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. Gantes 1 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) procedures with analytical or numerical methods [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: [email protected](G.E. Mageirou), [email protected](C.J. Gantes). 1 Tel.: +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 c b and c t , 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 c br . 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 model has been used by several investigators (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 converging at the bottom and top ends, respectively: c b = i c b,i , c t = j c t , j . (1) A frame is characterized as non-sway if the stiffness c br of the bracing system is very large, as sway if this stiffness is negligible, and as partially-sway for intermediate values of this 0143-974X/$ - see front matter c 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2005.11.019
Transcript
Journal of Constructional Steel Research 62 (2006) 893–905www.elsevier.com/locate/jcsr
es
and top,-sway andt with the
Buckling strength of multi-story sway, non-sway and partially-sway framwith 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
Nowadays, the buckling strength of a member canevaluated using engineering software based on linear or alnon-linear (in terms of large displacements and/or mateyielding) procedures with analytical or numerical methods [15].Nonetheless, the large majority of structural engineersprefer analytical techniques such as the effective length annotional load methods [26]. These two methodologies arincluded in most modern structural design codes (for examEurocode 3 [9], LRFD [23]).
The objective of this work is to propose a simplifieapproach for the evaluation of critical buckling loads of mulstory frames with semi-rigid connections, for different levelsof framesway ability. To that effect, a model of a column inmulti-story frame is considered as individual. The contributiof members converging at the bottom and top ends of
column is taken into account by equivalent springs. Namthe restriction provided by the other members of the framto the rotations of the bottom and top nodes is modevia rotational springs with constantscb and ct , respectively,while the resistance provided by the bracing system torelative transverse translation of the end nodes is modvia a translational spring with constantcbr . This is shownschematically inFig. 1. The rotational stiffness of thespringsmust be evaluated considering the influence of the connenon-linearity. This model has been used by several investig(for example, Wood [27], Aristizabal-Ochoa [1], and Cheong-Siat-Moy [6]) for the evaluation of the critical buckling load othe member, and is adopted by most codes.
The stiffness of the bottom and top rotational sprinis estimated by summing up the contributions of memberconverging at the bottom and top ends, respectively:
cb =∑
i
cb,i , ct =∑
j
ct, j . (1)
A frame is characterized as non-sway if the stiffnesscbr ofthe bracing system is very large, as sway if this stiffnesnegligible, and as partially-sway for intermediate values of t
894 G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905
,
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encearyrall
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eads
Notations
A, B, C, D integration constantsE modulus of elasticityG distribution factor at the end nodes of the column
according to LRFDI moment ofinertiaK effective buckling length coefficientL span length of adjoining membersM bending momentN axial force of adjoining membersP compressive loadΦ factor of slope deflection methoda factor of slope deflection method, factor for the
effect of the boundary condition at the far endnodes of the member
c stiffness coefficientc ratio of flexural stiffness to spanc# dimensionless rotational stiffnessd factor for the effect of the axial forceh column heightk non-dimensional compressive load� effective buckling length coefficient, according to
EC3n ratio of member’s compressive force to Euler’s
buckling loadx longitudinal coordinatez dimensionless distribution factor at the end node
of the columnw transverse deflectionδ relative transverse deflection between the en
nodes of the memberη distribution factor at the end nodes of the column
according to EC3θ rotation at the end nodes of the member
Subscripts:
A bottom end node of the columnB top end node of the columnE EulerEC3 Eurocode 3FEM Finite Element MethodLRFD Load Resistance Factor Designc columncr criticalb bottombm beambr bracing systemi memberin noder rigid connectiont top
stiffness. Eurocode3 and LRFDprovide the effective lengthK h of columns in sway and non-sway frames via graphs
r
Fig. 1. (a) Multi-story steel frame; (b) proposed model of column uninvestigation.
analytical relations as functions of the rotational boundconditions without considering the connection non-lineaand the partially-sway behaviour of the frame. The critbuckling load is then defined as:
Pcr = π2E Ic
(K h)2(2)
whereE Ic is the flexural resistance.The main source of inaccuracyin the above process lie
in the estimation of the rotational boundary conditions. LRmakes no mention to the dependence of the rotational stiffof members converging at the ends of the column unconsideration on their boundary conditions at their far entheir axial load. Annex E of EC3 is more detailed in accounfor the contribution of converging beams and lower/upcolumns, but ignores several cases that are encounterpractice, and are often decisive for the buckling strength. Bcodes ignore the partially-sway behaviour of the frames asas the connection non-linearity.
This problem has been investigated by several researcThe work of Wood [27] constituted the theoretical basisEC3. Cheong-Siat-Moy [5] examined the k-factor paradoxfor leaning columns and drew attention to the dependof buckling strength not only on the rotational boundconditions of the member in question but also on the ovestructural system behavior. Bridge and Fraser [4] proposedan iterative procedure for the evaluation of the effeclength, which accounts for the presence of axial forcethe restraining members and thus also considers the negatvalues of rotational stiffness. Hellesland and Bjorhovde [11]proposed a new restraint demand factor considering the veand horizontal interaction in member stability terms. Kiet al. [14] proposed an analytical relation for the evaluationthe effective length of columns with semi-rigid joints in swaframes.Essa [8] proposed a design method for the evaluatof the effective length for columns in unbraced multi-stframes considering different story drift angles. AristizabOchoa examined the influence of uniformly distributed axiaload on the evaluation of the effective length of columns insway and partially-sway frames [2]. He then examined thbehavior of columns with semi-rigid connections under lo
G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905 895
ed.
Fig. 2. Model of column in (a) non-sway frame, (b) sway frame, and(c) partially-sway frame, and (d) the sign convention us
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ulti-s
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eeads
wn
ial
such as those produced by tension cables that alwaysthrough fixed points or loads applied by rigid links [3]. Whatis more, Kounadis [16] investigated the inelastic buckling origid-jointedframes.
Christopher and Bjorhovde [7] conducted analyses ofseries of semi-rigid frames, each with the same dimensioapplied loads and member sizes, but with different connecproperties, explaining how connection properties affmember forces, frame stability, and inter-story drift. Jaspand Maquoi [12] described the mode of application of thelastic and plastic design philosophies to braced frawith semi-rigid connections. The buckling collapse of stereticulated domes with semi-rigid joints was investigatedKato et al. [13] on thebasis of a nonlinear elastic–plastic hinanalysis formulated for three-dimensional beam–columns welastic, perfectly plastic hinges located at both endsmid-span for each member. Lau et al. [17] performed ananalytical study to investigate the behavior of subassemblwith a range of semi-rigid connections under different tconditions and loading arrangements. They showedsignificant variations in theM–ϕ response had a negligibeffect on the load carryingcapacity of the column and thbehavior of the subassemblage. A method for column dein non-sway bare steel structures which takes into accothe semi-rigid action of thebeam to column connectionwas proposed by Lau et al. [18]. In [19], closed-formsolutions of the second order differential equation of nouniform bars with rotational and translational springs wderived for eleven important cases. A simplified methfor estimating the maximum load of semi-rigid frames wproposed by Li and Mativo [20]. The method was in theform of a multiple linear regression relationship between tmaximum load and various parameters (frame and secproperties), obtained from numerous analyses of frames. Let al. [21] proposed a comprehensive set of moment-rotadata, in terms of stiffness and moment capacity, so thacomparative assessment of the frame performance dudifferent connection types could be undertaken. Reyes-Saand Haldar [24], using a nonlinear time domain seismicanalysis algorithm developed by themselves, excited threeframes with semi-rigid connections by thirteen earthquake thistories. They proposed a parameter called theT ratio inorder to define the rigidity of the connections. This parameis the ratio of the moment the connection would havecarry according to the beam line theory and the fixed
ss
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es
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moment of the girder. In [25], the equilibrium path was tracefor braced and unbraced steel plane frames with semi-connections with the aid of a hybrid algorithm that combinthe convergenceproperties of the iterative-incremental tangemethod, calculating the unbalancing forces by consideringelement rigid body motion. Yu et al. [28] described the details oa test program of three test specimens loaded to collapse antest observations for sway frames under the combined actof gravity and lateral loads.
However, all these studies mention nothing aboutdependence of the rotational stiffness of the membconverging on the column under consideration, fromboundary conditions at their far ends and from their axial loaThis dependence is investigated in the present work for mstory frames with semi-rigid connections for different levelof lateral stiffnesscbr . Easy to use analytical relations andcorresponding graphs are proposed for the estimation ocolumns’ effective length for sway, non-sway and partiaswayframe behaviour. Furthermore, analytical expressionsderived for the evaluation of the rotational springs’ stiffnescoefficients for different member boundary conditions anaxial loads accounting for the connection non-linearity. Resobtained via the proposed approach for sway, non-swaypartially-sway frames with semi-rigid connections are foundbe in excellent agreement with finite element results, while thapplication of design codes such as Eurocode 3 and LRFD lto significant inaccuracies.
2. Buckling strength of columns in multi-story frames
2.1. Non-sway frames
Consider the model of a column in a non-sway frame, shoin Fig. 2(a), resulting from the model ofFig. 1(b) by replacingthe translational spring with a roller support. Denoting byw
the transverse displacement and by′ the differentiation withrespect to the longitudinal coordinatex, and using the signconvention ofFig. 2(d), the equilibrium of this column in itsbuckled condition is described by the well-known differentequation:
w′′′′ (x) + k2w′′ (x) = 0 (3)
where:
k =√
Pcr
E Ic= π
K h. (4)
896 G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905
Fig. 3. Effective buckling length factorK for different levels of frame-sway ability.
ion
n-
th
of
in
s
The general solution of this differential equation is given by:
w (x) = A sin(kx) + B cos(kx) + Cx + D. (5)
The integration constantsA, B, C, and D can be obtained byapplying the boundary conditions at the two column ends:
Transverse displacement at the bottom:
w(0) = 0. (6)
Moment equilibrium at the bottom:
−E Icw′′ (0) = −cbw
′ (0) . (7)
Moment equilibrium at the top:
−E Icw′′ (h) = ctw
′ (h) . (8)
Transverse displacement at the top:
w (h) = 0. (9)
The four simultaneous equations(6)–(9) have a non-trivialsolution for the four unknownsA, B, C, and D if thedeterminant of the coefficients is equal to zero. This criteryields the buckling equation for the effective length factorK :
wherezb and zt are distribution factors obtained by the nodimensionalization of the end rotational stiffnessescb and ct
with respect to the column’s flexural stiffnesscc:
zb = cc
cc + cb, zt = cc
cc + ct(11)
where:
cc = 4E Ich
. (12)
Eq. (10) can be solved numerically for the effective lengfactorK , which is then substituted into Eq.(2) to provide thecritical buckling load. Alternatively, the upper left graphFig. 3, obtained from Eq.(10), can be used.
2.2. Sway frames
The simplified model of a column in a sway frame, shownFig. 2(b), is considered, resulting from the model ofFig. 1(b)by omitting the translational spring. Three boundary condition
G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905 897
r
in
um
y
thing
a
sive
s-darylose
t fore for
mi-
e
l
s
are described by Eqs.(6)–(8), while a fourth conditionexpresses horizontal force equilibrium at the top:
−E Icw′′′ (h) − Pw′ (h) = 0. (13)
Thus the buckling equation for the effective length factoKis derived, following the same procedure as above:
4 [zt (2zb − 1) − zb]π
Kcos
( π
K
)+
[zt zb
( π
K
)2 − 16(zt − 1) (zb − 1)
]sin
( π
K
)= 0. (14)
Alternatively, the bottom right graph ofFig. 3, obtained fromEq.(14), can be used.
2.3. Partially-sway frames
Finally, consider the simplified model of a columna partially-sway frame, shown inFig. 2(c). Similarly, theboundary conditions are described by Eqs.(6)–(8) and thefollowing equation, representing horizontal force equilibriat the top:
−E ICw′′′ (h) − Pw′ (h) = −cbrw (h) . (15)
The effective length factorK of a column in a partially-swaframe is then obtained from the following equation:
Easy to use graphs such as those presented inFig. 3 areobtained from the above equation for several values ofcbr .
3. Stiffness distribution factors
The rotational stiffness of each member converging attop or bottom node of the column in question is derived usthe slope-deflection method [22]. The momentsMAB andMB A
at the two ends of a memberAB with span L and flexuralstiffnessE I , without axial force or transverse load (Fig. 4), canbe obtained as a function of the end rotationsθA andθB and therelative transverse deflection of the end nodesδ from:
MAB = 2E I
L
(2θA + θB + 3δ
L
),
MB A = 2E I
L
(2θB + θA + 3δ
L
). (18)
e
Fig. 4. Undeformed (dotted line) and deformed (continuous line) state ofmemberAB, and the sign convention of the slope-deflection method.
If, in addition, the member is subjected to a compresaxial forceP, then Eq.(18)becomes:
MAB = 2E I
L
[anθA + af θB + (
an + af) δ
L
]
MB A = 2E I
L
[anθB + af θA + (
an + af) δ
L
](19)
where:
an = Φn
2(Φ2
n − Φ2f
) , af = Φ f
2(Φ2
n − Φ2f
) (20)
Φn = 1 − kL cotkL
k2L2, Φ f = 1
k2L2
(kL
sinkL− 1
). (21)
Using the above equations, the rotational stiffness expresions have been derived for members with all possible bounconditions at the far end and a semi-rigid connection at the cend, with or without axial force, and are shown inTable 1. Thederivation of the rotational stiffness factors is described nextwo characteristic cases: one for a member without and ona member with axial force.
3.1. Member with a fixed support at the far end and a serigid connection at the near end, without axial force
Consider the memberAB of Fig. 5(a), with spanLi andflexural stiffnessEi Ii , where A refers to thebottom or topnode of the column under investigation, whileB is the farnode, attached to a fixed support. The connection at nodAis considered as semi-rigid with a rotational stiffnesscn.
The slope-deflection equations are given by(18), withindices i referring to the specificmember. Firstly, theconnection at nodeA is considered as rigid. The rotationastiffnesscr,i of the memberAB with a rigid connection wasevaluated in previous work by the authors [10].
The moment at nodeA of the member with rigid connectionis given by the equation:
MAB = 2Ei Ii
Li(2θA + θB) . (22)
Furthermore, there is no rotation at nodeB:
θB = 0. (23)
898 G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905
Table 1Converging members’ rotational stiffness expressions for different boundary conditions at the far end and for a semi-rigid connection at the near end, with or withoutaxial force
Rotational boundary conditions of far end Without axial force With axial force
Fixed support ci = 4c̄i cn4c̄i +cn
ci = 4c̄i cn(1−0.33ni )cn+4c̄i −1.32c̄i ni
Roller fixed support ci = c̄i cnc̄i +cn
ci = c̄i cn(1−0.82ni )cn+c̄i −0.82c̄i ni
Pinned support ci = 3c̄i cn3c̄i +cn
ci = 3c̄i cn(1−0.66ni )cn+3c̄i −1.98c̄i ni
Simple curvature ci = 2c̄i cn2c̄i +cn
ci = 2c̄i cn(1−0.82ni )cn+2c̄i −1.64c̄i ni
Double curvature ci = 6c̄i cn6c̄i +cn
ci = 6c̄i cn(1−0.16ni )cn+6c̄i −0.96c̄i ni
Roller support ci = 0c̄i cn0c̄i +cn
ci = c̄i cn(0−0.97ni )cn+0c̄i −0.97c̄i ni
Rotational spring support ci = c̄i cnc#
(c̄i +cn)c#+cnci = c̄i cn[c#−(1.047ni +1.773)ni ]
cn(0.591ni c#+c#+1)+c̄i [c#−(1.047ni +1.773)ni ]
Pinned and rotational spring support ci = 4c̄i cn(c#+3)
G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905 899
.
e
on
gs
mi-
r
he
t athe
ce
to
Fig. 5. Model of a member with a fixed support at the far endB and a semi-rigid connection at the near endA: (a) without axial force, (b) with axial force
Then, by virtue of(23), Eq.(22)becomes:
MAB = 4Ei Ii
LiθA. (24)
Thus, the rotational stiffness representing the resistancmemberAB to the rotation of nodeA is given by:
cr,i = 4Ei Ii
Li(25)
which can be written as
cr,i = 4c̄i (26)
where:
ci = Ei Ii
Li. (27)
Two rotational springs with rotational stiffnesscr,i and cn
in series are considered, in order to evaluate the rotatistiffnessci of the memberAB with a semi-rigid connection.The total rotation is the sum of the rotations of the two sprinTherefore:
ϕi = ϕr,i + ϕn. (28)
Considering that the springs have the same moment, Eq.(29)is written:
1
ci= 1
cr,i+ 1
cn. (29)
Thus, the rotational stiffnessci of memberAB with a semi-rigid connection is evaluated from Eq.(29):
ci = cr,i · cn
cr,i + cn. (30)
By substituting(26) into (30), the rotational stiffnessci ofthe memberAB with a semi-rigid connection is evaluated:
ci = 4c̄i cn
4c̄i + cn. (31)
of
al
.
3.2. Member with a fixed support at the far end and a serigid connection at the close end, with axial force
Now consider the member ofFig. 5(b), with spanLi andflexural stiffnessEi Ii , whereA again refers to the bottom otop node of the column under investigation, whileB is thefar node, rotationally fixed. The memberAB is subjected toa compressive axial forceNi . The rotational stiffnessci of thememberAB with a semi-rigid connection is evaluated in tsamemanner as above.
Firstly, the member AB is considered having rigidconnections. The slope-deflection equations are given by(19),with indicesi referring to the specific member. The momennodeA of the member with rigid connections is given by tequation:
MAB = 2Ei Ii
Li
(αn,i θA + α f,i θB
). (32)
As there is no rotation at nodeB (θB = 0), the previousequation is rewritten:
MAB = 2Ei Ii
Liαn,i θA. (33)
Therefore, the rotational stiffness representing the resistanof memberAB to the rotation of nodeA is given by:
cr,i = 2Ei Ii
Liαn,i (34)
which, by means of(20) and(21) andk2i = N/Ei Ii , NE,i =
π2Ei Ii /L2i , becomes:
cr,i = 4Ei Ii
Liπ
√ni
π
√ni cot
(π
√ni
) − 1
4π√
ni − 8 tan(
12π
√ni
) (35)
where ni is the ratio of the member’s compressive forceEuler’s buckling loadNE,i :
ni = Ni
NE,i. (36)
A Taylor series expansion of Eq.(35)gives:
cr,i = 4Ei Ii
Li
(1 − π2
30ni − 11π4
25 200n2
i
− π6
108 000n3
i − 509π8
2328 480 000n4
i . . .
). (37)
Keeping the first two terms, Eq.(38) is obtained:
cr,i = 4Ei Ii
Li
(1 − π2
30ni
)(38)
which can be written as
cr,i = 4c̄i (1 − 0.33ni ) . (39)
By substituting(39) into (30), the rotational stiffnessci of thememberAB with a semi-rigid connection is evaluated:
ci = 4c̄i cn(1 − 0.33ni )
cn + 4c̄i − 1.32c̄i ni. (40)
900 G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905
hatse;glthepasis
th
s-s-
o b
rin
lus
isSC
gy.deth
t is
e inentamell asr the
ed in
qualnide
ingealedd byd as
ings.mined
isl
ss
Fig. 6. The frame of example 1.
4. Examples
This section gives four examples of a simple frame tconsists of a column and a beam with a variety of supporthe far end — namely, a three-story, single-bay sway framthree-story, single-bay non-sway frame; and a one-story, sinbay sway, non-sway and partially-sway frame — for whichproposed approach is demonstrated and its results are comto (i) buckling loads obtained via linearized buckling analyof finite element models, considered as “exact”, for verificationpurposes, and (ii) buckling loads calculated by applyingpertinent procedures of Eurocode 3 and LRFD.
4.1. Example 1
Consider the frame ofFig. 6 with a single spanL = 20 mand heighth = 10 m, having a column with HEB360 crossection(Ic = 43 190 cm4) and a beam with IPE400 crossection(Ibm = 23 130 cm4). The columnis considered to bepinned at the base. A concentrated loadP is imposed on thebeam–column joint. The beam–column joint is considered tsemi-rigid, with a rotational stiffness ofcn = 150 kN m/rad.The beam has restricted translation and a rotational spsupport with a rotational stiffness ofc = 500 kN m at thefar end. The frame is made of steel with Young’s moduE = 210 000 000 kN/m2.
Firstly, a linearized buckling analysis of the frameconducted using the commercial finite element software MNASTRAN. The critical buckling load obtained from thisanalysis isPcr,FEM = 8981.02 kN. Secondly, the bucklinstrength is evaluated by using the proposed methodologorder to do so, the frame is substituted by the equivalent moof Fig. 2(b). The rotational stiffness of the beam consideringsemi-rigid connection is evaluated from the last row ofTable 1:
cB B′ = 4c̄B B′cn(c#B B′ + 3)
4c̄B B′(c#B B′ + 3) + cn(c#
B B′ + 4)= 147.02 kN m (41)
where:
c̄B B′ = E Ibm
L= 2428.65 kN m (42)
c#B B′ = c
c̄B B′= 0.206. (43)
tatae-
red
e
e
g
-
Inle
Applying the proposed method, the distribution factorzb isequal to 1 due to the pinned support, whilezt is obtained from:
zt = cc
cc + cB B′= 0.996 (44)
where:
cc = 4E Ich
= 36 279.60 kN m. (45)
Then, the evaluation of the buckling length coefficienconducted by means of Eq.(10), giving K = 0.998. Thus, theEuler buckling load is equal to:
Pcr,prop = π2E Ic
(K h)2= 8981.01 kN. (46)
Therefore, the results of the proposed approach arexcellent agreement with the results of the finite elemmethod. The previous procedure is followed for the same frwith different supports at the far end of the beam as wefor a single span, one-story sway and non-sway frame foverification of the rotational stiffness ofTable 1. The results ofthe proposed approach and the FEM analysis are presentTable 2and are practically the same.
4.2. Example 2
Consider the three-story sway frame ofFig. 7 with a singlespan L = 20 m and uniform story heighth = 10 m,having columns with HEB360 cross-section(Ic = 43 190 cm4)
and beams with IPE400 cross-section(Ibm = 23 130 cm4).The columns are considered to be pinned at the base. Econcentrated loadsP/3 are imposed on all beam–columjoints. The beam–column joints are considered to be semi-rigwith a rotational stiffness ofcn = 150 kN m. The frame is madof steel with Young’s modulusE = 210 000 000 kN/m2.
At first, abuckling analysis of the frame is conducted usthe same finite element software. The first buckling modis obtained from this analysis, and the corresponding criticbuckling load is 22.02 kN. In order to verify the proposrotational stiffness coefficients, the frame is then substitutea series of equivalent models. The first among them, denoteequivalent model 1a, is illustrated inFig. 7(b). It is obtained bysubstituting the beams at the three levels by rotational sprAssuming that, in the first buckling mode, the beams deforwith a double curvature, the stiffness of the springs is obtafrom the corresponding row ofTable 1:
cbm = 6c̄bmcn
6c̄bm + cn= 148.47 kN (47)
where:
c̄bm = E Ibm
L= 2428.65 kN m. (48)
The first buckling mode of the equivalent model 1aobtained from FEM analysis, and the corresponding criticabuckling load is also 22.02 kN, thus verifying the correctneof this substitution.
G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905 901
Table 2Critical 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
8979.83 8979.86 0.001
9027.06 9027.30 0.003
10.98 10.97 −0.091
8981.02 8981.01 0.0001
902 G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905
1c.
Fig. 7. (a) The three-story sway frame of example 2, (b) equivalent model 1a, (c) equivalent model 1b, (d) equivalent model
d
d
e
d i
2
islingasre
ne
mn
D,
tive
The second equivalent model, denoted as 1b, is illustrateFig. 7(c). It is obtained by substituting the top columnCD by arotational spring with stiffnessc′
C D calculated from the seconlast row ofTable 1:
c′CD
=c̄CDcn,CD
[c#CD − (1.047nCD + 1.773)nCD
]cn,CD
(0.591nCDc#
CD + c#CD + 1
)+ c̄CD
[c#CD − (1.047nCD + 1.773)nCD
]= 70.18 kN m (49)
where:
c̄C D = E Ich
= 9069.90 kN m (50)
c#C D = cbm
c̄C D= 0.016 (51)
NC D = 1
322.02 kN = 7.34 kN. (52)
Alternatively, the axial design load can be used for theNC D,without any significant influence on the results:
NE,C D = π2E Ich2 = 8942.56 kN (53)
nC D = NC D
NE,C D= 0.001. (54)
Then, the total rotational stiffness at nodeC of model 1b is:
cC D = c′C D + cbm = 218.65 kN m. (55)
The critical buckling load of the first buckling mode of thequivalent model 1b, obtained from Nastran, is 22.00 kN.
The third equivalent model, denoted as 1c, is illustrateFig. 7(d). It is obtained by substituting columnBC of model1b by a rotational spring with stiffnessc′
BC calculated similarlyfrom the second last row ofTable 1:
c′BC = 100.98 kN m. (56)
in
n
Table 3Critical loads for model 1 and its equivalent models 1a, 1b, 1c of Example
Then, the total rotational stiffness at nodeB of model 1c is:
cBC = c′BC + cbm = 249.45 kN m. (57)
The first buckling mode of the equivalent model 1cobtained from Nastran, and the corresponding critical buckload is 21.96 kN. The critical loads of all models, as welltheir deviations from the critical load of the full model, asummarized inTable 3, demonstrating excellent agreement.
In addition, the critical buckling load of column AB isevaluated according to the provisions of EC3 and LRFD.
Following the procedure of Annex E of EC3, the distributiofactorη1 at nodeA is 1 due to the hinged support, while thdistribution factorη2 at nodeB has a contribution from beamB B′ assumed to deform with a double curvature and coluBC, and is found to be equal toη2 = 0.833. Then, for the swaybuckling condition, the effective buckling length coefficient isfound to be�
L = 3.996.Thus, the Euler buckling load is calculated as:
Pcr,EC3 = π2E Ic[(�L
)h]2
= 560.03 kN. (58)
In the same manner, following the provisions of LRFthe distribution factorGA at nodeA is 10 due to the pinnedsupport, and the distribution factorGB at nodeB is equal to7.469. Assuming uninhibited side-sway behavior, the effecbuckling coefficient is calculated to beK = 1.820.
G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905 903
is
areoo
bu
ofmov
(c)
theisthe
andes
Table 4Critical loads according to different methodologies of columnAB of example 2
Fig. 8. The three-story non-sway frame of example 3.
The Euler buckling load is equal to:
Pcr,LRFD = π2E Ic
(K h)2= 2702.46 kN. (59)
Applying the proposed method, the distribution factorzb isequal to 1 due to the pinned support, whilezt is obtained from:
zt = cc
cc + cBC= 0.994. (60)
Then, the evaluation of the buckling length coefficientconducted by means of Eq.(14) and givesK = 20.940. Thus,the Euler buckling load is equal to:
Pcr,prop = π2E Ic
(K h)2= 21.94 kN. (61)
The above results are summarized inTable 4, and comparedto the results of the linearized buckling analysis, whichconsidered to be “exact”. The proposed method is in very gagreement with thenumerical results.
4.3. Example 3
Next, consider the same three-story frame of example 2,with inhibited side-sway at all stories, shown inFig. 8.
The same procedure is followed for the verificationthe proposed approach. The critical buckling load of coluAB is evaluated according to the code provisions as ab
d
t
ne.
Table 5Critical loads according to different methodologies of columnAB of example 3
Fig. 9. The frames of example 4 with (a) partially-sway, (b) non-sway andsway behaviour.
The proposed method is in very good agreement withnumerical results, while EC3 is overconservative and LRFDunderconservative but with much smaller deviations than insway-case (Table 5).
4.4. Example 4
Lastly, consider the one-story, partially-sway, non-swayswayframes ofFig. 9 (a), (b) and (c), respectively. The framhave a single spanL = 20 m and a story heighth equal to 10 m,
904 G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905
Table 6Critical loads according to different methodologies of columnAB of the partially-sway frame of example 4
having columns with HEB360 cross-section(Ic = 43 190 cm4)
and beams with IPE400 cross-section(Ibm = 23 130 cm4). Thecolumns are considered to be pinned at the base. A compreconcentrated loadP is imposed on the beam–column joints.The beam–column joints are considered to be semi-rigid wa rotational stiffness ofcn = 150 kN m. A translation springwith a stiffness ofcbr = 1000 kN/m simulates the partially-sway behaviour of the first frame. The frame is made of stewith Young’s modulusE = 210 000 000 kN/m2.
The same procedure is followed for the verification of tproposed approach. Moreover, the critical buckling loadcolumn AB is evaluated according to the code provisioconsidering the first frame firstly to be sway and secondlybe non-sway. The proposed method is in excellent agreemwith the numerical results, while EC3 and LRFD with the swbehaviour consideration give overconservative results whilenon-sway behaviour consideration leads to underconservresults (Table 6). The results of the proposed methodology athe design codes are presented inTables6–8 for the frames ofFig. 9(a), (b) and (c), respectively.
5. Summary and conclusions
A simplified method for the evaluation of the criticabuckling load of multi-story sway, non-sway and partially-sway frames with semi-rigid connections has been presenteFirstly, three analytical expressions for the effective buckling
ive
h
l
f,ont
eve
.
length coefficient as a function of the end nodes’ distributfactors, as well as accompanying graphs, have been propfor different levels of sway ability. The rotational stiffnessthe members(columns and beams) converging at the bottand top ends of the column with semi-rigid connectiodepend heavily on the boundary conditions at their far eand on the existence of axial force in them. Thus, analytexpressions of the stiffness distribution factors accountingfor these issues have been derived. Examples of sway,sway and partially-sway structures with semi-rigid connectioand comparisons to finite element results have been useto establish the improved accuracy of the above mentioprocedure compared to the pertinent code provisions. Ibelieved that the proposed approach maintains the inhesimplicity of the effective length method by not significantlyincreasing the required workload, but at the same time improveits accuracy a lot and could thus be considered to bcompetitive alternative for practical applications.
Acknowledgments
Financial support for this work is provided by th“Pythagoras: Support of Research Groups in Universities”. Theproject is co-funded by the European Social Fund (75%)National Resources (25%) (EPEAEK II)–PYTHAGORAS.
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