Punching of flat slabs: Design example

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Trevor Long
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1 fi Mol Co 010 Pnhing of flat la: Dign apl Stfan Lip, Arlio Mttoni, Migl Frnánz Riz Eol Polthniq Fééral Laann, Switzrlan, Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 1

2 1 Bai ata 1.1 Gotr (inion in []) Plan viw Stion trogh la an oln Span: L = 6.00 an L = 5.60 Sla thin h: Covr onrt : 5 1. Matrial Th atrial proprti an fon in haptr 5 of ol o 010. Conrt C0 Stl B500S (flral an tranvr rinfornt) f 0 MPa f 5 MPa E 00 GPa g Dtilit la B 1. Loa Slf-wight of onrt la: 6.5 N/ Spripo a loa: N/ Liv loa: N/ g + q = 1.5(6.5 + ) + = 15.6N/ Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan

3 Lvl I of approiation (prliinar ign) Th goal of th prliinar ign i to h if th inion of th trtr ar raonal with rpt to th pnhing har trngth an if pnhing har rinfornt i n. Th ration for in th oln ar tiat ing ontritiv ara. Ration for Innr (C5): 69 N Cornr (C1, C): 9 N Eg (C): 65 N (C an C6 ar not govrning N) Th fftiv pth v i a to 00. Entriit offiint ( ) ar aopt fro th ontar of Control pritr Innr: ( ) ( ) = + π = π = v π v 00 π Cornr: = = + = π v 00 π Eg: = = + = Innr oln Cornr Coln Eg Coln (C5) (C1, C) (C, C, C6) =0.9 =0.65 =0.7 Aoring to th ontar, th itan to th point whr th raial ont i zro r an tiat a on th pan. B ing th Lvl I approah, on an tiat th rotation. Th ai aggrgat iz of la to a fator g of g 8 8 = = = g Rotation r = 0.L = = 1. r = 0.L = = 1.,, =, 1. 5 r f E = = govrning =, 1. 5 r f E = = = = = g Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan

4 Shar trngth withot har rinfornt Innr: Cornr: f 0 = = = 7 N < = 69 N R, 0 v f 0 = = = 80 N < = 9N R, 0 v Eg: f 0 = = = 10 N < = 65 N R, 0 v Th thin of th la ha to inra or th la ha to har rinfor. To h if har rinfornt an whih t an, on an allat th inial n val of fator. f = = R,a 0 v f R, 0 v pn on th prforan of th har rinfornt t. Th ol o propo a val of =.0 for t opliant with ol o tailing rl ( ). Highr val a if th ar printall vrifi. Shar rinfornt Innr: Cornr: Eg: 69 = =.5 7 R, 9 = = R, 65 = = R, Conlion Innr oln: Shar rinfornt i rtainl nar. It ight that a pial har rinfornt t ha to in fator n to largr than th val propo in th ol o ( =.0). Thi ha to h ing a highr lvl of approiation. Cornr oln: Shar rinfornt ight proal not nar. Thi ha to onfir a highr lvl of approiation. Eg oln: Shar rinfornt ight proal nar. Th thin of th la i ffiint if har rinfornt i. Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan

5 Lvl II of approiation (tpial ign) Th ont an th ration for hav n allat with a finit lnt oftwar. For th anali, a linar-lati ol ha n. Th ont M i th vtor aition of th ont in - an -irtion. M = M + M,, For a lvl II approiation, on ha to now th flral rinfornt. It wa ign on th ai of th prvio finit lnt anali.. Strtral anali an flral ign Sar of th oln ration Coln R [N] M,, [N] M, [N] M [N] C11 5 C 66 0 C 11 5 C C C6 6 5 Rinfornt th Th flral trngth an allat aoring to th Mol Co. In thi apl, howvr, th flral trngth ha n allat aing a rigiplati havior of onrt an tl: ρ f = ρ 1 f f R Flral trngth R = 5 N/ = 10 R = 69 N/ = 10 / R = 115 N/ = 0 Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 5

6 .5 Shar ign innr oln C5 Th ign har for i qal to th oln ration for N in th appli loa within th ontrol pritr (g + q ) A. In a of innr oln, th ntroi of th oln orrpon to th ntroi of th ontrol pritr. Thrfor, Δ = 0 Dign har for 0.0 A v = + + π = π = 0.1 v ( ) N = N g + q A = = Control pritr 6 M 810 = Δ = 0 = = = = ( ) ( ) A 51 = π = π = = = + π = π = v Rotation Th itan r, an r, ar th a a for th Lvl I approiation. r = 1., r r = 1., r = 1. = 1.91 = 1.,, M Δ 661 8, = + = + =, M Δ 66, = + = + =, N/ 8N/ r f, = = = E R, govrning g i allat at Lvl I. r f, = = = E R, = = = 0. < g Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 6

7 Th pnhing har trngth of th onrt i not ffiint. Conqntl, har rinfornt i nar. Firtl, on ha to h if th ign har for i allr than th ai pnhing trngth R,a. Thi i on aing =. Th ign har for i low th ai pnhing trngth R,a. Thrfor, th la an rinfor with har rinfornt opling with tailing rl fin in la To avoi a failr oti th har rinfor ara, th otr pritr n to hav a inial lngth. Th ign har for an r to aont for th loa appli ini th otr pritr. Thi fft i nglt a a af tiat. In thi apl, th allating val of th fftiv pth v i qal to th fftiv pth in th onrt ovr on th otto rfa of th la. = = 0 0 = 17 vot, Aing a irlar ontrol pritr for th tiation of th ntriit, fator an tiat a tail in th right han i oln. Poil har rinfornt laot: Pnhing trngth withot har rinfornt f 0 = = = 90 N < = 661N R, 0 v Pnhing trngth with har rinfornt f = = 90 = 781N = 1N R,a R, 0 v = 781N = 661N R,a σ A A w w E = = = MPa > f = 5MPa w 6 6 ( ) 10 ( ) = = = 66 σ inα in 90 w,in w R, = = = 77 f in α in 90 w ( ) ( ) = = = 58 0 f vot, r ot ot = π = π = 0 58 = = = ρ = 8@100@100 = 0.50% w ot govrning = r = = ( ) ( ) A = ρ + π π w w v v v v A = π ( 0.5 0) π w A = 16 > 77 w ot = π = 7 > 96 Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 7

8 .6 Shar ign ornr oln C1 an C Th ign har for i qal to th oln ration for N in th appli loa within th ontrol pritr (g + q ) A. Dign har for 0.10 A v v = + + π = π = 0.1 ( ) N = N g + q A = = Control pritr v 1 Δ = Δ = v + = Δ = Δ = v + = + = Δ = Δ = 1 = 0 6 M 10 = Δ = 0 = A 08 = π = π = = = = π v π = = = + = Th itan r, an r, ar th a a for th Lvl I approiation. In a of ornr oln, th with of th pport trip a liit th itan r. Rotation r = 1., r r r = 1., = = 1.91 = r,, = = 0.6 = 0.5 govrning M Δ , = + = + = 1N < = =, N M Δ , = + = + = 5N < = =, 55N r f,, = = = E R, govrning Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 8

9 g i allat at Lvl I. Th pnhing har trngth of th onrt i ffiint. Th, no har rinfornt will nar Sin no har rinfornt ha n an < R, intgrit rinfornt n to provi. For th ign of th intgrit rinfornt, th aintal loa a an. Th, th ign loa an r. ( g + q ) = 1.0(6.5 + ) = 10.1N/ a ( g + q ) a, a 10.1 = = 110 = 71N g + q 15.6 Th atrial proprti an fon in haptr 5 of ol o 010. Dtilit la B : (f t /f ) = 1.08 an ε = 0.05 It i a that onl traight ar will, th α =0. With rpt to intgrit rinfornt, two rtrition hol flfill: -th intgrit rinfornt hol at lat opo of for ar -th iatr of th intgrit ar ø int ha to hon h that ø int 0.1 r r f,, = = = E R, = = = 0.0 < g Pnhing trngth withot har rinfornt f 0 = = = > R 0 v Intgrit rinfornt N 110 N A = = = 916 oα o( 0) f ( f / f ) 1 t ε a, ø0 A = 157 ( in ah irtion) ( ) ( ) 0.1 = 0.1 h φ φ = = 19 0 r top otto Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 9

10 .7 Shar ign g oln C Th ign har for i qal to th oln ration for N in th appli loa within th ontrol pritr (g + q ) A. 1 v v v ( + ) v Δ = v ( + ) + v v v Δ = + v Dign har for 0.1 A π π 8 8 v v = + + = = 0.17 ( ) N = N g + q A = = Control pritr v v Δ = = = Δ = 0 v Δ =Δ = 1 6 M 10 = Δ = 1 = = = = A 61 = π = π = π v π = = = + = Th itan r, an r, ar th a a for th Lvl I approiation. In a of g oln, th with of th pport trip a liit th itan r. Rotation r = 1.,,, r = 1., = r r = = 1.91 = = 0.6 = 0.78 govrning, r = + = + = r, 1.09 govrning M Δ , = + = + =, 5N M Δ , = + = + = N < = =, 66N Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 10

11 r f,, = = = E R, g i allat at Lvl I. Th pnhing har trngth of th onrt i not ffiint. Sin th trngth to rathr lo to th ign loa, a lvl III approiation will prfor. r f,, = = = E R, = = = 0.7 < g Pnhing trngth withot har rinfornt f 0 = = = < R 0 v N 6N govrning Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 11

12 Th Lvl III allation ar a on th rlt of th linar-lati finit lnt anali. Fro th rlt of th flral anali, on an otain th itan twn th ntr of th oln an th point, at whih th ning ont ar zro. Th avrag ont in th pport trip an otain th intgration of th ont at th trip tion. Sin th flral ont, an, at th pport rgion ar ngativ, th aolt val of th twiting ont, n to trat o that th aolt val of, an, will aiiz. =,,, =,,, Th pnhing har trngth of th onrt i ffiint. Th, no har rinfornt will nar Sin no har rinfornt ha n an < R, intgrit rinfornt n to provi to prvnt a progriv ollap of th trtr. For th ign of th intgrit rinfornt, th aintal loa a an. Th th ign loa an r. ( g + q ) = 1.0(6.5 + ) = 10.1N/ a ( g + q ) a, a 10.1 = = 6 = 171N g + q 15.6 r, -r, =0 r= =0 r=1.18, [N/] ,, [N/] Lvl III of approiation (tail ign or ant of iting trtr) r = 0.6,,, r = 1.18, = r r = = 1.0 = = 0.6 = 0.78 r, = + = + = 0.78 r, = N/ = N/ (avrag val on pport trip),, r = 0.6 > = 0.5 r = 1.18 > = 0.5, r,, r, r f, E R, = = = r f, E R, = = = = = = 0.1 < g f 0 = = = > R 0 Intgrit rinfornt N 6N govrning A = = = oα o( 0) f ( f / f ) 1 t ε a, 19 ø0 + ø16 A = 88 ( in ah irtion) ( ) ( ) 0.1 = 0.1 h φ φ = = 19 0 r top otto Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 1

13 Th atrial proprti an fon in haptr 5 of ol o 010. Dtilit la B : (f t /f ) = 1.08 an ε = 0.05 It i a that onl traight ar will, th α = 0. With rpt to intgrit rinfornt, two rtrition hol flfill: -th intgrit rinfornt hol at lat opo of for ar -th iatr of th intgrit ar ø int ha to hon h that ø int 0.1 r Cornr of wall hol h following th a thoolog. Anowlgnt: Th athor ar vr appriativ of th ontrition of Dr. Jan Sagata Alajar an La Tainari Th athor wol alo li to than Cartn Sirg (RWTH Aahn, Gran) for th inpnnt h of th apl h prfor. Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 1

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