Punching of flat slabs: Design example

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1 i Mol Co 00 Pnhing o lat la: Dign ampl Stan Lip, Arlio Mttoni, Migl Frnánz Riz Eol Polthniq Fééral Laann, Switzrlan, 6..0 Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan

2 Bai ata. Gomtr (imnion in [m]) Plan viw Stion trogh la an olmn Span: L 6.00 m an L 5.60 m Sla thin h: Covr onrt : 5 m m. Matrial Th matrial proprti an on in haptr 5 o mol o 00. Conrt C0 Stl B500S (lral an tranvr rinormnt) 0 MPa 5 MPa E 00 GPa g mm Dtilit la B. Loa Sl-wight o onrt la: 6.5 N/m Sprimpo a loa: N/m Liv loa: N/m g + q.5(6.5 + ) + 5.6N/m Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan

Conrt C0 Stl B500S (lral an tranvr rinormnt) 0 MPa 5 MPa E 00 GPa g mm Dtilit la B.

3 Lvl I o approimation (prliminar ign) Th goal o th prliminar ign i to h i th imnion o th trtr ar raonal with rpt to th pnhing har trngth an i pnhing har rinormnt i n. Th ration or in th olmn ar timat ing ontritiv ara. Ration or Innr (C5): 69 N Cornr (C, C): 9 N Eg (C): 65 N (C an C6 ar not govrning N) Th tiv pth v i am to 00 mm. Entriit oiint ( ) ar aopt rom th ommntar o Control primtr Innr: ( π) ( π) mm 0 v π v 00 π Cornr: mm π v 00 π Eg: mm Innr olmn Cornr Colmn Eg Colmn (C5) (C, C) (C, C, C6) Aoring to th ommntar, th itan to th point whr th raial momnt i zro r an timat a on th pan. B ing th Lvl I approah, on an timat th rotation. Th maimm aggrgat iz o mm la to a ator g o g 0.67 < 0.75 g g Rotation r 0.L m r 0.L m,,,. 5 r 0.05 E govrning,. 5 r E g Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan

Entriit oiint ( ) ar aopt rom th ommntar o 7..5. Control primtr Innr: ( π) ( π) + 0.90 60 + 00 50mm 0 v π v 00 π Cornr: 0.65 60 0mm 0 + + π v 00 π Eg: 0.

4 Shar trngth withot har rinormnt Innr: Cornr: R R N, N, 69 N 9N Eg: R N, 65N Th thin o th la ha to inra or th la ha to har rinor. To h i har rinormnt an whih tm an, on an allat th minimal n val o ator. R,ma R, pn on th prorman o th har rinormnt tm. Th mol o propo a val o.0 or tm ompliant with mol o tailing rl ( 7..5.). Highr val (p to.8) an i mor rtritiv tailing rl ar aopt an i th plaing o th tranvr rinormnt i h at th ontrtion it. Shar rinormnt 69 Innr: R, 9 Cornr:. 7 7 R, 65 Eg: Conlion R, Innr olmn: Shar rinormnt i nar an iint (aonting or th val o ) to nr pnhing har trngth Cornr olmn: Shar rinormnt might proal not nar. Thi ha to onirm a highr lvl o approimation. Eg olmn: Shar rinormnt might proal nar. Th thin o th la i iint i har rinormnt i. Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan

Highr val (p to.8) an i mor rtritiv tailing rl ar aopt an i th plaing o th tranvr rinormnt i h at th ontrtion it. Shar rinormnt 69 Innr:. 78 9 R, 9 Cornr:. 7 7 R, 65 Eg:.

5 Lvl II o approimation (tpial ign) Th momnt an th ration or hav n allat with a init lmnt otwar. For th anali, a linar-lati mol ha n. Th momnt M i th vtor aition o th momnt in - an -irtion. M M + M,, For a lvl II approimation, on ha to now th lral rinormnt. It wa ign on th ai o th prvio init lmnt anali.. Strtral anali an lral ign Smmar o th olmn ration Colmn R [N] M,, [Nm] M, [Nm] M [Nm] C 5 C 66 0 C 5 C C C6 6 5 Rinormnt th Th lral trngth an allat aoring to th Mol Co. In thi ampl, howvr, th lral trngth ha n allat aming a rigiplati havior o onrt an tl: ρ ρ m R Flral trngth mm m R 5 Nm/m 0 mm mm m R 69 Nm/m 0 mm mm / mm m R 5 Nm/m 0 mm Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 5

. Strtral anali an lral ign Smmar o th olmn ration Colmn R [N] M,, [Nm] M, [Nm] M [Nm] C 5 C 66 0 C 5 C 5 6 6 C5 66 8 8 C6 6 5 Rinormnt th Th lral trngth an allat aoring to th Mol Co.

6 .5 Shar ign innr olmn C5 Th ign har or i qal to th olmn ration or N min th appli loa within th ontrol primtr (g + q ) A. In a o innr olmn, th ntroi o th olmn orrpon to th ntroi o th ontrol primtr. Thror, Δ 0 Dign har or 0.0 A m v + + π π 0. v ( ) N N g + q A Control primtr 6 M 80 Δ 0 mm ( ) ( ) 06 0 A 5mm π π + π π 6mm 0 v Rotation Th itan r, an r, ar th am a or th Lvl I approimation. r.m r.m,, r r...9m,, m m M Δ 66 8, + +, M Δ 66, + +, Nm/m 8Nm/m r, m E m R, govrning g i allat at Lvl I. r, m E m R, g Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 6

98 + + 5 ( ) ( ) 06 0 A 5mm π π + π 0.98 60 + 0 π 6mm 0 v Rotation Th itan r, an r, ar th am a or th Lvl I approimation. r.m r.m,, r r...9m,, m m M Δ 66 8, + +, 8 8.

7 Th pnhing har trngth o th onrt i not iint. Conqntl, har rinormnt i nar. Firtl, on ha to h i th ign har or i mallr than th maimm pnhing trngth R,ma. Thi i on aming. Th ign har or i low th maimm pnhing trngth R,ma. Thror, th la an rinor with har rinormnt ompling with tailing rl in in la Th on trngth i tan a MPa (aoring to MC 00 or orrgat ar). To avoi a ailr oti th har rinor ara, th otr primtr n to hav a minimal lngth. Th ign har or an r to aont or th loa appli ini th otr primtr. Thi t i nglt a a a timat. In thi ampl, th allating val o th tiv pth v i qal to th tiv pth min th onrt ovr on th ottom ra o th la: mm vot, Aming a irlar ontrol primtr or th timation o th ntriit, ator an timat a tail in th right han i olmn. Poil har rinormnt laot: Pnhing trngth withot har rinormnt R N, Pnhing trngth with har rinormnt σ σ 67 7 N N, 0 R,ma R v R,ma A w w w 7 N 66N E φ 6 w w 5 MPa > 5 MPa w σ w R, inα 0.98 σ w ( 66 67) 5 in( 90 ) ( 90 ) 690 mm A, 779 mm govrning w min inα in v, ot r ot ot 68 π π mm 505mm ρ 8@00@ % w 68 mm + r ot ( ) + A ρ + π π w w v v v v ( 55) ( ) 66N 0.99 A π π w A 6mm > 77mm w ot π 76 mm > 505 mm Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 7

To avoi a ailr oti th har rinor ara, th otr primtr n to hav a minimal lngth. Th ign har or an r to aont or th loa appli ini th otr primtr. Thi t i nglt a a a timat.

8 .6 Shar ign ornr olmn C an C Th ign har or i qal to th olmn ration or N min th appli loa within th ontrol primtr (g + q ) A. Dign har or 0.0 A m 6 6 v v + + π π 0. ( ) N N g + q A Control primtr v Δ Δ v + + Δ Δ 60 0 mm v + + Δ Δ 0mm 6 M 0 Δ 0 97 mm A 08mm π π π v π mm Th itan r, an r, ar th am a or th Lvl I approimation. In a o ornr olmn, th with o th pport trip ma limit th itan r. Rotation r. m r.m,, r r...9m,, r m m govrning M Δ N, + + <, N m M Δ N , + + <, r, m, E m R, 55N govrning Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 8

8 60 0 55 mm 0 + + Th itan r, an r, ar th am a or th Lvl I approimation. In a o ornr olmn, th with o th pport trip ma limit th itan r. Rotation r. m r.m,, r r...9m,, r m 0.

9 g i allat at Lvl I. Th pnhing har trngth o th onrt i iint. Th, no har rinormnt will nar Sin no har rinormnt ha n an m < m R, intgrit rinormnt n to provi. For th ign o th intgrit rinormnt, th aintal loa a an. Th, th ign loa an r. ( g + q ).0(6.5 + ) N/m a ( g + q ) a 0. a, 0 7N g + q 5.6 Th matrial proprti an on in haptr 5 o mol o 00. Dtilit la B : ( t / ).08 It i am that onl traight ar will, th α lt 0. With rpt to intgrit rinormnt, two rtrition hol lill: -th intgrit rinormnt hol at lat ompo o or ar -th iamtr o th intgrit ar ø int ha to hon h that ø int 0. r r m,, E m R, g Pnhing trngth withot har rinormnt N > 0 N R, Intgrit rinormnt A 7 0, a ( ) inα 5.08 in( 0 ) t lt mm N, a int r ø A 5 mm ( in ah irtion with a paing o 00 mm) h φ φ mm r top ottom π π int int + r mm φ int mm 0. r mm Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 9

Dtilit la B : ( t / ).08 It i am that onl traight ar will, th α lt 0.

10 .7 Shar ign g olmn C Th ign har or i qal to th olmn ration or N min th appli loa within th ontrol primtr (g + q ) A. v v v ( + ) v Δ v ( + ) + v v v Δ + v Dign har or 0. A π π 8 8 v v m ( ) N N g + q A Control primtr v v Δ mm Δ 0mm Δ Δ mm 6 M 0 Δ 5mm v 67 0 A 6mm π π π v π mm Th itan r, an r, ar th am a or th Lvl I approimation. In a o g olmn, th with o th pport trip ma limit th itan r. Rotation r. m r.m,, r r...9m,, m govrning, r r,.09m govrning m m M Δ , + +, 5N M Δ , + + N <, 66N Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan 0

+ 0.7 m ( ) 66 5.6 0.7 6 N N g + q A Control primtr + 6 + v v 60 + 6600 + 0 Δ mm + 60 + 0 Δ 0mm Δ Δ mm 6 M 0 Δ 5mm 6 0 0.9 + + 5 6 v 67 0 A 6mm π π π v π 0.

11 r, m, E m R, g i allat at Lvl I. Th pnhing har trngth o th onrt i not iint. Sin th trngth m to rathr lo to th ign loa, a lvl III approimation will prorm. r m,, E m R, g Pnhing trngth withot har rinormnt govrning N < 6N R, Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan

Sin th trngth m to rathr lo to th ign loa, a lvl III approimation will prorm. r m,,. 5 66 0.

12 Th Lvl III allation ar a on th rlt o th linar-lati init lmnt anali. From th rlt o th lral anali, on an otain th itan twn th ntr o th olmn an th point, at whih th ning momnt ar zro. Th avrag momnt in th pport trip an otain th intgration o th momnt at th trip tion. Sin th lral momnt m, an m, at th pport rgion ar ngativ, th aolt val o th twiting momnt m, n to trat o that th aolt val o m, an m, will maimiz. m m m,,, m m m,,, Th pnhing har trngth o th onrt i iint. Th, no har rinormnt will nar Sin no har rinormnt ha n an m < m R, intgrit rinormnt n to provi to prvnt a progriv ollap o th trtr. For th ign o th intgrit rinormnt, th aintal loa a an. Th th ign loa an r. ( g + q ).0(6.5 + ) N/m a ( g + q ) a, a N g + q 5.6 r, -r, m0 r0.6 m m, [Nm/m] m0 r.8 m m, [Nm/m] , Lvl III o approimation (tail ign or amnt o iting trtr) r 0.6m r.8 m,, r r m,, m, r m r, m Nm/m m Nm/m (avrag val on pport trip),, r 0.6m > 0.5m r.8m > 0.5m, r,, r, r, m E m R, r, m E m R, g N > 6N R, Intgrit rinormnt A 7 0, a ( ) inα 5.08 in( 0 ) t lt 06 mm govrning N, a int r ø A 85 mm ( in ah irtion with a paing o 00 mm) h φ φ mm r top ottom Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan

Sin th lral momnt m, an m, at th pport rgion ar ngativ, th aolt val o th twiting momnt m, n to trat o that th aolt val o m, an m, will maimiz. m m m,,, m m m,,, Th pnhing har trngth o th onrt i iint.

13 Th matrial proprti an on in haptr 5 o mol o 00. Dtilit la B : ( t / ).08 It i am that onl traight ar will, th α lt 0. With rpt to intgrit rinormnt, two rtrition hol lill: -th intgrit rinormnt hol at lat ompo o or ar -th iamtr o th intgrit ar ø int ha to hon h that ø int 0. r π π mm int int r φ int mm 0. r mm Cornr o wall hol h ollowing th am mthoolog. Anowlgmnt: Th athor ar vr appriativ o th ontrition o Dr. Jan Sagata Alajar an Dr. La Tainari Th athor wol alo li to than Cartn Sirg (RWTH Aahn, Grman) or th inpnnt h o th ampl h prorm. Lip / Mttoni / Frnánz Riz / Eol Polthniq Fééral Laann, Switzrlan

r π π + 00 + 66 86mm int int r φ int mm 0. r 0. 66 0 mm Cornr o wall hol h ollowing th am mthoolog. Anowlgmnt: Th athor ar vr appriativ o th ontrition o Dr.

5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:

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