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Buckling strength of multi-story with semi-rigid connections.

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Journal of Constructional Steel Research 62 (2006) 893–905www.elsevier.com/locate/jcsr
Buckling strength of multi-story sway, non-sway and partially-sway frameswith 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 underinvestigation are proposed for the calculation of the effective buckling length coefﬁcient for different levels of frame sway ability. In addition, acomplete set of rotational stiffness coefﬁcients is derived, which is then used for the replacement of members converging at the bottom and topends 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 andpartially-sway frames with semi-rigid connections are presented, where the proposed approach is found to be in excellent agreement with theﬁnite 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 beevaluated using engineering software based on linear or alsonon-linear (in terms of large displacements and/or materialyielding)procedureswithanalyticalornumericalmethods[15].
Nonetheless, the large majority of structural engineers stillprefer analytical techniques such as the effective length andnotional 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 simpliﬁedapproach for the evaluation of critical buckling loads of multi-story frames with semi-rigid connections, for different levelsof frame sway ability. To that effect, a model of a column in amulti-story frame is considered as individual. The contributionof 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 frameto the rotations of the bottom and top nodes is modeledvia rotational springs with constants
c
b
and
c
t
, respectively,while the resistance provided by the bracing system to therelative transverse translation of the end nodes is modeledvia a translational spring with constant
c
br
. This is shownschematically in Fig. 1. The rotational stiffness of the springsmust be evaluated considering the inﬂuence of the connectionnon-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 springsis estimated by summing up the contributions of membersconvergingat 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 isnegligible, 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
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 theeffect of the boundary condition at the far endnodes 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,accordingtoEC3
n
ratio of member’s compressive force to Euler’sbuckling load
x
longitudinal coordinate
z
dimensionless distribution factor at the end nodesof the column
w
transverse deﬂection
δ
relative transverse deﬂection between the endnodes 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
bottom end node of the column
B
top end node of the column
E
EulerEC3 Eurocode 3FEM Finite Element MethodLRFD Load Resistance Factor Design
c
column
cr
critical
b
bottom
bm
beam
br
bracing system
i
member
in
node
r
rigid connection
t
topstiffness. 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 underinvestigation.
analytical relations as functions of the rotational boundaryconditions without considering the connection non-linearityand the partially-sway behaviour of the frame. The criticalbuckling load is then deﬁned as:
P
cr
=
π
2
EI
c
(
Kh
)
2
(2)where
EI
c
is the ﬂexural resistance.The main source of inaccuracy in the above process liesin the estimation of the rotational boundary conditions. LRFDmakes no mention to the dependence of the rotational stiffnessof members converging at the ends of the column underconsideration on their boundary conditions at their far end ortheir axial load. Annex E of EC3 is more detailed in accountingfor the contribution of converging beams and lower/uppercolumns, but ignores several cases that are encountered inpractice, and are often decisive for the buckling strength. Bothcodes ignore the partially-sway behaviour of the frames as wellas 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 paradoxfor leaning columns and drew attention to the dependenceof buckling strength not only on the rotational boundaryconditions of the member in question but also on the overallstructural system behavior. Bridge and Fraser [4] proposedan iterative procedure for the evaluation of the effectivelength, which accounts for the presence of axial forces inthe restraining members and thus also considers the negativevalues of rotational stiffness. Hellesland and Bjorhovde [11]proposeda new restraint demand factor consideringthe verticaland horizontal interaction in member stability terms. Kishiet al. [14] proposed an analytical relation for the evaluation of the effective length of columns with semi-rigid joints in swayframes. Essa [8] proposed a design method for the evaluationof the effective length for columns in unbraced multi-storyframes considering different story drift angles. Aristizabal-Ochoa examined the inﬂuence of uniformly distributed axialload on the evaluation of the effective length of columns insway and partially-sway frames [2]. He then examined the
behavior of columns with semi-rigid connections under loads
G.E. Mageirou, C.J. Gantes / Journal of Constructional Steel Research 62 (2006) 893–905
895Fig. 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 passthrough 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 aseries of semi-rigid frames, each with the same dimensions,applied loads and member sizes, but with different connectionproperties, explaining how connection properties affectmember forces, frame stability, and inter-story drift. Jaspartand Maquoi [12] described the mode of application of theelastic and plastic design philosophies to braced frameswith semi-rigid connections. The buckling collapse of steelreticulated domes with semi-rigid joints was investigated byKato et al. [13] on the basis of a nonlinear elastic–plastic hingeanalysis formulated for three-dimensional beam–columns withelastic, perfectly plastic hinges located at both ends andmid-span for each member. Lau et al. [17] performed ananalytical study to investigate the behavior of subassemblageswith a range of semi-rigid connections under different testconditions and loading arrangements. They showed thatsigniﬁcant variations in the
M
–
ϕ
response had a negligibleeffect on the load carrying capacity of the column and thebehavior of the subassemblage. A method for column designin non-sway bare steel structures which takes into accountthe semi-rigid action of the beam to column connectionswas 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 werederived for eleven important cases. A simpliﬁed methodfor estimating the maximum load of semi-rigid frames wasproposed by Li and Mativo [20]. The method was in theform of a multiple linear regression relationship between themaximum load and various parameters (frame and sectionproperties), obtained from numerous analyses of frames. Liewet al. [21] proposed a comprehensive set of moment-rotationdata, in terms of stiffness and moment capacity, so that acomparative assessment of the frame performance due todifferent connection types could be undertaken. Reyes-Salazarand Haldar [24], using a nonlinear time domain seismic
analysis algorithm developed by themselves, excited three steelframes with semi-rigid connections by thirteen earthquake timehistories. They proposed a parameter called the
T
ratio inorder to deﬁne the rigidity of the connections. This parameteris the ratio of the moment the connection would have tocarry according to the beam line theory and the ﬁxed endmoment of the girder. In [25], the equilibrium path was traced
for braced and unbraced steel plane frames with semi-rigidconnections with the aid of a hybrid algorithm that combinesthe convergence properties of the iterative-incremental tangentmethod, calculating the unbalancing forces by considering theelementrigidbodymotion.Yuetal.[28]describedthedetailsof atest programofthreetest specimensloadedtocollapseandthetest observations for sway frames under the combined actionsof gravity and lateral loads.However, all these studies mention nothing about thedependence of the rotational stiffness of the membersconverging on the column under consideration, from theboundaryconditions 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 levelsof lateral stiffness
c
br
. Easy to use analytical relations andcorresponding graphs are proposed for the estimation of thecolumns’ effective length for sway, non-sway and partially-sway frame behaviour. Furthermore, analytical expressions arederived for the evaluation of the rotational springs’ stiffnesscoefﬁcients for different member boundary conditions andaxial loads accounting for the connection non-linearity. Resultsobtained via the proposed approach for sway, non-sway andpartially-sway frames with semi-rigid connections are found tobe in excellent agreement with ﬁnite element results, while theapplicationofdesigncodessuchas Eurocode3andLRFDleadsto signiﬁcant inaccuracies.
2. Buckling strength of columns in multi-story frames
2.1. Non-sway frames
Considerthe modelof a columnin a non-swayframe,shownin 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 withrespect to the longitudinal coordinate
x
, and using the signconvention of Fig. 2(d), the equilibrium of this column in its
buckled condition is described by the well-known differentialequation:
w
(
x
)
+
k
2
w
(
x
)
=
0 (3)where:
k
=
P
cr
EI
c
=
π
Kh
.
(4)
896
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
)
=
A
sin
(
kx
)
+
B
cos
(
kx
)
+
Cx
+
D
.
(5)The integration constants
A
,
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:
−
EI
c
w
(
0
)
= −
c
b
w
(
0
).
(7)Moment equilibrium at the top:
−
EI
c
w
(
h
)
=
c
t
w
(
h
).
(8)Transverse displacement at the top:
w (
h
)
=
0
.
(9)The four simultaneous equations (6)–(9) have a non-trivialsolution for the four unknowns
A
,
B
,
C
, and
D
if thedeterminant of the coefﬁcients is equal to zero. This criterionyields the buckling equation for the effective length factor
K
:32
K
3
(
z
t
−
1
)(
z
b
−
1
)
−
4
K
8
K
2
(
z
t
−
1
)(
z
b
−
1
)
+
(
z
t
+
z
b
−
2
z
t
z
b
)π
2
cos
π
K
+
π
−
16
K
2
+
20
K
2
(
z
t
+
z
b
)
+
z
t
z
b
π
2
−
24
K
2
sin
π
K
=
0 (10)where
z
b
and
z
t
are distribution factors obtained by the non-dimensionalization of the end rotational stiffnesses
c
b
and
c
t
with respect to the column’s ﬂexural stiffness
c
c
:
z
b
=
c
c
c
c
+
c
b
,
z
t
=
c
c
c
c
+
c
t
(11)where:
c
c
=
4
EI
c
h
.
(12)Eq. (10) can be solved numerically for the effective lengthfactor
K
, which is then substituted into Eq. (2) to provide thecritical buckling load. Alternatively, the upper left graph of Fig. 3, obtained from Eq. (10), can be used.
2.2. Sway frames
The simpliﬁed model of a column in a sway frame,shown inFig. 2(b), is considered, resulting from the model of Fig. 1(b)
by omitting the translational spring. Three boundaryconditions

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