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Nonlinear inelastic analysis of building frames with thin-walled cores
Thin-Walled Structures 37 (2000) 189–205 www.elsevier.com/locate/tws
Nonlinear inelastic analysis of building frames with thin-walled cores H. Chen, J.Y. Richard Liew, N.E. Shanmugam
*
Department of Civil Engineering, National University of Singapore, 10 Kent Ridge Crescent, 119260 Singapore Received 27 January 2000; accepted 10 March 2000
Abstract A second-order inelastic analysis by combining the theories of stability and plasticity is proposed for studying frames with thin-walled cores. In the proposed approach, steel frameworks surrounding the cores are modelled by using the plastic hinge beam-column approach, and core walls modelled by using the thin-walled beam-column approach. Transformation procedures are proposed to consider the kinematic relationship between beams, columns, core walls and floor diaphragm. Nonlinear solution procedures are incorporated for the incremental analysis. The proposed inelastic analysis is used to investigate the inelastic behaviour and ultimate strength of core-braced frames. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Inelastic analysis; Stability; Plastic hinge; Beam-column; Thin-walled core; Transformation matrix
1. Introduction Core-braced frame represents an efficient type of structural system up to a certain height where the predominant lateral load resistance is provided by concrete cores [1,2]. The steel frameworks surrounding the cores are generally designed only for gravity forces with simple shear connections. In high seismic areas, the steel frameworks would be required to resist some part of the earthquake loads as a moment frame to provide a second line of defence. The most rigorous way of modelling core walls is by the finite element method * Corresponding author. Tel.: +65-722-2288; fax: +65-779-1635. E-mail address: [email protected] (N.E. Shanmugam). 0263-8231/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 0 0 ) 0 0 0 1 6 - 1
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involving the use of plate and shell elements. However, this approach is computationally intensive and costly because it involves discretization of walls into a large number of elements. Having noted the proportional similarity of core walls to Vlasov’s thin-walled beam [3], Taranath proposed the thin-walled beam-column approach for the first-order elastic analysis of core-braced frames with computational efficiency and sufficient accuracy [2]. Although the first-order elastic analysis provides a simple estimate of the distribution of forces within the structural system, it does not provide any information about the influence of either stability or plasticity effect on the behaviour of frame system. This paper proposes a second-order inelastic analysis, which can capture the geometrical and material nonlinearities of frames with thin-walled cores up to the inelastic limit load. Steel frameworks surrounding the cores are modelled by using the plastic hinge beam-column approach, and core walls modelled by using the thinwalled beam-column approach. Kinematic relationships between beams, columns, thin-walled cores, and floor diaphragm are derived to establish the transformation matrices for the incremental analysis. Nonlinear solution procedures are incorporated for the incremental analysis. The elastic behaviour of a core tube model is analysed by using the proposed approach. Lastly, the inelastic behaviour and inelastic limit load of core-braced frames are investigated using the proposed analysis theory and computer program. 2. Modelling of steel frameworks Steel frameworks surrounding the cores are modelled by using the beam-column element as shown in Fig. 1. The beam-column formulation is based on the updated Lagrangian approach. Transverse displacements are calculated by using the stability interpolation functions, which satisfy the fourth-order differential equation of beamcolumn subjected to end forces [4]. The influence of axial force on transverse displacements can be accurately represented in the form of stability functions. The coupling effect between compression, bending and torsion is included by using the nonlinear strain relationships in the virtual work equation. Through the use of proposed bowing matrix [4], the beam-column formulation can capture the member bowing effect and initial out-of-straightness by modelling each physical member as one element. A two-surface plasticity model, which employs the yield surface and the bounding surface, is used to represent partial yielding and hardening in steel beam-columns [5,6]. The yield surface bounds the region of elastic cross-sectional behaviour, while the bounding surface defines the state of full plastification of cross-section. The bounding surface encloses the state of cross-sectional force and yield surface at any stage during the yielding process. The size of initial yield surface is described by the surface extension parameter. Suitable value of surface extension parameter can be selected to model the effect of initial residual stress in cross-section. Once yielding is initiated, the yield surface will translate so that the state of cross-sectional forces remains on the yield surface during subsequent loading. This type of approach is termed as “refined” plastic hinge analysis [6].
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Fig. 1.
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Beam-column element.
Inelastic member buckling can be predicted by allowing plastic hinge to occur between the member ends [4]. The analysis will automatically subdivide the original element into two sub-elements at the location of plastic hinge. The internal hinge is then modelled by an end hinge at one of the sub-elements. The stiffness matrices of the two sub-elements are determined. The inelastic stiffness properties for the original element are obtained by the static condensation of extra node at the location of internal plastic hinge.
3. Modelling of core walls Core-walls are modelled by using the thin-walled beam-column approach [2,7]. As shown in Fig. 2, the thin-walled beam-column element has an additional warping degree-of-freedom over beam-column element at each end. The local coordinate is chosen with axis x lying on the shear centre axis, and y and z axes paralleling to the principal y¯ and z¯ axes. Some force and displacement components are referred to the shear centre, whereas the remaining ones are referred to the centroid. However, before stiffness matrices are transformed into the global coordinate, it is necessary that all force and displacement components are referred to a single point. The shear centre is selected as the reference point.
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Fig. 2. Thin-walled beam-column element.
Geometrical nonlinearity of core walls is modelled through the use of geometric stiffness matrix of thin-walled beam-column element proposed by Conci [8], which contains more initial terms than the existing geometric matrices in the current literature. The geometric matrix is consisted of two parts: the first part for a doubly symmetric cross-section and the second part, which should be added when the crosssection is asymmetric. Material nonlinearity of the core wall is modelled approximately by using the concentrated plastic hinge approach [7]. Because the height-towidth ratio of core-wall is generally large in practical buildings, it is assumed that the plastic resistance of cross-section is controlled by bending action only. The locations of shear centre and centroid of cross-section are assumed not to change due to yielding in the cross-section. It is noted that the yielding in the core wall section depends on the combined action of compression, biaxial bending, torsion and warping. It is necessary to model these factors if core wall is subjected to significant torsional and warping effects.
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4. Transformation procedures The kinematic relationships between beams, columns, thin-walled cores and floor diaphragm are proposed for the global incremental analysis. The conventional beamcolumn transformation matrix is modified to transform the stiffness matrices of beams from the local coordinate to the global coordinate with some of the unknowns retaining their local coordinate reference. The proposed procedure can facilitate further modifications to account for modelling of semi-rigid connection at beam ends. 4.1. Floor transformation Fig. 3 shows a portion of channel-shaped core walls “J” and “K” connected by beams “1” and “2” between floors “L” and “N”. Beams are modelled as beam-column elements “ab” and “cd”, and core walls as thin-walled beam-column elements “AB” and “CD” with the longitudinal axes lying on the shear centre axes. (x2, y2, z2) and (xK, yK, zK) are the local coordinates of beam “2” and core wall “K”, respectively. In the case of rigid diaphragm model, which is assumed to provide infinite inplane stiffness but without any out-of-plane stiffness, the lateral response of floor “N” is characterised by two translational and one rotational degrees of freedom, (UM, VM, qZM), located at the floor master node “M” as shown in Fig. 3. The displacements along the global X and Y axes and the rotation about the Z axis of any point on the floor slab, e.g., node “D”, are independent. They are related to the degrees of freedom at the floor mater node “M”. Hence the displacements of node “D” is give as
Fig. 3.
A portion of core walls in a storey.
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冦冧 冤
冥冦 冧
UD
1 0 0 0 0 −(YD−YM) 0
VD
0 1 0 0 0 XD−XM
0
WD
0 0 1 0 0 0
0
WD
{uD}⫽ qXD ⫽[FD]{uD}C⫽ 0 0 0 1 0 0
0
qXD
UM
VM
qYD
0 0 0 0 1 0
0
qYD
qZD
0 0 0 0 0 1
0
qZM
⬘ D
0 0 0 0 0 0
1
q⬘D
q
(1)
in which [FD] is the floor transformation matrix of node “D”, (XD, YD) and (XM, YM) are the global X and Y coordinates of nodes “D” and “M”, respectively. The displacements in {uD}C are all independent. 4.2. Kinematic relationship of nodes in core wall section Using the assumptions employed in the thin-walled beam theory that thin-walled open cross-section remains undeformed and the middle surface is free from shearing deformations [3], the displacement functions for the middle surface can be derived. As shown in Fig. 4, the displacements at node “d” of thin-walled section are related to those at shear centre “D” in local coordinate “K” as:
Fig. 4.
Displacements of thin-walled section.
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udK⫽uD⫹zdqyD⫺ydqzD⫺wdq⬘xD
(2)
vdK⫽vD⫺zdqxD
(3)
wdK⫽wD⫹ydqxD
(4)
in which yd, zd and wd are y, z and warping coordinates of node “d”, respectively, qyD=⫺∂wD/∂x, and qzD=∂vD/∂x. Based on Eqs. (2)–(4), the relationship between displacements of node “d” and shear centre “D”, can be written in a matrix form:
冦冧 冤
zd −yd −wd
udK
1 0 0 0
vdK
0 1 0 −zd 0 0
0
wdK
0 0 1 yd
0 0
0
{ud}K⫽ qxdK ⫽[Rd]{uD}K⫽ 0 0 0 1
0 0
0
qydK
0 0 0 0
1 0
−yd
qzdK
0 0 0 0
0 1
−zd
⬘ xdK
0 0 0 0
0 0
1
q
冥冦 冧 uD
vD
wD
qxD
(5)
qyD qzD
q⬘xD
in which [Rd] is the cross-section transformation matrix of node “d”. 4.3. Transformation matrix of beam Fig. 5 shows beam “2” in local coordinate. Beam end “c” is rigidly connected to core wall. If beam end denoted as point “e” is pin-connected to core wall at node “d”, then the beam end rotation is different from that of node “d”. At beam end “e”, the relative rotation, qze, between end “e” and node “d” to which it is connected, is allowed for the major-axis rotational degree-of-freedom. For the convenience of matrix operation, warping degrees of freedom, q⬘xc and q⬘xe, are introduced at beam ends, and the 12×12 stiffness matrices of beam-column are extended to 14×14 matrices. The warping degrees of freedom are not included in global unknowns of the structure stiffness equation. Those items in the matrices associated with warping degrees of freedom are assigned to be of zero values. Based on Eqs. (1) and (5), the displacements of node “d” in coordinate “2”, {ud}2 are related to the controlling displacements of shear centre “D”, {uD}C: {ud}2⫽[ud2 vd2 wd2 qxd2 qyd2 qzd2 q⬘xd2]T ⫽[TRd]{uD}C
(6)
⫽[T2][TK]T [Rd][TK][FD]{uD}C where [TRd] is 7×7 transformation matrix of node “d”, and [T2] and [TK] are coordinate transformation matrices given by
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Fig. 5. Displacements of beam “2”.
冤
[gi] 0
[Ti]⫽ 0 0
冥 冤
0
冥
cosax cosbx cosdx
[gi] 0 , [gi]⫽ cosay cosby cosdy 0
1
cosaz cosbz cosdz
(7)
in which [gi] (i =2 or K) is the coordinate rotational matrix, a, b and d designate the angles between the subscriptal local axis and each of the global X, Y and Z axes, respectively. Similarly, displacements of node “c” in coordinate “2”, {uc}2, can be related to the controlling displacements of shear centre “B”, {uB}C: {uc}2⫽[uc vc wc qxc qyc qzc q⬘xc]T ⫽[TRc]{uB}C
(8)
⫽[T2][TJ]T [Rc][TJ][FB]{uB}C in which [TRc] is 7×7 transformation matrix of node “c”, [FB] floor transformation matrix of node “B”, [Rc] cross-section transformation matrix of node “c”, and [TJ] transformation matrix of coordinate “J”. Because beam end “e” is pin-connected to core wall, its displacements are the
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same as those of node “d” except the relative rotation, qze, about the major axis. The relative rotation is treated as an additional global unknown in the structure stiffness equation. Based on Eq. (6), the displacements of beam end “e”, {ue}2, can be related to the controlling displacements, {ue}C:
冤
[TR1d]5×7 0
{ue}2⫽[ue ve we qxe qye qze q⬘xe]T ⫽[TRe]{ue}C⫽ 0
1 2 d 1×7
[TR ]
冥再 冎
0
{uD}C
qze
(9)
in which [TRe] is 7×8 transformation matrix of beam end “e”, [TR1d] the first five rows of [TRd], and [TR2d] the seventh row of [TRd]. Based on Eqs. (8) and (9), the transformation matrix of beam “2” is given by [⌫2]14×15⫽
冋
[TRc] 0
0
[TRe]
册
(10)
which can be used to transform the stiffness matrix of beam “2” from the local coordinate to the global coordinate as [K2]⫽[⌫2]T [k2][⌫2]
(11)
in which [k2] is 14×14 tangent stiffness matrix of beam in the local coordinate. The transformation procedures demonstrated above can be modified for beams with pinned connections at both ends, and for beams connected to columns. All of these cases have been included in the computer implementation.
5. Solution procedures Iterative solution method combined with adaptive load increment control have been implemented to perform the nonlinear inelastic analysis [7]. The inelastic limit load of structure is reached if one or more negative terms in the diagonal of structure stiffness matrix are detected. To control exceedingly large incremental displacement for a softening structure, a maximum displacement increment may be specified. A characteristic displacement is calculated as a weighted sum of specified degrees of freedom. If a plastic hinge occurs during an incremental analysis, the load step is scaled so that the cross-sectional forces comply “exactly” with the plastic criteria. To avoid small step length in case of frequent occurrence of plastic hinges, a lower limit may be specified for the increment of the scaling. In this way, “exact” scaling to the plastic strength surface is not always possible and several plastic hinges may be inserted during one load increment. Nodal coordinates, and element forces and local coordinate system are updated at the end of each incremental step in the nonlinear analysis. Natural deformation
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approach proposed by Gattass and Abel [9] is adopted for the element force recovery. The element incremental displacements can be conceptually decomposed into two parts: the rigid body displacements and the natural deformations. The rigid body displacements serve to rotate the initial forces acting on element from the previous configuration to the current configuration; whereas the natural deformations constitute the only source for generating the incremental forces. The element forces at the current configuration can be calculated as the summation of incremental forces and the forces at the previous configuration.
6. Analysis of core tube model A 20-storey core tube model with the same storey height of 62.2 mm is subjected to unit torque, Mz, applied at the top. A plan view of the tube model with doubly symmetric section is shown in Fig. 6. Elastic modulus E =2965 N/mm2 and shear modulus G =1103 N/mm2. Pekau et al. carried out the first-order elastic analysis on the tube model by using the finite storey method [10]. The method is based on the nodal displacement fields obtained from the two-storey substructures to approximate the shear, bending and torsional components of the global deformations of tall building structures. There are five principal unknown displacement parameters in every floor. A rigid diaphragm is used to model the floor slab. Vlasov’s thin-walled beam theory is used to model the core tube. In the proposed analysis, each lintel beam is modelled as one beam-column element and a storey high core wall as one thin-walled beam-column element. The rigid diaphragm action is considered. The structure stiffness matrix is formed by using the transformation matrices considering the kinematics relationships between floor diaphragm, core walls and adjoining beams. First-order elastic analysis is per-
Fig. 6.
Plan view of core tube model.
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formed using the proposed computer program by switching off iteration in the analysis. For core tube model with lintel beams, the floor rotations about the vertical axis at every storey obtained by the proposed method are found to be consistent with those obtained from the finite storey method (see Fig. 7). The accuracy of the proposed method for modelling the core tube behaviour is therefore established. Analysis is carried out for core tube model without lintel beams. The floor rotation at the top of core tube model without lintel beams is seven times that of the core tube model with lintel beams as shown in Fig. 7. It is noted that lintel beams rigidly connected to core walls can increase the torsional stiffness of core significantly.
7. Analysis of frames with thin-walled cores Plan views of 24-storey frames with “strong” and “weak” reinforced concrete cores are shown in Figs. 8 and 9, respectively. The elevation is shown in Fig. 10. Thickness of “strong” core walls is 0.406 m. Thickness of “weak” core walls is 0.254 m. Concrete lintel beams of 1.219 m depth are rigidly connected to the two channel-shaped core walls to increase the torsional stiffness. For concrete cores, elastic modulus Ec =23 400 N/mm2, and compressive strength f ⬙c =23.4 N/mm2. A36 steel is used for all members in steel frameworks surrounding the cores. The structure is analysed for the most critical load combination of gravity loads 4.8 kN/m2, and wind loads 0.96 kN/m2 acting in the Y direction. All floors are assumed to be rigid in plan to account for the diaphragm action of concrete slabs.
Fig. 7.
Floor rotations about the vertical axis.
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Fig. 8.
Plan view of frame with “strong” cores.
Fig. 9.
Plan view of frame with “weak” cores.
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Fig. 10. Elevation of frame with thin-walled cores.
Each column is modelled as one beam-column element, and each beam as four beamcolumn elements. A storey high channel-shaped core wall is modelled as one thinwalled beam-column element. In the examples, core-walls are mainly subjected to axial force and bending moment about the principal z¯ axis, which is parallel to the global X axis. Plastic section modulus about the z¯ axis of channel-shaped section is Z =8.311 m3 for “strong” core wall, and Z =2.549 m3 for “weak” core wall. It is assumed that the bending capacity of core-wall sections is calculated to be Mz=0.8 Zf ⬙c, which has been reduced to account approximately for the effect of tensile cracking and axial force interaction. For “strong” core wall section Mz =1.57×105 kNm, and for “weak” core wall section Mz =4.8×104 kNm. 7.1. Frame with “strong” cores Second-order inelastic analysis carried out on frame with “strong” cores indicates that when plastic hinges form at the bottom of core walls, core-braced frame with
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pinned connections collapses at a load ratio of 2.082, as shown in Fig. 11. All beams satisfy the requirement of girder deflection, L/360, under the service live load. The load factor for wind load is 1.3, and the serviceability load ratio is 0.77. It can be seen from Fig. 11 that core-braced frame with pinned connections satisfies the lateral drift requirement, H/400, for service wind load. In order to assess the load resistance capacity of unbraced steel frameworks with rigid connections, elastic modulus and compressive strength of core walls are assigned to be of small values. The unbraced rigid frameworks collapse at a load ratio of 0.539, which is 26% of the limit load of core-braced frame with pinned connections. The load resistance capacity of cores is much higher than that of rigid steel frameworks. To study the effect of connection in steel frameworks, inelastic analysis is carried out on core-braced frame with rigid connections. First plastic hinges form in beams at a load ratio of 0.959, which is higher than the service load ratio. Shortly after plastic hinges form at the bottom of core walls at a load ratio of 2.567, the structure collapses at a load ratio of 2.633. Since rigid frameworks can provide lateral load resistance, the elastic lateral stiffness and inelastic limit load of core-braced frame with rigid connections are increased respectively by 85% and 26% over those of core-braced frame with pinned connections. Core-braced frame is normally designed to adopt “strong” cores to provide the lateral load resistance. The steel frameworks surrounding cores are generally designed only for gravity forces with simple shear connections. In building construc-
Fig. 11.
Load-displacement analysis of frame with “strong” cores.
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tion, all beam-to-column connections possess some stiffness that falls between the two extreme cases of fully rigid and ideally pinned. It is expected that connection effect will provide strength and stiffness reserve for core-braced system to resist the lateral load. 7.2. Frame with “weak” cores The space occupied by “strong” cores may lead to the loss of overall floor area. One solution is to adopt “weak” cores to reduce the dimension of core wall. The steel frames will need to participate in reducing the lateral deflection. Inelastic analysis carried out on frame with “weak” cores indicates that core-braced frame with pinned connections collapses at a load ratio of 0.641 as shown in Fig. 12. The elastic lateral stiffness and inelastic limit load of unbraced rigid frameworks are 40% and 2% higher than those of core-braced frame with pinned connections. If core-braced frame with rigid connections is adopted, first plastic hinges form in beams at a load ratio of 1.176. Plastic hinges form at the bottom of core walls at a load ratio of 1.527. The structure has strength reserve beyond the formation of plastic hinges in core walls. Plastic hinges spread in beams and columns with the increase of load. The structure reaches the inelastic limit load at a load ratio of 1.787
Fig. 12.
Load-displacement analysis of frame with “weak” cores.
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as shown in Fig. 12. Core-braced frame with rigid connections satisfies the lateral drift requirement for service wind load, and can be adopted. It is noted that the inelastic limit load of core-braced frame with rigid connections is 28% higher than the summation of limit load of unbraced rigid frameworks and limit load of core-braced frame with pinned connections. This is due to the interaction between the frameworks and the core walls. The frameworks basically deflect in a shear mode while the core walls predominantly respond by bending as a cantilever. Compatibility of horizontal deflection introduces interaction between the two systems, which tends to reduce the lateral deflection and increases the load resistance capacity. However in core-braced with “strong” cores, the interaction is not obvious because the stiffness of cores is much higher than that of frameworks, and the lateral load resistance is mainly provide by cores. In building construction, site assembly is easier with semi-rigid connections than with rigid connections. Thus, easy erection and speed of construction will lead to the early completion of project. From economical point of view, it is necessary to reduce the number of moment connections in steel frameworks. The use of semirigid connections in core-braced frames may increase the lateral stiffness and inelastic limit load, and at the same time reduce the cost of construction.
8. Conclusions This paper presents a second-order inelastic analysis for building frames with thinwalled cores. The direct second-order inelastic approach can consider both geometrical and material nonlinearities of steel beam-columns and core walls. It provides better insight into the structural behaviour up to failure, and the strength and stability interaction between members and structural system. By using nonlinear inelastic analysis in conjunction with the limit states design specifications, structural engineers will have the opportunity to base design on realistic estimate of the nonlinear behaviour, and hence to achieve design objectives such as economy, system ductility and uniform safety factor. Core-braced frame is normally designed to rely on “strong” cores to provide lateral stability. The steel frameworks are designed only for gravity forces with simple shear connections. The limit load of core-braced frame with “strong” cores is controlled by the resistance capacity of core walls at the bottom storey. If frame with “weak” cores is adopted to increase the interior space, the steel frameworks will need to participate in reducing the lateral deflection. Due to the interaction between frameworks and core walls, the inelastic limit load of rigid core-braced frame is 28% higher than the summation of limit load of unbraced rigid frameworks and limit load of core-braced frame with pinned connections. However for rigid core-braced frame with “strong” cores, the interaction is not obvious because the core stiffness is much higher than that of the steel frameworks. In steel construction, all beam-to-column connections possess some stiffness that falls between the two extreme cases of fully rigid and ideally pinned. For frames with “strong” cores, connections in steel frameworks can provide strength and stiff-
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ness reserve for core-braced frames to resist the lateral loads. For frames with “weak” cores, the use of proper semi-rigid connections in steel frameworks may provide the optimum balance between the dual objectives of buildability and functionality.
Acknowledgements The research work reported in this paper is funded by the research grants (RP 940661 and RP 960648) made available by the National University of Singapore.
References [1] Iyengar SH. Mixed steel-concrete high-rise systems. In: Sabnis GM, editor. Handbook of Composite Construction Engineering, Chapter 7. New York: Van Nostrand Reinhold Company, 1979. [2] Taranath BS. Structural Analysis and Design of Tall Buildings. Singapore: McGraw-Hill, 1988. [3] Vlasov VZ. Thin-walled elastic beams, 2nd ed. Moscow, USSR; English translation, Israel Program for Scientific Translation, Jerusalem, Israel, 1961. [4] Liew JYR, Chen H, Shanmugam NE. Stability functions for second-order inelastic analysis of space frames. Proceedings 4th International Conference on Steel and Aluminium Structures, Espoo, Finland, June 20–23 1999:19–26. [5] Hilmy AI, Abel JF. Material and geometric nonlinear dynamic analysis of steel frames using computer graphics. Comp Struct 1985;21(4):825–40. [6] Liew JYR, Tang LK. Nonlinear refined plastic hinge analysis of space frame structures, Research Report No. CE029/99. Department of Civil Engineering, National University of Singapore, 1998. [7] Liew JYR, Chen H, Yu CH, Shanmugam NE. Advanced inelastic analysis of thin-walled core-braced frames. Proceedings of the Second International Conference on Thin-Walled Structures, Singapore, December 2–4, 1998:485–492. [8] Conci A. Large displacement analysis of thin-walled beams with generic open section. Int J Numer Meth Eng 1992;33:2109–27. [9] Gattas M, Abel JF. Equilibrium considerations of the Updated Lagrangian formulation of beamcolumns with natural concepts. Int J Numer Meth Eng 1987;24:2119–41. [10] Pekau OA, Lin L, Zielinski ZA. Static and dynamic analysis of tall tube-in-tube structures by finite storey method. Eng Struct 1996;18(7):515–27.
Nonlinear inelastic analysis of building frames with thin-walled cores H. Chen, J.Y. Richard Liew, N.E. Shanmugam
*
Department of Civil Engineering, National University of Singapore, 10 Kent Ridge Crescent, 119260 Singapore Received 27 January 2000; accepted 10 March 2000
Abstract A second-order inelastic analysis by combining the theories of stability and plasticity is proposed for studying frames with thin-walled cores. In the proposed approach, steel frameworks surrounding the cores are modelled by using the plastic hinge beam-column approach, and core walls modelled by using the thin-walled beam-column approach. Transformation procedures are proposed to consider the kinematic relationship between beams, columns, core walls and floor diaphragm. Nonlinear solution procedures are incorporated for the incremental analysis. The proposed inelastic analysis is used to investigate the inelastic behaviour and ultimate strength of core-braced frames. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Inelastic analysis; Stability; Plastic hinge; Beam-column; Thin-walled core; Transformation matrix
1. Introduction Core-braced frame represents an efficient type of structural system up to a certain height where the predominant lateral load resistance is provided by concrete cores [1,2]. The steel frameworks surrounding the cores are generally designed only for gravity forces with simple shear connections. In high seismic areas, the steel frameworks would be required to resist some part of the earthquake loads as a moment frame to provide a second line of defence. The most rigorous way of modelling core walls is by the finite element method * Corresponding author. Tel.: +65-722-2288; fax: +65-779-1635. E-mail address: [email protected] (N.E. Shanmugam). 0263-8231/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 0 0 ) 0 0 0 1 6 - 1
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involving the use of plate and shell elements. However, this approach is computationally intensive and costly because it involves discretization of walls into a large number of elements. Having noted the proportional similarity of core walls to Vlasov’s thin-walled beam [3], Taranath proposed the thin-walled beam-column approach for the first-order elastic analysis of core-braced frames with computational efficiency and sufficient accuracy [2]. Although the first-order elastic analysis provides a simple estimate of the distribution of forces within the structural system, it does not provide any information about the influence of either stability or plasticity effect on the behaviour of frame system. This paper proposes a second-order inelastic analysis, which can capture the geometrical and material nonlinearities of frames with thin-walled cores up to the inelastic limit load. Steel frameworks surrounding the cores are modelled by using the plastic hinge beam-column approach, and core walls modelled by using the thinwalled beam-column approach. Kinematic relationships between beams, columns, thin-walled cores, and floor diaphragm are derived to establish the transformation matrices for the incremental analysis. Nonlinear solution procedures are incorporated for the incremental analysis. The elastic behaviour of a core tube model is analysed by using the proposed approach. Lastly, the inelastic behaviour and inelastic limit load of core-braced frames are investigated using the proposed analysis theory and computer program. 2. Modelling of steel frameworks Steel frameworks surrounding the cores are modelled by using the beam-column element as shown in Fig. 1. The beam-column formulation is based on the updated Lagrangian approach. Transverse displacements are calculated by using the stability interpolation functions, which satisfy the fourth-order differential equation of beamcolumn subjected to end forces [4]. The influence of axial force on transverse displacements can be accurately represented in the form of stability functions. The coupling effect between compression, bending and torsion is included by using the nonlinear strain relationships in the virtual work equation. Through the use of proposed bowing matrix [4], the beam-column formulation can capture the member bowing effect and initial out-of-straightness by modelling each physical member as one element. A two-surface plasticity model, which employs the yield surface and the bounding surface, is used to represent partial yielding and hardening in steel beam-columns [5,6]. The yield surface bounds the region of elastic cross-sectional behaviour, while the bounding surface defines the state of full plastification of cross-section. The bounding surface encloses the state of cross-sectional force and yield surface at any stage during the yielding process. The size of initial yield surface is described by the surface extension parameter. Suitable value of surface extension parameter can be selected to model the effect of initial residual stress in cross-section. Once yielding is initiated, the yield surface will translate so that the state of cross-sectional forces remains on the yield surface during subsequent loading. This type of approach is termed as “refined” plastic hinge analysis [6].
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Fig. 1.
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Beam-column element.
Inelastic member buckling can be predicted by allowing plastic hinge to occur between the member ends [4]. The analysis will automatically subdivide the original element into two sub-elements at the location of plastic hinge. The internal hinge is then modelled by an end hinge at one of the sub-elements. The stiffness matrices of the two sub-elements are determined. The inelastic stiffness properties for the original element are obtained by the static condensation of extra node at the location of internal plastic hinge.
3. Modelling of core walls Core-walls are modelled by using the thin-walled beam-column approach [2,7]. As shown in Fig. 2, the thin-walled beam-column element has an additional warping degree-of-freedom over beam-column element at each end. The local coordinate is chosen with axis x lying on the shear centre axis, and y and z axes paralleling to the principal y¯ and z¯ axes. Some force and displacement components are referred to the shear centre, whereas the remaining ones are referred to the centroid. However, before stiffness matrices are transformed into the global coordinate, it is necessary that all force and displacement components are referred to a single point. The shear centre is selected as the reference point.
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Fig. 2. Thin-walled beam-column element.
Geometrical nonlinearity of core walls is modelled through the use of geometric stiffness matrix of thin-walled beam-column element proposed by Conci [8], which contains more initial terms than the existing geometric matrices in the current literature. The geometric matrix is consisted of two parts: the first part for a doubly symmetric cross-section and the second part, which should be added when the crosssection is asymmetric. Material nonlinearity of the core wall is modelled approximately by using the concentrated plastic hinge approach [7]. Because the height-towidth ratio of core-wall is generally large in practical buildings, it is assumed that the plastic resistance of cross-section is controlled by bending action only. The locations of shear centre and centroid of cross-section are assumed not to change due to yielding in the cross-section. It is noted that the yielding in the core wall section depends on the combined action of compression, biaxial bending, torsion and warping. It is necessary to model these factors if core wall is subjected to significant torsional and warping effects.
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4. Transformation procedures The kinematic relationships between beams, columns, thin-walled cores and floor diaphragm are proposed for the global incremental analysis. The conventional beamcolumn transformation matrix is modified to transform the stiffness matrices of beams from the local coordinate to the global coordinate with some of the unknowns retaining their local coordinate reference. The proposed procedure can facilitate further modifications to account for modelling of semi-rigid connection at beam ends. 4.1. Floor transformation Fig. 3 shows a portion of channel-shaped core walls “J” and “K” connected by beams “1” and “2” between floors “L” and “N”. Beams are modelled as beam-column elements “ab” and “cd”, and core walls as thin-walled beam-column elements “AB” and “CD” with the longitudinal axes lying on the shear centre axes. (x2, y2, z2) and (xK, yK, zK) are the local coordinates of beam “2” and core wall “K”, respectively. In the case of rigid diaphragm model, which is assumed to provide infinite inplane stiffness but without any out-of-plane stiffness, the lateral response of floor “N” is characterised by two translational and one rotational degrees of freedom, (UM, VM, qZM), located at the floor master node “M” as shown in Fig. 3. The displacements along the global X and Y axes and the rotation about the Z axis of any point on the floor slab, e.g., node “D”, are independent. They are related to the degrees of freedom at the floor mater node “M”. Hence the displacements of node “D” is give as
Fig. 3.
A portion of core walls in a storey.
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冦冧 冤
冥冦 冧
UD
1 0 0 0 0 −(YD−YM) 0
VD
0 1 0 0 0 XD−XM
0
WD
0 0 1 0 0 0
0
WD
{uD}⫽ qXD ⫽[FD]{uD}C⫽ 0 0 0 1 0 0
0
qXD
UM
VM
qYD
0 0 0 0 1 0
0
qYD
qZD
0 0 0 0 0 1
0
qZM
⬘ D
0 0 0 0 0 0
1
q⬘D
q
(1)
in which [FD] is the floor transformation matrix of node “D”, (XD, YD) and (XM, YM) are the global X and Y coordinates of nodes “D” and “M”, respectively. The displacements in {uD}C are all independent. 4.2. Kinematic relationship of nodes in core wall section Using the assumptions employed in the thin-walled beam theory that thin-walled open cross-section remains undeformed and the middle surface is free from shearing deformations [3], the displacement functions for the middle surface can be derived. As shown in Fig. 4, the displacements at node “d” of thin-walled section are related to those at shear centre “D” in local coordinate “K” as:
Fig. 4.
Displacements of thin-walled section.
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udK⫽uD⫹zdqyD⫺ydqzD⫺wdq⬘xD
(2)
vdK⫽vD⫺zdqxD
(3)
wdK⫽wD⫹ydqxD
(4)
in which yd, zd and wd are y, z and warping coordinates of node “d”, respectively, qyD=⫺∂wD/∂x, and qzD=∂vD/∂x. Based on Eqs. (2)–(4), the relationship between displacements of node “d” and shear centre “D”, can be written in a matrix form:
冦冧 冤
zd −yd −wd
udK
1 0 0 0
vdK
0 1 0 −zd 0 0
0
wdK
0 0 1 yd
0 0
0
{ud}K⫽ qxdK ⫽[Rd]{uD}K⫽ 0 0 0 1
0 0
0
qydK
0 0 0 0
1 0
−yd
qzdK
0 0 0 0
0 1
−zd
⬘ xdK
0 0 0 0
0 0
1
q
冥冦 冧 uD
vD
wD
qxD
(5)
qyD qzD
q⬘xD
in which [Rd] is the cross-section transformation matrix of node “d”. 4.3. Transformation matrix of beam Fig. 5 shows beam “2” in local coordinate. Beam end “c” is rigidly connected to core wall. If beam end denoted as point “e” is pin-connected to core wall at node “d”, then the beam end rotation is different from that of node “d”. At beam end “e”, the relative rotation, qze, between end “e” and node “d” to which it is connected, is allowed for the major-axis rotational degree-of-freedom. For the convenience of matrix operation, warping degrees of freedom, q⬘xc and q⬘xe, are introduced at beam ends, and the 12×12 stiffness matrices of beam-column are extended to 14×14 matrices. The warping degrees of freedom are not included in global unknowns of the structure stiffness equation. Those items in the matrices associated with warping degrees of freedom are assigned to be of zero values. Based on Eqs. (1) and (5), the displacements of node “d” in coordinate “2”, {ud}2 are related to the controlling displacements of shear centre “D”, {uD}C: {ud}2⫽[ud2 vd2 wd2 qxd2 qyd2 qzd2 q⬘xd2]T ⫽[TRd]{uD}C
(6)
⫽[T2][TK]T [Rd][TK][FD]{uD}C where [TRd] is 7×7 transformation matrix of node “d”, and [T2] and [TK] are coordinate transformation matrices given by
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Fig. 5. Displacements of beam “2”.
冤
[gi] 0
[Ti]⫽ 0 0
冥 冤
0
冥
cosax cosbx cosdx
[gi] 0 , [gi]⫽ cosay cosby cosdy 0
1
cosaz cosbz cosdz
(7)
in which [gi] (i =2 or K) is the coordinate rotational matrix, a, b and d designate the angles between the subscriptal local axis and each of the global X, Y and Z axes, respectively. Similarly, displacements of node “c” in coordinate “2”, {uc}2, can be related to the controlling displacements of shear centre “B”, {uB}C: {uc}2⫽[uc vc wc qxc qyc qzc q⬘xc]T ⫽[TRc]{uB}C
(8)
⫽[T2][TJ]T [Rc][TJ][FB]{uB}C in which [TRc] is 7×7 transformation matrix of node “c”, [FB] floor transformation matrix of node “B”, [Rc] cross-section transformation matrix of node “c”, and [TJ] transformation matrix of coordinate “J”. Because beam end “e” is pin-connected to core wall, its displacements are the
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same as those of node “d” except the relative rotation, qze, about the major axis. The relative rotation is treated as an additional global unknown in the structure stiffness equation. Based on Eq. (6), the displacements of beam end “e”, {ue}2, can be related to the controlling displacements, {ue}C:
冤
[TR1d]5×7 0
{ue}2⫽[ue ve we qxe qye qze q⬘xe]T ⫽[TRe]{ue}C⫽ 0
1 2 d 1×7
[TR ]
冥再 冎
0
{uD}C
qze
(9)
in which [TRe] is 7×8 transformation matrix of beam end “e”, [TR1d] the first five rows of [TRd], and [TR2d] the seventh row of [TRd]. Based on Eqs. (8) and (9), the transformation matrix of beam “2” is given by [⌫2]14×15⫽
冋
[TRc] 0
0
[TRe]
册
(10)
which can be used to transform the stiffness matrix of beam “2” from the local coordinate to the global coordinate as [K2]⫽[⌫2]T [k2][⌫2]
(11)
in which [k2] is 14×14 tangent stiffness matrix of beam in the local coordinate. The transformation procedures demonstrated above can be modified for beams with pinned connections at both ends, and for beams connected to columns. All of these cases have been included in the computer implementation.
5. Solution procedures Iterative solution method combined with adaptive load increment control have been implemented to perform the nonlinear inelastic analysis [7]. The inelastic limit load of structure is reached if one or more negative terms in the diagonal of structure stiffness matrix are detected. To control exceedingly large incremental displacement for a softening structure, a maximum displacement increment may be specified. A characteristic displacement is calculated as a weighted sum of specified degrees of freedom. If a plastic hinge occurs during an incremental analysis, the load step is scaled so that the cross-sectional forces comply “exactly” with the plastic criteria. To avoid small step length in case of frequent occurrence of plastic hinges, a lower limit may be specified for the increment of the scaling. In this way, “exact” scaling to the plastic strength surface is not always possible and several plastic hinges may be inserted during one load increment. Nodal coordinates, and element forces and local coordinate system are updated at the end of each incremental step in the nonlinear analysis. Natural deformation
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approach proposed by Gattass and Abel [9] is adopted for the element force recovery. The element incremental displacements can be conceptually decomposed into two parts: the rigid body displacements and the natural deformations. The rigid body displacements serve to rotate the initial forces acting on element from the previous configuration to the current configuration; whereas the natural deformations constitute the only source for generating the incremental forces. The element forces at the current configuration can be calculated as the summation of incremental forces and the forces at the previous configuration.
6. Analysis of core tube model A 20-storey core tube model with the same storey height of 62.2 mm is subjected to unit torque, Mz, applied at the top. A plan view of the tube model with doubly symmetric section is shown in Fig. 6. Elastic modulus E =2965 N/mm2 and shear modulus G =1103 N/mm2. Pekau et al. carried out the first-order elastic analysis on the tube model by using the finite storey method [10]. The method is based on the nodal displacement fields obtained from the two-storey substructures to approximate the shear, bending and torsional components of the global deformations of tall building structures. There are five principal unknown displacement parameters in every floor. A rigid diaphragm is used to model the floor slab. Vlasov’s thin-walled beam theory is used to model the core tube. In the proposed analysis, each lintel beam is modelled as one beam-column element and a storey high core wall as one thin-walled beam-column element. The rigid diaphragm action is considered. The structure stiffness matrix is formed by using the transformation matrices considering the kinematics relationships between floor diaphragm, core walls and adjoining beams. First-order elastic analysis is per-
Fig. 6.
Plan view of core tube model.
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formed using the proposed computer program by switching off iteration in the analysis. For core tube model with lintel beams, the floor rotations about the vertical axis at every storey obtained by the proposed method are found to be consistent with those obtained from the finite storey method (see Fig. 7). The accuracy of the proposed method for modelling the core tube behaviour is therefore established. Analysis is carried out for core tube model without lintel beams. The floor rotation at the top of core tube model without lintel beams is seven times that of the core tube model with lintel beams as shown in Fig. 7. It is noted that lintel beams rigidly connected to core walls can increase the torsional stiffness of core significantly.
7. Analysis of frames with thin-walled cores Plan views of 24-storey frames with “strong” and “weak” reinforced concrete cores are shown in Figs. 8 and 9, respectively. The elevation is shown in Fig. 10. Thickness of “strong” core walls is 0.406 m. Thickness of “weak” core walls is 0.254 m. Concrete lintel beams of 1.219 m depth are rigidly connected to the two channel-shaped core walls to increase the torsional stiffness. For concrete cores, elastic modulus Ec =23 400 N/mm2, and compressive strength f ⬙c =23.4 N/mm2. A36 steel is used for all members in steel frameworks surrounding the cores. The structure is analysed for the most critical load combination of gravity loads 4.8 kN/m2, and wind loads 0.96 kN/m2 acting in the Y direction. All floors are assumed to be rigid in plan to account for the diaphragm action of concrete slabs.
Fig. 7.
Floor rotations about the vertical axis.
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Fig. 8.
Plan view of frame with “strong” cores.
Fig. 9.
Plan view of frame with “weak” cores.
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Fig. 10. Elevation of frame with thin-walled cores.
Each column is modelled as one beam-column element, and each beam as four beamcolumn elements. A storey high channel-shaped core wall is modelled as one thinwalled beam-column element. In the examples, core-walls are mainly subjected to axial force and bending moment about the principal z¯ axis, which is parallel to the global X axis. Plastic section modulus about the z¯ axis of channel-shaped section is Z =8.311 m3 for “strong” core wall, and Z =2.549 m3 for “weak” core wall. It is assumed that the bending capacity of core-wall sections is calculated to be Mz=0.8 Zf ⬙c, which has been reduced to account approximately for the effect of tensile cracking and axial force interaction. For “strong” core wall section Mz =1.57×105 kNm, and for “weak” core wall section Mz =4.8×104 kNm. 7.1. Frame with “strong” cores Second-order inelastic analysis carried out on frame with “strong” cores indicates that when plastic hinges form at the bottom of core walls, core-braced frame with
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pinned connections collapses at a load ratio of 2.082, as shown in Fig. 11. All beams satisfy the requirement of girder deflection, L/360, under the service live load. The load factor for wind load is 1.3, and the serviceability load ratio is 0.77. It can be seen from Fig. 11 that core-braced frame with pinned connections satisfies the lateral drift requirement, H/400, for service wind load. In order to assess the load resistance capacity of unbraced steel frameworks with rigid connections, elastic modulus and compressive strength of core walls are assigned to be of small values. The unbraced rigid frameworks collapse at a load ratio of 0.539, which is 26% of the limit load of core-braced frame with pinned connections. The load resistance capacity of cores is much higher than that of rigid steel frameworks. To study the effect of connection in steel frameworks, inelastic analysis is carried out on core-braced frame with rigid connections. First plastic hinges form in beams at a load ratio of 0.959, which is higher than the service load ratio. Shortly after plastic hinges form at the bottom of core walls at a load ratio of 2.567, the structure collapses at a load ratio of 2.633. Since rigid frameworks can provide lateral load resistance, the elastic lateral stiffness and inelastic limit load of core-braced frame with rigid connections are increased respectively by 85% and 26% over those of core-braced frame with pinned connections. Core-braced frame is normally designed to adopt “strong” cores to provide the lateral load resistance. The steel frameworks surrounding cores are generally designed only for gravity forces with simple shear connections. In building construc-
Fig. 11.
Load-displacement analysis of frame with “strong” cores.
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tion, all beam-to-column connections possess some stiffness that falls between the two extreme cases of fully rigid and ideally pinned. It is expected that connection effect will provide strength and stiffness reserve for core-braced system to resist the lateral load. 7.2. Frame with “weak” cores The space occupied by “strong” cores may lead to the loss of overall floor area. One solution is to adopt “weak” cores to reduce the dimension of core wall. The steel frames will need to participate in reducing the lateral deflection. Inelastic analysis carried out on frame with “weak” cores indicates that core-braced frame with pinned connections collapses at a load ratio of 0.641 as shown in Fig. 12. The elastic lateral stiffness and inelastic limit load of unbraced rigid frameworks are 40% and 2% higher than those of core-braced frame with pinned connections. If core-braced frame with rigid connections is adopted, first plastic hinges form in beams at a load ratio of 1.176. Plastic hinges form at the bottom of core walls at a load ratio of 1.527. The structure has strength reserve beyond the formation of plastic hinges in core walls. Plastic hinges spread in beams and columns with the increase of load. The structure reaches the inelastic limit load at a load ratio of 1.787
Fig. 12.
Load-displacement analysis of frame with “weak” cores.
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as shown in Fig. 12. Core-braced frame with rigid connections satisfies the lateral drift requirement for service wind load, and can be adopted. It is noted that the inelastic limit load of core-braced frame with rigid connections is 28% higher than the summation of limit load of unbraced rigid frameworks and limit load of core-braced frame with pinned connections. This is due to the interaction between the frameworks and the core walls. The frameworks basically deflect in a shear mode while the core walls predominantly respond by bending as a cantilever. Compatibility of horizontal deflection introduces interaction between the two systems, which tends to reduce the lateral deflection and increases the load resistance capacity. However in core-braced with “strong” cores, the interaction is not obvious because the stiffness of cores is much higher than that of frameworks, and the lateral load resistance is mainly provide by cores. In building construction, site assembly is easier with semi-rigid connections than with rigid connections. Thus, easy erection and speed of construction will lead to the early completion of project. From economical point of view, it is necessary to reduce the number of moment connections in steel frameworks. The use of semirigid connections in core-braced frames may increase the lateral stiffness and inelastic limit load, and at the same time reduce the cost of construction.
8. Conclusions This paper presents a second-order inelastic analysis for building frames with thinwalled cores. The direct second-order inelastic approach can consider both geometrical and material nonlinearities of steel beam-columns and core walls. It provides better insight into the structural behaviour up to failure, and the strength and stability interaction between members and structural system. By using nonlinear inelastic analysis in conjunction with the limit states design specifications, structural engineers will have the opportunity to base design on realistic estimate of the nonlinear behaviour, and hence to achieve design objectives such as economy, system ductility and uniform safety factor. Core-braced frame is normally designed to rely on “strong” cores to provide lateral stability. The steel frameworks are designed only for gravity forces with simple shear connections. The limit load of core-braced frame with “strong” cores is controlled by the resistance capacity of core walls at the bottom storey. If frame with “weak” cores is adopted to increase the interior space, the steel frameworks will need to participate in reducing the lateral deflection. Due to the interaction between frameworks and core walls, the inelastic limit load of rigid core-braced frame is 28% higher than the summation of limit load of unbraced rigid frameworks and limit load of core-braced frame with pinned connections. However for rigid core-braced frame with “strong” cores, the interaction is not obvious because the core stiffness is much higher than that of the steel frameworks. In steel construction, all beam-to-column connections possess some stiffness that falls between the two extreme cases of fully rigid and ideally pinned. For frames with “strong” cores, connections in steel frameworks can provide strength and stiff-
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ness reserve for core-braced frames to resist the lateral loads. For frames with “weak” cores, the use of proper semi-rigid connections in steel frameworks may provide the optimum balance between the dual objectives of buildability and functionality.
Acknowledgements The research work reported in this paper is funded by the research grants (RP 940661 and RP 960648) made available by the National University of Singapore.
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