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Experimental cyclic loading of concentric HSS braces
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Madhar Haddad, Tom Brown, and Nigel Shrive
Abstract: During earthquake ground motion, diagonal braces in braced frames are subject to a series of cyclic loadings, alternately tension and compression. The brace can buckle and deform plastically, dissipating energy with damage accumulating in the steel. Eventually a crack may form and the brace fractures. To optimize energy dissipation, the effects of brace and gusset plate dimensions (thickness and length of the gusset plate, size of the brace, length of the brace), and material properties, on brace behaviour, need to be understood. Ten concentric bracing members, designed according to the weak brace – strong gusset concept, were tested. The objective was to investigate the effects of displacement history, brace effective slenderness ratio, and brace width/thickness ratio, on the hysteresis behaviour of bracing members. Displacement history was found to affect energy dissipation and fracture life. The effects of increasing the gusset plate thickness on the energy dissipation and the fracture life is not the same as reducing the effective slenderness ratio of the bracing member resulted from reducing the length of the HSS. New fracture life and energy life equations are proposed. Key words: concentric braces, experimental, steel, seismic, energy dissipation, fracture life, energy life. Re´sume´ : Durant les mouvements du sol lors d’un se´isme, les contreventements en diagonale dans les charpentes contrevente´es sont soumis a` une se´rie de charges cycliques, alternativement en tension et en compression. Le contreventement peut flamber et se de´former plastiquement, dissipant de l’e´nergie et accumulant les dommages dans l’acier. Une fissure peut e´ventuellement se former et le contreventement peut se fracturer. Pour optimiser la dissipation de l’e´nergie, il faut bien comprendre les effets du contreventement et des dimensions des plaques-goussets (e´paisseur et longueur de la plaquegousset, taille du contreventement, longueur du contreventement) ainsi que les proprie´te´s des mate´riaux sur le comportement du contreventement. Dix pie`ces de contreventements concentriques conc¸us selon le concept de faible contreventement – plaque-gousset forte ont e´te´ mises a` l’e´preuve. L’objectif e´tait d’examiner les effets de l’historique des de´placements, du rapport d’e´lancement effectif du contreventement et du rapport largeur/e´paisseur du contreventement, sur le comportement hyste´re`se des pie`ces de contreventement. L’historique des de´placements affectait la dissipation de l’e´nergie et l’endurance a` la fracture. Les effets d’accroıˆtre l’e´paisseur des plaques-goussets sur la dissipation de l’e´nergie et l’endurance a` la fracture ne sont pas les meˆmes que ceux de la re´duction du rapport d’e´lancement effectif des pie`ces de contreventement de´coulant de la re´duction de la longueur des profile´s de charpente creux. De nouvelles e´quations d’endurance a` la fracture et du cycle de vie de l’e´nergie sont propose´es. Mots-cle´s : contreventement concentrique, expe´rimental, acier, sismique, dissipation d’e´nergie, endurance a` la fracture, cycle de vie de l’e´nergie. [Traduit par la Re´daction]
1. Introduction In contrast to the strong brace – weak gusset approach where the gusset plate is mainly responsible for absorbing earthquake generated energy (Nast et al. 1999), the brace is primarily responsible for dissipating the energy in the weak brace – strong gusset approach (Shaback and Brown 2003). The weak brace – strong gusset approach was used to design all the bracing members in this study. Received 31 March 2010. Revision accepted 27 October 2010. Published on the NRC Research Press Web site at cjce.nrc.ca on 8 December 2010. M. Haddad. Department of Civil Engineering, Jerash Private University, P.O. Box 10, P.C. 26150, Jerash, Jordan. T. Brown and N. Shrive.1 Department of Civil Engineering, University of Calgary, AB T2N 1N4, Canada. Written discussion of this article is welcomed and will be received by the Editor until 31 May 2011. 1Corresponding
author (e-mail:
[email protected]).
Can. J. Civ. Eng. 38: 110–123 (2011)
Jain et al. (1977, 1978, 1980), Gugerli and Goel (1982), Liu and Goel (1987, 1988), Walpole (1995), Archambault (1995), Hector (1997), and Shaback and Brown (2003) have all conducted tests on HSS sections subjected to reverse cyclic loading. Kahn and Hanson (1976), Jain et al. (1980), Popov and Black (1981), and Gugerli and Goel (1982) conducted tests on other brace sections. These studies revealed that the geometric and material properties of the brace–gusset assembly, and the loading patterns, affect the corresponding hysteresis behaviour and fracture life of the brace. The effective slenderness ratio of the brace–gusset assembly was shown to be the most important parameter affecting the hysteresis behaviour and fracture life. Generally, the higher the slenderness ratio, the more pinched are the hysteresis loops. However, increasing the brace slenderness ratio causes a delay in local buckling, resulting in less deterioration in the flexural strength at the mid-length plastic hinge and thus longer fracture life. Gugerli and Goel (1982) concluded that the effect of the HSS width/thickness ratio on the hysteresis loops is not
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Fig. 1. (a) Schematic showing typical half symmetric specimen components, (b) end plate.
Table 1. Geometric properties of the HSS specimens. No. 1 2 3 4 5 6 7 8 9 10
HSS section (mm) 1521528 1521528 1521528 1271278 1271278 1271278 1271278 12712713 12712713 12712713
Total specimen length, L (mm) 4950 4950 4950 4450 4450 4450 3150 4450 4450 3550
K 0.763 0.763 0.763 0.714 0.586 0.538 0.760 0.697 0.534 0.669
KL/r 64.67 64.67 64.67 66.20 54.33 49.88 49.88 67.87 52.00 52.00
b/t 13.75 13.75 13.75 10.63 10.63 10.63 10.63 4.23 4.23 4.23
Note: K is the effective length of the brace; KL/r is the effective slenderness ratio of the brace; b/t is the width to thickness ratio of the HSS.
very significant. However, decreasing this ratio had the same effect as increasing the slenderness ratio. Shaback and Brown (2003) indicated that the width/thickness ratio of the brace influences the fracture life but has no other effect on the hysteresis behaviour. Tremblay (2002) concluded that the effect of higher yield strength could be significant, and should be accounted for in design. In all the experimental tests, all specimens exhibited a decrease in the maximum compressive load capacity with in-
creasing number of cycles, with the largest reduction occurring after the first buckling cycle. All specimens exhibited residual elongation. The tests described here were performed to study the effects of displacement history, brace effective slenderness ratio, and brace width/thickness ratio, on the hysteresis behaviour of bracing members. The results were also used to determine whether the effect of increasing the gusset plate thickness is the same as reducing the effective slenderness ratio of the bracing member. Published by NRC Research Press
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Can. J. Civ. Eng. Vol. 38, 2011 Table 2. Geometric properties of the gusset plates.
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Gusset plate (mm) No. 1 2 3 4 5 6 7 8 9 10
Thickness, t 25.4 25.4 25.4 25.4 38.1 50.8 25.4 25.4 50.8 25.4
Fillet weld length, L1 (mm)
Width, W1 250 250 250 225 225 225 225 350 350 350
Length 400 400 400 350 350 350 350 350 350 350
Korol 317 317 317 260 246 231 260 260 231 260
Result = Korol 350 350 350 300 300 300 300 300 300 300
w (mm) 278.6 278.6 278.6 228.6 215.9 203.2 228.6 228.6 203.2 228.6
L1/w 1.26 1.26 1.26 1.31 1.39 1.48 1.31 1.31 1.48 1.31
Fillet weld size (mm) 10 10 10 8 8 8 12 12 12 12
Note: w is a parameter distance between parallel welds (E480XX Electrode); Fy (gusset plate) = 300 MPa (hot rolled); Fy (HSS) = 350 MPa (cold formed) (G40.21-M350W).
Table 3. Tensile and compressive resistances of the specimens. Tensile resistance, Tr (kN) No. 1 2 3 4 5 6 7 8 9 10 1Ba 2A 2B 3A 3B 3C 4A 4B
Korol (1996) 2123 2123 2123 1793 1692 1593 1793 3031 2695 3031 1571 2010 2387 1376 1571 2042 2010 2387
Adjusted Korol 2317 2317 2317 1991 1991 1991 1991 3368 3368 3368 1745 2193 2604 1529 1745 2269 2193 2604
Buckling load (kN) CSA (1994) 1181 1181 1181 987 1480 1974 987 987 1974 987 934.6 1143.8 1345.2 764.2 934.6 1094.7 1143.8 1345.2
Experimental 1935 2085 2316 1952 2000 1980 2008 2715 2693 2690 1647 2165 2624 1462 1632 2284 2132 2585
CSA (1994) 1201 1201 1201 990 1189 1267 1267 1429 1849 1849 1093 1419 1672 813 923 1240 1231 1510
Experimental 1310 1184 1360 1071 1229 1523 1539 1685 2184 1928 1159 1507 1721 864 952 1011 1381 1435
a
Shaback and Brown (2003).
2. Test program Ten specimens (Fig. 1) were tested with HSS and gusset plate dimensions as shown in Tables 1 and 2. Cambers were measured at the mid-length of the specimen (four sides) using a surveying level with an accuracy of 0.19 mm (1/132 in). The gusset plates were inserted so that their plane was normal to the direction of the maximum initial imperfection. All specimens were designed according to the CAN/CSAS16.1-94 (CSA 1994) standard except for the fillet weld which was designed according to Korol (1996) as shown in Table 2. The CAN/CSA-S16.1-94 standard overestimates the necessary weld length (Korol 1996). The effective length factors were computed according to the formulae presented by Jain et al. (1978) and are listed in Table 1. According to the CAN/CSA-S16.1-94 standard, the tensile resistance (Table 3) of the gusset plate had to be greater than or equal
to that of the brace, a requirement that has since been superseded by a more general connection strength requirement. ½1
fðAg Fy ÞGusset ðAg Fy ÞHSS f ¼ 0:9
where f is the resistance factor; Ag is the gross crosssectional area; Fy is the yield strength. From eq. [1], gusset plate width is computed for a given HSS and gusset plate thickness. The gusset plate width has to be greater than the parallel dimension of the HSS plus two times the weld size. The weld length (L1) was calculated using the shear reduction factor, a = 1, and L1/w was greater than 1.2 so that the HSS net section failure mode governed. According to Korol (1996), shear lag will probably not be a problem for L1/w greater than 1.2. Published by NRC Research Press
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Table 4. Applied displacement histories. Applied axial displacement (Actuators) (mm) Specimen Cycle No. 0 1 2
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3 4 5 6 7 8 9 10 11 12–16
1
2
3
0 –2.32 2.72 –2.32 2.72 –7.96 7.98 –10.56 10.58 –21.18 21.19 –21.18 0 –31.74 31.75 –35.93 35.95 –53.01 35.98 –63.59 35.98 –74.19 35.98 –74.19 35.98
0 –2.32 2.71 –2.32 2.72 –7.97 7.98 –10.56 10.58 –21.18 21.19 –31.74 31.76 –35.94 35.95 –53.01 35.99 –63.59 35.98
0 –2.32 2.72 –2.32 2.72 –7.97 7.98 –10.56 10.58 –21.18 21.19 –31.74 31.76 –35.93 32.98 –35.93 32.98 –35.93 32.98 –35.93 32.98 –35.93 32.98
17–30
4–7 0 –2.31 2.49 –2.31 2.49 –7.09 7.29 –9.48 9.68 –19.08 18.49 –28.66 28.87 –38.25 38.46 –47.73 44.06 –57.32 44.05 –61.81 44.04 –61.81 44.05 –61.81 44.05 –61.81 44.07
30–45 45–69
8 0 –2.31 2.49 –2.31 2.49 –7.09 7.29 –9.48 9.68 –19.08 16.00 –28.66 28.87 –38.25 36.00 –47.73 42.00 –57.32 42.00 –61.81 42.00 –61.81 42.00 –61.81 42.00 –61.81 42.00 –61.81 42.00
9 &10 0 –2.31 2.49 –2.31 2.49 –7.09 7.29 –9.48 9.68 –19.08 18.49 –28.66 26.00 –38.25 29.45 –47.73 34.20 –57.32 34.20 –61.81 34.20 –61.81 34.20 –61.81 34.20 –61.81 34.20 –61.81 34.20 –61.81 34.20
Cycle rate (mm/min) 0 1.5 1.5 5.0 5.0 15.0 15.0a 25.0b 25.0 35.0 40.0 45.0 45.0 45.0 45.0
a
Specimen 1. Specimen 2–10.
b
The gusset plates were welded with full penetration welds (weld length = W1) to thick end plates. The free length of the gusset plate (the length between HSS and the end plate) ranged between one and two times its thickness. The specimens were tested in a purpose-built frame. Strain gauges were placed at expected plastic hinge locations in the middle of the specimen and at both ends of the HSS. In addition, string potentiometers were deployed to measure the change in length of the specimen as confirmation of the displacement control of the actuators, and the lateral deflection at the mid-length of the brace. The first three specimens tested were the same as specimen 4A of Shaback and Brown (2003), but were subjected to different axial displacement histories. The applied displacement histories for all specimens are given in Table 4. The fourth specimen was essentially a repeat of specimen 3B of Shaback and Brown (2003). The objective of the repeated tests was to investigate the effects of the material and the geometric imperfections on the hysteresis behaviour and the fracture life of the bracing members.
Changing the gusset plate flexural stiffness by varying its thickness (24.5, 38.1, and 50.8 mm), as in specimens four to six, and eight and nine, had the effect of changing the brace slenderness ratio. A reverse check on the energy dissipation and the fracture life was investigated in specimens 7 and 10 by changing both the HSS length and gusset plate thickness while maintaining the same KL/r and b/t as specimens 6 and 9, respectively. These specimens were tested to determine if changing the connection stiffness influences the hysteresis behaviour in the same manner as changing the HSS length, as suggested by Jain et al. (1978).
3. Experimental tests Sample experimental results of the axial and lateral hysteresis curves for specimen 2 are shown in Figs. 2a and 2b, respectively. Load cycles are clockwise in the diagrams. All specimens were subjected to two initial elastic cycles. Partial yielding developed in the third and fourth cycles. The strains on each side of the specimen at its mid-length start Published by NRC Research Press
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to diverge at the onset of the fifth compressive cycle. The buckling loads, as determined from the CAN/CSA-S16.194, and from the experiments, are shown in Table 3, and the cycles in which buckling occurred are listed in Table 5. In all specimens except specimen 1, the HSS fully yielded in the tension side of the sixth cycle. The axial compression capacity can be seen to degrade with each cycle (Fig. 2a) with noticeable reduction in both the tension and compression stiffnesses at zero axial displacement. This is attributed to the residual elongation, and to the plastic hinges that formed at both ends of the specimens. Plastic hinges formed in the end connections first with the third, mid-span hinge, forming later for all specimens except specimens 6 and 9. Local buckling at the mid-length plastic hinge of all specimens occurred during the specific cycles listed in Table 5. Local buckling included inward and outward bulging of the specimens’ compressive flange and the webs, respectively. A small inward bow of the tension flange was also noticed. For all specimens, except 6 and 9, the shape of the buckled specimens was sinusoidal until the mid-length plastic hinges had formed. At that stage, the shape became triangular (rotation was localized to the three plastic hinges).
continued with increasing local buckling at the specimen mid-span. In later cycles, cracks appeared, mostly at the compressive side corners (Fig. 3a), or in the compressive flange (Fig. 3b, which also shows the local buckling). The HSS failed in the tensile portion of the cycles listed in Table 5. Fracture life was defined as the number of cycles before the load resistance dropped substantially to around 500 kN and (or) 50% of the cross-section had cracked. Cracks propagated either quickly (specimens 2, 3, 5, 6, and 7) or slowly (specimens 1, 4, 8, 9, and 10) through the compressive flange, then the webs, and finally to the tension flange. The bow shapes of specimens 6 and 9 were concentrated in the inner third of the specimen length. Plastic hinges formed in the free length of the gusset plates where a small out-of-plane rotation occurred. A noticeable kink at the inner connection of the gusset plates and the HSS was also present, indicating additional plastic deformation as shown in Figs. 4a and 4b. However, the out-of-plane free length rotation was greater than the rotation at the inner connection of the gusset plate and the HSS. The kink yielded fully during the sixth tensile cycle with surface cracking of the weld connecting the inner end of the gusset plates to the HSS. The free length plastic hinges formed completely during the seventh compressive cycle when the specimen buckled in a noticeable bow shape compared to the previous compressive cycle. The excessive kinking at the new inner plastic hinges triggered a major crack at the kink location, starting from the weld, causing one end of specimen 6 to fracture. Due to this premature failure, no plastic hinge or local buckling occurred at the middle of this brace. As a result, four plastic hinges formed, two in the free length of the gusset plates and another two at the inner connection of the gusset plates to the HSS. Specimen 9 buckled during the compressive side of cycle number six, forming four end plastic hinges: two in the gusset plates (outer end hinges), 30–100 mm from the end plates. These plastic hinges covered the free length of the gusset plate and extended into the HSS. The other two plastic hinges were in the HSS at the inner connection of the gusset plates and the HSS (inner end hinges). Progressive formation of a fifth plastic hinge at the specimen mid-length started during the following cycle (Figs. 4a and 4b). The end plate rotation in the outer plastic hinge regions increased during the compression side of the seventh cycle and all subsequent cycles. One mid-length kink and four end kinks at the plastic hinge locations were consistently visible. The ratio of the rotation of the outer hinges to the inner hinges during compression was approximately 6 to 1.
4. Cracking and fracture of the specimens
5. Analysis and discussion of results
The high axial loads and the excessive rotation of the free length of the gusset plates resulted in the formation of two cracks in the weld connecting the gusset plates to the HSS. The first was a small surface crack in the weld connecting the front end of the gusset plates to the HSS, while the second was at the far end of the weld. These cracks did not cause failure of the connection. A noticeable residual lateral deflection remained in the specimen upon reloading in tension for each cycle. Loading
The maximum compressive resistance was attained during the first buckling cycle (Table 5). If the brace did not fracture suddenly as in specimen 2, the compressive resistance would remain almost constant (specimens 4, 8, 9, and 10) during the last cycles when slow crack propagation was occurring. According to CAN/CSA S16.1-94, the degradation in the compressive load capacity should be constant for all cycles following buckling (a requirement that is no longer specified
Fig. 2. Experimental hysteresis loops for specimen 2: (a) axial, (b) lateral.
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Cycle number Initial buckling 5 5 5 5 5 6 5 5 6 6
Specimen 1 2 3 4 5 6 7 8 9 10
First local buckling 8 8 8 9 8 7 7 18 15 13
First cracking 10 9 9 9 9 7 8 22 18 18
Crack propagation 10–17 9 9–11 9–30 9–10 7 8–9 22–46 18–57 18–69
50% fracture 10 9 10 11 10 7 9 31 23 22
Final fracture 17 9 11 30 10 7 9 46 57 69
Fig. 3. Fracture: (a) compression corners, (b) flange into webs.
in the more recent edition of this standard, CAN/CSA S16.1-01 (CSA 2001)): 1 ½2 Cr0 ¼ Cr 1 þ 0:35l 1
½3
Cr ¼ fAFy ð1 þ l2n Þ n
½4
KL l¼ r
rffiffiffiffiffiffiffiffi Fy p2 E
where n is a constant equal to 1.34, and f is the resistance factor or the strength reduction factor, equal to 0.9. However, the experimental results indicated that the compressive load resistance decreased continuously after the first buckling cycle; and was only constant later, when the mid-length section crack was propagating slowly, as in specimens 4, 8, 9, and 10. The maximum compressive buckling load in each cycle was normalized with respect to the post buckling load capacity according to CAN/CSA S16.194 (eq. [2]) and plotted against the cycle number as shown in Fig. 5. Published by NRC Research Press
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Fig. 4. Specimen 9 hinging, (a) connection hinges, (b) five hinges.
Fig. 5. Comparison of the compressive resistances following the first buckling cycle.
The maximum normalized compressive buckling load for specimen 6 was 1.06 during the seventh compressive cycle. Specimen 6 fractured during the seventh tensile cycle and was not included in Fig. 5. The degradation of the compressive resistance was greater for constant compressive displacement history, and less for increasing compressive displacement history. The intervening tension displacement history also influenced the compressive resistance. The interaction between the maximum compressive and tensile resistances, according to Lee (1988), is Pcomp;max Cr Ptens;max ½5 ¼ 0:054 þ 0:51 Py Py Py
The interaction equation between the maximum compressive and tensile resistances proposed by Lee (1988) generally provides a good estimation of the compressive resistance during all cycles except in the first cycle following the initial buckling cycle, and the last cycle when the cross-section is 50% fractured. The experimental and the theoretical (eq. [5]) normalized maximum compressive capacities for specimens 1 and 2 are plotted in Figs. 6a and 6b. However, the fifth compressive cycle in specimen 1 was repeated, which resulted in a clear divergence between the experimental results and the normalized compressive resistance calculated from Lee’s equation. Lee’s equation matched the experimental results in the slow crack propagation zones in specimens 4, 8, 9, and 10. The experimental and theoretical normalized maximum compressive resistances for specimen 6 are 0.605 and 0.449, respectively. Archambault (1995) and Shaback and Brown (2003) also concluded that the application of eq. [5] depends on the displacement history. The maximum experimental tensile capacities of all specimens were compared to the corresponding values calculated from the tensile resistance formulae proposed by Korol (1996) and the CSA (1994). As shown in Table 3, the formula of Korol (1996) is closer to the experimental capacities than that of the CSA (1994). The weld metal factored resistance Vr has to be greater than or equal to the unfactored yield load of the HSS as follows: Published by NRC Research Press
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Fig. 6. Comparison of normalized compressive resistances, (a) specimen 1, (b) specimen 2.
½6
Vr ¼ 0:67fFy1 ð4L1 DÞ Fy ð2wt0 aÞ f ¼ 0:67
where D is the fillet weld size (Table 2); t0 is the HSS thickness; w is the perimeter distance between parallel welds; Fy1 and Fy are the yield strengths of the fillet weld and the HSS, respectively. From eq. [6] the weld length (L1) is calculated using the shear reduction factor, a = 1, and L1/w is 1.2 so that the HSS net section failure mode governs. According to Korol (1996) the shear lag problem seems unlikely for L1/w greater than 1.2. The shear lag tensile resistance of the HSS is defined as follows: ½7
Tr ¼ 0:85fFu ð2wt0 aÞ ¼ Fy ð2wt0 aÞ
where f = 0.9 and Fu is the ultimate (Fy/0.75 MPa) strength of the HSS. According to Korol, the tensile capacity of the brace will decrease as the gusset plate thickness is increased. However, for specimens 4, 5, and 6, with gusset thicknesses of 25.4, 38.1, and 50.8 mm, respectively, the tensile capacities were approximately equal. The Korol formula could be adjusted by adding the gusset plate thickness to the original parameter (w), between two parallel welds, defined by Korol. The maximum tensile capacity for specimens 8, 9, and 10 was limited by the load capacity of the actuators. These specimens started yielding but did not reach full plasticity.
117 Fig. 7. Comparison of out-of-plane displacements, (a) specimen 5, (b) specimen 9.
Shaback and Brown (2003) proposed two equations to predict the out-of-plane displacement at the mid-length of concentric bracing members. The first equation was based on simple geometry, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi d d L ½8 D¼ 2 2 where L is the specimen length, and d is the axial compressive displacement. The second equation involved relating the maximum lateral displacement to the compressive displacement ductility ratio, mc: ½9
D 1 ð%Þ ¼ ðmc Þ1:9 þ 2mc L 9
Tremblay (2002) proposed the following formula to predict the lateral displacement of the bracing members under any loading history, based on the elastic and inelastic behaviour of the brace: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½10 D ¼ 0:7 ðmc þ mt 1Þdy LH where mc and mt are the compressive and tension ductility ratios, respectively; dy is the yield displacement; and LH is axial distance between the end plastic hinges. The equations proposed by Tremblay (2002) and Shaback and Brown (2003) overestimated the lateral displacement during the pre-buckling cycles as shown in Figs. 7a and 7b. Equation [9] was the closest to the experimental results. For the inelastic cycles (above cycle 10), again eqs. [9] and [10] Published by NRC Research Press
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Can. J. Civ. Eng. Vol. 38, 2011 Table 6. Energy dissipation capacity of the specimens.
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50% fracture Specimen 1 2 3 4 5 6 7 8 9 10
Energy dissipation (kNm) 260 270 298 352 352 148 279 1359 1469 1003
Total fracture Normalized energy dissipation 9 9.3 10.3 18.5 18.5 7.8 20.7 39.3 42.5 36.4
Energy dissipation (kNm) 355 270 320 541 352 148 279 1471 1915 1325
Fig. 8. Normalized axial hysteresis loops, (a) specimens 6 and 7, (b) specimens 1 and 2, (c) specimens 4, 5, and 6.
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Fig. 9. Normalized lateral hysteresis loops, (a) specimens 6 and 7, (b) specimens 1 and 2, (c) specimens 4, 5, and 6.
gave good estimates of the out-of-plane displacements, except for specimen 3, which was subjected to reduced displacements in the later cycles. Energy dissipation capacity is defined as the cumulative energy dissipated by the specimen, subjected to applied axial displacements, up to 50% cross-section fracture. The energy dissipation capacities of all specimens are listed in Table 6. The variability in the material ductility was excluded by normalizing the axial displacements and the axial forces by the corresponding axial yield displacement and axial yield force obtained from stub-column tests. In all specimens, except for specimen 5, no strain hardening was observed from the normalized axial hysteresis loops.
6. Parametric effects The effective slenderness ratio was varied by changing either the length of the HSS (series 1, Shaback and Brown 2003), or
119
the thickness of the gusset plate (series 2, current tests). Specimens with higher slenderness ratios fractured at higher numbers of cycles than those with low slenderness ratios. The normalized axial and lateral hysteresis loops and envelopes of specimens 6 and 7 are compared in Figs. 8a and 9a, respectively, and are similar in shape and stiffnesses, being identical until the end of cycle four. The local buckling of specimen 7 occurred at the mid length plastic hinge, while the local buckling of specimen 6 occurred at the inner end plastic hinges, and was less severe. Specimen 6 was the only specimen that fractured at the inner end plastic hinge. The normalized dissipation energy (Table 6) of specimen 7 is 62.5% higher than that of specimen 6 calculated at 50% cross-section fracture but the two specimens dissipated the same energy until the end of the seventh cycle when specimen 6 fractured. Specimens 6 and 7 had the same slenderness ratio although specimen 7 was shorter and had a thinner gusset plate. Specimen 7 showed greater energy dissipation capacity and fracture life than specimen 6. In contrast, when comparing the results for specimens 9 and 10, which also had the same slenderness ratios, but were both stockier specimens, specimen 10 showed decreased energy dissipation capacity and fracture life. The gusset plate plastic moment capacity should always be below that at which the sudden inner end plastic hinge fracture occurs (Table 7). With respect to repeatability, specimens 2 and 4 of this series were nominally the same as specimens 4A and 3B of Shaback and Brown (2003). The normalized axial behaviour of the specimens was similar in shape, coordinates, and stiffness. Specimens 2 and 4 fractured one and two cycles less than specimens 4A and 3B, respectively. The normalized energy dissipation capacity of specimen 2 was 42.3% less than that of specimen 4A. The specimens in the second series had 15% higher yield strength (Table 7) and there would be different initial cambers and geometric imperfections in welding the gusset to the HSS end slots. Any of these factors could contribute to the difference in behaviour. On the basis of these results, and Tremblay’s (2002) conclusion that the effect of yield strength is significant, it might be appropriate to have an upper limit to the yield strength in Canadian specification CAN/CSA-40.21-98 (CSA 1998) Class C, Grade 350W steel, to ensure greater ductility. Specimens 1, 2, and 3 were similar in geometry and material properties. Specimen 1 was subjected to a different displacement history to specimen 2, in that the fifth compressive cycle was repeated. The normalized axial and lateral hysteresis loops and envelopes are compared in Figs. 8b and 9b, respectively. The two specimens had virtually the same normalized dissipation energy for all cycles, and fractures in the same cycle. Specimen 3 was subjected to lower displacements than specimen 2, the displacements of the seventh tension and eighth compression cycle being repeated in all subsequent cycles. The normalized dissipation energy and the fracture life of specimen 3 was slightly greater than that of specimen 2. The normalized axial and lateral hysteresis loops for specimens 4, 5, and 6 are compared in Figs. 8c and 9c, respectively. There is a significant change from the pinched hysteresis behaviour of specimen 4 to the full loop hysteresis behaviour of specimen 6 at zero axial force. Published by NRC Research Press
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Can. J. Civ. Eng. Vol. 38, 2011 Table 7. Plastic moments for the HSS and the gusset plates.
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Fy (MPa) Specimen 1 2 3 4 5 6 7 8 9 10 1B 2A 2B 3A 3B 3C 4A 4B
HSS 467 467 467 480.3 480.3 480.3 480.3 500 500 500 421 442 442 461 421 461 442 442
Mp (kNm) Gusset plate 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300
Gusset plate 12 12 12 11 25 44 11 17 68 17 11 12 15 10 11 12 12 15
HSS 110 110 110 80 80 80 80 110 110 110 70 110 120 60 120 70 110 120
MPðGussetÞ =MPðHSSÞ 0.11 0.11 0.11 0.14 0.32 0.57 0.14 0.15 0.60 0.15 0.16 0.12 0.12 0.16 0.09 0.17 0.12 0.12
Note: Fy is yield strength; Mp is plastic moment.
Fig. 10. Empirical fracture life prediction, definition of D1 and D2 (Lee 1988).
lower out-of-plane displacements than specimen 9. However, the normalized axial and lateral envelopes of the two specimens were similar during the buckling and post buckling phases. Increasing the gusset plate thickness significantly increased the compressive resistance (specimen 9). The results show that the use of thick gusset plates caused the brace to fracture at a lower number of cycles. Increasing the gusset plate thickness did not increase the cumulative energy dissipation capacity of the brace. However, using a thick plate did increase the energy dissipation capacity per cycle for a lower number of cycles.
7. Fracture life All fracture life predictions to date have excluded the compression deformations and only included parts (Tang and Goel 1987; Lee 1988; Hassan and Goel 1991; Archambault 1995; Shaback and Brown 2003) or all of the axial tension deformations (Shaback and Brown 2003). Archambault (1995) adjusted the Lee and Goel (1987) equation by introducing the effective slenderness ratio: ð317=Fy Þ1:2 4ðb=dÞ þ 1 0:8 ½11 Df ¼ Cs ð70Þ2 0:5 5 ðb 2t0 Þ=t0 Specimens 4 and 5 dissipated the same normalized amount of energy (18.5), while specimen 6 dissipated only 7.8. Increasing the gusset plate thickness significantly increased the compressive resistance, but caused premature fracture. Specimens 8 and 9 demonstrated similar relative behaviour as specimens 4, 5, and 6; with pinched (specimen 8) to the full loop (specimen 9) hysteresis behaviour. Specimen 9 had a normalized energy dissipation capacity 7.5% higher than specimen 8. During elastic buckling, specimen 8 had 2C s
for KL=r < 70 ½12
Df ¼ Cs
ð317=Fy Þ1:2 4ðb=dÞ þ 1 0:8 ðKL=rÞ2 0:5 5 ðb 2t0 Þ=t0 for KL=r 70
where Df is the fracture life, Cs is equal to 0.01842, determined from the experimental results, d is the cross-section
(reported) = 0.0257, Cs (used) = 0.0184. Published by NRC Research Press
Haddad et al.
121
Table 8. Experimental and predicted fracture life results.
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Fracture life, Df (mm)
Specimen 1 2 3 4 5 6 7 8 9 10 1B 2A 2B 3A 3B 3C 4A 4B
Lee (1988) 9.0 9.0 9.0 12.1 12.1 12.1 12.1 29.1 29.1 29.1 14.1 9.6 13.1 8.5 14.1 17.4 9.6 13.1
Archambault (1995)
Shaback and Brown (2003)
Experimental
Cs = 0.0257 19.2 19.2 19.2 20.5 20.5 20.5 20.5 26.2 26.2 26.2 24.1 20.5 22.6 19.0 24.1 23.8 20.5 22.6
1 24.0 24.0 24.0 33.8 33.8 33.8 33.8 77.9 77.9 77.9 21.3 19.8 25.0 21.6 21.3 37.1 19.8 25.0
(0.1D1+D2) 8.6 10.2 9.8 16.4 16.9 7.8 17.6 17.9 31.8 21.2 24.3 16.8 19.6 11.8 21.6 31.5 17.2 20.5
Cs = 0.0184 13.7 13.7 13.7 14.7 14.7 14.7 14.7 18.7 18.7 18.7 17.2 14.7 16.2 13.6 17.2 17.1 14.7 16.2
Fig. 11. Theoretical and experimental, (a) fracture lives, (b) energy lives.
2 74.5 74.5 74.5 120.3 120.3 120.3 120.3 492.5 492.5 492.5 137.5 78.8 128.0 66.9 137.5 206.4 78.8 1278.0
Theoretical (D1+D2) 23.0 23.8 22.7 48.4 40.3 13.1 39.9 74.9 93.4 81.5 60.5 33.7 43.2 33.3 60.1 118.8 42.6 51.5
Equation [19] 32.2 32.2 32.2 41.2 39.7 39.0 39.0 99.3 94.4 94.4 39.2 30.7 40.2 32.0 40.7 54.8 31.7 41.2
Table 9. Experimental and predicted energy life results. Energy life, Ef (kNm) Specimen 1 2 3 4 5 6 7 8 9 10 1B 2A 2B 3A 3B 3C 4A 4B
depth of the HSS, b is the cross-section width of the HSS, t0 is the thickness of the HSS, ðb 2t0 Þ=t0 is the width to thickness ratio of the compression flange, and KL/r is the effective slenderness ratio of the brace.
Experimental 260 270 298 352 352 148 279 1359 1469 1003 357 300 443 178 406 739 366 352
Theoretical (eq. [20]) 281 281 281 403 351 331 331 1470 1220 1220 336 235 346 275 386 280 265 379
Shaback and Brown (2003) readjusted Archambault’s (1995) fracture life equation based on the new experimental test (series 1) as follows (Shaback 1): ð350=Fy Þ3:5 4ðb=dÞ 0:5 0:55 ½13 Df ¼ Cs ð70Þ2 1:2 5 ðb 2t0 Þ=t0 for KL=r < 70 Published by NRC Research Press
122
½14
Can. J. Civ. Eng. Vol. 38, 2011
ð350=Fy Þ3:5 4ðb=dÞ 0:5 0:55 Df ¼ Cs ðKL=rÞ2 1:2 5 ðb 2t0 Þ=t0
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for KL=r 70 where Cs is equal to 0.065 Contrary to all other fracture life equations, the fracture life here is directly proportional to the yield strength. All the previous fracture life equations were obtained from the normalized axial hysteresis loops according to the following equation: X ½15 Df exp ¼ ð0:1D1 þ D2 Þ where D1 is the tension displacement from Py/3 to the load reversal point, D2 is tension displacement from Py/3 to the unloading point. D1 and D2 are shown in Fig. 10. Shaback and Brown (2003) also suggested that the fracture of the bracing member be dependent on the total accumulation of the tension extrusions. X ½16 Df exp ¼ ðD1 þ D2 Þ Based on this assumption Shaback and Brown (2003) proposed another fracture life equation as follows (Shaback 2): ½17
Df ¼ Cs
ð350=Fy Þ1:01 4ðb=dÞ 0:5 0:55 ð70Þ2 2:5 5 ðb 2t0 Þ=t0 for KL=r < 70
½18
Df ¼ Cs
ð350=Fy Þ1:01 4ðb=dÞ 0:5 0:55 ðKL=rÞ2 2:5 5 ðb 2t0 Þ=t0 for KL=r 70
where Cs is a constant equal to 29.5 The empirical fracture life equations were proportional to the aspect ratio (width/depth) and the effective slenderness ratio of the brace and inversely proportional to the width/ thickness ratio. The fracture life equations of Shaback and Brown (2003) do not fit the current experimental fracture life results well (Table 8). The variation of the experimental fracture life with the width/thickness ratio showed that the experimental fracture life is high for low width/thickness ratios and vice versa. The latter is valid with the exception of specimen 3C, which fractured at 17 cycles. The axial force remained above Py/3, shown in Fig. 10, during the tension phase of the last cycles. Therefore, specimen 3C has the highest experimental fracture life. The following fracture life equation is proposed: ½19
Df ¼ 378ðl0:19 Þðb=tÞ0:94
where Df is the sum of the absolute displacements (Haddad 2004), D1 and D2. The theoretical and experimental fracture lives are presented in Table 8, and shown in Fig. 11a in which the 1:1 relationship has R2 = 0.49.
8. Energy life Energy life is defined as the cumulative energy dissipated by the specimen, up to 50% fracture of the cross section. The effect of lambda and the width/thickness ratio on the energy life is similar to their effect on the experimental fracture life. Based on both series of tests, the following equation is proposed to quantify energy life as a function of lambda and the width/thickness ratio. ½20
Ef ¼ 9:62 106 ðl0:7034 Þðb=tÞ1:36
The theoretical and experimental energy lives are presented in Table 9, and shown in Fig. 11b in which the 1:1 relationship has R2 = 0.89.
9. Conclusions The experimental results allow the following conclusions to be drawn: (1) The initial imperfections and yield strength of the HSS influence the hysteresis behaviour and fracture life. Increasing the yield strength of the HSS results in a reduction in the ductility and fracture life. (2) Increasing the gusset plate thickness reduces the fracture life while maintaining or reducing the cumulative energy dissipation capacity of the brace. However, increasing the gusset plate thickness increases the energy dissipation per cycle for the early inelastic cycles especially for the first and second buckling cycles. (3) Increasing the gusset plate thickness reduces the effective slenderness ratio of the brace, but has a different influence on the energy dissipation capacity and the fracture life, compared to reducing the length of the HSS member. (4) The gusset plate plastic moment capacity should always be below that at which the inner end plastic hinge can fracture. (5) Reducing the applied axial tension and compression displacements during later post buckling cycles increased the fracture life and energy dissipation capacity. (6) Half the overall parametric distance of the HSS should be used instead of the parametric distance between parallel welds in Korol’s (1996) tensile capacity formula. Consequently, the slight underestimation of Korol’s formula of the tensile capacity is adjusted. (7) The experimental fracture life and energy life are inversely proportional to the width/thickness ratio of the HSS specimen and to a lesser degree directly proportional to lambda, for the range of lambda values used in the current test series. (8) The upper limit to yield strength of the cold-formed steel should be checked to ensure the capacity design approach.
Acknowledgements The authors thank TSE Steel Ltd. for their generosity in donating the steel specimens. The assistance of the technical staff of the Department of Civil Engineering, University of Calgary, especially Mr. Dan Tilleman, is greatly appreciated. The work was performed with the financial support of the Natural Sciences and Engineering Research Council of Canada, NSERC: many thanks. Published by NRC Research Press
Haddad et al.
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