COLUMN RESTRAINING EFFECTS IN POST-TENSIONED SELF-CENTERING MOMENT FRAMES

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1 Otoer -7, 8, Beijing, China COLUN RERAINING EFFECTS IN PO-TENSIONED SELF-CENTERING OENT FRAES C.C. Chou and J.H. Chen Assoiate Professor, Dept. of Civil Engineering, Taiwan University, Taipei, Chinese Taiwan Graduate Student Researher, Dept. of Civil Engineering, Chiao Tung University, Hsinhu, Chinese Taiwan ABRACT : ehou@ntu.edu.tw Cyi tests on post-tensioned (PT onnetions have demonstrated self-entering apailities with gap opening, osing at the eam-to-olumn interfae. When the gap opens at the eam-to-olumn interfae in a real frame with more than one olumn, this gap opening is onstrained y the olumns. The olumns provide flexural restraint to the eams, leading to the ompression fore different from the strand fore in the eams. This study presents a methodology to evaluate the ending stiffness of the olumns and the ompression fore in the eams y deforming a uilding-height olumn in aordane with gap opening responses at all onnetion levels. The predited ompression fore in the eams is validated y a detailed 3-story PT frame analytial model and yi tests of a full-sale, two-ay y first-story PT frame. The proposed model shows that the eam ompression fore is ineased at the st story ut deeased at the nd and 3rd stories due to deformation ompatiility of the whole olumn. The PT frame tests demonstrate that the proposed model reasonaly predits the eam ompression fore and strand fore, and that the eam ompression fore at a 3% drift is % and 6% larger than the strand fore with respetive to a little restraint and pin-supported olumn oundary ondition. EYWORDS: Post-tensioned Frame, Steel, Column Restraint, Cyi Frame Test. INTRODUCTION A post-tensioned (PT self-entering moment frame that uses post-tensioning steel to ompress the steel eams against the olumns has reently een developed as an alternative to a steel speial moment-resisting frame (SRF. any researhers (Ries et al.,, et al., Garlok, Garlok et al. 5, Chou et al. 6, Tsai et al. 8 have experimentally validated the self-entering ehaviors of the PT onnetions with either energy yielding or frition damped devies. The issues of olumn and sla restraint raised y et al. ( and Garlok ( have een hallenging sujets in this system. Chou et al. (8a experimentally showed that the proposed PT onnetion with a ontinuous omposite sla self-enters with low residual deformation as long as negative onnetion moments provided y sla reinforements are onsidered in design. Chou et al. (8 also demonstrated similar yi responses etween a are PT onnetion and a omposite PT onnetion with a disontinuous omposite sla, whih opens freely along with the gap opening at the eam-to-olumn interfae. As the gap widens at the eam-to-olumn interfae, leading to an expansion in the PT frame and ending in the olumns, the ompression fore in the eams is affeted y this restraint. et al. ( proposed a pin-pin supported olumn oundary ondition for the upper stories and a pin-fixed supported olumn oundary ondition for the first story to estimate the ending stiffnesses of the olumns. The oundary ondition does not inude effets of the olumn aove and elow the story that is eing onsidered, and deformation ompatiility of the whole olumn is not onsidered, leading to an overestimation of the olumn ending stiffness. Instead, this study presents a methodology to evaluate the ending stiffness of the olumns and ompression fore in the eams y deforming a whole olumn in aordane with gap opening responses at all onnetion levels. The predited ompression fores in the eams are ose to those otained y a detailed 3-story PT frame omputer model and yi tests of a full-sale, two-ay y first-story PT frame.

2 Otoer -7, 8, Beijing, China. DESIGN OF A 3-ORY PO-TENSIONED SELF-CENTERING BUILDING Figs. (a and ( show the plan and elevation of the prototype uilding, whih was assumed to e loated on stiff soil in Los Angeles, California. The three two-ay PT frames provided lateral load resistane in the east-west diretion; eah PT frame was omposed of three PT reinfored onete olumns and three PT steel eams. A redued flange plate (RFP proposed y Chou et al. (6, 8a was inorporated at eah eam-to-olumn onnetion to inease energy dissipation (Fig. (a. The RFP moment onnetions (Chou and Wu 7 were also demonstrated to eliminate steel eam ukling as oserved in traditional welded moment onnetions sujet to yi loads. No energy dissipation devie was used at the PT olumn ase (Fig. (. The design dead loads were 5.8 kpa ( psf and 4.3 kpa (9 psf for the floors and the roof while the live loads for the floors and the roof were.39 kpa (5 psf. Effetive seismi weights for the floors and the roof were 3 kn and 896 kn, respetively, resulting in an effetive seismi weight of the uilding of 6536 kn. The strutural period, T, and the seismi response oeffiient, C s, alulated y the odified method (IBC were.6 se and.5, respetively, so the seismi design ase shear, V des, for one PT frame was 7 kn. The eam and olumn sizes, RFP thikness (t R and narrowest dimension ( R, strand and PT ar area (A, and initial PT fore (T in are given in Tale. High strength Dywidag (DSI ar was speified to the olumn PT ar, and A A 76 steel was speified for the olumn transverse and longitudinal reinforements. The speified 8-day onete strength, f n, was 8 Pa. A57 Grade 345 (5 steel was used for the steel eams, and A A46 Grade 7 strands with eah diameter of 5 mm were passed along the eam wes and anhored outside the exterior PT olumns. oment demands at the eam-to-olumn interfae and the olumn ase due to the seismi load ( E, dead load ( D, and live load ( L are also given in Tale. The deompression moment of PT onnetions, d, listed in Tale is omposed of the moments provided y the strands and the RFPs. The deompression moment is larger than the moment due to dead load and live load, ut is slightly less than the omined moment demand, dem. The onnetion moment at the onset of RFP yielding, y, is larger than α y dem, where α y. (Tale, indiating that the PT frame remains elasti under the ode-ased seismi load. Following the onnetion design proedure proposed y Chou et al. (8a, the onnetion moment at a roof drift of 4%, 4%, reahed aout.9, in whih R.3 was provided y the RFPs and.6 was provided y the strands. A notation represents the nominal plasti moment apaity of the eam. Story Size (mm Tale 3-story PT frame dimension and moment A T in t R R E D L (mm (kn (mm (mm 3rd H nd H st H Column RC dem d y 4% 4@5.5m 4@5m PT Frame E RC Column Tested Portion PT PT H H5 6 H5 6 CL CC CR RFP FRP Cover Plate Jaket Column (a Plan ( Elevation (unit: mm (a PT onnetion ( PT olumn Figure Prototype uilding Figure PT onnetion and olumn PT Bar Column Footing

3 Otoer -7, 8, Beijing, China 3. ODELING OF A 3-ORY PO-TENSIONED SELF-CENTERING FRAE et al. ( used a numer of axial springs at the eam-to-olumn onnetion to apture the self-entering ehavior of the frame. This axial spring ( modeling an apture the effets of onstrained eams. Figs. 3(a and ( show the monotoni and yi fore-deformation relationships of the 3-story PT frame ased on this model. The normalized ase shear was otained y dividing the ase shear y the value of V des = 7 kn. The PT frame deompressed after the odified ase shear V des. The PT olumn ase first opened (step A, followed y a onseutive gap opening of the eam-to-olumn interfae at the 3rd, st, and nd stories, respetively. The yield strength of the PT frame was 49 kn (=.8 V des at a roof drift aout.5%. At a roof drift of % (DBE level for the SRF, the maximum ase shear reahed 3V des. Fig. 3( shows the expansion of the exterior olumns CL and CR (Fig. (, whih were omputed y sutrating the exterior olumn lateral deformation y the entral olumn lateral deformation. This expansion is aused y the gap opening response at the onnetion level, and is more pronouned for the first story than other stories due to fixity at the ase. As an e seen from the analytial results, the strand fore at the st story is -3% smaller than the eam ompression fore (Fig. 4(a. However, the strand fores at the nd and 3rd stories are -4% and 7-8% larger than the eam ompression fore (Figs. 4( and (. Base Shear (kn Compression Fore (kn Tension Fore (kn 8 4 V des el B A Restraint No Restraint V/V des Base Shear (kn 8 4 Restraint No Restraint Story 3 CR CL % % 3% 4% Lateral Displaement, (mm (a onotoni pushover analysis ( Cyi analysis ( Frame expansion Figure 3 PT frame response (BL (BR Eq. (4.5 Eq. ( Eq. (4.7 Compression Fore (kn Tension Fore (kn (BL (BR Eq. (4.5 Eq. ( Eq. (4.7 Compression Fore (kn 8 6 (BL (BR Eq. (4.5 Eq. ( InterstoryDrift (% (a st Story ( nd Story ( 3rd Story Figure 4 ompression fore and strand fore 4. COPRESSION FORCE PREDICTION IN THE PT BEA 4.. Rotational Spring odel Tension Fore (kn 8 6 Eq. (4.7 Instead of using axial springs at the PT onnetion, Chou et al. (5 also proposed a rotational spring sheme to apture the self-entering response of the PT onnetion. As an e seen in Fig. 5(a, the intersetion of the eam and olumn enterlines has three nodes j, m, and n. Two zero-length spring elements, onneting the nodes j and m, were used to model the ilinear elasti ehavior of the PT strands (SC spring and the ilinear elastoplasti ehavior of the RFPs (RFP spring, respetively. A omination of these two rotational springs

4 Otoer -7, 8, Beijing, China predits well the experimental results of a PT onnetion (Fig. 5(. However, the PT fore in the strands, the ompression fore in the eams, and the olumn restraining effets ould not e disovered in the prior study. The rotational spring sheme ould also e adopted to model the self-entering ehavior of the PT olumn. Fig. 6(a shows two nodes j and k at the PT olumn ase. One zero-length rotational spring, onneting the nodes j and k, was used to model the ilinear elasti ehavior of the PT olumn. Before deompression, the elasti rotational stiffness of the PT olumn,, is approximated using that of a fully restrained olumn. After reahing the deompression moment of the PT olumn, d,, in Fig. 6( the rotational stiffness,, is = (4. + ar where the rotational stiffness, ar, is provided y the PT ars in the olumn d A ar d ar = Ear A ar L ar Aar A + g (4. where d is the olumn depth, E ar is the elasti modulus of the PT ar, A ar is the PT ar area, L ar is the PT ar length, and A g is the olumn setional area. 4. Bending Stiffness of a PT Column at Connetion Levels Fig. 7(a shows the 3-story PT frame in a deformed position. As the gap opens at the eam-to-olumn interfae (,, and 3 at the st, nd, and 3rd stories, respetively, the strands in the eam elongate and result in axial shortening of the eams and ending of the exterior olumns CL and CR. In order to develop ending stiffness of the exterior olumn aove the st story, et al. ( proposed a simple estimate y assuming that the olumn is pin-pin supported at stories aove and elow the story that is eing onsidered. For the st story, the olumn is PT to the ase and pin-supported at the nd story, so the ending stiffness,, is 36( h + h ( EI + ( h + h EI = (4.3 3 h h + + h EI h h 3 h + h where h is the st story height, h is the nd story height, E is the elasti modulus of the onete, and I is the moment inertia of the olumn. Considering that the ompression fore in the eams along the ays is symmetri with respet to the enter olumn (CC, the ompression fore, F, in the eams is otained y inuding the PT fore and the olumn restraining effets. Compared with those otained from the detailed PT frame model ( desied in the previous setion, this simple analytial model overestimates the eam ompression fores y 49 and 55% at the st and nd stories at a 4% drift (Figs. 4(a and (. Instead of using an assumed olumn oundary ondition, this study utilizes a deformed olumn to develop its ending stiffness at eah story. The deformed olumn shape onsiders: ( a gap-opening response at eah story aove the ground level, ( eam earing loations at the respetive olumn heights, and (3 olumn ase rigidity,, after the gap opens. Figs. 7( and ( show deformed shapes (shape 3 for the two exterior olumns CL and CR. A rotational spring with stiffness is positioned at the olumn ase to simulate restraining of the ase after the gap opens. At eah story, the olumn CL has a speified lateral displaement, (=θ g (d -t f, at a loation of the eam top flange inner side, and the olumn CR has the same speified lateral displaement,, at a loation of the eam ottom flange inner side, where d is the eam depth and t f is the flange thikness. Assuming the same gap opening angles (e.g. θ g =. rad. at the olumn ase and eah onnetion, numers, 4.84 and 3.7 mm marked in the figure, are the lateral displaements speified at eah story. The ending stiffnesses of the olumns CL and CR at eah story are and, respetively, whih are omputed y dividing the orresponding reation fore, F and F, y the lateral displaement. The olumn ending stiffnesses, and, at eah story are different and negative at the nd and 3rd stories due to a speified olumn deformation, leading to a redued ompression fore in the eams.

5 Otoer -7, 8, Beijing, China i i o SC RFP n m RE RE d/ d/ j k SC RFP l oment (kn-m Test Proposed odel -4-4 RE k Column SC j Column oment d, θ d, + g Column g RE Spring Interstory Drift, θ (a PT onnetion model ( oment-drift relationship (a Column model ( Column ehavior Figure 5 PT onnetion modeling Figure 6 PT olumn modeling 3 Initial L BL Column CL L BR Column CR Column CC h 3 θ gr θ gr 3 =3.7 mm =4.84 mm F 3 F H l3 F 3 H r3 F 3 =3.7 mm =4.84 mm H3 H H θ g = θ g θ g θ g = θ g h h θ g θ gr θ g =4.84 mm θ g F H l H l H r H r F θ g =4.84 mm g g (a PT frame deformation ( Column CL ( Column CR Figure 7 3-story PT frame and exterior olumn deformation 4.3. PT Compression Load The exterior olumns CL and CR ear against opposite sides of the eams BL and BR at the same drift, resulting in different ending stiffnesses (or restraints of the exterior olumns to the PT eams. At any story, onsidering inemental equilirium equations of horizontal fore for the olumns CL and CR: Fl = F + T (4.4 Fr = F + T (4.5 where F l and F r are the inemental ompression fores in the eams BL and BR, respetively; F and F are the inemental restraining fores of the olumns CL and CR, respetively, and T is the inemental strand fore. The inemental shortening of the left eam BL due to the ineased ompression fore is: l = + (4.6 where is the omponent of the eam BL shortening due to the olumn CL inemental restraining fore ( F, and is the omponent of the eam shortening due to the inemental strand fore T. The olumn CL inemental restraining fore is F = = ( (4.7 Where is the axial stiffness of the eam. Rearranging Eq. (4.7, the omponent of the eam BL shortening due to the olumn CL inemental restraining is = ( (4.8 + The omponent of the eam BR shortening due to the olumn CR inemental restraining is also expressed as = ( (4.9 + The ratio of the two eam shortenings due to olumn inemental restraining effets is

6 Otoer -7, 8, Beijing, China ( + ( + ς (4. = = Sine the olumn ending stiffnesses, and, are different, the ratio,ζ, ranges from The strand fore inement, T, is T = = [ ( + ς ] (4. where is the axial stiffness of the strands. Rearranging Eq. (4., the omponent of the eam shortening due to the inemental strand fore is = [ ( + ς ] + (4. Sustituting Eq. (4. into Eqs. (4.8 and (4.9, the omponents of the eam BL and BR shortenings due to the olumn inemental restraining fores are: = (4.3 ( + ( + + ( + ς ς = ( ς ( ( ( For a speifi gap opening, the eam BL and BR fores, F l and F r, and the strand fore, T, are: Fl + Fl + ( + (4.5 Fr + Fr + ( + (4.6 T + (4.7 Fig. 4 shows preditions ased on Eqs. (4.5-(4.7, whih are ose to those otained from the detailed frame model using axial springs (s. The predited eam ompression fore is larger than the strand fore at the first story and smaller than the strand fore at the nd and 3rd stories due to negative olumn ending stiffnesses. The redution of the eam ompression fore from the applied strand fore at the nd and 3rd stories ould not e otained from the simple estimate proposed y et al. ( eause the olumn deformation shape was not onsidered in developing ending stiffness. It was also found that although the ompression toes in eams BL and BR differ y a eam depth, the ompression fore variation, and, show minor different ompared to the strand fore, T, indiating that the eam-to-olumn enterline intersetion an e used as a ompression loation for simpliity. In this ase, the olumn ending stiffness at eah story,, an e omputed y deforming the olumn at the speified lateral displaement,, and assoiated reation fores. The resulting eam axial fore is % larger than the strand fore at the first story and 3-4% lower than the strand fore at the nd and 3rd stories. 5. PT FRAE TE To evaluate the effets of olumn restraining on the frame expansion, a full-sale, two-ay y first-story PT frame (marked in Fig. ( was yially tested. A total of twelve A A46 Grade 7 strands with eah diameter of 5 mm were passed along the eam we, through three PT olumns, and anhored outside the exterior olumns. The initial PT fores in the olumns and eams were aout kn and 96 kn, respetively; a total of four yi tests were onduted on this frame (Fig. 8. Eah olumn was extended to the mid-height of the seond story, at whih two kn atuators (laeled as At and At were positioned etween the reation wall and the frame and one kn atuator (laeled as At3 and At 4 was positioned etween eah eam span. Quasi-stati yi loading with ineasing displaement amplitude in aordane to AISC (5 for onnetion tests was adopted. The displaement of the olumn CC was ontrolled as a target displaement; the interstory drift was defined as the horizontal displaement at the loading point relative to the olumn height of 5.66 m. Two loading shemes were adopted in the test program. For the first loading sheme, the fores in At 3 and At 4 were slaved, respetively, to three-quarter and one-quarter the summation of the fores in At and At. Therefore, the shear applied to the olumns CL and CR would e half of that applied to the olumn CC at the loading point to simulate little restraint on the top of the olumns. This loading sheme was arried out for the first three tests, in whih RFPs for energy dissipation were only inuded in the

7 Otoer -7, 8, Beijing, China onnetions for the first two tests. For the seond loading sheme, no relative lateral deformation was allowed etween the olumns to simulate a full restraint on the top of the olumns. No energy dissipation devies were provided at the olumn ase. In the first yi test, two out of eight RFPs fratured when the frame moved towards an interstory drift of 4% (Fig. 9. The frame was retested using the same loading protool, and no more RFPs fratured in the nd tests. Fig. (a shows the ase shear versus olumn CC deformation for the first two tests; it appears that the PT frame under the st test dissipated larger energy than the nd test. Six RFPs were removed from the frame after the nd test in order to evaluate the frame response without energy dissipating devies. The ilinear elasti ehavior of the PT frame was oserved y omparing the hystereti loops etween the st and 3rd tests (Fig. (. For the 4th test, the PT frame was loaded with no relative olumn deformation at the level of atuators, so the post-yielding stiffness of the frame was % higher in the 4th test than in the 3rd test. Considering horizontal and vertial fore equiliriums in the three olumns and taking moment equiliriums aout olumn ompression toes, the ompression fore in the eams BL and BR (Fig. were otained from the tests. Fig. shows strand fores and ompression fores in the eam for the 3rd and 4th tests. The eam ompression fore is similar to the eam strand fore in the 3rd test (Figs. (a and (, in whih the olumn top an expand during the test. Two tests resulted in similar strand fore in the eams, ut the eam ompression fore is 6% larger than the eam strand fore in the 4th test (Fig. ( eause the distane etween eah olumn top does not vary during the test. Following the same proedure desied earlier, the exterior olumn ending stiffnesses,, are 86 and kn/m for the 3rd and 4th tests, respetively. The resulting eam ompression fores ased on Eqs. (4.5 and (4.6 with for oth olumn ending stiffnesses agree well the test results (Fig Negative Diretion 3 Positive Diretion 5 3 At At Steel (5 6 CL At 3 At 4 Jaket CC CR RC Column ( Figure 8 Test setup (unit: mm Figure 9 PT frame deformation (4% drift Base Shear (kn Tension Fore (kn Test Test Base Shear (kn Test Test Drift (% Drift (% Drift (% (a Test versus test ( Test versus test 3 ( Test versus predition Figure Hystereti responses of PT frame tests Compression Fore (kn Base Shear (kn Test Proposed odel Test Test Test 5 Proposed 5 Proposed 5 Proposed odel odel odel Drift (% Drift (% Drift (% (a Strand fore (3rd test ( ompression fore (3rd test ( ompression fore (4th test Figure strand and ompression fore in 3rd and 4th PT frame tests Compression Fore (kn

8 Otoer -7, 8, Beijing, China 6. CONCLUSIONS This paper presents a methodology to take into aount the PT frame expansion. The proedure is aimed at ( deforming a uilding-height olumn in aordane with gap opening responses at all onnetion levels, and ( omputing the olumn ending stiffness at eah story y the reation fore divided y the respetive lateral displaement. Beause this deformed olumn shape inudes effets of the olumn aove and elow the story that is eing onsidered, the olumn ending stiffness at eah story is more realisti and smaller than those developed ased on a pin-pin supported olumn oundary ondition. The following onusions are made: ( For the 3-story PT frame, the eam ompression fore is larger than the eam strand fore at the st story, ut smaller than the eam strand fore at the nd and 3rd stories due to olumn deformation ompatiility. The variation of the eam ompression fore an e reasonaly predited ased on the proposed model in this study. However, the simple estimate ased on an assumed olumn oundary ondition always predits the ineased ompression fore in the eam and the overestimation of the eam ompression fore is aout 5% at the st and nd stories. ( A full-sale, two-ay y first-story PT frame was yially tested. Two loading shemes were onduted on the test frame to evaluate the olumn restraints. The first loading sheme produed the shear in the exterior olumns half of that in the enter olumn to simulate little restraint from the olumn to the PT eam. The seond loading sheme produed no relative lateral deformation etween olumns to simulate a pin-supported olumn oundary at the olumn top. The two loading shemes resulted in similar eam strand fore ut signifiant different ompression fore in the eam; the eam ompression fore is 6% larger than the eam strand fore in the seond loading sheme. ACNOWLEDGEENTS The test program was supported y the NCREE with Prof.. C. Tsai as the program diretor. The writers are grateful to Prof. H.L. Hsu of NCU and Dr..C. Lin of NCREE for orporation on design of the tested frame. REFERENCES, C., Filiatrault, A., and Uang, C-. (. Self-entering post-tensioned energy dissipating (PTED steel frames for seismi regions. Report No. SSRP-/6, Dept.of Strutural Eng., University of California, San Diego, CA., C., Filiatrault, A., Uang, C-, and Folz, B. (. Posttensioned energy dissipating onnetions for moment-resisting steel frames. Journal of Strutural Engineering, 8(9, -. Chou, C-C, Tsai, -C, Chen, J-H, Chen, Y-C, and Jhuang, S-C. (5. Cyi ehavior of post-tensioned steel onnetions with redued flange plate and sla. st International Conferene on Advanes in Experimental Strutural Engineering, Nagoya, Japan. Chou, C-C, Chen, J-H, Chen, Y-C, and Tsai, -C. (6. Evaluating performane of post-tensioned steel onnetions with strands and redued flange plates. Earthquake Engineering and Strutural Dynamis, 35(9, Chou, C-C and Wu, C-C. (7. Performane evaluation of steel redued flange plate moment onnetions. Earthquake Engineering and Strutural Dynamis, 36, Chou, C-C, Wang, Y-C, Chen, J-H. (8a. Seismi design and ehavior of post-tensioned steel onnetions inuding effets of a omposite sla. Engineering Strutures. (availale online ay 8. Chou, C-C, Tsai, -C, and Yang, W-C. (8. Self-entering steel onnetions with steel ars and a disontinuous omposite sla. Earthquake Engineering and Strutural Dynamis. (aepted for puliation. Garlok,.. (. Full-sale testing, seismi analysis, and design of post-tensioned seismi resistant onnetions for steel frames. Ph.D. dissertation, Civil and Environmental Eng. Dept., Lehigh Univ., PA. Garlok,., Ries,.J, and Sause, R. (5. Experimental studies of full-sale posttensioned steel onnetions. Journal of Strutural Engineering, 3(3, Ries, J.., Sause, R., Garlok,.. and Zhao, C. (. Posttensioned seismi-resistant onnetions for steel frames. Journal of Strutural Engineering, 7(, 3. Ries, J., Sause, R., Peng, S.W., and Lu, L.W. (. Experimental evaluation of earthquake resistant posttensioned steel onnetions. Journal of Strutural Engineering, 8(7, Tsai, -C, Chou, C-C, Lin, C-L, Chen, P-C, Jhang, S-J. (8. Seismi self-entering steel eam-to-olumn moment onnetions using olted frition devies. Earthquake Engineering and Strutural Dynamis, 37,