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Design of Tension-Only Concentrically Braced Steel Frames for seismic induced impact loading
PII: S0141-0296(97)00205-8
Engineering Structures, Vol. 20, No. 12, pp. 1087–1096, 1998 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0141–0296/98 $19.00 + 0.00
Design of Tension-Only Concentrically Braced Steel Frames for seismic induced impact loading Andre´ Filiatrault and Robert Tremblay EPICENTRE Research Group, Department of Civil Engineering, Ecole Polytechnique, University of Montreal Campus, P.O. Box 6079, Station “Centre-Ville”, Montreal, Canada H3C 3A7
Tension-Only Concentrically Braced Frames (TOCBFs) exhibit deteriorating pinched hysteretic behaviour during strong earthquakes. It has long been suspected that the sudden increase in tensile forces in the braces of TOCBFs creates detrimental impact loading on the connections and other structural elements. This paper addresses this issue through shake table tests of half scale, twostorey, TOCBF models. By normalizing the hysteresis loops of braces obtained from shake table tests to the yield strength of steel obtained from static tests, the increase in tensile forces in the braces was obtained. Results of dynamic tensile tests on steel coupons under similar strain rates as observed during the shake table tests showed that this increase in tensile forces is caused by a yield strength increase of the steel under high strain rate. A procedure is proposed to predict this increase in tensile forces in the braces at the design stage. The use of this procedure is illustrated by a design example. 1998 Elsevier Science Ltd. All rights reserved Keywords: braced frames, impact, seismic
Slender braces in TOCBFs transit between an elastic buckling state, a restraightening state, in which they carry almost no load, an elastic tensile loading state as they are suddenly straightened and, finally, a tensile yielding state. North American seismic provisions associate the sudden increase in tensile forces in the braces with detrimental impact loading on the connections and other structural elements1–4. The latest edition of the NEHRP seismic regulations in the U.S. recommends to pretension diagonal members in TOCBFs to prevent loose diagonals during strong earthquakes1. In California, designers are warned that the force demand on a brace in tension is expected to exceed the brace tensile yield strength2. The latest edition of the Canadian steel standard4 specifies an explicit value of 1.10 for an impact factor to be used in the design of the brace connections. No experimental evidence, however, has been provided so far to confirm, or to quantify, this impact phenomenon. The main objective of this paper is to shed some light on the behaviour of TOCBFs through shake table tests of half scale, two-storey, TOCBF models. The bracing elements were instrumented to measure their hysteretic loops under seismic conditions. By comparing these hysteresis loops with quasi-static and dynamic monotonic test
1. Introduction Tension-Only Concentrically Braced Frames (TOCBFs) incorporate very slender bracing members, such as steel rods or flat plates, which are unable to dissipate much energy in compression. A characteristic of TOCBFs is their deteriorating pinched hysteretic behaviour during strong earthquakes. Alternating tension yielding and elastic compression buckling of the bracing elements induce slackness in the lateral load resisting system around its point of zero displacement. This slackness translates into severely pinched hysteresis loops and potential impact loading in the structure. This unfavourable behaviour was quickly recognized as a major fault of TOCBFs and, therefore, structural engineers have shied away from this system for medium and high-rise buildings located in active seismic areas. The TOCBF system, however, continues to be used extensively for low-rise industrial and commercial steel buildings because it is inexpensive and simple to design, fabricate and erect. Therefore, these structures constitute a large portion of the building stock in North America. This extensive use of TOCBFs, however, has not been accompanied by an adequate research effort to understand fully their behaviour and rationalize their design.
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results, the increase in tensile forces in the braces was quantified. A procedure is proposed to predict this increase in brace forces at the design stage.
2. Impact loading and strain rate effect in TOCBFs The sudden increase in tensile forces in the braces of TOCBFs has long been suspected to cause detrimental impact loading on the connections and other structural components of the lateral load resisting system. In fact, two mechanisms can contribute to amplify the forces in the braces during tensile loading. The first mechanism is a true impact phenomenon caused by a rapid transfer of energy from the floor masses, being suddenly halted in their motion, to the braces becoming suddenly taut. The second mechanism is associated with the strain rate effect on the yield strength of steel. The yield strength of steel increases with the rate of loading while its tensile strength and strain-hardening moduli remain almost unchanged. Therefore during elastic tensile loading, the braces are subjected to a high strain rate demand and a portion of the increase in tensile forces in the braces may be attributed to the increase of the yield strength of steel. The experimental investigation conducted by Wakabayashi et al.5 on strain rate effects is the most applicable to TOCBFs under seismic conditions. They proposed the following expression to estimate the increase of yield strength of steel with strain rate:
冉冊
Fyd ⑀˙ = 1 + 0.0473 log Fys ⑀˙ o
Figure 1 Test frame
(1)
where Fyd is the dynamic yield strength for a strain rate ⑀˙ and Fys is the quasi-static yield strength under a strain rate ⑀˙ o = 50 × 10−6 /s.
3. Shake table test program 3.1. Test frame Half scale, two-storey, TOCBF models were tested on the uniaxial, 3.4 m × 3.4 m earthquake simulation facility at Ecole Polytechnique in Montreal, Canada6. A general view of the test frame is presented in Figure 1. To represent a simply supported base, each base column was linked to a true pin mechanism which, in turn, was attached to a foundation beam on the shake table. The two pin-ended floor beams were made of hollow structural steel (HSS) 127 × 76 × 4.8 sections. The mass of the model was simulated by four concrete blocks (30 kN each) linked to the floor beams. These concrete blocks were supported vertically by an independent peripheral frame which was completely pinned in the direction of the shake table motion and braced in the opposite direction to ensure no motion of the test plane frame in that direction. This supported structure carried all the vertical loads from the concrete blocks so the test frame had only to resist the lateral inertia loads. Figure 2 shows a photograph of the test frame and supporting structure after a seismic test on the shake table. In the direction of excitation, the test frame was braced at each level by grade 300W steel (nominal Fy = 300 MPa and Fu = 450 MPa) round tie-rods having a diameter of
Figure 2 Test and supporting structures after a seismic test on shake table
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12.7 mm. These braces had a slenderness ratio of 800 and, thus, exhibited negligible buckling strength. The force timehistories of both first floor braces could be recorded with axial load cells placed adjacent to the bottom first floor brace connections. 3.2. Preliminary system identification tests The dynamic characteristics of the model frame were estimated from low amplitude ambient vibration simulations and free vibration tests. For the ambient vibration simulations, the test frame was excited by a 0–25 Hz flat white noise, with a RMS (Root Mean Square) amplitude of 0.03 g. This base motion was large enough to cause buckling in all compression braces at each cycle, but was sufficiently low to ensure that the tension braces remained in the elastic range of the steel. A dedicated ambient vibration analysis software7 was then used to determine the natural periods of the test frame from power spectral density plots of the absolute floor horizontal acceleration records. In the free vibration tests, the structure was excited by a sinusoidal input at its first natural period. When a steady-state response was obtained, the input was suddenly stopped and the floor relative displacements were recorded. The first modal damping ratio of the structure was then established by the logarithmic relative displacement decrement at each floor8. The first mode of vibration was measured at a period of 0.470 ± 0.005 s and represented a classical shear mode involving tensile and buckling deformations of the braces. The second mode of vibration, at a period of 0.073 ± 0.005 s, represents a flexural mode of deformation of the columns. The damping ratio in the first mode of vibration was measured at 2.4% critical. 3.3. Earthquake ground motions Three different historical ground motions, recorded in Western North America, were chosen as input for the shake table studies. The characteristics of these records are summarized in Table 1. These records present ratios of Peak Horizontal Acceleration (PHA in g) to Peak Horizontal Velocity (PHV in m/s) close to unity which is compatible with seismic zoning maps of Western Canada9. Each accelerogram was scaled to mobilize a displacement ductility level representative of code designed structures. This scaling procedure was based on equating, for each accelerogram, the base shear at first yield specified in the 1995 edition of the National Building Code of Canada (NBCC)9 to the actual yield base shear of the structure. This procedure yielded scaling factors of 1.3, 1.6 and 1.2 for the Puget Sound, San Fernando and El Centro accelerograms, respectively. Figure 3 presents the absolute acceleration response spectra of the three scaled seismic events for 2.4% damping (corresponding to the first modal damping ratio of the structure).
Figure 3 Absolute acceleration response damping) for the three scaled accelerograms
spectra
(2.4%
4. Shake table test results 4.1. Hysteretic behaviour of first floor braces Figure 4 presents, for the three seismic events considered, the stress–strain hysteresis loops for the first floor brace located on the east side of the frame (see Figure 1). The stress was normalized to the yield strength of each brace, Fys, obtained from quasi-static ( ⑀˙ = 50 × 10−6 /s) monotonic tensile tests performed on 150 mm long specimens of the brace material. The behaviour of the brace is consistent for the three earthquake records. The brace yields in tension at a strain of about 0.2% and exhibits ductility ratios ranging from 2.4 (for Puget Sound) to 7.7 (for El Centro). The maximum stress recorded in the brace exceeds the pseudostatic yield strength by 14%. This peak value occurs when the brace is stretched significantly, for the first time, in the inelastic range. During the tensile loading state before the first yield excursion, the structure vibrated elastically which caused a maximum strain rate demand in the tension braces. Following the maximum stress, the stress reduces slightly as the brace elongates in the inelastic range and the rate of straining progressively reduces. To quantify the strain rate demand on the braces, the time required to increase the tensile stress from zero to the static yield strength before each inelastic excursion was computed from the experimental data. The results of these calculations are summarized, for all the six first floor braces, in Table 2. The strain rate in the braces varies from 10 × 10−3 /s to 40 × 10−3 /s with a mean value of 22 × 10−3 /s for the three different excitations. The maximum normalized stress observed for each inelastic excursion is also given in Table 2. A good correlation can be observed between the strain rate in the elastic tensile loading portion of the stress–strain curve and the maximum normalized stress reached in the inelastic range of the steel. Figure 5 compares graphically this correlation with the predictions
Table 1 Historical seismic events considered for shake table tests Record name
Earthquake name Station
Date
Component
PHA (g)
PHV (m/s)
Puget Sound
13-04-49
N04W
0.16
0.21
San Fernando
Western Washington San Fernando
9-02-71
N90E
0.21
0.21
El Centro
Imperial Valley
18-05-40
S00E
0.34
0.33
Olympia highway testing laboratory Hollywood storage P.E. lot Los Angeles El Centro site Imperial Valley Irrigation District
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Figure 4 Stress–strain hysteresis loops for first floor brace on east side of frame
Table 2 Strain rate demand and maximum normalized stress in first floor braces Excitation
Brace No.
Inelastic excursion No.
Puget Sound
1 (East)
San Fernando
2 (West) 3 (East)
1 2 3 4 1 1 2 3 4 1 2 3 4 1 2 3 4 5
4 (West)
El Centro
5 (East)
6 (West) Mean strain rate for all inelastic excursions
Strain rate ⑀· (10⫺3 /s)
Maximum normalized stress
18 1.07 18 1.06 23 1.06 30 1.14 20 1.01 20 1.06 22 1.14 18 1.10 24 1.09 26 1.10 29 1.11 17 1.10 23 1.10 10 1.01 40 1.14 17 1.05 35 1.10 20 1.04 No significant inelastic excursion 22
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6. Predicting the dynamic amplification factor for brace forces
Figure 5 Influence of strain rate on maximum normalized stress in first floor braces
given by equation (1). This empirical equation represents a conservative estimate of the experimental results.
5. Dynamic tensile tests on steel bar specimens 5.1. Objectives and procedure Monotonic dynamic tensile tests were conducted on steel bars taken from the first floor bracing members used for the shake table investigation. These tests were performed at a prescribed strain rate of 22 × 10−3 /s corresponding to the mean strain rate demand on the braces during the shake table tests. The specimens, 150 mm long, were tested on a hydraulic servo-controlled testing system. 5.2. Results Characteristic results of the dynamic tensile tests are shown in Table 3 for all the braces used in the shake table tests. The ratios of dynamic to static yield and tensile strengths are also given in the table. The increase in yield strength due to the strain rate effect is very similar to the maximum normalized stress recorded on the braces during the shake table study, which confirms that the strain rate effect is the main contributor to the increase in tensile forces in braces of TOCBF. The mean value of 1.12 correlates very well with the prediction of 1.13 given by equation (1) for a strain rate of 22 × 10−3 /s, which indicates that equation (1) is adequate to predict the dynamic yield strength of braces in TOCBFs. Table 3 indicates also that the tensile strength of the steel is not as much affected as its yield strength under similar seismic strain rate conditions. The mean tensile strength amplification is only 4% with a maximum of 7%.
The experimental results reported above indicate clearly that the strain rate effect is directly responsible for the increase in tensile forces in braces of TOCBFs. This force increase can be well predicted by equation (1) and, therefore, a design dynamic force amplification factor could be applied to the brace forces if the maximum strain rate demand in the braces could be predicted for the design base earthquake. The expected strain rate demand can be obtained directly from nonlinear dynamic analyses performed on the structure under an ensemble of ground motion accelerograms representative of the design base earthquake. This direct approach, however, is not suitable for design purposes and it is proposed herein to use a combination of response spectrum and substitute structure techniques to estimate the strain rate demand in braces. By definition, the strain rate demand in a bracing element located in the ith storey of a TOCBF, ⑀˙ i, is equal to the rate of elongation of the element, ␦˙ i, divided by its length, Li.
⑀˙ i =
␦˙ i . Li
(2)
Assuming a predominant shear mode of deformation for low rise braced frames, the rate of elongation of the bracing element is directly proportional to the rate of interstorey drift, ⌬˙ i, for the storey i hosting the bracing element under consideration.
␦˙ i = Ci⌬˙ i
(3)
where Ci is a constant relating the geometry of the bracing element and the floors below and above the ith storey. Considering that the structure is vibrating in a particular mode of vibration, j, with an effective period of vibration, Tj eff, and assuming equal elastic and inelastic maximum displacements, the maximum interstorey drift of the ith storey can be simply expressed by a response spectrum approach8. (j) (j) ⌬i = ␣j (A(j) i − Ai − 1 )S D
(4)
and where ␣j is the participation factor for mode j, A(j) i are the jth mode shape components for floors i and i A(j) i−1 − 1 respectively and S (j) D is the relative spectral displacement corresponding to the elastic period of vibration and damping ratio in mode j, Tj and j. The mode shape compo(j) nents, A(j) i and Ai − 1, and the modal participation factor, ␣j, can be obtained from a free vibration dynamic analysis.
Table 3 Dynamic* yield and tensile strength of first floor braces Value
Yield strength Fyd (MPa)
Fyd / Fys†
Tensile strength Fud (MPa)
Fud / Fus†
Mean Maximum Minimum
354 361 340
1.12 1.14 1.09
488 495 481
1.04 1.07 1.02
*Under a constant strain rate of 22 × 10⫺3 /s. †Fys and Fus: Static yield and tensile strength.
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Alternatively, assuming the structure responds in its first mode of vibration (j = 1), these parameters can be estimated from the static lateral displacements at floors i−1 and i under the code lateral force distribution. From its initial position, the time necessary for the structure to reach this maximum interstorey drift (representing 1/4 cycle of vibration) must be equal to one-quarter of the effective period of vibration. Therefore, the average rate of interstorey drift can be estimated as follows: 4␣j (A ⌬˙ i =
− A )S Tj eff
(j) i
(j) i−1
(j) D
(5)
The inelastic response of the structure must be reflected by the appropriate value for the effective period of vibration to be used in equation (5). For this purpose, the substitute structure method10–12 is considered. This substitute structure approach is a procedure where an inelastic system is modelled as an equivalent elastic system as shown in Figure 6. For a TOCBF vibrating in a particular mode j of vibration, the stiffness, Kj, corresponds to the elastic stiffness of the tensile braces before yielding. The effective stiffness, Kj eff, is the secant stiffness corresponding to a maximum inelastic displacement characterized by a displacement ductility factor, . With these assumptions, the effective period of vibration, Tj eff, can then be simply related to the elastic period of vibration, Tj. Tj eff = √ Tj .
(6)
Substituting equations (3), (5) and (6) into equation (2), yields an expression for the average strain rate demand in the braces at a particular storey, i, in the structure.
⑀˙ i =
(j) (j) 4Ci␣j (A(j) i − Ai−1 )S D √ Tj Li
(7)
The average strain rate demand can also be written as a function of the pseudo absolute spectral acceleration, S (j) A .
⑀˙ i =
(j) (j) Ci␣j (A(j) i − Ai − 1 )Tj S A 2 √ Li
(8)
Finally, by substituting equation (8) into equation (1), leads to an expression for the dynamic amplification factor
Figure 6 Substitute structure approach for seismic response of a TOCBF
for the braces located in the ith storey, Dbi, of a TOCBF vibrating in a particular mode j.
冤
Dbi = 1 + 0.0473 log
冥
(j) (j) Ci␣j (A(j) i − Ai−1 )Tj S A . −6 2 50 × 10 √ Li
(9)
Table 4 presents the application of equation (9) to the first floor braces of the TOCBF shake table test model. The calculations were performed for the first elastic mode of vibration of the structure (j = 1). The coefficient C1 is equal to the cosine of the inclination angle of the braces from the horizontal. A linear free vibration analysis was performed on the test frame to determine the first modal participation factor, ␣1, and the first floor component of the first mode (1) is equal to shape, A(1) 1 . The mode shape component A0 zero for the first storey. The spectral accelerations in the first mode of vibration, S (1) A , were read from Figure 3 while ductility ratios, , were obtained from Figure 4. The results of the calculations shown in Table 4 agree well with the experimental data shown in Table 2. The maximum strain rate demands are reasonably well predicted by this simple approach. The resulting force amplification factors are, on the other hand, virtually identical to the maximum normalized stress measured in the first floor braces during the seismic tests (see Figure 4). Equation (9) is therefore adequate for estimating, at the design stage, the increase in tensile forces in braces of TOCBFs due to strain rate effects. The logarithmic function in equation (1) makes the force amplification factor relatively insensitive to the strain rate. The maximum strain rates that can be generated by seismic excitation is conservatively estimated as 10−1 /s. According to equation (1) and as shown in Figure 5, the maximum value for the dynamic amplification factor of a bracing member would be equal to 1.15. This conservative estimate could be also used for code design purposes.
7. Capacity design considerations Brace connections must be designed for the maximum expected brace force to maintain the integrity of the load resisting system during an earthquake. Similarly, all other elements along the lateral load path must be sized to resist, in the elastic range of the material, the gravity loads (if any) plus the forces induced during yielding of the bracing elements. The maximum brace force corresponds to its specified yield tensile resistance multiplied by an amplification factor to account for strain rate effects as described earlier. This amplification factor can be taken as 1.15 or, alternatively, can be based on the approximate dynamic response and anticipated ductility demand on the braced frame as given by equation (9). Other elements of the lateral load resisting system will also be subjected to dynamic loading during strong ground shaking and, therefore, their resistance can be increased also because of strain rate effects. For instance, the resistance of a structural component could be established based on its dynamic yield strength, Fyd, or tensile strength, Fud, depending whether yielding or fracture is the governing failure mode. If an element is to fail by inelastic buckling, then its compressive strength can also be increased to
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Table 4 Dynamic amplification factor ( Db1 ) for first floor braces of test frame Excitation Puget Sound San Fernando El Centro
␣1
C1*
0.87
1.27
A(1) 1
0.53
T1† (s)
0.47
2 S (1) A ‡ (m/s ) §
8.7
2.4
6.0
4.9
13.3
7.7
L1 (m)
2.5
⑀· 1 (10⫺3 /s)
Db1
64
1.15
31
1.13
42
1.14
*Cosine of angle of inclination of braces from horizontal. †See Table 2. ‡See Figure 3. §See Figure 4.
account for strain rate effects. In this case, the increase would be a function of the slenderness of the element, which can be expressed by the slenderness factor, , used to evaluate the compressive strength of columns13.
=
KL r
冪 E Fys 2
(10)
where KL/r is the slenderness ratio of the element under consideration and E is the elastic modulus. If = 0, the member can reach its yield strength and its resistance can be based on Fyd. If ⱖ 1.5, failure of the member is characterized by a nearly elastic buckling13 and no increase in strength should be applied. For intermediate values of , a linear interpolation could be used between Fys and Fyd. The designer must be aware, however, that strain rates will be generally higher in bracing members than in adjacent structural elements of the lateral load resisting system. In capacity design, these adjacent elements are designed to remain essentially elastic while the bracing elements yield. In this case, bracing members may still experience high strain rates and, thus, exhibit a higher yield resistance, while the elastic stress in the adjacent elements would remain constant with no strain rate effect. In addition, some of these adjacent elements could be larger in size than merely required to resist the forces induced by the braces and, thereby, would experience only a fraction of the stress variation induced in the braces. This would be the case of columns of braced bays which are designed to carry, both, gravity and lateral loads. Therefore, unless numerical investigations are performed to evaluate the strain rate demand in adjacent elements, their resistance should be based on static strength properties.
Figure 7 Floor plan view of the building considered for the design example
The design of the structure was performed according to the static method of the 1995 edition of the NBCC9 and the CSA-S16.1-94 Standard for the design of steel structures in Canada4. The gravity loads considered in the calculations were: roof dead load of 1.2 kPa, floor dead load of 3.7 kPa, roof snow load of 1.48 kPa, and an occupancy floor live load of 2.4 kPa. The weight of the exterior walls was assumed equal to 1.2 kPa. The bracing system was designed for a total lateral seismic force V given by:
冉冊
V=vSIFW
8. Design example A typical four-storey office building assumed to be located in Vancouver, B.C. is used herein to illustrate the application of the proposed method to determine the dynamic amplification factor for braces at the design stage. The plan lay-out of the structure is shown in Figure 7. The building is regular in shape, with a rectangular foot print and the vertical bracing bents symmetrically distributed in each direction. In this example, the resistance to lateral seismic loads is investigated in the E–W direction only. As shown in Figure 8, an X-bracing configuration with a pair of opposite slender steel braces at each floor was selected for these bracing bents. This bracing arrangement is very common for tension-only braced frames where bracing members would typically be made of steel plates.
U R
(11)
where v is the design ground velocity for the site (0.21 m/s), S is the seismic response factor, I is the importance factor, F is the foundation factor, W is the seismic weight of the structure, U is a calibration factor (U = 0.6) and R is the force modification factor. The value of S varies with the fundamental period of the structure and the seismicity at the site. For this building, S was equal to 1.68. The factors I and F were set equal to 1.0 (structure of normal importance on stiff soil). The weight of the structure included the roof dead weight, plus 25% of the roof snow load. The total weight of each floor, Wx, is given in Table 5. The R factor mainly accounts for the capacity of the system to absorb and dissipate energy during earthquakes. It varies from 4.0 for a ductile moment resisting frame to 1.0 for unreinforced masonry structures. Properly detailed
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brace, Tr, was equal to the brace forces induced by the forces Fx. The resistance Tr is given by: Tr = AFy
Figure 8 Bracing bent in the E–W direction of the building considered for the design example
tension-only bracing systems can classify under the nominally ductile braced frames for which a R = 2.0 can be used. That value was retained in this example and the total base shear V = 2248 kN. The seismic force at each level, FX, applied to each vertical bracing is given by: Fx = 0.6VWx
hx
(12)
冘W h n
i i
i=1
where hx refers to the height of the level under consideration and n is the total number of levels. Since the building is regular and symmetrical, each bracing bent is designed to resist 50% of the total lateral seismic loads plus the effects of the accidental torsional moments, which represent an additional 10% of the load. Thus, 60% of the total lateral load is applied to each bracing bent and the number 0.6 in equation (12) reflects this value. The values of hx and Fx are given in Table 5. At each level, the cross-sectional area of the braces, A, was selected so that the factored tensile resistance of the
(13)
where = 0.9 and Fy is the yield strength of the steel material (300 MPa). The cross sections of the braces, A, as obtained with equation (13) are presented in Table 5. For simplicity, P-delta effects have not been included in the design of the structure. The columns in the bracing bents must sustain the gravity loads plus the forces induced by the bracing members upon yielding. According to the 1995 edition of the NBCC9, the gravity loads to be considered with seismic effects must include 100% of the dead load and 50% of the live load. The column gravity loads corresponding to that load combination, CD + 0.5L, are given in Table 5. The column loads due to the occupancy live load were reduced as a function of the supported tributary floor area. The vertical component of the cumulative brace tensile resistance at each floor, AFy sin, (excluding any impact factor) is also given in Table 5 along with three different design axial loads in the columns: a design axial load without strain rate effect (CD + 0.5L + AFy sin ), a design axial load with a dynamic amplification factor of 1.10 for the braces (CD + 0.5L + 1.1AFy sin ) and a design axial load with a dynamic amplification factor of 1.15 for the braces (CD + 0.5L + 1.15AFy sin ). Preliminary design of the columns of the bracing bent was done assuming a dynamic amplification factor of 1.15 for the braces and 1.0 for the columns. The selected column sizes are shown in Figure 8. Dynamic properties in the E–W direction were obtained using the Drain-2DX computer program14. The analytical model that was used included one bracing bent and half of the gravity load carrying columns of the building. The gravity columns were included as they are continuous over two storeys and thus contribute significantly to the higher modes of vibrations. The in-plane stiffness of the floor and roof diaphragms was assumed to be very large and thus only one horizontal degree-of-freedom per storey was assigned to the model. The computed fundamental period, mode shape and participation factor of the building in the E–W direction are shown in Figure 9. The design absolute acceleration response spectrum normalized for the site is shown in Figure 10. The value of S (1) A corresponding to the fundamental period of the building in the E–W direction is also indicated on the spectrum. Table 6 presents the results of calculations of the dynamic amplification factors, Dbi, for the braces of the studied braced frame assuming only a dynamic response in the first mode of vibrations. The ductility ratio, , was assumed to be equal to the force modification factor, R, in
Table 5 Seismic design data for the studied braced frame Storey
4 3 2 1
Wx (kN)
hx (m)
Fx (kN)
A (mm2 )
CD + 0.5L (kN)
AFy sin* (kN)
CD + 0.5L + AFy sin (kN)
2678 6187 6187 6187
15.2 11.4 7.6 3.8
503 872 582 291
1250 3420 4860 5590
79 301 510 716
168 627 1279 2029
247 928 1789 2745
* is the brace angle from the horizontal = 26.6°.
CD + 0.5L + CD + 0.5L + 1.1AFy sin 1.15AFy sin (kN) (kN) 264 991 1917 2948
272 1022 1981 3049
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Table 6 Calculations of dynamic amplification factors ( Dbi ) for braces of studied braced frame Storey i 4 3 2 1
Ci*
␣1
0.89
0.93
A(1) i † 1.00 0.72 0.46 0.22
T1† (s)
1.05
2 S (1) A † (m/s )
2.51
‡
2.0
Li (m)
8.50
⑀· i (10⫺3 /s) 5.2 4.8 4.4 4.1
Dbi 1.10 1.09 1.09 1.09
*Cosine of angle of inclination from horizontal of braces at each storey. †See Figure 9. ‡Taken to be equal to the force reduction factor R.
rate effect would result in a reduction of 9.2% in the design axial load of the third and fourth storey columns (928 kN vs 1022 kN) and a reduction of 10.0% in the first and second storey columns (2745 kN vs 3049 kN). In this case, the first and second storey W310 × 107 column section would be replaced by a W310 × 97 section, which would be unconservative.
9. Conclusion Based on the experimental and analytical results obtained in this study, the following conclusions can be drawn:
Figure 9 Computed fundamental period, mode shape and participating factor of the building in the E–W direction
• The increase in tensile forces in the braces of TOCBFs is mainly caused by a yield strength increase of the steel under a high strain rate demand. • A simple equation, based on a combination of response spectrum and substitute structure techniques, has been proposed to estimate this increase in brace forces at the design stage. The predictions of this proposed equation correlates well with the shake table results obtained herein. Alternatively, a conservative dynamic amplification factor for the braces of 1.15 could be used for preliminary code design applications. • The adjacent elements of the lateral load resisting system should be designed based on their static strength properties unless strain rate effects in these elements can be evaluated through nonlinear dynamic analyses.
Acknowledgements
Figure 10 Design absolute acceleration response spectrum normalized for the building’s site
equation (11). The results indicate that a dynamic amplification factor of 1.09 should be used for the braces in the first three bottom storeys, while a value of 1.10 should be used for braces in the fourth storey. Note that these values agree well with the impact factor of 1.10 specified in the Canadian steel standard4. As shown in Table 5, a reduction of the dynamic amplification factor for the braces from 1.15 to 1.10 yields a reduction in the design axial load of 3.3% for the third and fourth storey columns (2948 kN vs 3049 kN); the corresponding reduction for the first and second floor columns is 3.0% (991 kN vs 1022 kN). For this particular example, no reduction in column cross-section could be made as a result of these reductions of the design axial loads. Note, however, that neglecting strain
This research project was funded by the Natural Science and Engineering Research Council of Canada (NSERC), and the “Fonds pour la Formation de Chercheurs et l’Aide a` la Recherche” (FCAR) of Quebec which provided research grants in support of this project. The assistance of the Group CANAM MANAC of Montreal, which supplied the testing material, is also gratefully acknowledged. The authors acknowledge also the assistance of Mrs Nathalie Robert for the design of the building example described in the paper. The Steel Structures Education Foundation provided a scholarship for this work.
References 1
2
FEMA, NEHRP recommended provisions for seismic regulations for new buildings Part 1—provisions. Federal Emergency Management Agency, FEMA 222A. Washington, DC: Building Seismic Safety Council, 1994. SEAOC, Recommended lateral force requirements and commentary. Seismology Committee. Sacramento, CA: Structural Engineers Association of California, 1990.
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3 AISC, Seismic provisions for structural steel buildings. American Institute of Steel Construction, 1992. 4 CSA, Limit state design of steel structures. National Standard of Canada, CAN/CSA-S16.1-94. Rexdale, Ontario: Canadian Standards Association, 1994. 5 Wakabayashi, M., Nakamura, T., Iwai, S., and Hayashi, Y. Effects of strain rate on the behavior of structural members. In: 8th World Conference on Earthquake Engineering, vol. 4. San Francisco, 1994, 491–498. 6 Filiatrault, A., Tremblay, R., Thoen, B. K., and Rood, J. A second generation earthquake simulation system in Canada: description and performance. In: 11th World Conference on Earthquake Engineering, Acapulco, 1996. 7 EDI, U2 and V2 manual. Vancouver: Experimental Dynamic Investigations Ltd, 1993. 8 Clough, R. W., and Penzien, P. Dynamics of structures, 2nd ed. New York: McGraw–Hill, 1993.
9 10
11 12 13 14
NRCC, National building code of Canada. Associate Committee on the National Building Code. Ottawa: National Research Council of Canada, 1995. Tremblay, R. and Filiatrault, A., Seismic impact loading in inelastic tension-only concentrically braced steel frames: myth or reality?, Earthquake Engineering and Structural Dynamics 1996, 25, 1373– 1389 Gulkan, P. and Sosen, M., Inelastic response of reinforced concrete structures to earthquake motions, ACI Journal 1974, 71, 604–610 Shibata, A. and Sozen, A., Substitute structure method for seismic design in R/C, Journal of the Structural Division, ASCE 1976, 102, 1–8 Salmon, C. G., and Johnson, J. E. Steel structures—design and behaviour. New York: Harper & Row, 1990. Allahabadi, R., and Powell, G. H. DRAIN-2DX user guide. Report No. UCB/EERC-88/06. Berkeley, CA: Earthquake Engineering Research Center, University of California, 1988.
Engineering Structures, Vol. 20, No. 12, pp. 1087–1096, 1998 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0141–0296/98 $19.00 + 0.00
Design of Tension-Only Concentrically Braced Steel Frames for seismic induced impact loading Andre´ Filiatrault and Robert Tremblay EPICENTRE Research Group, Department of Civil Engineering, Ecole Polytechnique, University of Montreal Campus, P.O. Box 6079, Station “Centre-Ville”, Montreal, Canada H3C 3A7
Tension-Only Concentrically Braced Frames (TOCBFs) exhibit deteriorating pinched hysteretic behaviour during strong earthquakes. It has long been suspected that the sudden increase in tensile forces in the braces of TOCBFs creates detrimental impact loading on the connections and other structural elements. This paper addresses this issue through shake table tests of half scale, twostorey, TOCBF models. By normalizing the hysteresis loops of braces obtained from shake table tests to the yield strength of steel obtained from static tests, the increase in tensile forces in the braces was obtained. Results of dynamic tensile tests on steel coupons under similar strain rates as observed during the shake table tests showed that this increase in tensile forces is caused by a yield strength increase of the steel under high strain rate. A procedure is proposed to predict this increase in tensile forces in the braces at the design stage. The use of this procedure is illustrated by a design example. 1998 Elsevier Science Ltd. All rights reserved Keywords: braced frames, impact, seismic
Slender braces in TOCBFs transit between an elastic buckling state, a restraightening state, in which they carry almost no load, an elastic tensile loading state as they are suddenly straightened and, finally, a tensile yielding state. North American seismic provisions associate the sudden increase in tensile forces in the braces with detrimental impact loading on the connections and other structural elements1–4. The latest edition of the NEHRP seismic regulations in the U.S. recommends to pretension diagonal members in TOCBFs to prevent loose diagonals during strong earthquakes1. In California, designers are warned that the force demand on a brace in tension is expected to exceed the brace tensile yield strength2. The latest edition of the Canadian steel standard4 specifies an explicit value of 1.10 for an impact factor to be used in the design of the brace connections. No experimental evidence, however, has been provided so far to confirm, or to quantify, this impact phenomenon. The main objective of this paper is to shed some light on the behaviour of TOCBFs through shake table tests of half scale, two-storey, TOCBF models. The bracing elements were instrumented to measure their hysteretic loops under seismic conditions. By comparing these hysteresis loops with quasi-static and dynamic monotonic test
1. Introduction Tension-Only Concentrically Braced Frames (TOCBFs) incorporate very slender bracing members, such as steel rods or flat plates, which are unable to dissipate much energy in compression. A characteristic of TOCBFs is their deteriorating pinched hysteretic behaviour during strong earthquakes. Alternating tension yielding and elastic compression buckling of the bracing elements induce slackness in the lateral load resisting system around its point of zero displacement. This slackness translates into severely pinched hysteresis loops and potential impact loading in the structure. This unfavourable behaviour was quickly recognized as a major fault of TOCBFs and, therefore, structural engineers have shied away from this system for medium and high-rise buildings located in active seismic areas. The TOCBF system, however, continues to be used extensively for low-rise industrial and commercial steel buildings because it is inexpensive and simple to design, fabricate and erect. Therefore, these structures constitute a large portion of the building stock in North America. This extensive use of TOCBFs, however, has not been accompanied by an adequate research effort to understand fully their behaviour and rationalize their design.
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results, the increase in tensile forces in the braces was quantified. A procedure is proposed to predict this increase in brace forces at the design stage.
2. Impact loading and strain rate effect in TOCBFs The sudden increase in tensile forces in the braces of TOCBFs has long been suspected to cause detrimental impact loading on the connections and other structural components of the lateral load resisting system. In fact, two mechanisms can contribute to amplify the forces in the braces during tensile loading. The first mechanism is a true impact phenomenon caused by a rapid transfer of energy from the floor masses, being suddenly halted in their motion, to the braces becoming suddenly taut. The second mechanism is associated with the strain rate effect on the yield strength of steel. The yield strength of steel increases with the rate of loading while its tensile strength and strain-hardening moduli remain almost unchanged. Therefore during elastic tensile loading, the braces are subjected to a high strain rate demand and a portion of the increase in tensile forces in the braces may be attributed to the increase of the yield strength of steel. The experimental investigation conducted by Wakabayashi et al.5 on strain rate effects is the most applicable to TOCBFs under seismic conditions. They proposed the following expression to estimate the increase of yield strength of steel with strain rate:
冉冊
Fyd ⑀˙ = 1 + 0.0473 log Fys ⑀˙ o
Figure 1 Test frame
(1)
where Fyd is the dynamic yield strength for a strain rate ⑀˙ and Fys is the quasi-static yield strength under a strain rate ⑀˙ o = 50 × 10−6 /s.
3. Shake table test program 3.1. Test frame Half scale, two-storey, TOCBF models were tested on the uniaxial, 3.4 m × 3.4 m earthquake simulation facility at Ecole Polytechnique in Montreal, Canada6. A general view of the test frame is presented in Figure 1. To represent a simply supported base, each base column was linked to a true pin mechanism which, in turn, was attached to a foundation beam on the shake table. The two pin-ended floor beams were made of hollow structural steel (HSS) 127 × 76 × 4.8 sections. The mass of the model was simulated by four concrete blocks (30 kN each) linked to the floor beams. These concrete blocks were supported vertically by an independent peripheral frame which was completely pinned in the direction of the shake table motion and braced in the opposite direction to ensure no motion of the test plane frame in that direction. This supported structure carried all the vertical loads from the concrete blocks so the test frame had only to resist the lateral inertia loads. Figure 2 shows a photograph of the test frame and supporting structure after a seismic test on the shake table. In the direction of excitation, the test frame was braced at each level by grade 300W steel (nominal Fy = 300 MPa and Fu = 450 MPa) round tie-rods having a diameter of
Figure 2 Test and supporting structures after a seismic test on shake table
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12.7 mm. These braces had a slenderness ratio of 800 and, thus, exhibited negligible buckling strength. The force timehistories of both first floor braces could be recorded with axial load cells placed adjacent to the bottom first floor brace connections. 3.2. Preliminary system identification tests The dynamic characteristics of the model frame were estimated from low amplitude ambient vibration simulations and free vibration tests. For the ambient vibration simulations, the test frame was excited by a 0–25 Hz flat white noise, with a RMS (Root Mean Square) amplitude of 0.03 g. This base motion was large enough to cause buckling in all compression braces at each cycle, but was sufficiently low to ensure that the tension braces remained in the elastic range of the steel. A dedicated ambient vibration analysis software7 was then used to determine the natural periods of the test frame from power spectral density plots of the absolute floor horizontal acceleration records. In the free vibration tests, the structure was excited by a sinusoidal input at its first natural period. When a steady-state response was obtained, the input was suddenly stopped and the floor relative displacements were recorded. The first modal damping ratio of the structure was then established by the logarithmic relative displacement decrement at each floor8. The first mode of vibration was measured at a period of 0.470 ± 0.005 s and represented a classical shear mode involving tensile and buckling deformations of the braces. The second mode of vibration, at a period of 0.073 ± 0.005 s, represents a flexural mode of deformation of the columns. The damping ratio in the first mode of vibration was measured at 2.4% critical. 3.3. Earthquake ground motions Three different historical ground motions, recorded in Western North America, were chosen as input for the shake table studies. The characteristics of these records are summarized in Table 1. These records present ratios of Peak Horizontal Acceleration (PHA in g) to Peak Horizontal Velocity (PHV in m/s) close to unity which is compatible with seismic zoning maps of Western Canada9. Each accelerogram was scaled to mobilize a displacement ductility level representative of code designed structures. This scaling procedure was based on equating, for each accelerogram, the base shear at first yield specified in the 1995 edition of the National Building Code of Canada (NBCC)9 to the actual yield base shear of the structure. This procedure yielded scaling factors of 1.3, 1.6 and 1.2 for the Puget Sound, San Fernando and El Centro accelerograms, respectively. Figure 3 presents the absolute acceleration response spectra of the three scaled seismic events for 2.4% damping (corresponding to the first modal damping ratio of the structure).
Figure 3 Absolute acceleration response damping) for the three scaled accelerograms
spectra
(2.4%
4. Shake table test results 4.1. Hysteretic behaviour of first floor braces Figure 4 presents, for the three seismic events considered, the stress–strain hysteresis loops for the first floor brace located on the east side of the frame (see Figure 1). The stress was normalized to the yield strength of each brace, Fys, obtained from quasi-static ( ⑀˙ = 50 × 10−6 /s) monotonic tensile tests performed on 150 mm long specimens of the brace material. The behaviour of the brace is consistent for the three earthquake records. The brace yields in tension at a strain of about 0.2% and exhibits ductility ratios ranging from 2.4 (for Puget Sound) to 7.7 (for El Centro). The maximum stress recorded in the brace exceeds the pseudostatic yield strength by 14%. This peak value occurs when the brace is stretched significantly, for the first time, in the inelastic range. During the tensile loading state before the first yield excursion, the structure vibrated elastically which caused a maximum strain rate demand in the tension braces. Following the maximum stress, the stress reduces slightly as the brace elongates in the inelastic range and the rate of straining progressively reduces. To quantify the strain rate demand on the braces, the time required to increase the tensile stress from zero to the static yield strength before each inelastic excursion was computed from the experimental data. The results of these calculations are summarized, for all the six first floor braces, in Table 2. The strain rate in the braces varies from 10 × 10−3 /s to 40 × 10−3 /s with a mean value of 22 × 10−3 /s for the three different excitations. The maximum normalized stress observed for each inelastic excursion is also given in Table 2. A good correlation can be observed between the strain rate in the elastic tensile loading portion of the stress–strain curve and the maximum normalized stress reached in the inelastic range of the steel. Figure 5 compares graphically this correlation with the predictions
Table 1 Historical seismic events considered for shake table tests Record name
Earthquake name Station
Date
Component
PHA (g)
PHV (m/s)
Puget Sound
13-04-49
N04W
0.16
0.21
San Fernando
Western Washington San Fernando
9-02-71
N90E
0.21
0.21
El Centro
Imperial Valley
18-05-40
S00E
0.34
0.33
Olympia highway testing laboratory Hollywood storage P.E. lot Los Angeles El Centro site Imperial Valley Irrigation District
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Figure 4 Stress–strain hysteresis loops for first floor brace on east side of frame
Table 2 Strain rate demand and maximum normalized stress in first floor braces Excitation
Brace No.
Inelastic excursion No.
Puget Sound
1 (East)
San Fernando
2 (West) 3 (East)
1 2 3 4 1 1 2 3 4 1 2 3 4 1 2 3 4 5
4 (West)
El Centro
5 (East)
6 (West) Mean strain rate for all inelastic excursions
Strain rate ⑀· (10⫺3 /s)
Maximum normalized stress
18 1.07 18 1.06 23 1.06 30 1.14 20 1.01 20 1.06 22 1.14 18 1.10 24 1.09 26 1.10 29 1.11 17 1.10 23 1.10 10 1.01 40 1.14 17 1.05 35 1.10 20 1.04 No significant inelastic excursion 22
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6. Predicting the dynamic amplification factor for brace forces
Figure 5 Influence of strain rate on maximum normalized stress in first floor braces
given by equation (1). This empirical equation represents a conservative estimate of the experimental results.
5. Dynamic tensile tests on steel bar specimens 5.1. Objectives and procedure Monotonic dynamic tensile tests were conducted on steel bars taken from the first floor bracing members used for the shake table investigation. These tests were performed at a prescribed strain rate of 22 × 10−3 /s corresponding to the mean strain rate demand on the braces during the shake table tests. The specimens, 150 mm long, were tested on a hydraulic servo-controlled testing system. 5.2. Results Characteristic results of the dynamic tensile tests are shown in Table 3 for all the braces used in the shake table tests. The ratios of dynamic to static yield and tensile strengths are also given in the table. The increase in yield strength due to the strain rate effect is very similar to the maximum normalized stress recorded on the braces during the shake table study, which confirms that the strain rate effect is the main contributor to the increase in tensile forces in braces of TOCBF. The mean value of 1.12 correlates very well with the prediction of 1.13 given by equation (1) for a strain rate of 22 × 10−3 /s, which indicates that equation (1) is adequate to predict the dynamic yield strength of braces in TOCBFs. Table 3 indicates also that the tensile strength of the steel is not as much affected as its yield strength under similar seismic strain rate conditions. The mean tensile strength amplification is only 4% with a maximum of 7%.
The experimental results reported above indicate clearly that the strain rate effect is directly responsible for the increase in tensile forces in braces of TOCBFs. This force increase can be well predicted by equation (1) and, therefore, a design dynamic force amplification factor could be applied to the brace forces if the maximum strain rate demand in the braces could be predicted for the design base earthquake. The expected strain rate demand can be obtained directly from nonlinear dynamic analyses performed on the structure under an ensemble of ground motion accelerograms representative of the design base earthquake. This direct approach, however, is not suitable for design purposes and it is proposed herein to use a combination of response spectrum and substitute structure techniques to estimate the strain rate demand in braces. By definition, the strain rate demand in a bracing element located in the ith storey of a TOCBF, ⑀˙ i, is equal to the rate of elongation of the element, ␦˙ i, divided by its length, Li.
⑀˙ i =
␦˙ i . Li
(2)
Assuming a predominant shear mode of deformation for low rise braced frames, the rate of elongation of the bracing element is directly proportional to the rate of interstorey drift, ⌬˙ i, for the storey i hosting the bracing element under consideration.
␦˙ i = Ci⌬˙ i
(3)
where Ci is a constant relating the geometry of the bracing element and the floors below and above the ith storey. Considering that the structure is vibrating in a particular mode of vibration, j, with an effective period of vibration, Tj eff, and assuming equal elastic and inelastic maximum displacements, the maximum interstorey drift of the ith storey can be simply expressed by a response spectrum approach8. (j) (j) ⌬i = ␣j (A(j) i − Ai − 1 )S D
(4)
and where ␣j is the participation factor for mode j, A(j) i are the jth mode shape components for floors i and i A(j) i−1 − 1 respectively and S (j) D is the relative spectral displacement corresponding to the elastic period of vibration and damping ratio in mode j, Tj and j. The mode shape compo(j) nents, A(j) i and Ai − 1, and the modal participation factor, ␣j, can be obtained from a free vibration dynamic analysis.
Table 3 Dynamic* yield and tensile strength of first floor braces Value
Yield strength Fyd (MPa)
Fyd / Fys†
Tensile strength Fud (MPa)
Fud / Fus†
Mean Maximum Minimum
354 361 340
1.12 1.14 1.09
488 495 481
1.04 1.07 1.02
*Under a constant strain rate of 22 × 10⫺3 /s. †Fys and Fus: Static yield and tensile strength.
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Alternatively, assuming the structure responds in its first mode of vibration (j = 1), these parameters can be estimated from the static lateral displacements at floors i−1 and i under the code lateral force distribution. From its initial position, the time necessary for the structure to reach this maximum interstorey drift (representing 1/4 cycle of vibration) must be equal to one-quarter of the effective period of vibration. Therefore, the average rate of interstorey drift can be estimated as follows: 4␣j (A ⌬˙ i =
− A )S Tj eff
(j) i
(j) i−1
(j) D
(5)
The inelastic response of the structure must be reflected by the appropriate value for the effective period of vibration to be used in equation (5). For this purpose, the substitute structure method10–12 is considered. This substitute structure approach is a procedure where an inelastic system is modelled as an equivalent elastic system as shown in Figure 6. For a TOCBF vibrating in a particular mode j of vibration, the stiffness, Kj, corresponds to the elastic stiffness of the tensile braces before yielding. The effective stiffness, Kj eff, is the secant stiffness corresponding to a maximum inelastic displacement characterized by a displacement ductility factor, . With these assumptions, the effective period of vibration, Tj eff, can then be simply related to the elastic period of vibration, Tj. Tj eff = √ Tj .
(6)
Substituting equations (3), (5) and (6) into equation (2), yields an expression for the average strain rate demand in the braces at a particular storey, i, in the structure.
⑀˙ i =
(j) (j) 4Ci␣j (A(j) i − Ai−1 )S D √ Tj Li
(7)
The average strain rate demand can also be written as a function of the pseudo absolute spectral acceleration, S (j) A .
⑀˙ i =
(j) (j) Ci␣j (A(j) i − Ai − 1 )Tj S A 2 √ Li
(8)
Finally, by substituting equation (8) into equation (1), leads to an expression for the dynamic amplification factor
Figure 6 Substitute structure approach for seismic response of a TOCBF
for the braces located in the ith storey, Dbi, of a TOCBF vibrating in a particular mode j.
冤
Dbi = 1 + 0.0473 log
冥
(j) (j) Ci␣j (A(j) i − Ai−1 )Tj S A . −6 2 50 × 10 √ Li
(9)
Table 4 presents the application of equation (9) to the first floor braces of the TOCBF shake table test model. The calculations were performed for the first elastic mode of vibration of the structure (j = 1). The coefficient C1 is equal to the cosine of the inclination angle of the braces from the horizontal. A linear free vibration analysis was performed on the test frame to determine the first modal participation factor, ␣1, and the first floor component of the first mode (1) is equal to shape, A(1) 1 . The mode shape component A0 zero for the first storey. The spectral accelerations in the first mode of vibration, S (1) A , were read from Figure 3 while ductility ratios, , were obtained from Figure 4. The results of the calculations shown in Table 4 agree well with the experimental data shown in Table 2. The maximum strain rate demands are reasonably well predicted by this simple approach. The resulting force amplification factors are, on the other hand, virtually identical to the maximum normalized stress measured in the first floor braces during the seismic tests (see Figure 4). Equation (9) is therefore adequate for estimating, at the design stage, the increase in tensile forces in braces of TOCBFs due to strain rate effects. The logarithmic function in equation (1) makes the force amplification factor relatively insensitive to the strain rate. The maximum strain rates that can be generated by seismic excitation is conservatively estimated as 10−1 /s. According to equation (1) and as shown in Figure 5, the maximum value for the dynamic amplification factor of a bracing member would be equal to 1.15. This conservative estimate could be also used for code design purposes.
7. Capacity design considerations Brace connections must be designed for the maximum expected brace force to maintain the integrity of the load resisting system during an earthquake. Similarly, all other elements along the lateral load path must be sized to resist, in the elastic range of the material, the gravity loads (if any) plus the forces induced during yielding of the bracing elements. The maximum brace force corresponds to its specified yield tensile resistance multiplied by an amplification factor to account for strain rate effects as described earlier. This amplification factor can be taken as 1.15 or, alternatively, can be based on the approximate dynamic response and anticipated ductility demand on the braced frame as given by equation (9). Other elements of the lateral load resisting system will also be subjected to dynamic loading during strong ground shaking and, therefore, their resistance can be increased also because of strain rate effects. For instance, the resistance of a structural component could be established based on its dynamic yield strength, Fyd, or tensile strength, Fud, depending whether yielding or fracture is the governing failure mode. If an element is to fail by inelastic buckling, then its compressive strength can also be increased to
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Table 4 Dynamic amplification factor ( Db1 ) for first floor braces of test frame Excitation Puget Sound San Fernando El Centro
␣1
C1*
0.87
1.27
A(1) 1
0.53
T1† (s)
0.47
2 S (1) A ‡ (m/s ) §
8.7
2.4
6.0
4.9
13.3
7.7
L1 (m)
2.5
⑀· 1 (10⫺3 /s)
Db1
64
1.15
31
1.13
42
1.14
*Cosine of angle of inclination of braces from horizontal. †See Table 2. ‡See Figure 3. §See Figure 4.
account for strain rate effects. In this case, the increase would be a function of the slenderness of the element, which can be expressed by the slenderness factor, , used to evaluate the compressive strength of columns13.
=
KL r
冪 E Fys 2
(10)
where KL/r is the slenderness ratio of the element under consideration and E is the elastic modulus. If = 0, the member can reach its yield strength and its resistance can be based on Fyd. If ⱖ 1.5, failure of the member is characterized by a nearly elastic buckling13 and no increase in strength should be applied. For intermediate values of , a linear interpolation could be used between Fys and Fyd. The designer must be aware, however, that strain rates will be generally higher in bracing members than in adjacent structural elements of the lateral load resisting system. In capacity design, these adjacent elements are designed to remain essentially elastic while the bracing elements yield. In this case, bracing members may still experience high strain rates and, thus, exhibit a higher yield resistance, while the elastic stress in the adjacent elements would remain constant with no strain rate effect. In addition, some of these adjacent elements could be larger in size than merely required to resist the forces induced by the braces and, thereby, would experience only a fraction of the stress variation induced in the braces. This would be the case of columns of braced bays which are designed to carry, both, gravity and lateral loads. Therefore, unless numerical investigations are performed to evaluate the strain rate demand in adjacent elements, their resistance should be based on static strength properties.
Figure 7 Floor plan view of the building considered for the design example
The design of the structure was performed according to the static method of the 1995 edition of the NBCC9 and the CSA-S16.1-94 Standard for the design of steel structures in Canada4. The gravity loads considered in the calculations were: roof dead load of 1.2 kPa, floor dead load of 3.7 kPa, roof snow load of 1.48 kPa, and an occupancy floor live load of 2.4 kPa. The weight of the exterior walls was assumed equal to 1.2 kPa. The bracing system was designed for a total lateral seismic force V given by:
冉冊
V=vSIFW
8. Design example A typical four-storey office building assumed to be located in Vancouver, B.C. is used herein to illustrate the application of the proposed method to determine the dynamic amplification factor for braces at the design stage. The plan lay-out of the structure is shown in Figure 7. The building is regular in shape, with a rectangular foot print and the vertical bracing bents symmetrically distributed in each direction. In this example, the resistance to lateral seismic loads is investigated in the E–W direction only. As shown in Figure 8, an X-bracing configuration with a pair of opposite slender steel braces at each floor was selected for these bracing bents. This bracing arrangement is very common for tension-only braced frames where bracing members would typically be made of steel plates.
U R
(11)
where v is the design ground velocity for the site (0.21 m/s), S is the seismic response factor, I is the importance factor, F is the foundation factor, W is the seismic weight of the structure, U is a calibration factor (U = 0.6) and R is the force modification factor. The value of S varies with the fundamental period of the structure and the seismicity at the site. For this building, S was equal to 1.68. The factors I and F were set equal to 1.0 (structure of normal importance on stiff soil). The weight of the structure included the roof dead weight, plus 25% of the roof snow load. The total weight of each floor, Wx, is given in Table 5. The R factor mainly accounts for the capacity of the system to absorb and dissipate energy during earthquakes. It varies from 4.0 for a ductile moment resisting frame to 1.0 for unreinforced masonry structures. Properly detailed
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1094
brace, Tr, was equal to the brace forces induced by the forces Fx. The resistance Tr is given by: Tr = AFy
Figure 8 Bracing bent in the E–W direction of the building considered for the design example
tension-only bracing systems can classify under the nominally ductile braced frames for which a R = 2.0 can be used. That value was retained in this example and the total base shear V = 2248 kN. The seismic force at each level, FX, applied to each vertical bracing is given by: Fx = 0.6VWx
hx
(12)
冘W h n
i i
i=1
where hx refers to the height of the level under consideration and n is the total number of levels. Since the building is regular and symmetrical, each bracing bent is designed to resist 50% of the total lateral seismic loads plus the effects of the accidental torsional moments, which represent an additional 10% of the load. Thus, 60% of the total lateral load is applied to each bracing bent and the number 0.6 in equation (12) reflects this value. The values of hx and Fx are given in Table 5. At each level, the cross-sectional area of the braces, A, was selected so that the factored tensile resistance of the
(13)
where = 0.9 and Fy is the yield strength of the steel material (300 MPa). The cross sections of the braces, A, as obtained with equation (13) are presented in Table 5. For simplicity, P-delta effects have not been included in the design of the structure. The columns in the bracing bents must sustain the gravity loads plus the forces induced by the bracing members upon yielding. According to the 1995 edition of the NBCC9, the gravity loads to be considered with seismic effects must include 100% of the dead load and 50% of the live load. The column gravity loads corresponding to that load combination, CD + 0.5L, are given in Table 5. The column loads due to the occupancy live load were reduced as a function of the supported tributary floor area. The vertical component of the cumulative brace tensile resistance at each floor, AFy sin, (excluding any impact factor) is also given in Table 5 along with three different design axial loads in the columns: a design axial load without strain rate effect (CD + 0.5L + AFy sin ), a design axial load with a dynamic amplification factor of 1.10 for the braces (CD + 0.5L + 1.1AFy sin ) and a design axial load with a dynamic amplification factor of 1.15 for the braces (CD + 0.5L + 1.15AFy sin ). Preliminary design of the columns of the bracing bent was done assuming a dynamic amplification factor of 1.15 for the braces and 1.0 for the columns. The selected column sizes are shown in Figure 8. Dynamic properties in the E–W direction were obtained using the Drain-2DX computer program14. The analytical model that was used included one bracing bent and half of the gravity load carrying columns of the building. The gravity columns were included as they are continuous over two storeys and thus contribute significantly to the higher modes of vibrations. The in-plane stiffness of the floor and roof diaphragms was assumed to be very large and thus only one horizontal degree-of-freedom per storey was assigned to the model. The computed fundamental period, mode shape and participation factor of the building in the E–W direction are shown in Figure 9. The design absolute acceleration response spectrum normalized for the site is shown in Figure 10. The value of S (1) A corresponding to the fundamental period of the building in the E–W direction is also indicated on the spectrum. Table 6 presents the results of calculations of the dynamic amplification factors, Dbi, for the braces of the studied braced frame assuming only a dynamic response in the first mode of vibrations. The ductility ratio, , was assumed to be equal to the force modification factor, R, in
Table 5 Seismic design data for the studied braced frame Storey
4 3 2 1
Wx (kN)
hx (m)
Fx (kN)
A (mm2 )
CD + 0.5L (kN)
AFy sin* (kN)
CD + 0.5L + AFy sin (kN)
2678 6187 6187 6187
15.2 11.4 7.6 3.8
503 872 582 291
1250 3420 4860 5590
79 301 510 716
168 627 1279 2029
247 928 1789 2745
* is the brace angle from the horizontal = 26.6°.
CD + 0.5L + CD + 0.5L + 1.1AFy sin 1.15AFy sin (kN) (kN) 264 991 1917 2948
272 1022 1981 3049
Design of Tension-Only Concentrically Braced Steel Frames: A. Filiatrault and R. Tremblay
1095
Table 6 Calculations of dynamic amplification factors ( Dbi ) for braces of studied braced frame Storey i 4 3 2 1
Ci*
␣1
0.89
0.93
A(1) i † 1.00 0.72 0.46 0.22
T1† (s)
1.05
2 S (1) A † (m/s )
2.51
‡
2.0
Li (m)
8.50
⑀· i (10⫺3 /s) 5.2 4.8 4.4 4.1
Dbi 1.10 1.09 1.09 1.09
*Cosine of angle of inclination from horizontal of braces at each storey. †See Figure 9. ‡Taken to be equal to the force reduction factor R.
rate effect would result in a reduction of 9.2% in the design axial load of the third and fourth storey columns (928 kN vs 1022 kN) and a reduction of 10.0% in the first and second storey columns (2745 kN vs 3049 kN). In this case, the first and second storey W310 × 107 column section would be replaced by a W310 × 97 section, which would be unconservative.
9. Conclusion Based on the experimental and analytical results obtained in this study, the following conclusions can be drawn:
Figure 9 Computed fundamental period, mode shape and participating factor of the building in the E–W direction
• The increase in tensile forces in the braces of TOCBFs is mainly caused by a yield strength increase of the steel under a high strain rate demand. • A simple equation, based on a combination of response spectrum and substitute structure techniques, has been proposed to estimate this increase in brace forces at the design stage. The predictions of this proposed equation correlates well with the shake table results obtained herein. Alternatively, a conservative dynamic amplification factor for the braces of 1.15 could be used for preliminary code design applications. • The adjacent elements of the lateral load resisting system should be designed based on their static strength properties unless strain rate effects in these elements can be evaluated through nonlinear dynamic analyses.
Acknowledgements
Figure 10 Design absolute acceleration response spectrum normalized for the building’s site
equation (11). The results indicate that a dynamic amplification factor of 1.09 should be used for the braces in the first three bottom storeys, while a value of 1.10 should be used for braces in the fourth storey. Note that these values agree well with the impact factor of 1.10 specified in the Canadian steel standard4. As shown in Table 5, a reduction of the dynamic amplification factor for the braces from 1.15 to 1.10 yields a reduction in the design axial load of 3.3% for the third and fourth storey columns (2948 kN vs 3049 kN); the corresponding reduction for the first and second floor columns is 3.0% (991 kN vs 1022 kN). For this particular example, no reduction in column cross-section could be made as a result of these reductions of the design axial loads. Note, however, that neglecting strain
This research project was funded by the Natural Science and Engineering Research Council of Canada (NSERC), and the “Fonds pour la Formation de Chercheurs et l’Aide a` la Recherche” (FCAR) of Quebec which provided research grants in support of this project. The assistance of the Group CANAM MANAC of Montreal, which supplied the testing material, is also gratefully acknowledged. The authors acknowledge also the assistance of Mrs Nathalie Robert for the design of the building example described in the paper. The Steel Structures Education Foundation provided a scholarship for this work.
References 1
2
FEMA, NEHRP recommended provisions for seismic regulations for new buildings Part 1—provisions. Federal Emergency Management Agency, FEMA 222A. Washington, DC: Building Seismic Safety Council, 1994. SEAOC, Recommended lateral force requirements and commentary. Seismology Committee. Sacramento, CA: Structural Engineers Association of California, 1990.
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3 AISC, Seismic provisions for structural steel buildings. American Institute of Steel Construction, 1992. 4 CSA, Limit state design of steel structures. National Standard of Canada, CAN/CSA-S16.1-94. Rexdale, Ontario: Canadian Standards Association, 1994. 5 Wakabayashi, M., Nakamura, T., Iwai, S., and Hayashi, Y. Effects of strain rate on the behavior of structural members. In: 8th World Conference on Earthquake Engineering, vol. 4. San Francisco, 1994, 491–498. 6 Filiatrault, A., Tremblay, R., Thoen, B. K., and Rood, J. A second generation earthquake simulation system in Canada: description and performance. In: 11th World Conference on Earthquake Engineering, Acapulco, 1996. 7 EDI, U2 and V2 manual. Vancouver: Experimental Dynamic Investigations Ltd, 1993. 8 Clough, R. W., and Penzien, P. Dynamics of structures, 2nd ed. New York: McGraw–Hill, 1993.
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NRCC, National building code of Canada. Associate Committee on the National Building Code. Ottawa: National Research Council of Canada, 1995. Tremblay, R. and Filiatrault, A., Seismic impact loading in inelastic tension-only concentrically braced steel frames: myth or reality?, Earthquake Engineering and Structural Dynamics 1996, 25, 1373– 1389 Gulkan, P. and Sosen, M., Inelastic response of reinforced concrete structures to earthquake motions, ACI Journal 1974, 71, 604–610 Shibata, A. and Sozen, A., Substitute structure method for seismic design in R/C, Journal of the Structural Division, ASCE 1976, 102, 1–8 Salmon, C. G., and Johnson, J. E. Steel structures—design and behaviour. New York: Harper & Row, 1990. Allahabadi, R., and Powell, G. H. DRAIN-2DX user guide. Report No. UCB/EERC-88/06. Berkeley, CA: Earthquake Engineering Research Center, University of California, 1988.