NON LINEAR ANALYSIS OF STRUCTURES ACCORDING TO NEW EUROPEAN DESIGN CODE
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1 October 1-17, 008, Bejng, Chna NON LINEAR ANALYSIS OF SRUCURES ACCORDING O NEW EUROPEAN DESIGN CODE D. Mestrovc 1, D. Czmar and M. Pende 3 1 Professor, Dept. of Structural Engneerng, Faculty of Cvl Engneerng, Zagreb. Croata Assstant, Dept. of Structural Engneerng, Faculty of Cvl Engneerng, Zagreb. Croata 3 Graduate student, Dept. of Structural Engneerng, Faculty of Cvl Engneerng, Zagreb. Croata Emal: czmardean@yahoo.com, dmestrovc@grad.hr ABSRAC : Structures desgned n sesmcally actve regons must comply wth two basc demands: frst, structure must be desgned for loads durng usage (ultmate lmt state and servceablty lmt state) and second, structure must be sound enough to avod collapse durng earthquake (ultmate lmt state). Except from lnear-elastc calculatons, very often are used non-lnear methods. In ths artcle smple plan frame concrete structure wll be analyzed usng N method from Eurocode 8 (EN :004). N s smple non-lnear method used for calculaton of structures durng earthquakes. It combnes mult degree pushover analyss wth spectrum analyss of equvalent sngle degree of freedom (SDOF) system. It s formulated n acceleraton-dsplacement format, whch s very sutable for vsual overvew of basc varables that account for sesmc response of the structure. N method can be consdered as combnaton of pushover analyss and spectrum analyss. Inelastc demand spectrum s obtaned from elastc spectrum. Results obtaned are accurate enough f structure has domnant frst mode of oscllaton. For now, t s used only for plane structures. hs paper wll gve numercal example of N method. It s concluded that nelastc structural response s crucal n earthquake engneerng. Modern methods, supported wth usage of computers and strct desgn codes ensure better understandng of structural response durng earthquakes and at the same tme sesmc resstant structures. KEYWORDS: sesmc analyss, Eurocode, pushover, non lnear analyss 1. INRODUCION Structures desgned n sesmcally actve regons must comply wth two basc demands: frst, structure must be desgned for loads durng usage (ultmate lmt state and servceablty lmt state) and secondly, structure must be sound enough to avod collapse durng earthquake (ultmate lmt state). For purpose of the sesmc calculaton, we can use both lnear elastc structural analyss and non lnear analyss. In European norm EN :004 calculaton methods are gven n part of [1]. hese two methods are as follows: a) Lateral force method of analyss - ths type of analyss may be appled to structures whose response s not sgnfcantly affected by contrbutons from modes of vbraton hgher than the fundamental mode n each prncpal drecton. b) Modal response spectrum analyss whch can be used for all structures whose respone s/or can be sgnfcantly affected by contrbutons from modes of vbraton hgher than the fundamental mode n each prncpal drecton. Need for the non lnear sesmc analyss dates a long tme ago. It was not used prmarly because of nsuffcent computer power, software lmtatons and nsuffcent research n ths feld. As software and computaton power ncreases rapdly, so ths knd of analyss s becomng more and more popular. New generaton of procedures for desgn of the new and rehabltaton of the damaged structures s now avalable (performance based engneerng concept). It s now evdent that durng desgn phase more attenton must be gven to damage control. hs can not
2 October 1-17, 008, Bejng, Chna be effcently ncorporated f non lnear methods are not used. Except of lnear methods mentoned prevously, non lnear are as follows: c) Non-lnear statc (pushover) analyss mentoned s eurocode part d) Non lnear dnamc analyss (tme hstory) In ths artcle smple plan frame concrete structure wll be analyzed usng N method from Eurocode 8 (EN :004). N s smple non-lnear method used for calculaton of structures durng earthquakes. It combnes mult degree pushover analyss wth spectrum analyss of equvalent sngle degree of freedom (SDOF) system. It s formulated n acceleraton-dsplacement format, whch s very sutable for vsual overvew of basc varables that account for sesmc response of the structure. N method can be consdered as combnaton of pushover analyss and spectrum analyss. Inelastc demanded spectrum s obtaned from elastc spectrum. Results obtaned are accurate enough f structure has domnant frst mode of oscllaton. For now, t s used only for plane structures. Results gven here are based on calculatons and research from []. Sesmc load (demand) n N method s defned n the shape of elastc acceleraton spectrum (fgure 1). For better vsualzaton sesmc demand n N method s defned as elastc spectrum n acceleraton-dsplacement format (fgure 1 rght pcture). Fgure 1 Acceleraton spectrum (N method) As ths method s non lnear, nelastc spectrum must be defned. Only two factors are needed: ductlty factor and reducton factor. hs knd of nelastc spectrum n acceleraton-dsplacement format s called demand spectrum. ypcal demand spectrum s shown n fgure. Fgure Inelastc spectrum Structure s modeled as plan frame model wth multple degrees of freedom (MDOF) model. Wth pushover method
3 October 1-17, 008, Bejng, Chna characterstc non lnear force-dsplacement relaton for MDOF can be calculated (usually base shear and dsplacement n hghest pont are used). Usng transformaton factor Γ transfer to equvalent sngle degree of freedom (SDOF) s made. Non lnear force dsplacement relaton s smplfed usng deal elastc plastc relaton as shown n fgure 3. Fgure 3 Idealzaton of force dsplacement relaton As the fnal result capacty dagram n acceleraton dsplacement format s obtaned (see fgure 4). Demand spectrum and capacty dagram are always on the same fgure. Intersecton of radal lne whch corresponds to elastc perod of dealzed blnear system wth elastc demand spectrum ( =1) defnes demand elastc dsplacement S de. Inelastc demand, related to acceleraton S ay and dsplacement S d corresponds to ntersecton of capacty dagram wth demand spectrum (wth demanded ductlty ). For medum and short range perods structures ( C ) rule of equal dsplacement can be appled (demand nelastc dsplacement S d equals to demand elastc dsplacement S de ). Fgure 4 Elastc and demand spectrum n relaton wth capacty dagram pcture above and for < pcture below) ( C C
4 October 1-17, 008, Bejng, Chna. N MEHOD For the purpose of better understandng of N method wll be gven n steps..1. Step 1 Besde the data needed for usual elastc analyss, data concernng non lnear relaton force dsplacement must be gven. Blnear or tr-lnear dagram are often used. Sesmc demand s defned n the shape of elastc spectrum. Spectral acceleraton s gven n relaton to perod ()... Step For elastc system wth sngle degree of freedom followng relaton apples: = Sae (.1) 4π Sde ω Sae = where S ae S de are values from elastc spectrum (acceleraton and dsplacement) for perod and correspondng dampng. For nonlnear SDOF wth blnear relatonshp spectral acceleraton S a and spectral dsplacement S d can be calculated as n Eqns.. and.3: Sae Sa = (.) R = S (.3) S d Sde = Sae = R R 4π 4π a where s ductlty factor (defned as rato between maxmal dsplacement and yeld dsplacement) and reducton factor R. For reducton factor several propostons are gven as n Eqns..4 and.5. R = ( 1) + 1, C C < (.4) R =, C (.5) where C s characterstc perod defned as ntermedate value between short and medum perods (between constant acceleraton and constant velocty). Startng from elastc spectrum and usng equatons defned prevously demand spectrum can be calculated (see fgure 5)..3. Step 3 Pushover s conducted n a way that structure s subjected to monotone lateral loads, whch correspond to nertal forces durng sesmc actons. By gradually ncreasng lateral forces, constructve elements are falng consecutvely and stffness of the structure decreases. Due to lmtaton of ths artcle equatons wll be omtted. All equatons can be found n []. Base shear wll correspond to force and roof dsplacement wll correspond to dsplacement..4. Step 4 Sesmc demand must be obtaned form spectrum whch s usually made for SDOF system. hat mples that
5 October 1-17, 008, Bejng, Chna structure must be modeled as equvalent SDOF system and than ths two model must be compared. For the purpose of obtanng SDOF system varous procedures are avalable. One of them wll be gven n short n ths chapter. Start pont s the equaton of shear buldng (only translatons are taken nto account as n Eqn..6). Fgure 5 Demand spectrums for constant ductlty values m u'' + R= m 1α (.6) In Eqn..6. u and R are vectors (dsplacements and nternal forces), 1 s unt vector whle α. Dampng s not ncluded n ths Eqn. but t wll be ncluded n desgn spectrum. V s shear force (see Eqn..8). Constant Γ can be obtaned as n Eqn..9. Elastc perod of SDOF system ( ) can be calculated as n Eqn..10., where F and D are yeld strength and yeld dsplacement, respectvely. Eqn..11 represents capacty dagram whch s gven as force and equvalent mass rato. V F = (.7) Γ V = P = m Φ m1 Γ = = Φ mφ Φ m1p = p m Φ = p (.8) m Φ m Φ = m m Φ y (.9) y π = π ω = m D y Fy (.10) F S a = (.11) m.5. Step 5 Sesmc demand for equvalent SDOF system can be obtaned graphcally as n fgure 6 for structures wth short and medum perods. Fgure 6 represents demand and capacty spectrum. For structures wth short perods fgure 4 mples. Intersecton of radal lne whch corresponds to elastc perod of dealzed blnear system, wth
6 October 1-17, 008, Bejng, Chna elastc demand, defnes demand acceleraton (strength) S ae for elastc behavor and correspondng elastc dsplacement S de. Acceleraton at yeld strength represents both demand acceleraton and capacty of nelastc system. Fgure 6 Demand and capacty spectrum.6. Steps 6 and 7 Demand dsplacement of SDOF model S d s transformed nto global demanded maxmal dsplacement D t usng transform equatons. hat maxmal dsplacement D t corresponds to target dsplacement (part B of []). Local sesmc demand can also be obtaned by means of pushover analyss. Due to monotoncally ncrease of lateral loads, structure s pushed untl target dsplacement s not reached..7. Step 8 In the last step, predcted behavor of the structure can be estmated by comparson of local and global sesmc demands from chapter 7 wth the capactes of dfferent levels. Global behavor can also be estmated by comparson of dsplacement capacty and dsplacements accordng to sesmc demand. 3. NUMERICAL EXAMPLE As example, response of four story renforced concrete plane frame wll be gven. Structure s subjected to three dfferent earthquakes. Buldng s also tested n European Laboratory for Structural Assessment (ELSA). Results of experment are used to verfy mathematcal model. Structure s desgned accordng to European norms as structure wth peak ground acceleraton of 0,3g. Masses of floors (from frst floor) are 87, 86, 86 and 83 tons. Resultng shear factor has a value of 0,15. Addtonal detals are gven n []. Fgure 7 represents bearng structure. In fgure 8 same curve defne two relatons: V D t for MDOF system and F D rato of equvalent SDOF system. In must be noted that scales are dfferent, however ( Γ = 1, 34 ). Blnear representaton of the pushover curve s represented n fgure 8 (left part of the pcture, see thck lne) and values of strength and dsplacement at yeld pont can be determned: ( F = y 830kN and D y = 6,1 cm ). Elastc perod can be calculated as n Eqn. 3.1: m Dy π = 0,79s (3.1) F = y Capacty dagram (fgure 8, left part) s determned by Eqn. 3.:
7 October 1-17, 008, Bejng, Chna Fy S 830 ay = = = 3,8ms 0, 39g m 17 = (3.) Capacty dagram and demanded spectrum are compared n fgure 8 (rght part of the pcture). Fgure 7 Overvew of the structure Fgure 8 Pushover curve and capacty dagram (left part of the pcture); demand spectrum for 3 dfferent ground acceleratons and capacty dagram (rght part of the pcture) Pushover analyss preformed on MDOF model, wth maxmal dsplacement D t (roof) provdes dsplacements for whole structure, local sesmc demands (n term of relatve floor dsplacements and jont rotatons) as gven n fgure 9. Maxmal rotaton n fgure 9 s,%. Other rotatons are proportonal to maxmal. Resultng envelopes are obtaned by pushover from left to rght and vce versa. Expermental results from ELSA are smlar to ths model.
8 October 1-17, 008, Bejng, Chna Fgure 9 Dsplacements, relatve dsplacements and rotatons of frame 4. CONCLUSION In ths paper overvew of N method s gven. hs s smple non-lnear method used for calculaton of structures durng earthquakes. It combnes mult degree pushover analyss wth spectrum analyss of equvalent sngle degree of freedom (SDOF) system. It s formulated n acceleraton-dsplacement format, whch s very sutable for vsual overvew of basc varables that account for sesmc response of the structure. he nelastc structural response s crucal n earthquake engneerng as all concrete buldngs behave nelastc durng sesmc extaton. Modern methods lke N, supported wth usage of computers and strct desgn codes wll ensure better understandng of structural response durng earthquakes and at the same tme sesmc resstant structures. REFERENCES [1] Eurocode 8: Desgn of Structures for Earthquake Resstance - Part 1: General Rules, Sesmc Actons Rules for Buldngs (004). CEN, Brussels, Belgum. [] Zahenter, E. Non lnear sesmc analyss of structures accordng to European norm EN :004. (006). MS thess, Unversty of Rjeka, Rjeka, Croata. [3] Chopra, A. K. (1995). Dynamcs of Structures, heory and Applcatons to Earthquake Engneerng, New Yersey: Prentce Hall, USA [4] M. Causevc. (001). Earthquake Engneerng, Unversty of Rjeka, Rjeka, Croata. [5] Eurocode : Desgn of concrete structures - Part 1-1: Basc rules and regulatons for buldngs (HRN ENV ). (004), Zagreb, Croata. [6] Radć, J., (006), Concrete structures manual, Croatan Unversty Press, Zagreb, Croata
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