conditions and

axial loads accounting for the connection non-linearity. Results

obtained via the proposed approach for sway, non-sway and

partially-sway frames with semi-rigid connections are found to

be in excellent agreement with finite element results, while the

applicationofdesigncodessuchas Eurocode3andLRFDleads

to significant inaccuracies.

2. Buckling strength of columns in multi-story frames

2.1. Non-sway frames

Considerthe modelof a columnin a non-swayframe,shown

in Fig. 2(a), resulting from the model of Fig. 1(b) by replacing

the translational spring with a roller support. Denoting by w

the transverse displacement and by?the differentiation with

respect to the longitudinal coordinate x, and using the sign

convention of Fig. 2(d), the equilibrium of this column in its

buckled condition is described by the well-known differential

equation:

w????(x) + k2w??(x) = 0

where:

?

EIc

(3)

k =

Pcr

=

π

Kh.

(4)

Page 4

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G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905

Fig. 3. Effective buckling length factor K for different levels of frame-sway ability.

The general solution of this differential equation is given by:

w(x) = Asin(kx) + B cos(kx) + Cx + D.

The integration constants A, B,C, and D can be obtained by

applying the boundary conditions at the two column ends:

Transverse displacement at the bottom:

(5)

w(0) = 0.

Moment equilibrium at the bottom:

−EIcw??(0) = −cbw?(0).

Moment equilibrium at the top:

−EIcw??(h) = ctw?(h).

Transverse displacement at the top:

(6)

(7)

(8)

w(h) = 0.

The four simultaneous equations (6)–(9) have a non-trivial

solution for the four unknowns A, B,C, and D if the

determinant of the coefficients is equal to zero. This criterion

yields the buckling equation for the effective length factor K:

?

(9)

32K3(zt− 1)(zb− 1) − 4K

8K2(zt− 1)(zb− 1)

+ (zt+ zb− 2ztzb)π2?

+20K2(zt+ zb) + ztzb

where zband zt are distribution factors obtained by the non-

dimensionalization of the end rotational stiffnesses cband ct

with respect to the column’s flexural stiffness cc:

cc

cc+ cb,

where:

cc=4EIc

h

Eq. (10) can be solved numerically for the effective length

factor K, which is then substituted into Eq. (2) to provide the

critical buckling load. Alternatively, the upper left graph of

Fig. 3, obtained from Eq. (10), can be used.

cos

?

?π

K

?

+ π

?

−16K2

sin

π2− 24K2???π

K

?

= 0 (10)

zb=

zt=

cc

cc+ ct

(11)

.

(12)

2.2. Sway frames

The simplified model of a column in a sway frame,shown in

Fig. 2(b), is considered, resulting from the model of Fig. 1(b)

by omitting the translational spring. Three boundaryconditions

Page 5

G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905

897

are described by Eqs. (6)–(8), while a fourth condition

expresses horizontal force equilibrium at the top:

−EIcw???(h) − Pw?(h) = 0.

Thus the buckling equation for the effective length factor K

is derived, following the same procedure as above:

?π

+

K

(13)

4[zt(2zb− 1) − zb]π

?

Alternatively,thebottomrightgraphofFig. 3,obtainedfrom

Eq. (14), can be used.

Kcos

K

?

ztzb

?π

?2− 16(zt− 1)(zb− 1)

?

sin

?π

K

?

= 0. (14)

2.3. Partially-sway frames

Finally, consider the simplified model of a column in

a partially-sway frame, shown in Fig. 2(c). Similarly, the

boundary conditions are described by Eqs. (6)–(8) and the

following equation, representing horizontal force equilibrium

at the top:

−EICw???(h) − Pw?(h) = −cbrw(h).

The effectivelengthfactor K of a columnin a partially-sway

frame is then obtained from the following equation:

(15)

−32K5cbr(zt− 1)(zb− 1) + 4K

+ K2cbr(zt+ zb− 2ztzb)π2+ (−zt− zb+ 2ztzb)π4?

× cos

K

−16K2π2(1 − zt− zb+ ztzb)

− K2cbrπ2ztzb+ π4ztzb

where:

cbr=cbrh3

EI

Easy to use graphs such as those presented in Fig. 3 are

obtained from the above equation for several values of cbr.

?

8K4cbr(zt− 1)(zb− 1)

?π

?

+ π

?

4K4cbr(4 − 5zt− 5zb+ 6ztzb)

?

sin

?π

K

?

= 0(16)

.

(17)

3. Stiffness distribution factors

The rotational stiffness of each member converging at the

top or bottom node of the column in question is derived using

the slope-deflectionmethod [22]. The moments MABand MBA

at the two ends of a member AB with span L and flexural

stiffness EI, without axial force or transverse load (Fig. 4), can

be obtained as a function of the end rotationsθAand θBand the

relative transverse deflection of the end nodes δ from:

MAB=2EI

LL

MBA=2EI

L

?

2θA+ θB+3δ

?

?

,

2θB+ θA+3δ

L

?

.

(18)

Fig. 4. Undeformed (dotted line) and deformed (continuous line) state of a

member AB, and the sign convention of the slope-deflection method.

If, in addition, the member is subjected to a compressive

axial force P, then Eq. (18) becomes:

?

MBA=2EI

L

MAB=2EI

L

anθA+ afθB+?an+ af

anθB+ afθA+?an+ af

? δ

? δ

L

?

??

L

(19)

where:

an=

Φn

2

?

Φ2

n− Φ2

f

?,

af =

Φf

2

?

k2L2

Φ2

n− Φ2

?

f

?

(20)

Φn=1 − kL cotkL

k2L2

,

Φf =

1

kL

sinkL− 1

?

.

(21)

Using the above equations, the rotational stiffness expres-

sionshavebeenderivedformemberswith allpossibleboundary

conditionsatthefarendandasemi-rigidconnectionat theclose

end, with or without axial force, and are shown in Table 1. The

derivationof the rotationalstiffness factors is describednextfor

two characteristic cases: one for a member without and one for

a member with axial force.

3.1. Member with a fixed support at the far end and a semi-

rigid connection at the near end, without axial force

Consider the member AB of Fig. 5(a), with span Li and

flexural stiffness EiIi, where A refers to the bottom or top

node of the column under investigation, while B is the far

node, attached to a fixed support. The connection at node A

is considered as semi-rigid with a rotational stiffness cn.

The slope-deflection equations are given by (18), with

indices i referring to the specific member. Firstly, the

connection at node A is considered as rigid. The rotational

stiffness cr,i of the member AB with a rigid connection was

evaluated in previous work by the authors [10].

Themomentatnode A ofthememberwithrigidconnections

is given by the equation:

MAB=2EiIi

Furthermore, there is no rotation at node B:

Li

(2θA+ θB).

(22)

θB= 0.

(23)

Page 6

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G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905

Table 1

Converging members’ rotational stiffness expressions for different boundary conditions at the far end and for a semi-rigid connection at the near end, with or without

axial force

Rotational boundary conditions of far endWithout axial forceWith axial force

Fixed support

ci=

4¯ cicn

4¯ ci+cn

ci=

4¯ cicn(1−0.33ni)

cn+4¯ ci−1.32¯ cini

Roller fixed support

ci=

¯ cicn

¯ ci+cn

ci=

¯ cicn(1−0.82ni)

cn+¯ ci−0.82¯ cini

Pinned support

ci=

3¯ cicn

3¯ ci+cn

ci=

3¯ cicn(1−0.66ni)

cn+3¯ ci−1.98¯ cini

Simple curvature

ci=

2¯ cicn

2¯ ci+cn

ci=

2¯ cicn(1−0.82ni)

cn+2¯ ci−1.64¯ cini

Double curvature

ci=

6¯ cicn

6¯ ci+cn

ci=

6¯ cicn(1−0.16ni)

cn+6¯ ci−0.96¯ cini

Roller support

ci=

0¯ cicn

0¯ ci+cn

ci=

¯ cicn(0−0.97ni)

cn+0¯ ci−0.97¯ cini

Rotational spring support

ci=

¯ cicnc#

(¯ ci+cn)c#+cn

ci=

¯ cicn[c#−(1.047ni+1.773)ni]

cn(0.591nic#+c#+1)+¯ ci[c#−(1.047ni+1.773)ni]

Pinned and rotational spring support

ci=

4¯ cicn(c#+3)

4¯ ci(c#+3)+cn(c#+4)

ci=

2¯ cicn[(c#(c#+9)+24)niπ2−30(c#+3)(c#+4)]

2¯ ci(c#(c#+9)+24)niπ2−15(c#+4)[4¯ ci(c#+3)+cn(c#+4)]

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G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905

899

Fig. 5. Model of a member with a fixed support at the far end B and a semi-

rigid connection at the near end A: (a) without axial force, (b) with axial force.

Then, by virtue of (23), Eq. (22) becomes:

MAB=4EiIi

Li

θA.

(24)

Thus, the rotational stiffness representing the resistance of

member AB to the rotation of node A is given by:

cr,i=4EiIi

Li

(25)

which can be written as

cr,i= 4¯ ci

where:

(26)

ci=EiIi

Li

.

(27)

Two rotational springs with rotational stiffness cr,i and cn

in series are considered, in order to evaluate the rotational

stiffness ci of the member AB with a semi-rigid connection.

The total rotation is the sum of the rotations of the two springs.

Therefore:

ϕi= ϕr,i+ ϕn.

Consideringthatthespringshavethesame moment,Eq.(29)

is written:

(28)

1

ci

=

1

cr,i

+1

cn.

(29)

Thus, the rotational stiffness ciof member AB with a semi-

rigid connection is evaluated from Eq. (29):

ci=

cr,i· cn

cr,i+ cn.

By substituting (26) into (30), the rotational stiffness ci of

the member AB with a semi-rigid connection is evaluated:

(30)

ci=

4¯ cicn

4¯ ci+ cn.

(31)

3.2. Member with a fixed support at the far end and a semi-

rigid connection at the close end, with axial force

Now consider the member of Fig. 5(b), with span Li and

flexural stiffness EiIi, where A again refers to the bottom or

top node of the column under investigation, while B is the

far node, rotationally fixed. The member AB is subjected to

a compressive axial force Ni. The rotational stiffness ciof the

member AB with a semi-rigid connection is evaluated in the

same manner as above.

Firstly, the member AB is considered having rigid

connections. The slope-deflection equations are given by (19),

with indices i referring to the specific member. The moment at

node A of the member with rigid connections is given by the

equation:

MAB=2EiIi

As there is no rotation at node B (θB = 0), the previous

equation is rewritten:

Li

?αn,iθA+ αf,iθB

?.

(32)

MAB=2EiIi

Therefore,the rotational stiffness representingthe resistance

of member AB to the rotation of node A is given by:

Li

αn,iθA.

(33)

cr,i=2EiIi

which, by means of (20) and (21) and k2

π2EiIi/L2

Li

αn,i

(34)

i= N/EiIi, NE,i =

i, becomes:

cr,i=4EiIi

Li

π√ni

π√nicot?π√ni

?− 1

4π√ni− 8tan

?

1

2π√ni

?

(35)

where ni is the ratio of the member’s compressive force to

Euler’s buckling load NE,i:

Ni

NE,i.

A Taylor series expansion of Eq. (35) gives:

?

−

Keeping the first two terms, Eq. (38) is obtained:

?

which can be written as

ni=

(36)

cr,i =4EiIi

Li

1 −π2

30ni−

11π4

25200n2

509π8

2328480000n4

i

π6

108000n3

i−

i...

?

.

(37)

cr,i=4EiIi

Li

1 −π2

30ni

?

(38)

cr,i= 4¯ ci(1 − 0.33ni).

By substituting (39) into (30), the rotational stiffness ciof the

member AB with a semi-rigid connection is evaluated:

(39)

ci=

4¯ cicn(1 − 0.33ni)

cn+ 4¯ ci− 1.32¯ cini.

(40)

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Fig. 6. The frame of example 1.

4. Examples

This section gives four examples of a simple frame that

consists of a column and a beam with a variety of supports at

the far end — namely, a three-story, single-bay sway frame; a

three-story,single-baynon-swayframe;anda one-story,single-

bay sway, non-sway and partially-sway frame — for which the

proposedapproachis demonstratedandits results arecompared

to (i) buckling loads obtained via linearized buckling analysis

of finite element models, considered as “exact”, for verification

purposes, and (ii) buckling loads calculated by applying the

pertinent procedures of Eurocode 3 and LRFD.

4.1. Example 1

Consider the frame of Fig. 6 with a single span L = 20 m

and height h = 10 m, having a column with HEB360 cross-

section (Ic = 43190 cm4) and a beam with IPE400 cross-

section (Ibm = 23130 cm4). The column is considered to be

pinned at the base. A concentrated load P is imposed on the

beam–columnjoint. Thebeam–columnjoint is consideredto be

semi-rigid, with a rotational stiffness of cn = 150 kN m/rad.

The beam has restricted translation and a rotational spring

support with a rotational stiffness of c = 500 kN m at the

far end. The frame is made of steel with Young’s modulus

E = 210000000 kN/m2.

Firstly, a linearized buckling analysis of the frame is

conducted using the commercial finite element software MSC-

NASTRAN. The critical buckling load obtained from this

analysis is Pcr,FEM = 8981.02 kN. Secondly, the buckling

strength is evaluated by using the proposed methodology. In

order to do so, the frame is substituted by the equivalent model

ofFig. 2(b).Therotationalstiffness ofthe beamconsideringthe

semi-rigid connection is evaluated from the last row of Table 1:

cBB? =

4¯ cBB?cn(c#

BB?+ 3) + cn(c#

BB?+ 3)

4¯ cBB?(c#

BB?+ 4)= 147.02 kN m (41)

where:

¯ cBB? =EIbm

c#

BB? =

L

c

= 2428.65 kN m

¯ cBB?= 0.206.

(42)

(43)

Applying the proposed method, the distribution factor zbis

equal to 1 due to the pinned support, while ztis obtained from:

cc

cc+ cBB?= 0.996

where:

cc=4EIc

h

Then, the evaluation of the buckling length coefficient is

conducted by means of Eq. (10), giving K = 0.998. Thus, the

Euler buckling load is equal to:

zt=

(44)

= 36279.60 kN m.

(45)

Pcr,prop=π2EIc

(Kh)2= 8981.01 kN.

(46)

Therefore, the results of the proposed approach are in

excellent agreement with the results of the finite element

method.The previousprocedureis followedfor the same frame

with different supports at the far end of the beam as well as

for a single span, one-story sway and non-sway frame for the

verification of the rotational stiffness of Table 1. The results of

the proposed approach and the FEM analysis are presented in

Table 2 and are practically the same.

4.2. Example 2

Consider the three-story sway frame of Fig. 7 with a single

span L = 20 m and uniform story height h = 10 m,

having columns with HEB360 cross-section (Ic= 43190 cm4)

and beams with IPE400 cross-section (Ibm = 23130 cm4).

The columns are considered to be pinned at the base. Equal

concentrated loads P/3 are imposed on all beam–column

joints. The beam–column joints are considered to be semi-rigid

witharotationalstiffnessofcn= 150 kN m.Theframeismade

of steel with Young’s modulus E = 210000000 kN/m2.

At first, a buckling analysis of the frame is conducted using

the same finite element software. The first buckling mode

is obtained from this analysis, and the corresponding critical

buckling load is 22.02 kN. In order to verify the proposed

rotational stiffness coefficients, the frame is then substituted by

a series of equivalent models. The first among them, denoted as

equivalent model 1a, is illustrated in Fig. 7(b). It is obtained by

substituting the beams at the three levels by rotational springs.

Assuming that, in the first buckling mode, the beams deform

with a double curvature, the stiffness of the springs is obtained

from the correspondingrow of Table 1:

cbm=

6¯ cbmcn

6¯ cbm+ cn

= 148.47 kN (47)

where:

¯ cbm=EIbm

The first buckling mode of the equivalent model 1a is

obtained from FEM analysis, and the corresponding critical

buckling load is also 22.02 kN, thus verifying the correctness

of this substitution.

L

= 2428.65 kN m.

(48)

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901

Table 2

Critical loads according to the proposed and finite element methods for a variety of structural systems with semi-rigid connections of example 1

Frame

Pcr,FEM(kN)

Pcr,prop(kN)

Pcr,prop−Pcr,FEM

Pcr,FEM

(%)

8981.58 8981.16

−0.005

8979.838979.860.001

9027.069027.300.003

10.9810.97

−0.091

8981.028981.010.0001

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Fig. 7. (a) The three-story sway frame of example 2, (b) equivalent model 1a, (c) equivalent model 1b, (d) equivalent model 1c.

The second equivalent model, denoted as 1b, is illustrated in

Fig. 7(c). It is obtained by substituting the top column CD by a

rotational spring with stiffness c?

last row of Table 1:

CDcalculated from the second

c?

CD

=

¯ cCDcn,CD

CD+ c#

?

c#

CD− (1.047nCD+ 1.773)nCD

CD+ 1

?

cn,CD

?

0.591nCDc#

?

+ ¯ cCD

?

c#

CD− (1.047nCD+ 1.773)nCD

?

(49)

= 70.18 kN m

where:

¯ cCD=EIc

c#

h

= 9069.90 kN m

= 0.016

(50)

CD=cbm

NCD=1

Alternatively,the axial design load can be used for the NCD,

without any significant influence on the results:

¯ cCD

322.02 kN = 7.34 kN.

(51)

(52)

NE,CD=π2EIc

h2

= 8942.56 kN(53)

nCD=

NCD

NE,CD

= 0.001.

(54)

Then, the total rotational stiffness at node C of model 1b is:

cCD= c?

The critical buckling load of the first buckling mode of the

equivalent model 1b, obtained from Nastran, is 22.00 kN.

The third equivalent model, denoted as 1c, is illustrated in

Fig. 7(d). It is obtained by substituting column BC of model

1b by a rotational spring with stiffness c?

from the second last row of Table 1:

CD+ cbm= 218.65 kN m.

(55)

BCcalculated similarly

c?

BC= 100.98 kN m.

(56)

Table 3

Critical loads for model 1 and its equivalent models 1a, 1b, 1c of Example 2

Pcr(kN)

Pcr−Pcr,model1

Pcr,model1

(%)

Model 1

Model 1a

Model 1b

Model 1c

22.02428

22.01921

22.00429

21.96301

0

−0.02

−0.09

−0.28

Then, the total rotational stiffness at node B of model 1c is:

cBC= c?

The first buckling mode of the equivalent model 1c is

obtained from Nastran, and the corresponding critical buckling

load is 21.96 kN. The critical loads of all models, as well as

their deviations from the critical load of the full model, are

summarized in Table 3, demonstrating excellent agreement.

In addition, the critical buckling load of column AB is

evaluated according to the provisions of EC3 and LRFD.

Followingthe procedureof AnnexE of EC3, the distribution

factor η1at node A is 1 due to the hinged support, while the

distribution factor η2at node B has a contribution from beam

BB?assumed to deform with a double curvature and column

BC, and is found to be equal to η2= 0.833. Then, for the sway

buckling condition, the effective buckling length coefficient is

found to be?

L= 3.996.

Thus, the Euler buckling load is calculated as:

BC+ cbm= 249.45 kN m.

(57)

Pcr,EC3=

π2EIc

???

L

?h?2= 560.03 kN.

(58)

In the same manner, following the provisions of LRFD,

the distribution factor GAat node A is 10 due to the pinned

support, and the distribution factor GB at node B is equal to

7.469. Assuming uninhibited side-sway behavior, the effective

buckling coefficient is calculated to be K = 1.820.

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903

Table 4

Critical loads according to different methodologies of column AB of example 2

Pcr(kN)

Pcr−Pcr,FEM

Pcr,FEM

(%)

FEM

EC3

LRFD

Proposed

22.02428

560.60

2702.46

21.9399

0

2445.37

12170.40

−0.38

Fig. 8. The three-story non-sway frame of example 3.

The Euler buckling load is equal to:

Pcr,LRFD=π2EIc

(Kh)2= 2702.46 kN.

(59)

Applying the proposed method, the distribution factor zbis

equal to 1 due to the pinned support, while ztis obtained from:

cc

cc+ cBC

Then, the evaluation of the buckling length coefficient is

conducted by means of Eq. (14) and gives K = 20.940. Thus,

the Euler buckling load is equal to:

zt== 0.994.

(60)

Pcr,prop=π2EIc

(Kh)2= 21.94 kN.

(61)

The above results are summarized in Table 4, and compared

to the results of the linearized buckling analysis, which are

considered to be “exact”. The proposed method is in very good

agreement with the numerical results.

4.3. Example 3

Next, consider the same three-story frame of example 2, but

with inhibited side-sway at all stories, shown in Fig. 8.

The same procedure is followed for the verification of

the proposed approach. The critical buckling load of column

AB is evaluated according to the code provisions as above.

Table 5

Critical loads according to different methodologies of column AB of example 3

Pcr(kN)

Pcr−Pcr,FEM

Pcr,FEM

(%)

FEM

EC3

LRFD

Proposed

11237.75

9358.89

11745.60

11274.80

0

−16.72

4.52

0.33

Fig. 9. The frames of example 4 with (a) partially-sway, (b) non-sway and (c)

sway behaviour.

The proposed method is in very good agreement with the

numerical results, while EC3 is overconservative and LRFD is

underconservativebut with much smaller deviations than in the

sway-case (Table 5).

4.4. Example 4

Lastly, consider the one-story, partially-sway, non-sway and

sway frames of Fig. 9 (a), (b) and (c), respectively. The frames

havea singlespan L = 20 m anda storyheighth equalto 10m,

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Table 6

Critical loads according to different methodologies of column AB of the partially-sway frame of example 4

Pcr(kN)

Pcr−Pcr,FEM

Pcr,FEM

(%)

FEM

EC3 (lower limit assuming sway behaviour)

EC3 (upper limit assuming non-sway behaviour)

LRFD (lower limit assuming sway behaviour)

LRFD (upper limit assuming non-sway behaviour)

Proposed

5000.636

898.78

9980.74

3441.23

11821.70

5000.01

0

−82.03

99.59

−31.18

136.40

−0.01

Table 7

Critical loads according to different methodologies of column AB of the non-

sway frame of example 4

Pcr(kN)

Pcr−Pcr,FEM

Pcr,FEM

(%)

FEM

EC3

LRFD

Proposed

8980.67

9980.74

11821.70

8980.67

0

11.14

31.64

0

Table 8

Critical loads according to different methodologies of column AB of the sway

frame of example 4

Pcr(kN)

Pcr−Pcr,FEM

Pcr,FEM

(%)

FEM

EC3

LRFD

Proposed

14.77

898.78

3441.23

14.77

0

5983.56

23192.60

0

havingcolumns with HEB360 cross-section (Ic= 43190 cm4)

andbeams with IPE400cross-section(Ibm= 23130 cm4). The

columnsare consideredto be pinnedat the base. A compressive

concentrated load P is imposed on the beam–column joints.

The beam–column joints are considered to be semi-rigid with

a rotational stiffness of cn = 150 kN m. A translation spring

with a stiffness of cbr = 1000 kN/m simulates the partially-

sway behaviour of the first frame. The frame is made of steel

with Young’s modulus E = 210000000 kN/m2.

The same procedure is followed for the verification of the

proposed approach. Moreover, the critical buckling load of

column AB is evaluated according to the code provisions,

considering the first frame firstly to be sway and secondly to

be non-sway. The proposed method is in excellent agreement

with the numerical results, while EC3 and LRFD with the sway

behaviourconsiderationgive overconservativeresults while the

non-sway behaviour consideration leads to underconservative

results (Table 6). The results of the proposed methodology and

the design codes are presented in Tables 6–8 for the frames of

Fig. 9 (a), (b) and (c), respectively.

5. Summary and conclusions

A simplified method for the evaluation of the critical

buckling load of multi-story sway, non-sway and partially-

sway frames with semi-rigid connections has been presented.

Firstly, three analytical expressions for the effective buckling

length coefficient as a function of the end nodes’ distribution

factors, as well as accompanying graphs, have been proposed

for different levels of sway ability. The rotational stiffness of

the members (columns and beams) converging at the bottom

and top ends of the column with semi-rigid connections

depend heavily on the boundary conditions at their far end

and on the existence of axial force in them. Thus, analytical

expressions of the stiffness distribution factors accounting

for these issues have been derived. Examples of sway, non-

sway and partially-sway structures with semi-rigid connections

and comparisons to finite element results have been used

to establish the improved accuracy of the above mentioned

procedure compared to the pertinent code provisions. It is

believed that the proposed approach maintains the inherent

simplicity of the effective length method by not significantly

increasingtherequiredworkload,butat thesametimeimproves

its accuracy a lot and could thus be considered to be a

competitive alternative for practical applications.

Acknowledgments

Financial support for this work is provided by the

“Pythagoras:SupportofResearchGroupsinUniversities”. The

project is co-funded by the European Social Fund (75%) and

National Resources (25%) (EPEAEK II)–PYTHAGORAS.

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