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An economical structural system for wind and earthquake loads
Engineering Structures 23 (2001) 491–501 www.elsevier.com/locate/engstruct
An economical structural system for wind and earthquake loads T. Balendra a
a,*
, C.H. Yu a, F.L. Lee
b
Department of Civil Engineering, National University of Singapore, Kent Ridge, Singapore 117576, Singapore b Furnovus Innovations, Singapore Received 23 February 1999; accepted 26 July 1999
Abstract An economical structural system which would mitigate undesirable vibrations due to wind and earthquake loads is being developed. In this proposed system, the inherent hysteretic damping of the knee–brace-frame (KBF) with shear yielding knee is tapped to maintain the structural integrity in the event of a severe earthquake, while the frictional damping (in the form of slotted bolted connection, SBC) in the brace of KBF is utilised to dissipate energy in order to meet the serviceability requirements during severe wind storms or moderate earthquakes. As SBC dissipates energy without causing permanent damage, part of the strength required for serviceability design can be reduced by activating the SBC at an appropriate slip force. This would economise the material required for serviceability design. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Steel frame; Seismic; Wind
1. Introduction Generally, the design of buildings must satisfy two criteria. First, under frequently occurring low to moderate earthquakes or severe wind storm, the structure should have sufficient strength and stiffness to control deflection and prevent any structural damage. Second, under rare and severe earthquakes, the structure must have sufficient ductility to prevent collapse. The commonly used structural systems such as a momentresisting frame (MRF), concentric braced frame (CBF), eccentric braced frame [1,2] (EBF) and knee–braceframe [3] (KBF) are designed to resist the moderate loadings in the linear elastic range while the severe loadings are resisted through plastic hinges. The conventional MRFs and CBFs do not satisfy both criteria economically. A MRF is an excellent energy dissipating system, but it has to be designed with larger beam sections in order to develop sufficient stiffness to prevent excessive drift. The CBF is much stiffer than an MRF with similar sections, but it has poor energy dissipating capability due to the buckling of the brace. By connecting the brace eccentrically to the beam– column joint, EBF [1,2] combines the good features of * Corresponding author. Tel.: +65-874-2149; fax: +65-779-1635.
MRF and CBF. It dissipates energy through the development of shear hinges in the beam. As the beam is a principal member of the frame, the repair is very costly and difficult. The other system, KBF proposed by Balendra et al. [3], also combines the advantages of the MRF and CBF. In this system, the brace is connected to the knee, which is a secondary member, instead of the beam–column joint. The energy is dissipated through shear yielding of the knee while the brace is designed to prevent buckling. Since it is the knee that yields when a severe earthquake strikes, the cost of repairing the structure is limited to replacing the knee member only. Various types of energy dissipating devices, utilising friction as a means of energy dissipation, have been tested and studied by researchers [4–6]. These devices require precision workmanship and exotic materials for manufacturing, and specialised training in installation. As a consequence, they are not widely accepted in engineering practice. The slotted bolted connection (SBC) proposed by Grigorian et al. [7] as an energy dissipator represents an attempt to overcome the abovementioned shortcomings of these devices. The SBCs require only a slight modification of standard construction practice, and require materials that are widely available commercially. In this study, a SBC is incorporated into the brace of a KBF. The behaviour of the KBF with slotted bolted
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connections is investigated for its performance under seismic and wind loadings. One storey KBF with slotted bolted connections was tested using a pseudo-dynamic test method [8]. The SBC in the brace was activated based on predetermined slip force. During severe seismic excitation, the knee was brought into inelastic action by preventing the slip as the drift approached the serviceability limit. Based on the experimental results, an analytical model for trilinear force–displacement behaviour of the brace with SBC is developed and incorporated as a subroutine in DRAIN-2D [9], a general purpose structural dynamic analysis program.
2. General characteristics 2.1. Knee braced frame The diagonal brace of the KBF is connected to a knee element instead of a beam–column joint as shown in Fig. 1. The knee element is designed to dissipate energy under severe dynamic loading and therefore brace buckling or yielding of other members of the frame is prevented. A short knee member will yield in shear, while a longer knee yields in flexure. A KBF with a knee designed to yield in shear has larger ductility than one designed to yield in flexure [3]. In this study, the KBF with a shear yielding knee is investigated when the brace is incorporated with a SBC. 2.2. Slotted bolted connection In the slotted bolted connection shown in Fig. 2, the elongated holes or slots in the main connecting plate are parallel to the line of loading. The main connecting plate is placed between two outer members. A friction lining pad is ‘sandwiched’ between each outer member and the main plate, and the lining pad always moves with the
Fig. 1.
Details of the test frame.
Fig. 2.
Basic set-up of the slotted bolted connection.
outer member. All the components are clamped together through bolts. Upon tightening the bolts, frictions develop between the contact surfaces of the lining pads and the slotted plate. When the tensile or compressive force applied to the connection exceeds the impending frictional forces, the slotted plate slips relative to the lining pads. Energy is dissipated by means of friction between the sliding surfaces. The knee braced frame with a slotted bolted connection retains the main features of KBF except the brace– column connection. The brace is connected to the column through a gusset plate with a slotted bolted connection. When the brace force exceeds the impending frictional force of the SBC, the brace slips with respect to the SBC. The impending frictional force is always kept below the brace force required to yield the knee. In this way, the knee is always elastic when the SBC is slipping. The maximum slip distance of the SBC is governed by the slot length. Once the SBC hits the slot ends, the brace force builds up. The knee yields when the brace force exceeds the shear yielding capacity of the knee. Therefore, by altering the slot length of the SBC, the drift at which the energy dissipating capability of the knee is activated, can be controlled.
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3. Design of the test frame The test frame is the same as the test 2 specimen investigated by Balendra et al. [3] except for the inclusion of SBC. The design and section sizes of various members of the test frame are described below. 3.1. Beam, column and brace The beam and columns are wide flange sections 100×100×17.2 kg/m and 125×125×23.8 kg/m, respectively. These members are designed to remain elastic even when the knee member is loaded up to failure. The brace is designed to take cyclic loadings without buckling. It is made up of two C-channels (100×50×5×9.36) connected back-to-back with a 16 mm gap in between by 100×100×12 mm thick batten at 500 mm spacing.
Details of the knee.
3.3. Slotted bolted connection
3.2. Knee To ensure that the knee member yields in shear instead of moment, the longer of the two segments of the knee member generated by the intersection of the diagonal brace and the knee member, denoted as lk, should satisfy the following condition M∗p lk⬍2 Vp
(1)
where M∗p is the reduced plastic moment contributed by flanges only and Vp the plastic shear force. They are defined as follows M∗p ⫽tfb(d⫺tf)sy Vp⫽tw(d⫺tf)
Fig. 3.
sy
冑3
(2)
The SBC is located at the lower corner of the frame, serving the connection between the brace and the column. The brace is connected to the column by being bolted to a gusset plate which is a 310×330×10 mm thick stainless steel plate with six elongated slots parallel to the brace as shown in Fig. 4. The gusset plate is bolted to the column flange and base for the ease of removing it for alteration. It serves as the main plate in the SBC. And the brace, which is made up of two C-channels, acts as the outer member in the SBC. Between each C-channel and the gusset plate there is a 3 mm thick brass shim inserted as the friction lining pad. The gusset plate, C-channels and brass shims are bolted together through the two narrower slots in the gusset plate. In such a way, friction develops between the contact surfaces of the brass shim and the gusset
(3)
in which sy, d, tf, b and tw are, respectively, the yield stress, depth, flange thickness, flange width and web thickness of the knee member. A built-up 50×50 mm I-section which has a flange thickness of 6.0 mm and web thickness of 4.4 mm is used for the knee member. The yield stress of the steel used is 300 N/mm2. By employing Eqs. (1)–(3), the limiting value of lk, for the knee member to yield in shear is found to be 236 mm. The longer portion of the knee is the lower segment which is taken to be 180 mm. It is smaller than the limiting value of 236 mm, and so the knee member yields in shear. The section chosen is also prevented from local and lateral buckling. The knee is designed to avoid premature buckling of the web by adding web stiffeners with maximum spacing of 90 mm. Fig. 3 shows the details of the knee.
Fig. 4.
Details of the modified SBC for the test frame.
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plate. The amount of friction developed is proportional to the tightness of the bolts which in turn is proportional to the tightening torque. Thus, by adjusting the torque applied to the bolts, the desired impending frictional force can be achieved. When the brace force exceeds the impending frictional forces, the brace slips together with the brass shim from the gusset plate. There are stopper blocks to control the maximum slip distance of the SBC. Although, theoretically, the maximum slip distance can be controlled by cutting the slot to the desired length, practically, it is very difficult especially for a tolerance of 1 mm. It can be seen from Fig. 4 that three stopper blocks are bolted to each side of the gusset plate. These blocks can be adjusted along the four bigger slots in the gusset plate. A 20 mm thick steel block, which acts as the contact point with the stopper blocks, is welded to the end of each C-channel. Thus, the gap between the steel block and the stopper blocks is the maximum distance the brace can slip with respect to the gusset plate. Once the steel blocks hit the stoppers, the brace force builds up and the knee yields when the brace force exceeds the shear yielding capacity of the knee.
4. Experimental set-up and instrumentation For the dynamic testing of the frame, the pseudodynamic test is used. The schematic diagram for the pseudodynamic test set-up is shown in Fig. 5. The PC functions as a central controller. It calculates the displacement to be imposed at each time step and sends it as a digital signal to the digital–analogue converter. This device converts the digital signal into an analogue voltage which is fed into the actuator controller to command the actuator movement. The movement of the actuator within a time step is broken down into several small steps. The actual movement of the frame is monitored by a displacement transducer with an accuracy of 0.005 mm. A load cell which is mounted on the shaft of the actuator is used to measure the restoring force of the test frame when the desired displacement is reached. The analogue voltage of this force feedback is converted to a digital signal before being fed into the PC. This measured force is used to calculate the displacement for the next time step. This process is repeated until the entire test is completed. To study the behaviour of the brace and SBC, four linear strain gauges and a displacement transducer with an accuracy of 0.002 mm are installed at the locations shown in Fig. 6. The displacement transducer is used to monitor the distance the SBC has slipped. The strain gauges are used to record the strain level in the brace at every time step. From the strain readings, the slip force of the SBC can be deduced. The slip forces are parti-
cularly useful when comparing the experimental with the theoretical results using DRAIN-2D [9]. They are to be used as the input for the brace tensile and compression strengths in DRAIN-2D [9]. There are several other strain gauges (linear and rectangular rosettes) installed at the location shown in Fig. 6. From these strain gauge readings, the moment, shear stress and strain, axial force and strain of the various members can be deduced.
5. Experimental procedures and test results The determination of the impending frictional force in the SBC is the priority before any test is to be conducted. By trial and error, the bolts in the SBC are tightened with various torques in increasing order. For every setting, the frame is pulled and pushed statically, until the SBC slips. The maximum brace force when a particular torque is applied, is obtained by calculating the average of the four strain gauge readings in the brace. This process is repeated until the maximum brace force deduced is approximately 50 kN, which is less than the brace force of 57 kN at which the knee is expected to yield. A free vibration test was performed before the elastic and inelastic tests to determine the coulomb damping in the frame. The free vibration test was conducted assuming that the viscous damping of the frame is zero. If there is no coulomb damping, the plot of the restoring force vs frame displacement will be linear as shown in Fig. 7(a). On the other hand, the plot will show hysteretic behaviour as shown in Fig. 7(b) if there is some coulomb damping. The coulomb force is obtained by taking half of the difference in force, F. All the tests were monitored closely so that none of the members will yield before and after the SBC slips, that is, the brace force is kept below 57 kN. Before every test, the SBC was released and the frame was brought back to the position where the actuator load cell registers zero force. The SBC was then tightened to the prescribed torque. This is to ensure that there is no residual stress in the brace from the previous test. 5.1. Elastic response The frame elastic stiffness was found to be 24.6 kN/mm. At the floor level, an analytical lumped mass of 23,125 kg was assumed. This results in the period of the test frame being 0.193 s. In the pseudo dynamic test, the central difference integration scheme is used. The scheme is stable when the time step is smaller than 1/p of the period of the frame. A viscous damping of 2% of the critical damping was prescribed for all the tests. In this series of tests, only the SBC was allowed to slip. Other structural elements were to remain elastic. The SBC was tightened such that it slips when the brace
T. Balendra et al. / Engineering Structures 23 (2001) 491–501
Fig. 5.
Schematic diagram for the pseudo-dynamic test.
force is around 30 kN, so the knee would not yield provided the SBC slips freely. Tests were conducted for three types of excitations, viz. sinusoidal base excitation, earthquake and simulated wind excitations. In the first test, sinusoidal ground acceleration with a frequency of 20 rad/s and an initial amplitude of 0.2 m/s2 was used. The amplitude was increased by 0.2 m/s2 after every four cycles. Then, the N–S component of the 1940 El Centro earthquake, as shown in Fig. 8(a), was chosen for the earthquake excitation. Its maximum peak ground acceleration was normalised to 0.2g. For the third test, the simulated wind velocity shown in Fig. 8(b) was used. The turbulent component of the wind velocity–time history is simulated from Simiu’s [10] wind velocity spectrum using the Shinozuka [11] digital simulation method. The wind turbulence is assumed to be fluctuating about a mean wind of 22 m/s. The simulation of the velocity–time history of the wind turbulence is given by
冑
v(t)⫽ 2
冘冑
k⫽400
k⫽1
¯ k)⌬w ¯ cos(w ¯ kt⫹fk) S0(w
495
(4)
¯ =0.0125 Hz, w ¯ k=0.0125k Hz, fk=a random where ⌬w phase angle, uniformly distributed from 0 to 2p. ¯ k) is given by Simiu’s wind velocity spectrum, S0(w the following expression, u2∗(200f) ¯ k)⫽ S0(w (5) ¯ k(1+50f)5/3 w where ¯ kz w f⫽ ¯ ; V(z) z=2.8 m, the height at which the simulation is desired;
冉冊
z a , 10 the mean wind velocity in m/s at height z, using logarithmic law and z=10 m, if it is smaller than 10 m; V¯(10) =22 m/s, the mean wind velocity at 10 m height; a=0.33 for city centres; V¯(10) u ∗⫽ , 10 2.5 log z0
V¯(z)⫽V¯(10)
冉冊
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T. Balendra et al. / Engineering Structures 23 (2001) 491–501
Fig. 6.
Location of strain gauges, rectangular rosettes and displacement transducer.
Fig. 7.
Estimation of coulomb force in the test frame.
the friction velocity in m/s; and z0=0.7 m, the surface roughness at city centres. The wind velocity data is generated from 0 s at 0.1 s interval up to 40 s, and the random phase angles fk are generated from uniform distribution in the inteval (0,2p), by a multiplicative congruential method with constant seed numbers, using the standard mathematical library, NAG, available in the UNIX system of the National University of Singapore. The velocity–time history was converted to equivalent force created by a ground acceleration of 0.15g on a mass of 23,125 kg.
The experimental frame displacement–time histories of the three tests are shown in Fig. 9. For the symmetrical ground excitation such as the sinusoidal and earthquake excitations, the frame deflects around the zero position [refer to Fig. 9(a) and (b)]. When the frame is under wind excitation, it deflects to one side only as shown in Fig. 9(c). The frame maximum deflection is greater than that under earthquake excitation even though the peak acceleration is smaller for wind excitation.
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Fig. 8. (a) First 10 s of the El Centro earthquake record; (b) simulated wind excitation record.
5.2. Inelastic response In order to study the contribution from the inelastic behaviour of the knee to the energy dissipation capability of the frame, the test frame was stressed into the inelastic range using a sinusoidal base excitation with a frequency of 20 rad/s and an initial amplitude of 0.5 m/s2. The amplitude is increased by 0.1 m/s2 after every three cycles. A viscous damping of 2% of the critical damping is prescribed for the test. From free vibration tests, the coulomb damping is found to be 450 N. The maximum slip distance of the SBC is restricted to ±3 mm from the original position by the stopper blocks. The slip force of the SBC is adjusted to around 40 kN. Figs. 10 and 11 show the frame displacement–time history and the brace force–time history respectively. The SBC first slipped when the frame drift was 1/1500 (at time=4.62 s) which is during sinusoidal ground excitation with an amplitude of 0.9m/s2. It can be seen from Fig. 11 that the maximum brace force remained constant after that. At a drift of 1/560 (at time=9.42 s), the brace hit the stopper blocks and the maximum brace force picked up. The corresponding peak ground excitation is 1.4 m/s2. At the later part of the brace force–time history, the maximum brace force remained constant again. This was caused by the knee yielding. The test stopped when the drift was 1/280. The hysteretic curve for the test (see
Fig. 9. Frame displacement–time history for the inelastic test. (a) sinusoidal excitation, (b) El Centro earthquake, and (c) simulated wind excitation.
Fig. 12) is unpinched and no significant deterioration of strength and stiffness is observed up to this drift. The hysteretic curve is made up of two parts. In the region between ±40 kN the SBC dissipated the energy. Outside this range, the knee dissipated energy (refer to Fig. 12). Fig. 13(a) shows the hysteretic curve for the SBC. The slip force is reasonably constant and the brace force shot up when the SBC hit the 3 mm slip distance. The damaged knee is shown in Fig. 14. The peeling off of the paint from the web of the knee indicates the yielding of the knee.
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Fig. 10.
Frame displacement–time history for the inelastic test.
Fig. 11.
Fig. 12.
Brace force–time history for the inelastic test.
Base shear–displacement hysteresis for the inelastic test.
6. Analytical comparison DRAIN-2D [9] is used to compare the experimental results with analytical predictions. A subroutine for the shear yielding element developed by Roeder and Popov
[2] is used to model the knee. For the modelling of SBC, a new subroutine was developed and added to DRAIN2D [9]. The SBC is assumed to be located at one end of a truss element which does not buckle. Thus, a trilinear model as shown in Fig. 15 is adopted for the truss element instead of the ordinary bi-linear model. It is characterised by the parameters E, A, L, Fyt, Fyc, Dslip,t and Dslip,c which is the elastic modulus, cross section area, length of the truss, tension slip force, compression slip force, maximum slip distance in tension and maximum slip distance in compression respectively. The SBC slips when the axial force in the truss is Fyt or Fyc. When the SBC slips, the axial force remains constant at Fyt or Fyc depends on the direction at which the SBC slips. The axial force builds up again after the SBC hits the maximum slip distance Dslip,t or Dslip,c. The analytical model of the test frame is shown in Fig. 16. The beam and columns are modelled as beam– column elements, the brace is modelled as a tri-linear truss element and the knee is modelled as a shear element. Rigid links are included in the model to account
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Fig. 15. Tri-linear truss model.
Fig. 13.
Brace force–displacement hysteresis for the inelastic test.
Fig. 16.
Fig. 14. Yielded knee at the end of the inelastic test.
for the rigidity of the knee–beam, knee–brace and knee column joints. The stiffness of the rigid element is adjusted so that the elastic stiffness of the model matches the measured stiffness. Nodes 9, 10 and 11 are assumed to have the same horizontal displacement, as the axial
Analytical model for the test frame.
deformation of the beam is expected to be small. The mass is assumed to be lumped at any one of these nodes. From the inelastic test results, knee yielding shear is estimated to be 60 kN and the shear strain hardening ratio to be 0.1. These values are used in the analytical model to simulate the inelastic behaviour of the knee. The tension and compression slip forces are required to model the tri-linear truss element. It can be seen from Fig. 13 that the maximum slip force is not perfectly constant. Therefore, an energy averaging method is used to quantify the slip forces to be input for the model, where the friction force is obtained by adding the total work done by the friction force in a particular direction and dividing it by the total distance the specimen travelled during the work done. The value thus obtained is the desired input slip force for the model of that particular
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Table 1 Input slip forces for the analytical models of various test Test
Elastic test Test 1 Test 2 Test 3 Inelastic test
Tension slip force (kN)
Compression slip force (kN)
31.92 — 30.46 35.00
33.15 29.43 31.62 38.00
test. Table 1 shows the slip forces used for the modelling of the various tests. The comparisons of the displacement–time histories are shown in Fig. 9 for the three elastic tests and Fig. 10 for the inelastic test. Experimental and analytical results agree reasonably well. Some discrepancies are observed because of the difference in the actual slip force and the one used for modelling. From the comparisons of the experimental and theoretical hysteretic curves [Fig. 13(a) and (b)], it can be seen that in the experiment the SBC did not always slip at the same force whereas for the analytical model a constant slip force is assumed. The discrepancies can be seen more clearly in the inelastic test. As shown in Fig. 10, before the SBC is activated at 4.62 s, the experimental and analytical results agree very well. But between 4.62 s and 9.42 s when the SBC is governing the behaviour of the frame, the discrepancy
Fig. 17.
is large. When the knee starts governing after 9.42 s, the experimental and analytical results agree very well again. Cycle-by-cycle comparisons for the inelastic test are shown in Fig. 17. The analytical curves agree reasonably well with the experimental curves.
7. Conclusions From the large scale dynamic test results, the ability of the proposed system in dissipating energy at two different excitation levels is proven. For this particular study, the SBC is activated when the frame drift is 1/1500, It ceases at a drift of 1/560 which is within the serviceability limit of the frame. When the frame is subjected to stronger excitation the inherent hysteretic damping of the KBF is tapped to maintain the structural integrity. The frame hysteretic loops are unpinched and with no deterioration in strength and stiffness. This implies that the system can dissipate a large amount of energy before failure. Further more, as the SBC dissipates energy without causing permanent damage, part of the strength required for serviceability design can be reduced by activating the SBC at an appropriate slip force. During severe earthquakes the damage is localised to the knee member, and thus retrofitting the structural frame would not be costly.
Cycle by cycle comparison between experimental and analytical hysteretic loops for the inelastic test.
T. Balendra et al. / Engineering Structures 23 (2001) 491–501
References [1] Roeder CW, Popov EP. Inelastic behaviour of eccentrically braced steel frames under cyclic loadings. In: EERC Report 7788. Berkeley, CA: University of California, 1977. [2] Roeder CW, Popov EP. Eccentrically braced steel frames for earthquakes. J Struct Div, ASCE 1978;104:391–411. [3] Balendra T, Lim EL, Lee SL. Ductile knee braced frames with shear yielding knee for seismic resistant structures. Engng Struct 1994;16:263–9. [4] Pall AS, Verganalakis V, Marsh C. Response of friction damped braced frames. J Struct Div, ASCE 1982;108:1313–23. [5] Constantinou MC, Reinhorn AM, Mokha A, Watson R. Displacement control devices for base isolated bridges. Earthquake Spectra 1991;7. [6] Aiken ID, Kelly JM. Earthquake simulator testing and analytical
[7]
[8]
[9]
[10] [11]
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studies of two energy absorbing system for multistorey structures. In: Report No. UCB/EERC-90/03. Berkeley, CA: University of California, 1990. Grigorian CE, Yang TS, Popov EP. Slotted bolted connection energy dissipators. In: Report No. UCB/EERC-92/10. Berkeley, CA: University of California, 1992. Balendra T, Lam KY, Liaw CY, Lee SL. Behaviour of eccentrically braced frame by pseudo-dynamic test. J Struct Eng, ASCE 1987;113:673–88. Kanaan AE, Powell GH. DRAIN-2D: a general purpose computer program for the dynamic analysis of inelastic plane structure. In: EERC Report 73-6 and 73-22. Berkeley, CA: University of California, 1973. Simiu E. Wind spectra and dynamic response. J Struct Div, ASCE 1974;100:1897–910. Shinozuka M, Jan CM. Digital simulation of random process and its application. J Sound Vibr 1972;25:111–28.
An economical structural system for wind and earthquake loads T. Balendra a
a,*
, C.H. Yu a, F.L. Lee
b
Department of Civil Engineering, National University of Singapore, Kent Ridge, Singapore 117576, Singapore b Furnovus Innovations, Singapore Received 23 February 1999; accepted 26 July 1999
Abstract An economical structural system which would mitigate undesirable vibrations due to wind and earthquake loads is being developed. In this proposed system, the inherent hysteretic damping of the knee–brace-frame (KBF) with shear yielding knee is tapped to maintain the structural integrity in the event of a severe earthquake, while the frictional damping (in the form of slotted bolted connection, SBC) in the brace of KBF is utilised to dissipate energy in order to meet the serviceability requirements during severe wind storms or moderate earthquakes. As SBC dissipates energy without causing permanent damage, part of the strength required for serviceability design can be reduced by activating the SBC at an appropriate slip force. This would economise the material required for serviceability design. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Steel frame; Seismic; Wind
1. Introduction Generally, the design of buildings must satisfy two criteria. First, under frequently occurring low to moderate earthquakes or severe wind storm, the structure should have sufficient strength and stiffness to control deflection and prevent any structural damage. Second, under rare and severe earthquakes, the structure must have sufficient ductility to prevent collapse. The commonly used structural systems such as a momentresisting frame (MRF), concentric braced frame (CBF), eccentric braced frame [1,2] (EBF) and knee–braceframe [3] (KBF) are designed to resist the moderate loadings in the linear elastic range while the severe loadings are resisted through plastic hinges. The conventional MRFs and CBFs do not satisfy both criteria economically. A MRF is an excellent energy dissipating system, but it has to be designed with larger beam sections in order to develop sufficient stiffness to prevent excessive drift. The CBF is much stiffer than an MRF with similar sections, but it has poor energy dissipating capability due to the buckling of the brace. By connecting the brace eccentrically to the beam– column joint, EBF [1,2] combines the good features of * Corresponding author. Tel.: +65-874-2149; fax: +65-779-1635.
MRF and CBF. It dissipates energy through the development of shear hinges in the beam. As the beam is a principal member of the frame, the repair is very costly and difficult. The other system, KBF proposed by Balendra et al. [3], also combines the advantages of the MRF and CBF. In this system, the brace is connected to the knee, which is a secondary member, instead of the beam–column joint. The energy is dissipated through shear yielding of the knee while the brace is designed to prevent buckling. Since it is the knee that yields when a severe earthquake strikes, the cost of repairing the structure is limited to replacing the knee member only. Various types of energy dissipating devices, utilising friction as a means of energy dissipation, have been tested and studied by researchers [4–6]. These devices require precision workmanship and exotic materials for manufacturing, and specialised training in installation. As a consequence, they are not widely accepted in engineering practice. The slotted bolted connection (SBC) proposed by Grigorian et al. [7] as an energy dissipator represents an attempt to overcome the abovementioned shortcomings of these devices. The SBCs require only a slight modification of standard construction practice, and require materials that are widely available commercially. In this study, a SBC is incorporated into the brace of a KBF. The behaviour of the KBF with slotted bolted
0141-0296/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 1 - 0 2 9 6 ( 0 0 ) 0 0 0 6 1 - 4
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connections is investigated for its performance under seismic and wind loadings. One storey KBF with slotted bolted connections was tested using a pseudo-dynamic test method [8]. The SBC in the brace was activated based on predetermined slip force. During severe seismic excitation, the knee was brought into inelastic action by preventing the slip as the drift approached the serviceability limit. Based on the experimental results, an analytical model for trilinear force–displacement behaviour of the brace with SBC is developed and incorporated as a subroutine in DRAIN-2D [9], a general purpose structural dynamic analysis program.
2. General characteristics 2.1. Knee braced frame The diagonal brace of the KBF is connected to a knee element instead of a beam–column joint as shown in Fig. 1. The knee element is designed to dissipate energy under severe dynamic loading and therefore brace buckling or yielding of other members of the frame is prevented. A short knee member will yield in shear, while a longer knee yields in flexure. A KBF with a knee designed to yield in shear has larger ductility than one designed to yield in flexure [3]. In this study, the KBF with a shear yielding knee is investigated when the brace is incorporated with a SBC. 2.2. Slotted bolted connection In the slotted bolted connection shown in Fig. 2, the elongated holes or slots in the main connecting plate are parallel to the line of loading. The main connecting plate is placed between two outer members. A friction lining pad is ‘sandwiched’ between each outer member and the main plate, and the lining pad always moves with the
Fig. 1.
Details of the test frame.
Fig. 2.
Basic set-up of the slotted bolted connection.
outer member. All the components are clamped together through bolts. Upon tightening the bolts, frictions develop between the contact surfaces of the lining pads and the slotted plate. When the tensile or compressive force applied to the connection exceeds the impending frictional forces, the slotted plate slips relative to the lining pads. Energy is dissipated by means of friction between the sliding surfaces. The knee braced frame with a slotted bolted connection retains the main features of KBF except the brace– column connection. The brace is connected to the column through a gusset plate with a slotted bolted connection. When the brace force exceeds the impending frictional force of the SBC, the brace slips with respect to the SBC. The impending frictional force is always kept below the brace force required to yield the knee. In this way, the knee is always elastic when the SBC is slipping. The maximum slip distance of the SBC is governed by the slot length. Once the SBC hits the slot ends, the brace force builds up. The knee yields when the brace force exceeds the shear yielding capacity of the knee. Therefore, by altering the slot length of the SBC, the drift at which the energy dissipating capability of the knee is activated, can be controlled.
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3. Design of the test frame The test frame is the same as the test 2 specimen investigated by Balendra et al. [3] except for the inclusion of SBC. The design and section sizes of various members of the test frame are described below. 3.1. Beam, column and brace The beam and columns are wide flange sections 100×100×17.2 kg/m and 125×125×23.8 kg/m, respectively. These members are designed to remain elastic even when the knee member is loaded up to failure. The brace is designed to take cyclic loadings without buckling. It is made up of two C-channels (100×50×5×9.36) connected back-to-back with a 16 mm gap in between by 100×100×12 mm thick batten at 500 mm spacing.
Details of the knee.
3.3. Slotted bolted connection
3.2. Knee To ensure that the knee member yields in shear instead of moment, the longer of the two segments of the knee member generated by the intersection of the diagonal brace and the knee member, denoted as lk, should satisfy the following condition M∗p lk⬍2 Vp
(1)
where M∗p is the reduced plastic moment contributed by flanges only and Vp the plastic shear force. They are defined as follows M∗p ⫽tfb(d⫺tf)sy Vp⫽tw(d⫺tf)
Fig. 3.
sy
冑3
(2)
The SBC is located at the lower corner of the frame, serving the connection between the brace and the column. The brace is connected to the column by being bolted to a gusset plate which is a 310×330×10 mm thick stainless steel plate with six elongated slots parallel to the brace as shown in Fig. 4. The gusset plate is bolted to the column flange and base for the ease of removing it for alteration. It serves as the main plate in the SBC. And the brace, which is made up of two C-channels, acts as the outer member in the SBC. Between each C-channel and the gusset plate there is a 3 mm thick brass shim inserted as the friction lining pad. The gusset plate, C-channels and brass shims are bolted together through the two narrower slots in the gusset plate. In such a way, friction develops between the contact surfaces of the brass shim and the gusset
(3)
in which sy, d, tf, b and tw are, respectively, the yield stress, depth, flange thickness, flange width and web thickness of the knee member. A built-up 50×50 mm I-section which has a flange thickness of 6.0 mm and web thickness of 4.4 mm is used for the knee member. The yield stress of the steel used is 300 N/mm2. By employing Eqs. (1)–(3), the limiting value of lk, for the knee member to yield in shear is found to be 236 mm. The longer portion of the knee is the lower segment which is taken to be 180 mm. It is smaller than the limiting value of 236 mm, and so the knee member yields in shear. The section chosen is also prevented from local and lateral buckling. The knee is designed to avoid premature buckling of the web by adding web stiffeners with maximum spacing of 90 mm. Fig. 3 shows the details of the knee.
Fig. 4.
Details of the modified SBC for the test frame.
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plate. The amount of friction developed is proportional to the tightness of the bolts which in turn is proportional to the tightening torque. Thus, by adjusting the torque applied to the bolts, the desired impending frictional force can be achieved. When the brace force exceeds the impending frictional forces, the brace slips together with the brass shim from the gusset plate. There are stopper blocks to control the maximum slip distance of the SBC. Although, theoretically, the maximum slip distance can be controlled by cutting the slot to the desired length, practically, it is very difficult especially for a tolerance of 1 mm. It can be seen from Fig. 4 that three stopper blocks are bolted to each side of the gusset plate. These blocks can be adjusted along the four bigger slots in the gusset plate. A 20 mm thick steel block, which acts as the contact point with the stopper blocks, is welded to the end of each C-channel. Thus, the gap between the steel block and the stopper blocks is the maximum distance the brace can slip with respect to the gusset plate. Once the steel blocks hit the stoppers, the brace force builds up and the knee yields when the brace force exceeds the shear yielding capacity of the knee.
4. Experimental set-up and instrumentation For the dynamic testing of the frame, the pseudodynamic test is used. The schematic diagram for the pseudodynamic test set-up is shown in Fig. 5. The PC functions as a central controller. It calculates the displacement to be imposed at each time step and sends it as a digital signal to the digital–analogue converter. This device converts the digital signal into an analogue voltage which is fed into the actuator controller to command the actuator movement. The movement of the actuator within a time step is broken down into several small steps. The actual movement of the frame is monitored by a displacement transducer with an accuracy of 0.005 mm. A load cell which is mounted on the shaft of the actuator is used to measure the restoring force of the test frame when the desired displacement is reached. The analogue voltage of this force feedback is converted to a digital signal before being fed into the PC. This measured force is used to calculate the displacement for the next time step. This process is repeated until the entire test is completed. To study the behaviour of the brace and SBC, four linear strain gauges and a displacement transducer with an accuracy of 0.002 mm are installed at the locations shown in Fig. 6. The displacement transducer is used to monitor the distance the SBC has slipped. The strain gauges are used to record the strain level in the brace at every time step. From the strain readings, the slip force of the SBC can be deduced. The slip forces are parti-
cularly useful when comparing the experimental with the theoretical results using DRAIN-2D [9]. They are to be used as the input for the brace tensile and compression strengths in DRAIN-2D [9]. There are several other strain gauges (linear and rectangular rosettes) installed at the location shown in Fig. 6. From these strain gauge readings, the moment, shear stress and strain, axial force and strain of the various members can be deduced.
5. Experimental procedures and test results The determination of the impending frictional force in the SBC is the priority before any test is to be conducted. By trial and error, the bolts in the SBC are tightened with various torques in increasing order. For every setting, the frame is pulled and pushed statically, until the SBC slips. The maximum brace force when a particular torque is applied, is obtained by calculating the average of the four strain gauge readings in the brace. This process is repeated until the maximum brace force deduced is approximately 50 kN, which is less than the brace force of 57 kN at which the knee is expected to yield. A free vibration test was performed before the elastic and inelastic tests to determine the coulomb damping in the frame. The free vibration test was conducted assuming that the viscous damping of the frame is zero. If there is no coulomb damping, the plot of the restoring force vs frame displacement will be linear as shown in Fig. 7(a). On the other hand, the plot will show hysteretic behaviour as shown in Fig. 7(b) if there is some coulomb damping. The coulomb force is obtained by taking half of the difference in force, F. All the tests were monitored closely so that none of the members will yield before and after the SBC slips, that is, the brace force is kept below 57 kN. Before every test, the SBC was released and the frame was brought back to the position where the actuator load cell registers zero force. The SBC was then tightened to the prescribed torque. This is to ensure that there is no residual stress in the brace from the previous test. 5.1. Elastic response The frame elastic stiffness was found to be 24.6 kN/mm. At the floor level, an analytical lumped mass of 23,125 kg was assumed. This results in the period of the test frame being 0.193 s. In the pseudo dynamic test, the central difference integration scheme is used. The scheme is stable when the time step is smaller than 1/p of the period of the frame. A viscous damping of 2% of the critical damping was prescribed for all the tests. In this series of tests, only the SBC was allowed to slip. Other structural elements were to remain elastic. The SBC was tightened such that it slips when the brace
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Fig. 5.
Schematic diagram for the pseudo-dynamic test.
force is around 30 kN, so the knee would not yield provided the SBC slips freely. Tests were conducted for three types of excitations, viz. sinusoidal base excitation, earthquake and simulated wind excitations. In the first test, sinusoidal ground acceleration with a frequency of 20 rad/s and an initial amplitude of 0.2 m/s2 was used. The amplitude was increased by 0.2 m/s2 after every four cycles. Then, the N–S component of the 1940 El Centro earthquake, as shown in Fig. 8(a), was chosen for the earthquake excitation. Its maximum peak ground acceleration was normalised to 0.2g. For the third test, the simulated wind velocity shown in Fig. 8(b) was used. The turbulent component of the wind velocity–time history is simulated from Simiu’s [10] wind velocity spectrum using the Shinozuka [11] digital simulation method. The wind turbulence is assumed to be fluctuating about a mean wind of 22 m/s. The simulation of the velocity–time history of the wind turbulence is given by
冑
v(t)⫽ 2
冘冑
k⫽400
k⫽1
¯ k)⌬w ¯ cos(w ¯ kt⫹fk) S0(w
495
(4)
¯ =0.0125 Hz, w ¯ k=0.0125k Hz, fk=a random where ⌬w phase angle, uniformly distributed from 0 to 2p. ¯ k) is given by Simiu’s wind velocity spectrum, S0(w the following expression, u2∗(200f) ¯ k)⫽ S0(w (5) ¯ k(1+50f)5/3 w where ¯ kz w f⫽ ¯ ; V(z) z=2.8 m, the height at which the simulation is desired;
冉冊
z a , 10 the mean wind velocity in m/s at height z, using logarithmic law and z=10 m, if it is smaller than 10 m; V¯(10) =22 m/s, the mean wind velocity at 10 m height; a=0.33 for city centres; V¯(10) u ∗⫽ , 10 2.5 log z0
V¯(z)⫽V¯(10)
冉冊
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Fig. 6.
Location of strain gauges, rectangular rosettes and displacement transducer.
Fig. 7.
Estimation of coulomb force in the test frame.
the friction velocity in m/s; and z0=0.7 m, the surface roughness at city centres. The wind velocity data is generated from 0 s at 0.1 s interval up to 40 s, and the random phase angles fk are generated from uniform distribution in the inteval (0,2p), by a multiplicative congruential method with constant seed numbers, using the standard mathematical library, NAG, available in the UNIX system of the National University of Singapore. The velocity–time history was converted to equivalent force created by a ground acceleration of 0.15g on a mass of 23,125 kg.
The experimental frame displacement–time histories of the three tests are shown in Fig. 9. For the symmetrical ground excitation such as the sinusoidal and earthquake excitations, the frame deflects around the zero position [refer to Fig. 9(a) and (b)]. When the frame is under wind excitation, it deflects to one side only as shown in Fig. 9(c). The frame maximum deflection is greater than that under earthquake excitation even though the peak acceleration is smaller for wind excitation.
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Fig. 8. (a) First 10 s of the El Centro earthquake record; (b) simulated wind excitation record.
5.2. Inelastic response In order to study the contribution from the inelastic behaviour of the knee to the energy dissipation capability of the frame, the test frame was stressed into the inelastic range using a sinusoidal base excitation with a frequency of 20 rad/s and an initial amplitude of 0.5 m/s2. The amplitude is increased by 0.1 m/s2 after every three cycles. A viscous damping of 2% of the critical damping is prescribed for the test. From free vibration tests, the coulomb damping is found to be 450 N. The maximum slip distance of the SBC is restricted to ±3 mm from the original position by the stopper blocks. The slip force of the SBC is adjusted to around 40 kN. Figs. 10 and 11 show the frame displacement–time history and the brace force–time history respectively. The SBC first slipped when the frame drift was 1/1500 (at time=4.62 s) which is during sinusoidal ground excitation with an amplitude of 0.9m/s2. It can be seen from Fig. 11 that the maximum brace force remained constant after that. At a drift of 1/560 (at time=9.42 s), the brace hit the stopper blocks and the maximum brace force picked up. The corresponding peak ground excitation is 1.4 m/s2. At the later part of the brace force–time history, the maximum brace force remained constant again. This was caused by the knee yielding. The test stopped when the drift was 1/280. The hysteretic curve for the test (see
Fig. 9. Frame displacement–time history for the inelastic test. (a) sinusoidal excitation, (b) El Centro earthquake, and (c) simulated wind excitation.
Fig. 12) is unpinched and no significant deterioration of strength and stiffness is observed up to this drift. The hysteretic curve is made up of two parts. In the region between ±40 kN the SBC dissipated the energy. Outside this range, the knee dissipated energy (refer to Fig. 12). Fig. 13(a) shows the hysteretic curve for the SBC. The slip force is reasonably constant and the brace force shot up when the SBC hit the 3 mm slip distance. The damaged knee is shown in Fig. 14. The peeling off of the paint from the web of the knee indicates the yielding of the knee.
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Fig. 10.
Frame displacement–time history for the inelastic test.
Fig. 11.
Fig. 12.
Brace force–time history for the inelastic test.
Base shear–displacement hysteresis for the inelastic test.
6. Analytical comparison DRAIN-2D [9] is used to compare the experimental results with analytical predictions. A subroutine for the shear yielding element developed by Roeder and Popov
[2] is used to model the knee. For the modelling of SBC, a new subroutine was developed and added to DRAIN2D [9]. The SBC is assumed to be located at one end of a truss element which does not buckle. Thus, a trilinear model as shown in Fig. 15 is adopted for the truss element instead of the ordinary bi-linear model. It is characterised by the parameters E, A, L, Fyt, Fyc, Dslip,t and Dslip,c which is the elastic modulus, cross section area, length of the truss, tension slip force, compression slip force, maximum slip distance in tension and maximum slip distance in compression respectively. The SBC slips when the axial force in the truss is Fyt or Fyc. When the SBC slips, the axial force remains constant at Fyt or Fyc depends on the direction at which the SBC slips. The axial force builds up again after the SBC hits the maximum slip distance Dslip,t or Dslip,c. The analytical model of the test frame is shown in Fig. 16. The beam and columns are modelled as beam– column elements, the brace is modelled as a tri-linear truss element and the knee is modelled as a shear element. Rigid links are included in the model to account
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Fig. 15. Tri-linear truss model.
Fig. 13.
Brace force–displacement hysteresis for the inelastic test.
Fig. 16.
Fig. 14. Yielded knee at the end of the inelastic test.
for the rigidity of the knee–beam, knee–brace and knee column joints. The stiffness of the rigid element is adjusted so that the elastic stiffness of the model matches the measured stiffness. Nodes 9, 10 and 11 are assumed to have the same horizontal displacement, as the axial
Analytical model for the test frame.
deformation of the beam is expected to be small. The mass is assumed to be lumped at any one of these nodes. From the inelastic test results, knee yielding shear is estimated to be 60 kN and the shear strain hardening ratio to be 0.1. These values are used in the analytical model to simulate the inelastic behaviour of the knee. The tension and compression slip forces are required to model the tri-linear truss element. It can be seen from Fig. 13 that the maximum slip force is not perfectly constant. Therefore, an energy averaging method is used to quantify the slip forces to be input for the model, where the friction force is obtained by adding the total work done by the friction force in a particular direction and dividing it by the total distance the specimen travelled during the work done. The value thus obtained is the desired input slip force for the model of that particular
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Table 1 Input slip forces for the analytical models of various test Test
Elastic test Test 1 Test 2 Test 3 Inelastic test
Tension slip force (kN)
Compression slip force (kN)
31.92 — 30.46 35.00
33.15 29.43 31.62 38.00
test. Table 1 shows the slip forces used for the modelling of the various tests. The comparisons of the displacement–time histories are shown in Fig. 9 for the three elastic tests and Fig. 10 for the inelastic test. Experimental and analytical results agree reasonably well. Some discrepancies are observed because of the difference in the actual slip force and the one used for modelling. From the comparisons of the experimental and theoretical hysteretic curves [Fig. 13(a) and (b)], it can be seen that in the experiment the SBC did not always slip at the same force whereas for the analytical model a constant slip force is assumed. The discrepancies can be seen more clearly in the inelastic test. As shown in Fig. 10, before the SBC is activated at 4.62 s, the experimental and analytical results agree very well. But between 4.62 s and 9.42 s when the SBC is governing the behaviour of the frame, the discrepancy
Fig. 17.
is large. When the knee starts governing after 9.42 s, the experimental and analytical results agree very well again. Cycle-by-cycle comparisons for the inelastic test are shown in Fig. 17. The analytical curves agree reasonably well with the experimental curves.
7. Conclusions From the large scale dynamic test results, the ability of the proposed system in dissipating energy at two different excitation levels is proven. For this particular study, the SBC is activated when the frame drift is 1/1500, It ceases at a drift of 1/560 which is within the serviceability limit of the frame. When the frame is subjected to stronger excitation the inherent hysteretic damping of the KBF is tapped to maintain the structural integrity. The frame hysteretic loops are unpinched and with no deterioration in strength and stiffness. This implies that the system can dissipate a large amount of energy before failure. Further more, as the SBC dissipates energy without causing permanent damage, part of the strength required for serviceability design can be reduced by activating the SBC at an appropriate slip force. During severe earthquakes the damage is localised to the knee member, and thus retrofitting the structural frame would not be costly.
Cycle by cycle comparison between experimental and analytical hysteretic loops for the inelastic test.
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