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 . "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~. 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. 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