SEISMIC VULNERABILITY ASSESSMENT OF ASYMMETRIC STRUCTURES

Transcription

1 13 th World Conference on Earthquake Engneerng Vancouver, B.C., Canada August 1-6, 2004 Paper No SEISMIC VULNERABILITY ASSESSMENT OF ASYMMETRIC STRUCTURES Ajay Kumar SINHA 1 and Pratma Ran BOSE 2 SUMMARY Earthquakes fnd weaknesses left behnd by us n structures. Sesmc damage concentrated at weak zones of structure speaks of as a report of an expert. Obvously, the expert here s the nature, whch s often treated as the man reason of such dsasters, by even termng such dsasters natural dsasters. But the fact s beyond doubt that the wde-spread damage to structures and mmense losses are not due to just natural act. We leave the weaknesses n our structures ether by our neglgence (ether techncal or non-techncal), or by our gnorance. The dscusson mght seem to be phlosophcal but our never-endng research actvty s fueled by ths phlosophy only. There are mysteres to be unraveled, and there are thngs to be understood and explaned. Lots of research effort has been drected n recent tmes towards sesmc vulnerablty assessment. Ths s a part of sesmc dsaster mtgaton. Earthquake Engneers can contrbute n the noble cause by understandng and predctng structural behavour under sesmc acton as close as the actual one and provdng the structure safe capactes aganst all the possble demands durng strong moton. Sesmc vulnerablty assessment exercse poses several dffcultes as demand and capacty calculatons depend not only on earthquake parameters but also on the structures. The present study s n the drecton dscussed above. Due to several reasons structures acqure asymmetry. Asymmetry n structures makes analyss of the sesmc behavour very complcated. Sesmc demand n perpheral elements s enhanced. Unformty n load dstrbuton gets dsturbed. The paper frst concentrates n understandng the complex behavour of structure under asymmetrc form. Then a smplfed nonlnear pushover analyss has been used to fnd structural descrptors requred n sesmc vulnerablty assessment. Deformaton demand for dfferent story for low-medum rse framed buldng has been found. Sesmc load has been dstrbuted n stores usng trangular form. Then, force on frame has been dstrbuted as per the dstance from the center of rgdty and stffness of the frame. Results have been produced n form of structural descrptors and eccentrcty. The relatonshp wth the damage s then establshed wth eccentrcty. The results are ndcatve of sesmc vulnerablty of asymmetrc structure. The method proposed s an approxmate method and the fndngs should be 1 Assstant Professor (Str), College of Mltary Engneerng, Mnstry of Defence, Pune, INDIA 2 Professor, Delh College of Engneerng, Delh , INDIA

2 subjected to detaled evaluaton and valdaton. The paper provdes a smplfed conceptual base to deal wth the, otherwse, complcated analyss. INTRODUCTION Torson has been the cause of major damage to buldngs subjected to strong earthquakes, rangng from vsble dstorton of the structure to structural collapse. Torson occurs under the acton of earthquake forces when the center of mass of a buldng does not concde wth ts center of rgdty. Some of the stuatons that can gve rse to ths stuaton n the buldng plan are postonng the stff elements asymmetrcally wth respect to the center of gravty of the story; The placement of large masses asymmetrcally wth respect to stffness. A combnaton of the two, mass as well as stffness dstrbuton results n the stuatons descrbed above. It should be kept n mnd that the nflls attached to the panel are usually very stff and, therefore, partcpate n the structural response to an earthquake and can cause torson. The buldng rotates about ts center of rgdty. Ths causes large ncrease n the lateral forces and dsplacement demands n lateral load resstng elements, n proporton to ther dstance from the center of rotaton. The conventonal analyss for torson smply gves the force due to moment produced by an eccentrc statc force. It takes no account of the torsonal vbratons and the assocated acceleratons. Quanttatvely, an eccentrcty between the centers of mass and stffness s consdered sgnfcant when t exceeds 10% of the horzontal plane dmensons under study. In such cases, correctve measures should be taken n the structural desgn of the buldng. Torson may become even more complcated when there are vertcal rregulartes, such as setbacks. In effect, the upper part of the buldng transmts an eccentrc shear to the lower part, whch causes downward torson of the transton level regardless of the structural symmetry or asymmetry of the upper and lower floors. Non symmetrc or torsonally unbalanced buldngs are prone to earthquake damage due to coupled lateral and torsonal movements producng non-unform dsplacement demands n buldng elements and concentratons of stresses and forces on structural members. Current codes fall short of provdng recommendatons for rregular structures. Thus, there s an apparent need to develop a smple analyss procedure based on rgorous analytcal and expermental nformaton on the nelastc sesmc response of rregular structures. REASONS OF ECCENTRIC BUILDINGS The compromse between structural desgn and archtecture s always a debated matter. But, n case of sesmc performance of a buldng, archtecture has a greater effect. Not beng conducve t can hamper sesmc performance beyond magnaton. Snce, sesmc actvtes are sparse and uncertan, such detrmental results are seen only event of such actons. Hence, these consderatons are easly overlooked. The buldng archtecture must permt sesmc desgn as smple and effectve as possble. On the other hand desgn complcaces should accommodate the functonal and aesthetc themes of the buldng realzed by the archtect. There could be several reasons for eccentrcty creepng n the structure. Some of theses are: 1. Archtectural reasons 2. Confguraton ) Sze, shape and proportons of the 3-D form of the buldng ) Locaton, shape and sze of major structural elements 3. Addton and alteraton 4. Desgn assumptons 5. Functonal requrement: due to demand for certan settng and space dvson, a crculaton pattern for movement of people, supples and equpments

3 6. Ste constrants 7. Desre to produce unqueness 8. Nature, locaton, shape and approxmate sze of mportant elements 9. Nature, sze, locaton, connecton to structural members for non-structural elements. 10. Elements lke walls, columns, beams, constructon and expanson jonts, servce core, starcase well, lght core 11. Quantty (length) and type of nteror partton and nature of exteror walls 12. Non-structural components. 13. Urban desgn and plannng requrement of local legslaton. The fnal confguraton s the result of a decson process, whch balances the varyng requrements and nfluences. Confguraton determnes the ways and the magntude n whch the sesmc forces are to be dstrbuted n the buldng. The buldng mass nfluences the nertal forces mparted on the buldng due to ground moton. Sze and shape of the buldng, dstrbuton of structural elements, and materals used establsh the mass. Integrty of the buldng and completeness of the load path enable the transfer of sesmc force to the ground back. Hence, dentfcaton of parameters related to confguraton becomes an mportant aspect n sesmc vulnerablty assessment. These parameters can be seen as devaton from a favourable qualty regularty. Devaton from regular structure causes devaton n results of analyss as the assumpton n smplfed analyss whether wth earthquake load, structure model, consttutve laws etc., get volated more and more wth ncrease n rregularty. Smplfed ELF cannot represent sesmc response any more. Irregularty affects dstrbuton of sesmc force and mode of vbraton does not reman predomnantly fundamental. Stress concentraton, dfferental vbraton of varous parts, nternal poundng, torson and many other problems surface and need to be specfcally tackled. ECCENTRICITY IN ASYMMETRIC STRUCTURES General In order to desgn buldngs n earthquake prone regons, sesmc codes present dfferent torsonal provsons accordng to the sesmcty of the regon. Sesmc provsons ntroduce desgn eccentrcty to estmate the value of the torson n buldngs as accurately as possble. The dynamc eccentrcty results from the rregular mass, resstance or stffness dstrbuton of the system, whle the accdental eccentrcty s expected to account for factors not explctly consdered, such as uncertan estmaton of the mass, stffness and rotatonal component, whch s beleved to play a hghly mportant role. Ths accdental eccentrcty, e a, whch s a fracton of the plan dmenson, b, s consdered n desgn to be on ether sde of the center of stffness. The coeffcent β, proposed by dfferent sesmc buldng codes s equal to 0.05 or 0.1. The desgn eccentrcty e d() s generally presented as e d1 = α e s + β b e d2 = δ e s - β b where, e s s the structural eccentrcty calculated at the th storey and s defned as the dstance between center of mass (CM) and the centre of rgdty (CR). For each element, the larger value of the found forces s selected usng the two desgn eccentrcty equatons. b s the lateral floor dmenson as compared to the prncpal drecton of earthquake ground moton. In the above equaton for the desgn eccentrcty the frst part takes care of the translatonal and torsonal coupled response of asymmetrc buldng. The frst part n the equatons s dynamc eccentrcty. Ths s an enhancement over statc eccentrcty. The force based concept wants that the ductlty capacty of structure be never exceeded. For ths the statc eccentrcty s ncreased by a pre-decded factor. In the above equatons α and δ are the dynamc amplfcatons.

4 Durng an earthquake, dfferent parts of the buldng foundaton are excted due to the spatally nonunform motons. Ths rregularty n the foundaton motons s due to the phase lag of the random ground motons, whch create the rotatonal earthquake component. These phenomena lead to the occurrence of torsonal motons n the system. In sesmc buldng codes, the effects of earthquake rotatonal component are ndrectly taken under consderaton as the accdental eccentrcty parameter. Therefore, the accdental eccentrcty s selected n a way that makes t possble to consder the ncrease of the structural response due to rotatonal component of earthquake. The second part accounts for the accdental eccentrcty, manly consderng rotatonal component of ground moton, uncertanty n stffness values, dstrbuton of sesmc mass etc. Accdental eccentrcty s consdered to compensate the unavodable dfferences between assumed and the actual dynamc characterstcs of the structure. Accdental eccentrcty may be taken as the ncluson of addtonal shears n desgn of lateral resstng elements of the buldng. Such allowance s bass of elastc desgn amng to keep the structure wthn elastc range. But, most of the structures are expected to yeld under strong sesmc acton. Varaton n defnng eccentrcty: Ideally, centre of rgdty cannot be defned for a storey or stores of a multstory structure, even for structures wth rgd floor daphragms. The mportant step n determnng torsonal moment usng eccentrcty approach lacks clear defnton of eccentrcty. Generally CR defned for sngle storey buldng wth regard to rgd daphragm s extended to multstory buldngs. But CR for multstory buldng s dfferent from the CR n sngle story. A clear understandng of the ssue s requred here for careful handlng of such extensons. A careful probe of the prevous works n the feld leads the headway n the study of response of multstory asymmetrc buldngs. The characterstcs of CR n sngle storey structure are traced here. It can be termed as centre of resstance or centre of rgdty as applcaton of equvalent statc load at ths pont does not cause any rotaton to the floor. It s called shear centre as t s the locaton for the shear resultant n case of no rotaton of the floor. It remans statonary n case of torque appled at ths pont. Ths defnes t as the most mportant pont centre of twst. In fact, centre of twst or centre of rotaton s mportant so that nodal DOFs can be reduced n the analyss by treatng deformaton of load resstng elements wth regard to ths statonary pont whle floor rotates. CR s located as the rato of the sum of the frst moment of lateral stffness to total lateral stffness. Wth CR as reference pont, the equatons for translatonal and torsonal equlbrum of the floor become uncoupled. Hence, t can be termed as centre of decouplng. In case of sngle storey structure, all the ponts are same and hence have been used nterchangeably by authors wthout dffculty. Dffculty n establshng CR n a multstory buldng: In sngle storey model, locaton of these centers are ndependent of load dstrbuton, snce there can be only one sngle load resultant actng at the roof level. The stuaton changes n case of multstory structures. Contrary to CR n sngle story buldng, determnng the CR n case of multstory buldng poses several problems and thus needng a careful handlng for better analytcal results n study of response n asymmetrc structures under sesmc acton. Ths stuaton aggravates further n case of nelastc behavour warrantng post-yeld dynamc behavour as well as stffness calculatons. Let us frst see the defntons and the processes gven by varous authors. CR of a floor s the locaton of the resultant of shear forces of varous resstng elements n the story Some of the defntons for CR n multstory have been reproduced here. 1. Poole (1977) carred hs study takng CR of a storey as the pont of the resultant of the shear forces n varous lateral load-resstng elements such that there would be no rotaton n any of the floors. 2. But, Hoomer (1984) propounded that CR at any floor s the pont where applcaton of the statc sesmc force wll not rotate that floor whle other floors may rotate. 3. In further smplfcaton (Smth and Vezna (1985) took CR as localzed ponts where lateral load applcaton wll not cause rotaton n the whole structure

5 4. Cheung and Tso (1986) and Hejal and Chopra (1987) canvassed that only proportonal buldng has CR defned for multstory buldngs. 5. Cheung and Tso (1986) defned the CRs as the set of ponts located on the floors through whch applcaton of lateral force would not cause any rotaton of any of the floors. 6. Storey eccentrcty; ths s the dstance between CM and shear centre 7. Floor eccentrcty: ths s the dstance between CM and centre of stffness The sesmc load n statc form s defned to act at the centre of mass. Agan, the equvalent lateral load depends on heght wse dstrbuton. Ths heght wse load dstrbuton, n fact, depends on the vbraton characterstcs of the buldngs, whch keep on changng wth tme. Hence the change n CR locaton wth tme s a possblty, whch cannot be gnored. Ths problem s somewhat avoded here as Pushover analyss (POA) assumes tme-nvarant vertcal load dstrbuton pattern. Ths nherent weakness of POA makes devaton n results for study of sesmc response of asymmetrc structures. As other smplfed methods also neglect ths effect, POA seems to be a better than remanng as t oversmarts them n predcton of nelastc response of a asymmetrc structure. One more problem s there unque to multstory buldngs. Vertcal load dstrbuton does not affect the locaton of ths pont n sngle storey buldng, as there would be just one lateral load to be appled. Important parameters The mass rotatonal nerta plays an mportant role n the response of asymmetrc structures. The strength eccentrcty and dstrbuton of mass affect the rotaton of the system. For the structure modeled as an equvalent sngle degree of freedom,.e, ESDOF system, the system dsplacement ductlty capacty becomes a smple functon of the dsplacement ductlty capacty of the crtcal element. STIFFNESS OF ELEMENT AND STRUCTURE As per the theory of elastcty, stffness (k) of an element s based on the flexural rgdty (EI), where E s the modulus of elastcty of the materal and I s the second moment of area of the cross secton. In most renforced concrete components crackng extends throughout a sgnfcant porton of the length. Therefore, stffness based on the gross secton propertes can lead to msleadng values. In the past stffness was assumed to be ndependent of element strength but, stffness s strength dependent. An mplcaton of the element stffness beng strength dependent s that the locaton of the centre of rgdty, CR, could be qute dfferent from that found wth the tradtonal approach. As the stffness s related to the strength, element stffness can be wrtten as y k = V (1) y where, V y s the nomnal strength and y s the nomnal yeld dsplacement of the th element. Now, the structure s strength and stffness are defned as the sum of strength and stffness of all the resstng elements, respectvely. Thus, the structure s yeld dsplacement s, ΣVy S = (2) Σk The centre of strength, C V, s defned as the locaton about whch the frst moment of the element strength s zero..e. Σ V n x =0 when taken about CV. The dstance between the centre of mass, CM and the CV s referred to as the strength eccentrcty, e v. Ths eccentrcty s expressed as, ΣVn x e v = (3) ΣV n

6 The radus of gyraton of strength, r v, whch s a measure of the dstrbuton of strength n the structure, s defned as, 2 ΣVn x r v = (4) ΣVn where, x s the dstance of the element from the CM. The total mass, M, and the polar moment of nerta of the dstrbuted mass, Im, also affect the dynamc response n asymmetrc structures. These two parameters are consdered through the radus of gyraton of mass, r m, defned as, I m m = M In equaton (5) I m s defned wth respect to the CM. The rato of the radus of gyraton of strength to radus of gyraton of mass, r v /r m, s an mportant parameter that accounts for the system rotaton. The CR of elastc systems s defned as the locaton about whch the frst moment of stffness of the elements s zero. Thus, t s the pont through whch the applcaton of a statc lateral force causes no rotaton. The dstance between the CM and the CR s referred to as the stffness eccentrcty, e r, as shown n fgure 1. Ths eccentrcty s, Σ k x e r = (6) Σ k The system uncoupled translatonal perod, Ty, as expressed by eq. (7), consders the system stffness as the sum of the stffness of all the elements,.e., r (5) m Ty = 2π (7) Σ K Ths defnton of perod mples that the CM and CR are concdent. Thus, no couplng occurs between the translatonal and the rotatonal mode shapes. Effect of the mass rotatonal nerta In case of asymmetrc systems, the mass nerta and the mass rotatonal nerta generate both translaton and rotaton, whereas only translaton s generated n symmetrc systems. The mportance of the mass rotatonal nerta n the dynamc response of the system s explaned by combnng statcs wth a dynamcally nduced torque. SIMPLIFIED APPROACH Only the weght of masonry s consdered n desgn of frames, and no consderaton of the stffenng effect of the walls and parttons are taken nto consderaton. As a result the dstrbuton of the lateral load assumed n the analyss s based on the rgdtes of the bare frame of the bracng bent. Ths s qute dfferent from the actual sesmc load dstrbuton. Research n ths area s more of academc nterest rather beng reflected n practce. System descrpton The n-plane floor daphragm s consdered nfntely rgd and structure s fxed at ground level. Only flexural deformatons are consdered. The elasto-plastc hysteretc rule s used to represent the non-lnear response of the elements. 2D analyss

7 results, but for rregular buldngs, torson requres a complete 3-D nonlnear tme-hstory analyss. Such analyss nvolves great amount of effort and stll perfect results are llusve. A realstc approach to asesmc torsonal analyss should nclude the effect of dynamc magnfcaton of the torsonal moments, the effect of smultaneous acton of the two horzontal components of the ground moton, and accdental torson. The accdental torsonal moments are ntended to account for the possble addtonal torson arsng from the varaton n the relatve rgdtes, uncertan estmates of dead and lve loads, addton of wall panels and parttons after completon of the buldng, varaton of stffness wth tme, and nelastc actons. For smplcty, these effects should be accounted for by use of realstc modfcaton factors. The Indan code uses reserve strength of buldng to meet ts well-defned sesmc resstant desgn as done by codes of other countres. Structure gong n nelastc zone durng strng moton but not collapsng s assumed whle desgn. Ths results n reducton of forces calculated on the bass of elastc structural concepts for desgn. Ths s requred for the purpose of economy. The ductlty ensures safety n case of structure movng n nelastc zone. The man pont to ponder s that all the codal provsons are based on the elastc structural response. Hence the relevance to ths scenaro of current code recommendatons, based on elastc response requres nvestgaton and possble modfcaton. The effort s to search a better but practce- orented smplfed approach. In-plan load dstrbuton Falures n buldngs durng earthquake are often due to nablty of ts elements to work together n resstng lateral force. Ths requrement seeks contnuty n load transfer system,.e., sesmc load path. A contnuous path s necessary for the transmsson of lateral force from one part to the other, so that load s transferred back to sol. Horzontal elements or daphragms that te the structure together and dstrbute all the lateral forces dstrbute lateral forces to vertcal members. For buldngs havng RCC or steel deck as floors, t s assumed that the daphragm s rgd under the acton of the forces n the plane of daphragm but flexble normal to t. Ths assumpton s acceptable as there s no sgnfcant n-plane deformaton for the floor. Hence, forces n frame supportng the rgd daphragm can be dstrbuted by consderng rgd body movement of floor slab under sesmc acton. Ths s true for symmetrc as well as asymmetrc structures. In symmetrc structures, there wll be no rotaton. All the frames wll have same storey drft. Hence, load wll be shared n relaton to ther stffness. Asymmetrc structures wll have translatonal as well as rotaton of the floor. Load sharng wll be done accordngly. Snce stffness comes under study, the queston regardng determnaton of ts value come next. Renhorn has carred hs study n four stages. The load transfer scheme actve durng sesmc acton s essentally a 3-D phenomenon. The sesmc acton, the resultng nerta force and the structure all are 3-dmensonal n nature. Hence t requres 3-D analyss for predcton of the response of buldng under earthquake. But, n smplfed analyses, structural elements parallel to prncpal drecton (frames) are 2-D n character and 2-D analyss s used. Larger the eccentrcty between the centre of rgdty and the centre of mass, larger wll be the torsonal effect. Because of the eccentrcty between load and the resstance, floor experences rotaton along wth translaton. Ths results n enhanced strength and ductlty demand at certan part of the structure. Ths has been ncluded by consderng rgd daphragm under sesmc load. A nonlnear fnte element modelng of 4 storey buldng has been subjected to nonlnear pushover analyss and drft rato between maxmum deformaton and average deformaton have been found to ndcate enhanced sesmc demands for sx cases startng wth no accdental eccentrcty to statc eccentrcty of 0.20 b. The man emphass of analyss s on deformaton capacty. Results are shown n fg 1, fg 2 and table 1. Sx cases for RC MRF buldng: No accdental eccentrcty Wth accdental eccentrcty e s /b = accdental eccentrcty e s /b = accdental eccentrcty e s /b = accdental Earthquake load as per IS Desgn bass earthquake Maxmum consdered earthquake Response modfcaton R=3 Response modfcaton R=5

8 Seres1 Seres2 Fg 1 Maxmum storey drft at Ground storey for the sx eccentrcty stuatons at DBE wth Seres 1(R=5) and Seres 2 (R=3) Seres1 Seres Fg 1 Maxmum storey drft at roof for the sx eccentrcty stuatons at DBE wth Seres 1(R=5) and Seres 2 (R=3) Storey 1 Storey 2 Storey 3 Storey 4 case case case case case

9 case Table 1: Drft rato for all the sx cases CONCLUSION In modern constructons we encounter buldngs, whch exhbt some degree of plan asymmetry. Ths asymmetry can be caused by an uneven dstrbuton of strength, stffness and/or mass. In addton, these structures are supposed to be affected by accdental eccentrcty. Smplfed plane pushover analyss can be used for quck sesmc vulnerablty assessment for low-rse asymmetrc buldng. Storey drft rato ncreases wth ncrease n desgn eccentrcty. Drft raton for dfferent eccentrcty gves ndcaton of the enhanced sesmc demand on the structure. REFERENCES 1. Paulay, T. (1997), Are Exstng Sesmc Torson Provsons Achevng the Desgn Ams?, Earthquake Spectra, vol. 13, no. 2, Moghadam, A.S. & Tso, W.K. (1996), Damage Assessment of Eccentrc Multstory Buldngs usng 3-D Pushover Analyss, Eleventh World Conference on Earthquake Engneerng, Mexco 3. Klar, V. & Fafjar, P. (1997), Smple Push-Over Analyss of Asymmetrc Buldng, Earthquake Engneerng and Structural Dynamcs, Vol.- 26, Bertero, R.D. (1995), Inelastc Torson For Prelmnary Sesmc Desgn, Journal of Structural Engneerng, A.S.C.E., Vol. 121, no. 8, SesmoSoft (2003) "SesmoStruct - A computer program for statc and dynamc nonlnear analyss of framed structures" 6. Chopra A.K. (1995), Dynamcs of Structures: Theory and Applcatons to Earthquake Engneerng, Prentce-Hall. 7. Annger, S. and A.K. Jan (1994), Torsonal Provsons for Asymmetrcal Multstory Buldngs n IS-1893, Internatonal Journal Of Structures, Vol. 14, No IS: , Crtera for Earthquake Resstant Desgn of Structures, B.I.S., Inda. 9. A. K. Snha, & Dr. P. R. Bose (2002), Analytcal Approach n Sesmc Vulnerablty Assessment, 12 th Symposum on Earthquake Engneerng, Pp , Vol-II, Dec , IIT, Roorkee.