Advances in Steel Structures - part 21 pdf

180 J.Y.R. Liew et al.
with 'EEP' connections and the frame with 'DWA' connections. It can be concluded that if proper
semi-rigid connections are used, the frame can be constructed much faster and cheaper than the rigid
frame, at the same time satisfying the strength and serviceability limit states.
CONCLUSIONS
The basic principles of the proposed advanced inelastic analysis program have been presented.
Inelastic analysis has been applied to study the roof truss system and emphasises the importance of
lateral brace to assure the system's stability, which is important for the safe erection of such
structure. Inelastic analyses on core-braced frame with semi-rigid connections show that
construction with proper selection of connections can satisfy limit states design and achieve fastrack
construction. When properly formulated and executed, the advanced analysis can be used to assess
the interdependence of member and system strength and stability, the actual failure mode and the
maximum strength of the overall framework, and, hence, efficient and cost-effective design
solutions can be obtained. This is in line with the modem design codes such as Eructed, which
allows the use of advanced analysis for designing steel structures.
REFERENCES
Chen, W.F., Goto, Y., and, Liew, J.Y.R. (1996), Stability Design of Semi-Rigid Frames, John
Wiley& Sons, NY.
Hsieh, S.H. (1990), Analysis of three-dimensional steel frames with semi-rigid connections,
Structural Eng. Report 90-1, School of Civil and Environmental Eng., Comell University, NY.
Liew, J.Y.R., White, D.W., and Chen, W.F. (1993), Limit-states design of semi-rigid frames using
advanced analysis: Part 1: Connection modelling and classification, J. Construct. Steel Res., 26,
1-27.
Liew, J.Y.R., Chen, H., Yu, C.H., Shanmugam, N.E., and Tang, L.K. (1997), Second-order inelastic
analysis of three-dimensional core-braced frames, Research Report No: CE024/97, Dept. of Civil
Eng., National University of Singapore.
Liew, J.Y.R., Chen, H., Yu, C.H., and Shanmugam, N.E. (1998), Advanced inelastic analysis of
thin-walled core-braced frames, Proc. of the 2nd International Conference on Thin-Walled
Structures, Dec. 2-4, 1998, Singapore.
Liew, J.Y.R., Chen, H., and Shanmugam, N.E. (1999), Stability functions for second-order inelastic
analysis of space frames, Proc. of 4th International Conference on Steel and Aluminium

Structures, June 20-23, 1999, Espoo, Finland.
Liew, J.Y.R., and Tang, L.K. (1998), Nonlinear refined plastic hinge analysis of space frame
structures, Research Report No: CE029/99, Dept. of Civil Eng., National University of
Singapore.
Table 1. Parameters and Mn/M values for connections under in-plane bending moment
Mo' Ke' Kp' n Mn/Mpb
Connection
type
M0/Mn ~n Kp/Mn
DWA 1.03 301 5.0 1.06
TSAW 0.94 363 6.9 1.11 0.4
EEP 0.97 309 5.5 1.20 1.0
DWA: Double web-angle connection
TSAW: Top- and seat-angle connections with double web angles
EEP: Extended end-plate connection without column stiffeners
At the beam framing
about the major-axis
of column (see Fig. 3)
0.05
At the beam framing
about the minor-axis
of column (see Fig. 3)
0.025
0.2
0.5
Advanced Inelastic Analysis of Spatial Structures
181
Fig. 2 Four-parameter power model
Fig. 1 Thin-walled beam-column element
Fig. 3 Beam-to-column connections

Fig. 4 Plan view of roof truss system
Fig. 5 Elevation view of truss
Fig. 6 Load-lateral displacement curve under
gravity load
Fig. 7 Deformed shape of roof truss system at
collapse under gravity load
182
J.Y.R. Liew et al.
Fig. 8 Horizontal load-lateral displacement curve
Fig. 9 Deformed shape of roof truss
system at collapse under the horizontal
surge forces
Fig. 10 Plan view of core-braced frame
Fig. 12 Top-storey load-deflection curves
Fig. 11 Elevation view of core-braced
frame: (a) at axes 1, 2, 5, 6 (b) at axes 3, 4
Table 2. Comparison of limit loads and
initial lateral stiffness
Connection
types
Pin
connection
Limit
load
36%
Initial lateral
stiffness
21%
DWA 40% 30%
TSAW 65% 68%

EEP 93% 81%
All % values are compared with the core-
braced frame with rigid connections
STABILITY ANALYSIS OF MULTISTORY FRAMEWORK
UNDER UNIFORMLY DISTRIBUTED LOAD
Chen Haojun and Wang Jiqing
Department of Construction Engineering,
Changsha Communications University
45 Chiling Road, Changsha 410076 China
ABSTRACT
Problems of overall stability in a multistory framework become significant with the increase in its
height. This paper presents the stability analysis to a one-bay multistory framework under uniformly
distributed load by means of continuum model. Continuum model is a substituting column converted
from multistory framework. So, the analysis to multistory frame, which is an indeterminate structure,
is reduced to that to a determinate one. The formula of critical load is developed by Galerkin method.
The effect of the axial compressive deformation of framework column is taken into consideration.
KEYWORDS
Multistory framework, overall stability, continuum model, uniformly distributed load, critical load,
substituting column.
In the analysis to a multistory framework structure, one pays more attention to analysis to internal
forces of a multistory framework at vertical and horizontal loads, than to analysis to overall stability.
However, the problems of overall stability in a multistory framework become significant with the
increase in height. Generally, the exact stability analysis of multistory frames can be solved by finite
element method. This is an extremely complex procedure, even with the help of computer. The
higher the structure is, the more complicated the problem is to handle. The critical load is usually
obtained by determination of effective length factor of each framework column. In this paper, the
framework structure will be taken as a whole for determination of the critical load. A critical load for
183
184
C. Haojun and IV. Jiqing

a one-bay multistory framework subjected to uniformly distributed load at floor level is developed by
Galerkin method. The effect of axial compressive deformation on the critical load is taken into
account in following analysis.
1. BASIC ASSUMPTIONS
During the analysis, following assumptions will be used.
A). The material of the structure is homogeneous, isotropic and obeys Hook's law.
B). The loads are applied statically and maintain their direction during buckling.
C). The structure develops small deformation and the axial deformation in the beam is negligible when
the axial framework buckles.
D). All stories have the same height and the structure are at least four story high.
E). The structure has a rectangular net work with elements attached by rigid joints to each other.
F). The stiffness (El/l) of beams is the same.
G). The inflection point is on the middle of the beam when the framework buckles.
2. SUBSTITUTING COLUMN
The continuum model of multistory framework is a substituting column converted from the framework.
The substituting column is obtained from the original framework (Fig. 2.1a) in several steps. First,
the UDL on the beam is transferred to the columns at floor levels (Fig. 2.1b) in the form of
concentrated forces (the reactions on the beams). These concentrated forces are then distributed
along story height (Fig. 2.1 c), in fact along the height of the framework. The beams are cut through
at inflection points (Fig. 2.1 d) and finally the columns are added up into a single substitute cantilever
(Fig. 2.1 e).
P
46464+444
p
4~4~4+4~4
P
4F4F4~4~41 1
(D | @
l
4 4 t ~ !4 4 4~

4)
t ~
q=
ql)m
q:F 4q= qx4
' I
144 4)
t 4)
4)
1 ,' l
(~) (b) (c) (d) (e)
Fig. 2.1 Continuum Model
The bending stiffness of the substituting column is the sum of the bending stiffness of columns of the
framework. The load on the substituting column equals the total load on the original framework.
The distributed force along the height of substituting column is converted from the uniformly
Stability Analysis of Multistory Framework
185
distributed load at floor levels. The distributed moments along the substituting column are induced
from deformation of the framework during buckling. In doing so, the framework is converted into a
fixed-free column on which a distributed force and a distributed moment act. It should be noted that
the difference of axial compressive deformation between two framework columns makes the
framework have sway. This phenomenon is not shown in substituting column. Comparing actual
column with substituting column, it is known that the restraint moment acting at floor level due to
beam bending makes the column double-curvature between two beams for an actual framework. But
for a substituting column, the restraint moment due to beam bending is distributed along the
substituting column and does not make the substituting column double-curvature.
3. CRITICAL LOAD OF A ONE-BAY MULTISTORY FRAMEWORK
There is a one-bay multistory framework as shown in Fig. 3.1. The stiffness of beam of each floor
level is Eblb except the top one of Eblb/2; and the stiffness of framework columns is Eclc. There are
uniformly distributed loads at each floor level. According to preceding procedure, the substituting

column is shown in Fig. 3. lb. When the framework buckles, it can be in equilibrium both in original
configuration (undeformed configuration) and in slightly deformed configuration. Now, let us
consider the equilibrium of framework in slightly deformed configuration.
EcI~/2
P
IIIIIIIII
Edj2
Edb
IIIIIIIII
Edb
IIIIIIIII
Edb
IIIIIIIII
E~b
EcIc/2 h
h
h
h
~y
I
l
l
ql
I
I
t
I
)
)m
)

~y
H=eh
(a)
0o)
Fig.3.1 Substituting Column
3.1 Distributed Moment When Framework Bends
The separated body for analysis may be taken as shown in Fig. 3.2 when the framework bends. It is
cut at the middle point of beams (inflection points) and replaces with a shear force T. This shear
force T can be obtained by the condition that the deformation at the middle point of beams (inflection
points) is equal to zero,
y, l
TM (t/2) 3
- +~=0 (3.1)
2 3Ebl b
in which TM is the shear force in beams due to framework bending; l is the distance between the axes
of columns; Eb is the elastic modulus of beams; Ib is the moment of inertia of beam; Yb ~ is the first
186
derivative of framework.
C. Haojun and W. Jiqing
Eqn. 3.1 gives
12EbIb ,
TM = l 2
YM
(3.2)
The distributed force along the framework column due to bending is
tM TM 12Ejb , (3.3)
=-~-= h ~yM
Transfer of the shear forces at inflection points TM to the axis of the columns produces the concentrated
moments acting on the column at floor level,
l 6Eblb , (3.4)

M M = T M x - = ~ YM
2 l
Distribution of the concentrated moment M i along the column height leads to
mM MM 6Eblb ' (3.5)
if =
hl YM
3.2 Consideration of Axial Deformation of Column
Shear force TM at beams makes the axial forces in two columns different. The axial force increases in
TM in right column and decreases in left column. This variation of axial force causes an additional
axial deformation in left and right columns. It is denoted by the sign AN. This deformation consists
of two parts (Fig. 3.3). One (denoted by AN1) makes the beams bend and the other (denoted by Am)
makes the columns bend. Hence,
or A~
= AN1 + AN2
Y~v = Y~vl + Y~v2 (3.6)
The bending moment at beam end due to AN1 is
12Eblb (3.7)
MN1
= /2 AN1
Fig. 3.2 Separated Body When Buckling
Fig. 3.3 Compressive Deformation
Letting 2AN1/I=y'N1, one obtains the distributed moment along column
Stability Analysis of Multistory Framework
MN1 6Eblb ,
mN1 =~ = ~YN1
h lh
187
(3.8)
Variation of axial force in column due to AN~ is
TN1 = MN____A_I = _ 12Eblb

l
Distribution of the force
TN1
along the column leads to
TNI 12EbI b ,
=
~YN1
tN1 = h hl 2
y;,,
(3.9)
(3.10)
According to Figs. 3.3 and 3.4, the compressive deformation distant to z from original O is
AN(Z) ~ ~(tM-~'tN1)d(dz
E cAc
(3.11)
where A c is the cross-section area of column. Substitution of Eqns. 3.3 and 3.10 into Eqn. 3.11 leads
to
AN(z ) = f ~
12Eblb (y~ y'N,)d(dz
EcI c
(3.12)
Making use of Y'N =2AN/l, Ir=2Ac(l/2) 2, and differentiating twice, Eqn. 3.12 may be written in the form
,, 12Eblb
YN = Eclrhl
(Y~vl-YM) (3.13)
Integrating Eqn. 3.13 once and making use of the boundary condition, y"N(0)=0 and yN~(0)=yN2(0)=0,
one obtains
" 12Eblb(YNl_yg2 )
YU- Eclrhl
(3.14)

3.3
Equilibrium Differential Equation of Substituting Column
The Equilibrium differential equation is
Eclcy" + ~q(y- rl)d ~ -
~m(~:)d~: = 0
It is known that m=2(mM+mN0, and making use of Eqns. 3.5 and 3.8, one obtains
(3.15)
m
12Eblb (Y~ - Y'~I)
hl
(3.16)
Substituting Eqn. 3.16 into Eqn. 3.15, one obtains
Ec lc Y " + f q (Y - rl )d ~ - ~ 12 E b l b (Y " - Y 'N1 )dz = 0
hl
(3.17)
188
C. Haojun and W. Jiqing
The bending deformation of the substituting column is
Y = YM + YJv2
(3.18)
3.4
Solution of Differential Equations
I
I
I
r
i
Y
Z
Fig. 3.4 Coordinate for Calculation of

Compressive Deformation
Y o Y o
Z
Z Z
1
q
I
Fig. 3.5 Coordinate and Separated Body
of Substituting Column
Combination of Eqns. (3.17), (3.13), (3.6) and (3.18) gives
Eclcy" + ~ q(y - ~7)dr - ~ 12Eblb (Y'M - YN1)d~ = 0
hl
(3.19a)
. 12EbIb ,
YN Eclrhl (Ym - Y~t )
=0
(3.19b)
Y~
= YN1 +
Y~v2 (3.19c)
t p
Y'= YM + YN2
(3.19d)
Arrangement of above equations and letting Kb=12Eblb/hl leads to the equilibrium differential equation
Eci~ ] " Kbq
Eclcy + qz-x y
Eqn. (3.20) is solved by Galerkin method. Letting the approximate deflection curve be
(3.20)
7De
y = 6 sin~

2H (3.21)
which satisfies the geometric and mechanic boundary conditions
Stability Analysis of Multistory Framework
y(O)= y'q): y"(O)= y"(l)=O
one obtains the Galerkin equation
fL(y)sin az
dz=O
2H
in which
Eclc ~ . Kbq
L(y)= EclcY'V + qz- Kb - Kb ~cI~ ) y + qY'-~l~ ~(y- rl)dr
Substituting Eqns. (3.24) and (3.21) into Eqn. (3.23) and making use of the integration
~
sin 2 az dz=H
2H 2
f z sin 2 nz
dz = ( 1
I___]H 2
2H 4+re 2 )
f 7tz 7tz H
cos~sin dz =
2H 2H n"
~
=
1 -
cos ~ sin
2H 2H
H
dz= m
one obtains

I ~r 14H (rcl2(1 1 ~]H2 ( Eclcl(rc ]2H
8Ec lc ~ -~ - &t ~ + rc 2 ) + 6 K b + K b ~ ) k,-~ ) -2
+~
a
+ +a =o
E~I, EcI, rc zc
Letting
F c =rc2EcI~//(2H) 2
F o =rc2Ej,.(2H) 2
and substituting these into Eqn. 3.25, one obtains
189
(3.22)
(3.23)
(3.24)
(3.25)
(qH)o. Fc + Kb + Kb Fc/Fo
(3.26)
= 0.279(1+
Kb/Fo)
Eqn. 3.26 is the critical load of the one-bay multistory framework according to Galerkin method by