Seismic behavior of the special truss moment frame

Transcription

1 A Modified Equation for Expected Maximum Shear Strength of the Special Segment for Deign of Special Tru Moment Frame SHIH-HO CHAO and SUBHASH C. GOE Seimic behavior of the pecial tru moment frame (STMF) ytem ha been tudied both analytically and experimentally by Goel and Itani (1994) and Baha and Goel (1995) at the Univerity of Michigan during the pat 15 year and ha been incorporated into the AISC Seimic Proviion for Structural Steel Building (AISC, 2005), hereafter referred to a the AISC Seimic Proviion. Thi ytem conit of a tru girder with a pecial egment deigd to behave ilatically under evere earthquake while the other member, including girder-to-column conction, outide the pecial egment remain eentially elatic. The pecial egment can be made with X-diagonal or ierendeel web member, a hown in Figure 1. When an STMF i ubjected to lateral eimic force, the induced hear force in the middle of the joit girder i reited primarily by the chord member and the web diagonal of the pecial egment. After yielding and buckling of the diagonal member, platic hinge will form at the end of the chord member a hown in Figure 2. The yield mechanim of the frame i a combination of yielding of all pecial egment in the frame plu the platic hinge at the column bae (Figure 2). The platic hinge at the bae may be in the column or in the bae conction, whichever i practically feaible. Deign of STMF tart with deigning the pecial egment. Baed on a capacity deign approach, other element are then deigd to remain elatic under the hear force in the middle of the tru girder gerated by fully yielded and train harded pecial egment (in other word, the expected vertical hear trength, ), along with other external force. The expected maximum hear trength of the pecial egment,, i given in the current AISC Seimic Proviion [Equation 12-1 in AISC (2005)] a: 375. RM = EI (1) 3 + R P y( +03. P nt nc )inα ( ) R y = yield tre modification factor M nc = nominal flexural trength of the chord member of the pecial egment E I = flexural elatic tiff of the chord member of the pecial egment = pan length of the tru = length of the pecial egment, center-to-center of upport P nt = nominal axial tenion trength of diagonal member of the pecial egment P nc = nominal axial compreion trength of diagonal member of the pecial egment α = angle of diagonal member with the horizontal The firt two term of Equation 1 were derived baed on a tudy of a ierendeel pecial egment (without X-brace) (Baha and Goel, 1994). O of the aumption made in the derivation wa that the elatic moment at the end of Subhah Goel i profeor, department of civil and environmental engiering, Univerity of Michigan, Ann Arbor, MI. Shih-Ho Chao i aitant profeor, department of civil and environmental engiering, Univerity of Texa, Arlington, TX. Fig. 1. STMF with different confi guration of pecial egment. ENGINEERING JOURNA / SECOND QUARTER / 2008 / _Chao_Goel1_2008_2Q.indd 117 7/15/08 11:08:47 AM

2 chord member of the pecial egment reult from vertical tranlation only, in other word, the effect of end rotation i glected a hown in Figure 3a. Thi aumption lead to overetimation of the elatic tiff of the chord member, which in turn reult in a higher coefficient, 0.075, in the econd term of Equation 1. Neverthele, thi overetimation ha a mall influence on if the moment of irtia of the chord member i mall. However, for heavier chord member the overetimation can be quite large becaue of their large moment of irtia and the econd term of Equation 1 i directly related to the member moment of irtia. Becaue the member outide the pecial egment, uch a vertical member, diagonal member, conction and column are deigd baed on, any overetimation would reult in an overly corvative deign of thoe member. PROPOSED MODIFICATION A remedy to the aforementiod problem i to ue a more realitic flexural elatic tiff of the chord member. Thi i do by the approach that follow. If the chord member ha no end rotation (fixed end condition), the elatic moment at the chord end can be given by: θ EI M = 6 θ (2) = relative vertical diplacement at chord end divided by the length of pecial egment Hence, the elatic tiff i: M EI k θ = 6 = Thi elatic tiff will decreae by allowing the end rotation to occur. For the extreme cae, in other word, when the chord member ha pind end a hown in Figure 3b, the elatic tiff i equal to zero. True elatic tiff i ome between thee two extreme cae. It i aumed here that the true elatic tiff can be approximated by: (3) EI k = 3 (4) By uing thi formulation, the maximum chord end moment i determid a follow: Referring to Figure 4, the chord moment-rotation relation can be modeled by a biliar curve, in which the ilatic tiff i ηk. From Equation 4, the maximum elatic rotation i: M p θ e = (5) 3 EI (a) (b) Fig. 2. Mechanim of STMF with different pecial egment. Fig. 3. Deformation of chord of the ierendeel pecial egment: (a) no chord end rotation; (b) free chord end rotation. 118 / ENGINEERING JOURNA / SECOND QUARTER / _Chao_Goel1_2008_2Q.indd 118 7/15/08 11:08:47 AM

3 The maximum rotation of the chord member can be obtaid by uing the geometric relation. Thu, θ u = ( /h) = tory drift Hence the platic rotation i: θ = θ θ = p u e (6) h M p h 3EI The expreion for maximum moment can be written a (ee Figure 4): M = M + ηθ k max p p = ( 1 η) RM + 3EIη 2 (8) h η (7) = ratio of the pot-yield lope to the elatic lope of the aumed biliar moment-rotation model of the chord member The expected maximum hear trength of the pecial egment,, i then calculated a: It i noted that the lope, η, only account for train hardening; that i, the expected platic moment, M p, wa ued intead of the nominal trength, M nc, a hown in Figure 4. Baed on tet on STMF ubaemblage in which the pecial egment wa compoed of double-angle ection, Baha and Goel (1994) found that a value of η equal to 10% wa adequate for the pot-yield lope (ee Figure 5). Thi value wa alo found adequate for double-chanl ection baed on tet conducted by Parra-Monteino, Goel and Kim (2006). Figure 6 how the backbo curve of hyteretic loop for a typical doublechanl element, a well a the correponding biliar model. By uing η = 10%, and /h = 0.03 in Equation 8, Equation 9 become: 36. RM = EI (10) 3 M = 4 max (9) (a) (b) Fig. 4. Moment-rotation relationhip of chord member. Fig. 5. Moment veru drift plot of ubaemblage teted, chord member wa a built-up ection compoed of double-angle and plate (Baha and Goel, 1994). ENGINEERING JOURNA / SECOND QUARTER / 2008 / _Chao_Goel1_2008_2Q.indd 119 7/15/08 11:08:50 AM

4 The value of /h = 0.03 wa choen becaue current deign practice i gerally baed on a limiting drift ratio of around If the maximum rotation capacity, θ p, of a chord member i available, Equation 9 can alo be expreed by: 4RM = E I θ (11) 2 p The econd term in Equation 11 account for the contribution of train-hardening in the chord member. It i noted that Duicka, Itani and Sahai (2002) derived a imilar equation for. Their expreion, however, did not account for train-hardening effect. ERIFICATION OF THE MODIFIED EQUATION The propoed modification wa verified by previou tet reult a well a nonliar tatic and dynamic analye a decribed in the following. Previou Tet Reult Equation 10 wa firt verified by the tet reult (Parra- Monteino et al., 2006) on double-chanl (2C10 25, I = in. 4, Z = 46.2 in. 3, F y = 50 ki) element ued for chord member of STMF, in which the maximum hear capacity wa meaured at 150 kip when the correponding tory drift reached 3%. The value of calculated from Equation 10, by uing R y = 1.1, give the ame exact value, in other word, 150 kip. Equation 10 wa further verified uing the tet reult of a ubaemblage STMF (Baha and Goel, 1994), in which the chord member of the ierendeel pecial egment were made of a built-up ection compoed of double angle and a plate ( and P1 24). The built-up ection had a rather mall moment of irtia (I = 5.87 in. 4 ). The maximum end moment at the tory drift of 3% wa 440 kip-in., which correpond to = 26.2 kip. The etimated value from Equation 10 i 24.4 kip, which i about 7.3% lower than the experimental value. On the other hand, the AISC equation (Equation 1) give a value of 28.4 kip, about 8.5% higher than the teting data. Nonliar Static Analyi Equation 10 wa then verified by invetigating the maximum developed hear in the ierendeel pecial egment of a even-tory STMF (Chao and Goel, 2006). The chord member at the fifth floor of thi frame wa made of 2C Modeling of the moment-rotation relation for the chord member wa baed on that hown in Figure 6. Puhover analyi wa performed until the intertory drift at the fifth level reached 3%, a hown in Figure 7. It wa found that the average wa about 143 kip, which i cloe to the predicted value, 150 kip. While the propoed equation give good agreement with the tet and analyi reult, the a calculated by uing the AISC expreion (Equation 1) i kip, about 20% higher than that obtaid from Equation 10. It hould be noted that both the AISC Fig. 6. Modeling of the backbo curve of double-chanl element uing biliar curve. Fig. 7. Maximum developed hear in the fifth-fl oor pecial egment when the tudy STMF wa tatically puhed until the fifth fl oor reached 3% tory drift. 120 / ENGINEERING JOURNA / SECOND QUARTER / _Chao_Goel1_2008_2Q.indd 120 7/15/08 11:08:54 AM

5 expreion and Equation 10 are baed on a tory drift equal to 3%. If the expected maximum tory drift i well below 3%, the AISC expreion would be even more corvative. It hould alo be noted that the ued in Equation 10 i taken a center-to-center ditance of the vertical member at the end of the pecial egment. In reality, the ditance between the platic hinge, p, will be maller than. Thi length in the Univerity of Michigan double-chanl tet wa formed to be about 0.82 and the in the even-tory STMF analyi wa calculated baed on Both of them have almot the ame a that obtaid from Equation 10. Thi ugget that Equation 10 i till lightly corvative even when i ued intead of p in the denominator of Equation 10. Nonliar Dynamic Analyi Two ni-tory STMF with ierendeel type pecial egment (Figure 8), repreenting the cla of eential facilitie (in other word, hopital building) a well a ordinary office/reidential occupancy type, were deigd for evaluating the propoed modification (Chao and Goel, 2006). Double-chanl ection were ued for chord member. For ordinary building type, the deign target drift of 2% and 3% for 10% in 50 year and 2% in 50 year deign hazard level, repectively, were choen. The correponding number for eential facilitie were 1.5% and 2.25%. Deign pectral value were baed on NEHRP Proviion (FEMA, 2001) for the San Francico ite. After the final deign work wa completed, nonliar dynamic analye were conducted to tudy the repo. Ni, 10% in 50 year, and five, 2% in 50 year, SAC o Angele region ground motion repreenting the two deign hazard level were ued for the nonliar repo hitory analye. The reult howed that yielding wa limited to the pecial egment only, while the other element remaid elatic. Thi ugget that the propoed expreion for wa quite adequate to enure elatic performance of the element outide the pecial egment. Figure 9 and 10 how the maximum developed hear in the pecial egment for the ordinary and eential STMF, repectively. It can be een that the AISC equation ignificantly overetimate the expected hear trength, which would lead to undue over-deign of element outide the pecial egment. On the other hand, Fig. 8. Elevation of tudy ni-tory STMF. ENGINEERING JOURNA / SECOND QUARTER / 2008 / _Chao_Goel1_2008_2Q.indd 121 7/15/08 11:08:58 AM

6 the propoed expreion (Equation 10) i cloer to the actual developed hear while maintaining ome afety margin. Therefore, baed on the preceding dicuion, it i uggeted that the current AISC expreion (with the web diagonal alo included in the pecial egment) can be modified a: or 36. RM = EI + Ry( Pnt P 3 nc) inα (12-1) 4RM = E I θ + R P P 2 ( ) p y nt nc in α (12-2) IERENDEE SPECIA SEGMENTS WITH INTERMEDIATE ERTICA MEMBERS A pecial egment can contain multiple ierendeel pal by adding intermediate vertical member, a hown in Figure 11. O befit of uing multiple ierendeel pal i that the redundancy of the eimic ergy diipation mechanim increae. It alo ha the advantage of allowing more flexibility in mechanical and architectural layout, a well a reducing the rotational ductility demand on pecial egment chord (alley and Hooper, 2002). It i very likely that during minor to moderate earthquake event, ilatic deformation would only occur in the intermediate vertical member, which could be replaced relatively eaily. In addition, the ize of chord member can be reduced becaue of additional (a) (a) (b) Fig. 9. Maximum developed hear in pecial egment in the ni-tory ordinary STMF ubjected to 10%/50 and 2%/50 ground motion. (b) Fig. 10. Maximum developed hear in pecial egment in the ni-tory eential STMF ubjected to 10%/50 and 2%/50 ground motion. 122 / ENGINEERING JOURNA / SECOND QUARTER / _Chao_Goel1_2008_2Q.indd 122 7/15/08 11:09:00 AM

7 trength due to thoe intermediate vertical member. In the cae multiple ierendeel pal are preent, the calculation of hould include the contribution from intermediate vertical member. It can be hown that both the chord member and the intermediate vertical member have the ame platic rotation when the yield mechanim i reached (Figure 11). Thu the maximum moment developed in the chord member and intermediate vertical member can be obtaid from Equation 8, a follow: ( Mc) = ( ) RyMnc + EIc max 1 η 3 η (13) 2 h ( Mv) = ( ) RyMnv + EIv max 1 η 3 η (14) 2 h (M c ) max, (M v ) max = maximum expected developed moment in the chord member and intermediate vertical member, repectively I c = moment of irtia of the chord member I v = moment of irtia of the intermediate vertical member It hould be noted that the train-hardening ratio tend to increae when the member length decreae (Engelhardt and Popov, 1989). For a typical chord member, a 10% trainhardening ratio may be reaonable but the train-hardening ratio for the vertical member might be actually higher due to it much horter length. In thi tudy, the chord and intermediate vertical member were aumed to have the ame train-hardening ratio. The expected maximum hear trength of the pecial egment with o intermediate vertical member i then calculated a (Figure 12): 4 Mc max 2 Mv = ( ) + ( ) max (15) For a pecial egment with two intermediate vertical member (Figure 13): 4 Mc 4 M max v = ( ) + ( ) max (16) In geral, the expected maximum hear trength of a pecial egment with intermediate vertical member can be expreed a: m 4 Mc = ( ) + ( ) m 4 Mv 2 max max (17) = number of intermediate vertical member Fig. 12. Calculation of for two ierendeel pal. Fig. 11. Yield mechanim of STMF with multiple ierendeel pal. Fig. 13. Calculation of for three ierendeel pal. ENGINEERING JOURNA / SECOND QUARTER / 2008 / _Chao_Goel1_2008_2Q.indd 123 7/15/08 11:09:06 AM

8 By uing η = 0.1, and /h = 0.03, will be given by: or 36. RM = EI (18-1) c 3 m 36RM y + 2. nv EI v 3 4RM E I c = θ (18-2) 2 pc m 4RM y nv E I v θ 2 2 pv θ pc and θ pv = maximum rotation capacity of the chord and intermediate vertical member, repectively A NOTE ON THE DESIGN OF MUTIPE IERENDEE PANES It hould be noted that, a illutrated in Figure 11, while the vertical member gerally have maller length than the chord member, the vertical member experience the ame platic rotation with the chord member when the yield mechanim form. A pointed out by Engelhardt and Popov (1989), a member length decreae, flexural yielding tend to be confid to a maller region at the end of the member leading to larger curvature and bending train demand for the ame platic rotation. Thi higher demand on bending train in turn reult in a higher poibility of fracture at welded conction at the member end. In addition, the reduced length of the platic region can caue problem of flange buckling and lateral-torional buckling in flexural yielding member. A a coquence, until further experimental reult are available, it i uggeted that the intermediate vertical member hould be treated a econdary member to prevent poible intability of the frame in cae premature failure occur in the vertical member. The term econdary member i ued here in the e that mot of the tru trength and ergy diipation capacity hould be provided by the chord member. In addition, platic hinge mut be avoided in the chord member except at chord end; therefore, the moment capacity of vertical member ha to be limited o that the moment in the chord member at ection adjacent to the vertical member i le than the moment capacity of chord member after the vertical member yield. Therefore, to deign intermediate vertical member a econdary member, it i uggeted at thi time that at leat 70% of the input ergy be diipated by the chord member and the remainder by intermediate vertical member, unle further reearch can how that yielding of intermediate vertical member i not detrimental to the overall performance of an STMF. Therefore, the following equation can be applied at a given floor level (Chao and Goel, 2006): 30% 70% ( ) 2mM π / 2 φ i pvi ( 4M )( π / 2 φ pbi ) = ( ) (19) M pbi M (20) pvi mi M pvi = required platic moment capacity of the intermediate vertical member at the ith level M pbi = required platic moment capacity of the chord member at the ith level m i = number of intermediate vertical member at the ith level The term π/2 φ repreent the rotation of the chord and intermediate vertical member (ee Figure 11). It hould alo be mentiod that the deign of chord and intermediate vertical member can be eaily implemented by uing the platic deign method (Chao and Goel, 2006). In that cae, the internal work tored in a pecial egment can be expreed by: 4M 2m M 4M 2m M pbi + θ pbi i pvi pbi i θ (21) m ( ) = + = 6M θ p bi = External Work If the deign of chord member i performed baed on Equation 21, then the deign of intermediate vertical member hould follow Equation 20. SUMMARY AND CONCUSION A revied equation for maximum expected hear trength,, wa derived by uing a more realitic aumption and validated by experimental reult a well a nonliar tatic and dynamic analye. Baed on extenive nonliar dynamic analye, it wa found that the current AISC equation for in the pecial egment can ignificantly overetimate the expected hear trength, which in turn lead to undue overdeign of member outide the pecial egment, uch a vertical member, diagonal member, conction and column. The value given by the propoed equation were cloer to the actual developed hear while maintaining ome afety margin. A deign equation of for STMF uing multiple ierendeel pal in the pecial egment wa alo propoed. i 124 / ENGINEERING JOURNA / SECOND QUARTER / _Chao_Goel1_2008_2Q.indd 124 7/15/08 11:09:06 AM

9 ACKNOWEDGMENTS The author gratefully acknowledge partial financial upport provided by Nabih Youef & Aociate Structural Engier, and NUCOR Reearch and Development for thi tudy. The enior author alo received a tipend from the G.S. Agarwal Fellowhip Fund at the Department of Civil and Environmental Engiering while working on thi project. The opinion and view expreed in the paper are olely thoe of the author. REFERENCES AISC (2005), Seimic Proviion for Structural Steel Building, ANSI/AISC , American Intitute of Steel Contruction, Chicago, I. Baha, H.S. and Goel, S.C. (1994), Seimic Reitant Tru Moment Frame with Ductile ierendeel Segment, Report No. UMCEE 94-29, Department of Civil and Environmental Engiering, Univerity of Michigan, Ann Arbor, MI. Baha, H.S. and Goel, S.C. (1995), Special Tru Moment Frame with ierendeel Middle Pal, Engiering Structure, ol. 17, No. 5, pp Chao, S.-H. and Goel, S.C. (2006), Performance-Baed Platic Deign of Seimic Reitant Special Tru Moment Frame, Report No. UMCEE 06-03, Department of Civil and Environmental Engiering, Univerity of Michigan, Ann Arbor, MI. Duicka, P., Itani, A.M. and Sahai, R. (2002), Advance in the Seimic Deign of Special Tru Moment Frame, Proceeding, 71t Annual Convention of Structural Engier of California, Santa Barbara, CA, pp Engelhardt, M.D. and Popov, E.P. (1989), Behavior of ong ink in Eccentrically Braced Frame, Report No. UCB/ EERC-89/01, Earthquake Engiering Reearch Center, Univerity of California at Berkeley. Goel, S.C. and Itani, A.M. (1994), Seimic-Reitant Special Tru-Moment Frame, Journal of Structural Engiering, ASCE, ol. 120, No. 6, pp NEHRP (2001), Recommended Proviion for the Development of Seimic Regulation for New Building (FEMA 368), Federal Emergency Management Agency, Wahington, DC. Parra-Monteino, G.J., Goel, S.C. and Kim, K.Y. (2006), Behavior of Steel Double-Chanl Built-Up Chord of Special Tru Moment Frame under Revered Cyclic Bending, Journal of Structural Engiering, ASCE, ol. 132, No. 9, pp alley, M. and Hooper, J. (2002), Iue in the Deign of Special Tru Moment Frame in High-Seimic Region, Proceeding, 7th U.S. National Conference on Earthquake Engiering, Boton, MA, July 21 25, pp ENGINEERING JOURNA / SECOND QUARTER / 2008 / _Chao_Goel1_2008_2Q.indd 125 7/15/08 11:09:07 AM