SIMPLIFIED DESIGN MODEL FOR REINFORCED MASONRY

Transcription

1 1 h Inernaional Brick an Block Masonry Conference Amseram, July 4-7, 004 SIMPLIFIED DESIGN MODEL FOR REINFORCED MASONRY Carl-Alexaner Graubner 1, Chrisian Glock Absrac In regar o an economic esign of reince masonry wo new simplifie esign moels have been evelope. Hereby he varying non-linear maerial behavior of ifferen masonry ypes is being consiere by easy-o-use equaions. The simplifie meho he approximaion of laeral wall emaion accoring o heory of n orer is base on he heoreical moel use he esign of reince concree. Furhermore analyical equaions he imensioning of he require reincemen of masonry srucures are being presene. Accoringly esign iagrams an esign ables are no neee any more in fuure. In consieraion of an easy applicaion of he new mehos he esign of a reince masonry beam an a reince basemen wall is shown. Key Wors reince masonry, require reincemen, simplifie esign, non-linear 1 Inroucion In Germany reince masonry is no as common as in oher European counries oay. Due o he avanages of reincing masonry srucures his is likely o change in fuure. Reincemen oes no only increase he masonry's loa-bearing capaciy bu also enhances i's serviceabiliy, e.g. reuce crack iniiaion. Furhermore he laes rerafs of German an European sanars concerning win an earhquake esign propose increase loas some German regions. Due o higher laeral loaing reincemen will be necessary more ofen. Accoringly he ou-of-ae esign mehos of DIN 105-:1990 have o be replace by new mehos which mee he laes echnical sanars. Consiering he similariy of reince masonry an reince concree wo new simplifie esign moels have been evelope in Darmsa on he basis of he varying non-linear maerial behavior of ifferen masonry ypes (Graubner an Glock 004, Glock an Graubner 00). The simplifie concep he approximaion of laeral wall 1 Prof. Dr.-Ing. Carl-Alexaner Graubner, Darmsa Universiy of Technology, Insiue of Concree an Masonry Srucures, graubner@massivbau.u-armsa.e Dipl.-Ing. Chrisian Glock, Darmsa Universiy of Technology, Insiue of Concree an Masonry Srucures, glock@massivbau.u-armsa.e page 1 1

2 emaion accoring o heory of n orer has been base on DIN :001 an Eurocoe (DIN V ENV :199) reince concree. The new imensioning meho he require reincemen of masonry srucures is base on he non-linear sress isribuion in he cross secion epening on he provie masonry ype. Hereby he reincemen can be calculae on he basis of easy-o-use equaions. Consequenly esign iagrams an esign ables are unnecessary. Due o he general approach he new esign mehos guaranee he consisency of concree an masonry esign. Their applicaion o reince concree is possible, oo. The use of he new mehos is shown he esign of a reince masonry beam an a reince masonry basemen wall. Laeral wall emaion accoring o heory of n orer.1 General Due o he epenence of laeral wall emaion an acing bening momen he srucural analysis of slener walls has o consier effecs accoring o heory of n orer. Hereby he complex non-linear analysis is usually replace by simplifie mehos in various esign sanars where he aiional loa eccenriciy accoring o laeral wall emaion e II an he imperfecion e a is being approximae. Accoringly he esign of slener walls hen can be simplifie as a proof of he loabearing capaciy of he cross secion in he mile par of he wall on he basis of he oal bening momen M II =N e II : e e I e a e II II = e + e + e (1) I a II loa eccenriciy accoring o heory of 1 s orer loa eccenriciy accoring o geomerical imperfecions aiional loa eccenriciy accoring o heory of n orer. DIN 105- an Eurocoe 6 DIN 105-:1990 proposes a simplifie equaion he approximaion of e II common wall slenerness raios (h ef / 0). In case of 0<h ef / 5 DIN 1045:19 reince concree has o be applie. Accoring o DIN 105- he loa eccenriciy e II in he mile hir of he wall is eermine e a + e II, where e a =h ef /00. hef eii = ei + ea + eii = ei + () 46 h ef effecive heigh of he wall consiering he buckling lengh facor wall hickness The laes reraf auf Eurocoe 6 (pren :00) limis he maximum wall slenerness o h ef / 7. The loa eccenriciy accoring o geomerical imperfecion is eermine by e a =h ef /450. The oal loa eccenriciy hen is: e II hef = ei + ea + eii = ei + + () Simplifie approximaion meho A simplifie approximaion meho he laeral wall emaion accoring o heory of n orer has been evelope reince masonry walls (Graubner an Glock 004, Glock an Graubner 00) on he basis of Korina an Quas (00). Due o he page

3 ongoing iscussion abou srain limiaion in German sanarizaion commiees he simplifie equaions are mulae as a funcion of he reincemen's an masonry's ulimae srain. Accoringly he aiional loa eccenriciy e II is: s, max + u eii = h ef wih (4) 10 r r effecive eph of he cross secion s,max maximum olerae srain of reincemen or yiel srain if s,max > yk maximum srain of masonry u In case of limiing he reincemen's srain o he yiel srain s,max = yk =.5 if f yk =500N/mm² an he maximum masonry compression o u =.0 he criical curvaure (1/r) of he cross secion is illusrae in Figure 1. The aiional loa eccenriciy accoring o laeral wall emaion hen is: eii = (5) 00 s,max = yk =.5 u =-.0 Figure 1 Criical curvaure (1/r) of he cross secion Consiering he parial safey facor γ s =1.15 reincemen, he maximum yiel srain has o be limie o y =.5/1.15=.17 wall esign. The ulimae srain of compresse masonry has o regar he provie masonry ype. In case of masonry wih u =.0 (e.g. clay or verically perae calcium silicae bricks) e II is: eii = ( u =.0 ) (6) 400 In case of u =.5 (e.g. soli calcium silicae bricks) e II is: eii = ( u =.5 ) (7) 1750 If he presene approximaion mehos are compare, wo main ifferences urn ou. While he loa eccenriciy accoring o geomeric imperfecion e a is eermine inepenenly of e II in Eurocoe 6 an he new simplifie meho, DIN 105-:1990 (Eq. ()) consiers a fixe loa eccenriciy e a =h ef /00. Furhermore he aiional loa eccenriciy e II accoring o DIN 105-1:1990 an Eurocoe 6 has o be calculae as a funcion of he effecive heigh h ef an he wall hickness. In conras he new concep consiers he effecive eph of reince masonry walls. Figure exemplarily shows he comparison of he loa eccenriciy e a + e II on he basis of a unimly geomeric imperfecion e a =h ef /00 DIN 105-, Eurocoe 6 an he new concep accoring o Equaion (5). Hereby he effecive eph is varie beween he borer case /=1.0 an /=0.5 cenric reincemen accoring o Equaion (5). page

4 The significan influence of he raio / o he aiional loa eccenriciy e II poins ou he imporance of i's consieraion he esign of masonry walls. (e a + e II ) / ( e a =h ef /00) DIN 105- Eurocoe 6 Eq. (5) / = 0.50 Eq. (5) / = 0.60 Eq. (5) / = 0.70 Eq. (5) / = 0.0 Eq. (5) / = 0.90 Eq. (5) / = h ef / Figure Loa eccenriciy e a + e II accoring o ifferen approximaion mehos Require Reincemen.1 General Currenly he require reincemen of masonry an concree srucures is eermine on he basis of esign ables an esign iagrams, which are provie varying maerial behavior. In conras he new esign meho (Graubner an Glock 004, Glock an Graubner 00) makes possible he analyical calculaion of he require reincemen of asymmerically reince masonry an concree srucures. Hereby he reincemen can be calculae as a funcion of he bening momen, he axial loa, he geomeric parameers an he maerial properies of he provie masonry ype. In aiion esign iagrams an esign ables can be generae, oo.. Analyical imensioning meho The require reincemen of masonry srucures is eermine by he equilibrium coniions he cross secion. Hereby he sress-srain relaionship of he reincemen can be assume as linearly elasic wih a consan sress srains greaer han he yiel srain. In conras he maerial behavior of masonry epens on he provie masonry unis an he morar ype. Accoring o DIN 105-:1990 a parabolic sress-srain relaionship is assume. The laes reraf of Eurocoe 6 (pren :00) iffereniaes beween a linear, parabolic, parabolic-recangular an recangular sress-srain relaionship. Consiering he varying maerial behavior of masonry he new imensioning meho is mulae in a general way. Accoringly he require reincemen can be calculae any masonry ype an concree. Hereby he axial loa an he bening momen have o be sanarize accoring o Figure : N s ns = b f (N s is posiive compression loa) () m ss M s = + ns 1 (9) b f page 4 4

5 M N A s M S b Figure Cross secion wih asymmerical reincemen The specific maerial behavior of any masonry ype is being consiere by: δ = α k a R y = y u u = s,max u (10) δ parameer escribing he sress-srain relaionship compression: consan: δ=1.0 parabolic-recangular (as concree): δ= parabolic: δ=1.15 linearly elasic: δ=1. α R raio of resuling axial ce compare o consan sress σ=f k a sanarize locaion of resuling axial ce f compressive srengh of masonry (f =f k /γ m ) u ulimae srain of masonry y esign yiel srain of reincemen maximum olerae srain of reincemen s,max Accoring o he equilibrium coniions he require reincemen area A S is efine as a funcion of he sanarize loas m ss an n s an he maerial parameers: f req As = 1, b (11) f y = 1 1 δ (1) m ss 1 = n s δ k a + 1 y wih (1) s y y = k a 1 k a < k + 1 y a wih 0 < (14) s y For small axial compression loas he reincemen's srain usually is greaer han he yiel srain y, especially when consiering reince concree. In his case Equaion (1)-(1) have o be applie. Due o he sric srain limiaions masonry he compue srain of he reincemen accoring o he simplifie esign equaions occasionally is greaer han he olerable srain ( s > s, max < k a /( u +1)). In his case Equaion (1) is no exac any more, bu he iscrepancies o he exac resul is smaller han 6%. In case of s < y Equaion (14) has o be applie. In erms of an easy applicaion of he new esign meho he general Equaions (1)- (14) are being analyze he specific maerial behavior of ifferen masonry ypes accoring o Eurocoe 6. As a lower limi a linearly elasic sress-srain relaionship is being assume compresse masonry. Accoring o he parial safey facor of page 5 5

6 reincemen he esign yiel srain is y =f yk /(E γ s )=500/( )=.17. The maerial parameers of masonry are: u =.0, α R =1/ an k a =1/. Consiering hese maerial properies he require reincemen can be calculae on he basis of Equaion (11) an he sanarize reincemen raios 1 an : = 1 1 m ss (15) 1 = n s 4 10 = (16) 1 < (17) In case of a parabolic sress-srain relaionship of compresse masonry he imensioning equaions have o be base on he specific maerial parameers u =.0, α R =/ an k a =/. Hereby he sanarize reincemen area is: = (1) m ss 1 = n s 9 1 = (19) < (0) 4 If he sress-srain relaionship of masonry is assume o be parabolic-recangular he following equaions are vali reince concree, oo. The iealizaion of he loa emaion behavior by a parabolic curve.0 an a consan sress.0 <.5 is also use in DIN :001 an Eurocoe. Herby he specific maerial parameers are u =.5, α R =0.1 an k a =0.4: = 1 1 (1) m ss 1 = n s 0, 5 () = < 0. () 4 5 Exemplarily Figure 4 an 5 show esign iagrams he require reincemen on he basis of a linearly elasic an a parabolic-recangular sress-srain relaionship of masonry a effecive eph of =0.9. These iagrams are base on he new imensioning concep. The influence of varying maerial behavior can be seen when comparing he esign iagrams. For he sanarize reincemen area =0. an he axial loa n s =0.1 he resuling sanarize bening momen is m s =0.19. In case of a parabolic-recangular iealizaion of he maerial behavior he bening momen is m s =0.7. Regaring he wie range of m s ifferen sress-srain relaionships he maerial behavior has o be consiere he imensioning of reincemen. page 6 6

7 A s M N 0.5 n s = N s / (b f ) =0 =0.1 s < y =0. b A s = b (f /f y ) / = 0.9 y =.17 u = -.00 f y = 45 N/mm² linearly elasic σ- relaionship 0.05 s > y = m s = M s / (b ² f ) Figure 4 Design iagram /=0.9 an linearly elasic maerial behavior M N 0.60 A s n s = N s / (b f ) s < y parabolicrecangular 0.0 =0 =0.1 =0. σ- relaionship 0.10 s > y =0. = m s = M s / (b ² f ) b A s = b (f /f y ) / = 0.9 y =.17 u = -.50 f y = 45 N/mm² Figure 5 Design iagram /=0.9 an parabolic-recangular maerial behavior 4 Examples he esign of reince masonry srucures 4.1 Reince masonry beam In view of an easy applicaion of he new concep he imensioning of he require reincemen he masonry beam shown in Figure 6 is imensione. Accoring o Eurocoe 6 he require reincemen is calculae on he basis of a linear, parabolic an parabolic-recangular sress-srain relaionship. Masonry: e.g. verically perae clay brick wih general purpose morar f = 1.9 MN/m²; = 40 mm Srucural sysem: l ef.5 m; 0.40 m page 7 7

8 0.49 m.6 m reincemen 0.4 m Figure 6 Masonry beam wih horizonal reincemen Loas: g = 11.0 kn/m; q = 6.5 kn/m M s = 17.0 knm M ss (9) m ss = = = 0. b f Require reincemen on he basis of linearly elasic maerial behavior: (15) = 1 1 m ss = 0. 5 ( >1/) 10 (16),(17) 1 = = 0. 9 = 1 = f (11) A s = b = = 1.4 cm² f 45 Require reincemen on he basis of parabolic maerial behavior: y (1) = m ss = ( <0.75) (19) 1 = = f (11) A s = 1 b = = 1.16 cm² f 45 Require reincemen on he basis of parabolic-recangular maerial behavior: y (1) = 1 1 m ss = ( <0.5) () 1 = = f (11) A s = 1 b = = 1.1 cm² f 45 y Accoring o he presene masonry beam he require reincemen epens on he sress-srain relaionship of masonry an varies from A s =1.1 cm² parabolicrecangular o A s =1.4 cm² linearly elasic maerial behavior. This wie range unerlines he necessiy of applying he new esign meho, which regars he maerial behavior of ifferen masonry ypes. 4. Reince masonry basemen wall In orer o clarify he applicaion of he approximaion meho laeral wall emaion accoring o heory of n orer a basemen masonry wall wih verical page

9 reincemen is being esigne, oo. Hereby axial loas an laeral earh pressure have o be regare. The basemen wall which is illusrae in Figure 7 is verically reince in he cener of he wall m.00 m reincemen Figure 7 Reince masonry basemen wall subjece o laeral earh pressure Masonry: e.g. verically perae calcium silicae brick wih normal morar f = 5.0 MN/m²; =175 mm Srucural sysem: h ef =.00 m; = m Loas: N s = 5.0 kn/m; M s = 7.0 knm/m Loa eccenriciy accoring o heory of 1 s M s orer: ei = = 0.0 m N s hef Loa eccenriciy accoring o geomerical imperfecion: ea = = m 450 Aiional loa eccenriciy accoring o heory of n orer: (6) eii = = 0.04 m 400 (new meho u =,0 ) (7) eii = = m 1750 (new meho u =,5 ) () eii = = 0.06 m 000 (Eurocoe 6) () hef eii = ea 46 = 0.07 m (DIN 105-:1990) The aiional loa eccenriciy accoring o heory of n orer varies beween e II =0.06 m an e II =0.059 m epening on he ifferen maerial behavior an approximaion mehos. Due o he reincemen buil in he mile of he wall he effecs accoring o heory of n orer are uneresimae if he effecive eph is no consiere. Consequenly he oal bening momen is being calculae on he basis of Equaion (6): e II = e I + e a + e II, Eq.(6) = 0.50 m M s,ii =. knm/m page 9 9

10 Require reincemen on he basis of parabolic maerial behavior: N s 0.05 () n s = = = b f M s, II 0.00 (9) mss = + n 1 = = 0, 0 s b f (1) = m = ss (19) 1 = n s = f (11) A s = 1 b = = 1.9 cm²/m f 45 y 5 Conclusion The esign of a reince masonry wall wih axial loas requires a srucural analysis accoring o heory of n orer. In his conex a new approximaion meho, which is base on he heoreical backgroun of DIN :001 an Eurocoe reince concree has been evelope. Hereby he varying maerial behavior of ifferen masonry ypes an he effecive eph is being regare. The require reincemen can be imensione on he basis of he oal bening momen an he axial ce wih new simplifie analyical equaions. Accoringly he use of esign iagrams an esign ables is unnecessary in fuure. Due o he general mulaion of he esign equaions any maerial behavior of masonry can be consiere. Consequenly he esign meho is more economic an ransparen han he exising conceps. Furhermore he esign of reince concree an masonry is consisen, when using he new esign meho, because he ifferen srucural maerials can be regare by specific inpu parameers. The applicaion of his new easy-o-use meho is presene wih he esign of a reince masonry beam an a reince basemen wall. References DIN 1045: 19, Beon un Sahlbeon, Bemessung un Ausführung. DIN : 001, Tragwerke aus Beon, Sahlbeon un Spannbeon: Teil 1: Bemessung un Konsrukion. DIN 105-: 1990, Mauerwerk, Teil : Bewehres Mauerwerk, Berechnung un Ausführung. DIN V ENV : 199, Eurocoe : Planung von Sahlbeon- un Spannbeonragwerken - Teil 1: Grunlagen un Anwenungsregeln für en Hochbau. Glock, C., Graubner, C.-A., 00, Design of reince masonry walls, Darmsa Concree 00, Insiue of Concree an Masonry Srucures, Darmsa, Germany. Graubner, C.-A., Glock, C., 004, Bemessung bewehrer Mauerwerkswäne, Mauerwerk-Kalener 004, Erns & Sohn, Berlin, Germany. Korina, K., Quas, U., 00, Bemessung von schlanken Baueilen für en urch Tragwerksvermungen beeinflussen Grenzzusan er Tragfähigkei - Sabiliäsnachweis, Beon-Kalener 00, Erns & Sohn, Berlin, Germany. pren : 00, Eurocoe 6: Design of masonry srucures par 1-1: Common rules reince an unreince masonry srucures, raf. page 10 10