Advances in Steel Structures - part 20 pps

170
S. Motoyui and T. Ohtsuka
GENERALIZED PLASTIC HINGE MODEL CONSIDERING LOCAL BUCKLING
Precondition
The development of plastic displacements conforms to associate flow rule. In this paper, we consider
the strength surface for
Zone I
in Fig. 2 which is moving parallel to the initial full yield surface
according to equivalent strength parameter. The relationship equivalent strength parameter and
equivalent plastic displacement parameter is obtained from the results calculated with finite element
method. And the relationship hysteresis characteristic under monotonic loading and that under cyclic
loading is modeled by Kato and Akiyama (1973). However, structural member behaves without shear
yielding and shear buckling.
Evaluate plastic and damage progress
Considering strength decrease governed by local buckling, the strength function for
Zone I
defined in
Eqn. 3 for plus and minus ff is rewritten as follows:
~(~,m,~): I~1 +,lml- g: o
(10)
Assuming associate flow rule, the incremental generalized plastic displacement vector A~P can be
expressed as:
where
aa~/~=~/lffl=
v,8~/dm=~/Iml=
~,
and a2p is energy dimensional incremental equivalent
plastic displacement parameter, is condition OnA2p >__ 0, ~0_ 0, A2p~0 - 0 and A2pzx~o - o.
Then, we lead nodal displacement, nodal force and tangent stiffness matrix by using return mapping
algorithm, M. Oritz and J.C.Simo (1992). Fig. 5 shows the properties of a element with plastic hinge
at its two ends.

The displacement vector, its elastic vector and generalized plastic displacement vector at time
t +
At
are 1+~' u ,'+~'u e and '+~'~P respectively. If we know plastic displacement vector'u p and equivalent
plastic displacement parameter'2pat timet, elastic displacement vector'+~'ueand equivalent plastic
parameter 2p are expressed as follows:
'+AtAp
='2p + A2p
(12)
(13)
where At is incremental time. Firstly, we try to obtain a trial force vector 'n~ according to freezing
incremental plastic displacement during At.
Figure 5: Nodal displacement and nodal force
Generalized Plastic Hinge Model for the Collapse Behavior of Steel Frames
Elastic predictor
Au p
= O, trialgle t+Atl, l tl, ip, t N = K e tnal ue
AA v = 0 2,p = )tp = 'ri~t/l v
171
(14)
where elastic stiffness matrix K, is given as follows:
-~
0 o -k o o
kqq kqm 0 -kqq kqm kn n EA 12EI
1
EI (4 +y)
kii 0 - kqm kii =T' kqq L3 (l+r)' k,, = ~L
(1+7)
Ke= k 0
o

SYM. kqq - kq~ k q,, 6E1 1 E1
(2- Z)
12E1 L
J
-L O+r) , r=
' L (1+7")
GA.,
kii
If plastic displacement don't develop during At, 'ri~N and 'ri'S obtained from Eqn. 14 satisfy Eqn. 15.
That is, when'ri"tN and 'ri"tg; don't satisfy Eqn.15, plastic displacement develop, we evaluate
development of plastic displacement and correct trial force.
trial (~9( trial l.l, trial~m, trial-~t_~ j <~ O
Plastic corrector
'+~'ue='ri~'U ~ ~P, Aid p
= A2,pP-1t~},
'+~'N =
Ke(trialtl e -All p)
(16)
'+~';tp , '+~'g ~('ri~
= t~,p + AA.p = 2p + A/].p )
(15)
These correct forces at time t + At should conform the strength function, therefore
'+A' q)('+A'~,'+*'~,'+~'S) = 0 (17)
This equation is nonlinear for A2~ so that we solve this by Newton method. To put it concretely, since
the values of iteration step k are, Eqn. 17 can be expressed for node i and j at iteration step k + 1 as:
,ri~tn _k,~ (k) +k,,, (k)
' ~
vi
(X')A)~'Pi
Ny

Vj
(k) A~.pj
tri~Zmi - k~i (~)
' (k)A/]'pi ~pk~J
(k) l.t j
(k)A2p j
(k+l) ~/ =
+'C - Si ( l A pi + ( k) mApi )
Ny
,ri, t +k,~ (k) V~
(k)A2,v; -k,,,, (1,)
n j
N y -~y v j
(k) AA.pj
Mp
,ri~t _ k o. (k) i u (k)AAp i _ k~ (k)
m ) CT i /-I ) ( k ) AA'vj
Mv
(k+l) ~j
Ny Mp
(18)
Considering to the first order term ofTaylor's series ofEqn. 18 for SAip which is a variation
OfAA, p
(k+l) ~, =(k) ~.
_
a,; (k)6&,Zp,
-
a,j (k) 6A2pj (19)
(k+l) q) =(k) q)j _aii (k)gA2"pi air (k)gA2pj
where

_
k~ +r: k~
+
((k)Api , k,~ v: ku k,, v: kq k,~
+v: k;~
Ny Mp Ny Mp Ny Mp Ny
Mp
172
S. Motoyui and T. Ohtsuka
Equating Right-hand of Eqn. 19 with zero, then '"'8A2,,i and '"'"At,,, are obtained as follows:
where
/,,),s,,,x,,,
= p,, In ~
_ p,~/',) aS,
/',),SAX,,j
=
_p,, I,,)~ +p,/,,)r
(k+,) At,,i = (k)A2,,, + (k)6At ,,, (2 0 )
(k+,)/~,,j =(k)A~, N
+(k)6A]~,,, j
_ ~ij aji l~,jj
p,= a,, , p,j_ , p.,= , p,.=
~ i i l~, jj t~, O. ~ j i ~ i i ~ jj ~ ij ~ j i l~ H l~ jj ~ ij ~ j i ~ i i ~ jj ~ O. l~ j #
And elastic displacement vector and force vector of iteration step k + 1 are given in Eqn. 21, then we
repeat that until accuracy reach a established value.
Tangent stiffness matrix
(k+,) u" =(~)u" - (k) 6At, P -' { (k) (k)/xJ'v~
(k+l)N
=
K.(k+')u"

(21)
We will have tangent stiffness matrix as follow. Rewriting elastic displacement as shown in Eqn. 16-a
to the mention of rate, we have
du" = d f tri~ u " -
k
then the rate of nodal force vector is expressed as follow:
trial v e 0
'ri:' " 0 (23)
dN:KedllLe-'gedtrialu;[ i
0
L {'o'o;
/,,/M,
Beside, conforming to the rule as shown in Eqn. 24 during plastic flow.
a~ = ~__a~ + ~___am + ~__a~ : o
drd cGm OS
Then Eqn. 25 is given from Eqn. 23 and 24.
'ria'ue [V,/oNY 0
I 1 (j) t'Va'v~ :
"dNZ-dS2 __ i "Lei d -dZ~pi ~.~li/Mp -dA~pj -HidA~pi
=0
' '~~
[~,,/M,
"~ [~/o N o
9 dNj - dSj _ dA2v ~ v _ dA&z - Hj dA& z = 0
t(o-:-:-:-~/8-~)./
1 c~
'ri'u; v.j .,,
where
dS={;;;}, g. =[~:7
(24)

(25)
Generalized Plastic Hinge Model for the Collapse Behavior of Steel Frames
173
Rearranging Eqn. 25, we obtain the following equations:
aii(k)6A~,pi +aO.(k)6A,~,pj = "Leidtriatll e , aji(k)6A~,p# +ajj(k)6A~,pj = "Lejdtrialll e
(26)
t~ IM, ta. IM,
therefore, we can solve Eqn. 26 for
dA2p~
and dA2pj :
[
]
[
]
dSXp, = fls, .L.,-fl,j "Les d "~ , dA2, : -fl,, "L., +fl,, "L u d""u
(27)
{s'.IM, J t'.l M. {I'.IMpJ t~'.IM,
Substituting Eqn. 27 into Eqn. 23 and tangent stiffness matrix is given as follow:
where
[fljjVio/NY [Vi/oNY [-flJioi/NY 1
[ 0 0
aN= x. ~"%,-~ p ~,./M ~-p /M,.L
I -fl~vOINy .K, dt,~Q,u~l,61Mp_ .K, dt,~,u~ 0
t-n ~.lM, t P s,.IM,
J t~./M,
:["
- |
+P,,".:, |
"
/.:{ /u. o

IM.
o o o}'. :.:{o o o
".1". o IM.}"
(28)
Comparison the numerical results
Fig. 6 compares load-displacement curve subjected to static loading given by the proposed model and
the finite element method in which 0p is an elastic rotation angle corresponding to M~. It can be seen
that two solutions agree well regardless of loading types. What is more important is that the
relationship ~ and ;tv using in Fig. 6 is the same one for each loading type.
Figure 6: Load-displacement curve (static)
In dynamic loading, using Newmark solution scheme and the Newmark's parameters/7 and 7" taken as
0.25 and 0.5, without considering effect of damping. A mass point
m m =O.1046[MN.s2/m]is
added to
the free end, and mass density p=
7.81xlO-9[N.s2/mm4].
Firstly, only Pvis loading at the almost static
rate until Pv is equal to
0.4Ny.
Secondly, P~ keeps constant, P,, is cyclic loading as shown in Fig. 7 in
which Qpc =
Mpc/L
where Mvc is the full plastic moment in the present of axial force, P,, and
Opc
is the
elastic rotation angle corresponding to M v~, and T is the elastic first natural period of this structure. In
this situation, time increment At is
1.286 x 10-3[sec].
The vertical displacement and restoring force time
174

S. Motoyui and T. Ohtsuka
history are shown in Fig. 8 and Fig. 9. Fig. 10 shows the hysteresis characteristic under dynamic
loading given by the proposed model and finite element method. Though external vertical force Pv is
constant, the vertical restoring force N is variable, as shown in Fig. 9. Involving this, the hysteresis
characteristic is not smooth like in static but waving, as shown in Fig. 10. As shown in Fig. 8,9 and 10,
the results given by the proposed model correspond to the results given by finite element method.
Figure 7: Loading program
Figure 8: Vertical displacement
Figure 10: Load-displacement curve (dynamic)
Figure 9: Vertical force
CONCLUSIONS
We clarify the establishment which is to give the effect of local buckling based on plasticity theory
according to the numerical results calculated with finite element method for simple structural model
of steel member subjected to relatively high axial force ratio. Then according to these establishments,
we propose a generalized plastic hinge model which takes local buckling into account, and we
confirmed the proposed model can express the effect of local buckling by means of comparing with
the results calculated with finite element method.
REFERENCES
Ohi K., Takahashi K. and Meng L.H. (1991). Multi-Spring Joint Model for Inelastic Behavior of Steel
members with Local Buckling.
Bulletin of Earthquake Resistant Structure Research Center, Institute
of lndustrial Science, Univ. of Tokyo
24:March, 105-114
Yoda K., Kurobane Y., Ogawa K. and Imai K. (1991). Hysteretic Behavior and Earthquake Resistant
Design of Single Story Building Frames with Thin-Walled Welded I-Sections.
Journal of Struct.
Constr. Engng, AIJ
424:June, 79-89 (in Japanese)
Yamada S. and Akiyama H. (1996). Inelastic Response Analysis of Multi-Story Frames Based on the
Realistic Behaviors of Members

Proc. ICASS'96
1, 159-164
Kato B. and Akiyama H. (1973). Theoretical Prediction of the Load-Deflexion Relationship of Steel
Members and Frames
IABSE Symposium on Resistance and Ultimate Deformability of Structures
Acted on by Well Defined Repeated Loads,
23-28
Oritz M. and Simo J.C. (!986). An Analysis of a New Class of Integration Algorithms for
Elastoplastic Constitutive Relations.
Int. J. Num. Mech.
23:3, 353-366
ADVANCED INELASTIC ANALYSIS OF SPATIAL STRUCTURES
J Y Richard Liew, H Chen and L K Tang
Department of Civil Engineering, National University of Singapore
10 Kent Ridge Crescent, Singapore 119260
ABSTRACT
This paper describes the methodology of an advanced analysis program for studying the large-
displacement inelastic behaviour of steel frame structures. A brief review of the advanced inelastic
analysis theory is provided, placing emphasis on a two-surface plastic hinge model for steel beam-
columns, a thin-walled beam-column model for core-walls, and a four-parameter power model for
semi-rigid connections. Numerical examples are provided to illustrate the acceptability of the use of
the inelastic models in predicting the ultimate strength and inelastic behaviours of spatial
frameworks.
INTRODUCTION
With the advancement of computer technology in the recent years, research works are currently in
full swing to develop the advanced inelastic analysis methods and computer packages which can
sufficiently represent the behavioural effects associated with member primary limit states such that
the separated specification member capacity checks are not required. This paper presents the
nonlinear inelastic models that can be used for analysing space frame structures within the context
of advanced inelastic analysis. In the proposed approach, each steel framing member is modelled as

one beam-column element. Plastic hinges are allowed to form at the element ends and within the
element length. To allow for the gradual plastification effect, a two-surface model is adopted. The
initial yield surface bounds the region of elastic sectional behaviour, while the plastic strength
surface defines the state of full plastification of section. Smooth transition from the initial yield
surface, as the force state moves to the plastic strength surface, is assumed. Core-walls provide a
major part of the bending and torsional resistance in a building structure. They are modelled by
thin-walled frame elements. The centre line of the core-wall is located on the shear centre axis.
Any significant twisting action should be analysed to include both warping and torsional effects.
Beam-to-column and beam-to-core-wall connections are modelled as rotational spring elements
having the moment-rotation relationship described by the four-parameter power model. At last, the
advanced analysis program is applied to investigate the collapse of a roof truss system, and perform
nonlinear inelastic analysis of a core-braced frame with semi-rigid connections.
175
176
J.Y.R. Liew et al.
ADVANCED PLASTIC HINGE FORMULATION
The basic feature of the proposed plastic hinge formulation is to use one beam-column element per
member to model the nonlinear inelastic effects of steel beam-columns. The element stiffness
matrix is derived from the virtual work equation based on the updated Lagrangian formulation. The
elastic coupling effects between axial, flexural and torsional displacements are considered so that the
proposed element can be used to predict the axial-torsional and lateral-torsional instabilities. By
using the stability interpolation functions for the transverse displacements, the elastic flexural
buckling loads of columns and frames can be predicted by modelling each physical member as one
element. The member bowing effect and initial out-of-straightness are also considered so that the
nonlinear behaviour of frame structures can be captured more accurately (Liew et al., 1999).
Material non-linear behaviour is considered by introducing plastic hinges at the element ends and
within the element length if the sectional forces exceed the plastic criterion, which is expressed by
an interaction function. If a plastic hinge is formed within the element length, the element is divided
into two sub-elements at the plastic hinge location. The internal plastic hinge is modelled by an end
hinge at one of the sub-element. The stiffness matrices for the two sub-elements are determined.

The inelastic stiffness properties of the original element are obtained by static condensation of the
"extra" node at the location of the internal plastic hinge. To allow for gradual plastification effect,
the bounding surface theory in force space is adopted. Two interaction surfaces representing the
state of the stress resultants on a section are employed (Liew and Tang, 1998). The yield surface
bounds the region of elastic al behaviour, while the bounding surface defines the state of full
plastification of the section. The bounding surface encloses the sectional force state and the yield
surface at any stage during the plastic process. To avoid intersection of the surfaces, the yield and
bounding surfaces are given the same shape. When the section is loaded, the force point travels
through the elastic region and contacts the yield surface, which is given by
1-'y =f(S-[3/ =f( P-j31 QY-[32 Qz-~3
Mx_.~_ ~4
My-J35 Mz-[36/_l= 0 (1)
~ZySp ) ZyPy ' ZyQpy 'zyQp z ' ZyMpx ' ZyMpy ' ZyMpz
in which P, Qy, Qz, Mx, My, Mz are the sectional forces, Py, Qpy, Qpz, Mpx, Mpy, Mpz are the plastic
capacities for each force component, j3 is the position vector of the yield surface's origo in the force
space, and Zy is the yield surface size. The function Fy is defined that Fy = -1 corresponding to a
stress-free section, while Fy < 0 corresponds to a initial yielding or any subsequent yielding state.
When the further loading takes place, the yield surface starts to translate so that the current force
state remains on it during subsequent loading. For the advanced plastic hinge analysis, the plastic
hardening parameter and transition parameter, which are specific for each force component, are
crucial for the elasto-plastic behaviour of the element. They may be determined from experiments
or numerical calibrations, and the details of such calibration work and further verification studies are
demonstrated in Liew and Tang (1998).
MODELLING OF CORE-WALLS
Core-walls are modelled by the thin-walled beam-column element for their proportional similarity to
Vlasov's thin-walled beams and for their computational efficiency in the inelastic analysis (Liew et
al., 1998). As shown in Fig. 1, the thin-walled beam-column element has an additional warping
degree-of-freedom over the beam-column element at each end. The local coordinate is chosen: axis
x lies on the shear centre axis, and y and z axes parallel to the principal y and ~, axes. Some force
and displacement components are referred to the shear centre, whereas the remaining ones are

referred to the centroid of the section. However, before the element stiffness matrices are
transformed into the global coordinate, it is necessary that all the forces and displacements are
referred to a single point. The shear centre can be selected as the reference point. The detailed
derivation for the elastic and geometric matrices of the thin-walled beam-column element is given
Advanced Inelastic Analysis of Spatial Structures
177
by Liew et al. (1997). Because the height-to-width ratio of core-walls is large and the axial force
respective to the sectional area is small in practical building frames, material nonlinearity of core-
walls is considered approximately, assuming that the plastic strength is controlled by the bending
action only. The locations of the shear centre and the centroid of cross-section are assumed not to
change due to the inelastic effects.
MODELLING OF SEMI-RIGID CONNECTIONS
Beam-to-column connections can be modelled as rotational spring elements in the nonlinear analysis
of semi-rigid frames (Hsieh, 1990; Chen et al., 1996). Many connection models have been proposed
to describe the moment-rotation relationships of connections used in building steelworks (Liew et
al., 1993). The present work adopts a four-parameter power model to represent the moment-rotation
relationship of typical beam-to-column connections (Hsieh, 1990). The selection of this model is
guided by its simplicity and robustness for representing the basic behaviour of typical connections,
and for ease of implementation in the nonlinear inelastic analysis program. The four-parameter
power model has the following form:
(Ke -Kp~
M-[I+I(K _Kp)O/Moln]/n+KpO (2)
in which I~ is the initial stiffness of connection, Kp is the strain-hardening stiffness of connection,
M0 is a reference moment, and n is a shape parameter as shown in Fig. 2. The four-parameter model
can easily encompass the more simple models. For examples, Eq. 2 becomes a linear model if I~ =
Kp, a three-parameter power model if Kp=0, and a bilinear model when n is large.
In the structural design, it is unlikely that specific connection details will be known during the
preliminary design until the structural members have been sized in the final design. Since
connection flexibility will affect the structural response and therefore the required member sizes,
there is a need to develop some means to account for connection behaviour during the analysis and

design process before the final member sizes are selected. One solution is to use the standard
connection reference curves which are based on the connection test database. An optimisation
approach utilising the conjugate-gradient method is first used to find a set of parameters (M0, Ke, Kp,
and n) which gives the best curve-fit to the experimental connection response data. The moment-
rotation curves are then normalised with respect to the nominal connection capacity Mn, which
equals to the moment at a rotation of 0.02 radian as shown in Fig. 2. The standard reference curve is
calibrated by fitting a curve through the average of the normalised curves. The average values of
M'=M/Mn, K'e=Ke/Mn, K'p=Kp/Mn and n in the standard reference curves for nine types of
commonly used connections subjected to in-plane moment have been established (Hsieh, 1990).
Then, for the analysis of the overall structure, only the connection type and nominal connection
capacity would need to be defined without unnecessary concern over the final connection details.
Based on the connection test database, a survey of the ratio of Mn/Mpb for different types of
connections have been carried out, in which Mpb is the plastic bending capacity of beam where the
semi-rigid connection is located. The standard reference curve parameters and values of Mn/Mpb for
several types of connections are listed in table 1.
COLLAPSE ANALYSIS OF A ROOF TRUSS SYSTEM
An accident took place when a roof truss system was assembled on site. Advanced analysis was
carried out to investigate the cause of collapse. The roof truss system includes seven trusses
connected by eight purlins at their top chords and its plan view is shown in Fig. 4. The span and
height of each truss are L = 35.05m and h = 2.45m respectively, as shown in Fig. 5. All trusses are
restrained from the displacement at the supports of bottom chord. The truss at axis 1 is laterally
restrained at the mid-span of the top chord, while the other trusses are connected by purlins only.
178
J.Y.R. Liew et al.
The truss at axis 1 consists of initial out-of-straightness of double-curvature shape at the top chord,
with maximum magnitude of (0.5L)/500 = L/1000 =.35 mm. The top chords of other trusses (from
axes 2 to 7) consist of single-curvature initial out-of-straightness with a maximum magnitude L/500
= 70 mm at the mid-length. The lateral restraint and initial out-of-straightness of the top chords of
all trusses are illustrated in Fig. 4. The supporting ends of all trusses are constrained from
displacements in all directions and out-of-plane rotation, except that the rotational restraint of

support A, whose position is shown in Fig. 4, is released to simulate a careless mistake made during
the installation of the trusses.
The truss system is analysed for two loading conditions. Firstly the system is assumed to be
subjected to only vertical load, so that the safety factor for the overall system under gravity can be
evaluated. The vertical load at every truss includes (1) its self-weight, (2) eight concentrated load of
602.4N each on the connection with purlins to simulate the purlin weight, and (3) two concentrated
load of 2530N each at mid-span of the truss, one at the top chord and the other at the bottom chord,
to simulate the weight of Gusset plates and connections. This can be seen in Fig. 5. Subsequently
the system is studied under full self-weight plus horizontal surged force created by the crane. A
horizontal point load is applied at nodes B and C on the top chords of the truss at axis 7. Nodes B
and C are located at nearly one third of the truss span, as shown in Figs. 4 and 5. This is to evaluate
the horizontal surged forces required to cause the structural failure.
A separate analysis is also carried to evaluate the resistance of individual truss under two load
situations: (1) gravity only, and (2) both the gravity and the horizontal surged force created by the
crane. For the truss at axis 1, which has a lateral restraint at mid-span, its resistance is 1.49 times the
gravity or 1.0 times the gravity plus a horizontal load, supplied at nodes B and C, of 29.5kN each.
In contrast, for the truss at axis 2, without the lateral restraint, its capacity is only 0.38 times the total
gravity. In other words, during the erection, the individual truss cannot resist its self-weight if
lateral restraint is not provided. Since the restrained truss at axis 1 is required to provide the lateral
restraint to the other six trusses by purlins, the maximum resistance of the truss is expected to be less
than when it is acting alone.
When the gravity is applied progressively, the truss system collapse at the load factor 1.15. Fig. 6
shows the plots of applied load ratio versus lateral displacement at node B. The deformed shape of
the truss system at collapse is shown in Fig. 7. This safety factor appears to be very small for the
safe erection of steel structures. To investigate the effect of crane surge, the full self-weight of the
structure is applied first, followed by two horizontal surged forces each at nodes B and C. Fig. 8
shows the horizontal load - displacement plots at node B for the truss at axis 1. The total maximum
horizontal force that can be applied to cause the collapse of the overall truss system is 9.6 kN. The
deformed shape of the trusses at collapse is shown in Fig. 9. This lateral load resistance is
considered to be too small for practical viewpoint. Hence, a single point bracing at the mid-length

of truss at axis 1 is not adequate in providing lateral restraint against normal impact load due to
crane surge. The analysis concludes that more lateral restraints to the compression chord are
necessary for safe erection of the roof trusses.
INELASTIC ANALYSIS OF SEMI-RIGID CORE-BRACED FRAMES
Figures 10 &l 1 show a 24-storey core-braced frame with storey height h = 3.658 m and total height
H = 87.792 m (Liew et al., 1998). Thickness of concrete core-walls is 0.254 m. Depth of concrete
lintel beam is 1.219 m. A36 steel is used for all sections. Material properties of concrete are:
modulus of elasticity Ec = 23,400 N/mm 2, and compressive strength f~ = 23.4 N/mm 2. The
structure is analysed for the most critical load combination of gravity loads and wind loads that act
in the Y-direction. Core-walls are mainly subjected to the bending moment about the principle ~-
Advanced Inelastic Analysis of Spatial Structures
179
axis, which is parallel to the global X-axis. The bending moment about the principle ~-axis is
small. The plastic section modulus about the principle ~. axis of the channel-shaped core-wall
section is Z = 2.549 m 3. In this example, the height-to-width ratio of core-walls is 24:1. It is
assumed that the plastic resistance of core-walls is dominated by the plastic bending resistance about
the principle 7 axis, Mz = 0.8Z f" = 4.8x 10
4
kNm, only. The plastic resistance of core-walls has
been reduced to approximately account for the tensile cracking and axial force interaction effect.
In the nonlinear inelastic analysis, each steel column is modelled as one plastic hinge beam-column
element, and each beam is modelled as four beam-column elements. Core-walls are modelled as
thin-walled beam-column elements. Concrete lintel beams are rigidly connected to core-walls for
resisting the lateral and torsional loads. All floors are assumed to be rigid in plane to account for the
diaphragm action of concrete slabs. The gravity loads, which are equivalent to a uniform floor load
of 4.8 kN/m 2, are applied as concentrated loads at the beam quarter points and at core-walls of every
storey. The wind loads are simulated by applying the horizontal forces in the Y-direction at every
frame joints of the front elevation, and are equivalent to a uniform pressure of 0.96 kN/m 2.
Firstly, inelastic analysis is performed on rigid core-braced frame. The loads are proportionally
applied until the frame collapses at a load ratio of 1.787 when plastic hinges form at the bottom and

the top of core-walls in the first storey. To study the lateral resistance capacity of core-walls,
inelastic analysis is performed on core-braced frame with pin-connections. In this case, the whole
building relies core-walls to provide the lateral resistance only. The limit load and initial lateral
stiffness of the frame with pin connections are only 36% and 21% of those of the rigid frame.
Similarly, to study the lateral resistance capacity of the pure steel frameworks, the elastic modulus
and the compressive strength of concrete are assigned to be very small values. The frame collapses
at a load ratio of 0.654, which is similar to that of the frame with pin-connections. It is noted that
the inelastic lateral deflection behaviour of steel framework is more ductile than that of the frame
with pin-connection. It can found that the building frame cannot only rely on core-walls or steel
frameworks to provide the lateral resistance. Core-walls and steel frameworks must act together to
withstand the external loads.
Semi-rigid construction is faster and cheaper than rigid construction. For high-rise building design,
service wind drift is always the main concern. In order to reduce the number of moment
connections in high-rise building construction, the use of core-braced frames with semi-rigid
connections may provide optimum balance between the dual objectives of buildability and
functionality (Chen et al., 1996). Different types of beam-to-column and beam-to-core-wall
connections in the steel frameworks are assumed to study the connection effect on the inelastic limit
loads and lateral deflections of the frame. The connection properties are given in table 1. The
proposed semi-rigid formulation can model the torsional and both major- and minor-axis flexibility.
However, in this analysis, only the relative rotations about the major-axis of beam section are
allowed at the semi-rigid connections. This is due to two reasons: (1) at present there is little
experimental information on the torsional and out-of-plane behaviours of semi-rigid connection, and
(2) for typical framed structures with rigid floor, the torsional and out-of-plane effects of semi-rigid
connections are not significant.
Inelastic analyses are performed on core-braced frames with 'DWA', 'TSAW' and 'EEP'
connections. The inelastic limit loads and load - deflection curves are shown in Fig. 12. It can been
seen from table 2 that if 'EEP' connections are adopted, the load and lateral stiffness can reach to
93% and 81% of those of the rigid frame. The limit load and inelastic stiffness of frame with
'DWA' connections are only a little higher than those of the frame with pin-connections. The limit
load and inelastic behaviour of the frame with 'TSAW' connections are between those of the frame