Between Square and Circle

DOCTORAL T H E SIS Between Square and Circle A Study on the Behaviour of Polygonal Steel Profiles Under Compression Panagiotis Manoleas Steel Structures

Printed by Luleå University of Technology, Graphic Production 2018 ISSN 12-1544 ISBN 978-91-7790-220-1 (print) ISBN 978-91-7790-221-8 (pdf) Luleå 2018 www.ltu.se

n l t b θ b p b /εt r r a r A p A I I ω x, y, z u, v, w f ε f k σ n n l l Δw f N,, N,,

f y >

n r l n r r

l=3 m l=3 m N b,rd,plt [MN] 3 2 1 500 A B r [mm] c 250 C N b,rd,shl [MN] 3 2 1 500 r [mm] c 250 N b,rd,plt [MN] 3 2 1 l=5 m 500 r [mm] c A C B 250 N b,rd,shl [MN] 3 2 1 l=5 m 500 r [mm] c 250 A = n r N /N

rc 0 0 200 0.75 l =3m 5 1.5 1.75 10 20 30 0.75 l =5m 5 1.5 1.75 10 20 30 2.00 1.75 1.50 5 0 0.75 0 N,, N,, A = σb,rd,shl σb,rd,shl,draft 1.1 l =3m l =5m l =7m 200 0 0 r c 200 0 0 r c 200 0 0 r c

ε p n

0.4 rc 0 0 200 l =3m 2.0 1.6 2.4 10 20 30 2.8 0.4 l =5m 2.0 1.6 2.4 10 20 30 2.8 3.0 2.5 2.0 1.5 N,, N,, n =18 σ = σ f

σ f =1.34 8R, 0.03 < λ 0.3 λ = rπ L f E f R = = b 12 (1 ν 2 ) f σ t Eπ 2 k σ b/t =30 b/t =72.9

σ f = σ f = { 1, R 0.44 7/R, 0.44 <R 1.3 { 1, R 0.44 0.74/R 0.75, 0.44 <R 1.3 n = {8, 12, 16} ( ) σ = f 1+λ 2n 1/n n λ k σ =4

λ = n 12 p =61

n p f

Facet Edge Side Vertice

r n r θ r θ = π n b =2r θ b r b = r θ c

b b c b f b c r b θ r i rp θ b = b 2b b =2(r θ r θ) p = n (b + r θ) p = n b +2πr t r t r = a t

A = p t A = t (n b +2πr ) A = t [2n (r θ r θ)+2πr ] A =2t (n r θ n a t θ + πa t) r A =2πr t A = A 2πr t =2t (n r θ n a t θ + πa t) r = n r θ n a t θ + πa t π r = n π (r θ a t θ)+a t p = b εt

p = b εt p = 2(r θ a t θ) εt 2r θ = p ε +2a t θ r = t θ ( p ε 2 + a θ ) r = n π [ t ( p ε ) ] θ 2 + a θ θ a t θ + a t r = n tp ε + a t 2π ( n p ε ) r = t 2π + a β = t θ b I = n b 3 t 8 b = b + t θ ( 1 3 + 1 ) (1 3β +4β 2 2β 3) 2 θ

N, = Aσ, σ, σ, = k σ, σ π 2 Et 2 σ = 12 (1 ν 2 ) b 2 b N, = Aσ, σ, t σ, =0.5EC r c ( ) ( ) 1.83 2.07 1.36 +,, ω 1.7 w w 2 C = 1,, 1.7 <ω r [ t, 1+ 0.2 ( )] 1 2ωt,, ω > r C r t ω = l rt C C

ω>2.86 r /t b εt =42 d ε 2 t =90

A, = n A + A A =2πr t ρ 1, λ 73 ρ = λ 0.055 (3 + ψ), λ 2 λ > 73 λ f b λ = = σ, 28.4εt k σ k σ =4 N, = A f γ χ λ =0.2

N, = Aσ, σ, = σ, γ σ, = f χ γ =1.1 f χ f λ = σ, 1, ( ) λ λ η λ λ χ = 1 β, λ < λ λ λ α λ, λ 2 λ λ =0.2 α λ = 1 β 2 α = ( ) 1.44 Δw 1+1.91 t β = η =1 Δw

Q =[, 25, 16] Δw = t Q r t χ χ = χ λ λ (χ 1), λ λ, χ λ alpha α =3 α = α α 3 α =0.06 + ( Δw 1+2.7 t ) 5

σ, σ,

u v x y w z w x y w = f(x, y), 0 <x<l, 0 <y<p w(x, 0) = w(x, p ) w(x, 0) x = w(x, p ) x

l, n = p l,, n N n /n n /n =4 n /n f (x) =0.42 (2πx)+0.08 (4πx) 0 x 1

n = n 2 l = 2π n w (θ) = (n θ) l = l n ( 2π ( z l 2 w (z) = n l )) l = l n = n l 2π n = n l 4π ( w (θ, z) =Δw w w f θ ( z ) 2π) f l ( 2π ( )) z l ( 2 =Δw (n θ) n f θ ( z ) l 2π) f l

jxkx AKTH2K2Mi ibqm a jr b a b a - imperfection wave b - windowing l U V ~ i T ii2`mx U#V j. K2b?X 6B;m`2 jxr, SH i2 #m+fhbm; `2H i2/ BKT2`72+iBQMb QM /2+ ;QM H bt2+bk2mx jxkxk.bbiq`ibqm H BKT2`72+iBQMb b /BbiQ`iBQM H BKT2`72+iBQMb `2 mm/2`biqq/ i?qb2 T ii2`mb i? i i` Mbp2`b2 i?2 2/;2b Q7 i?2 _*SaX h?`22 ivt2b Q7 /BbiQ`iBQM H BKT2`72+iBQMb `2 7Q`KmH i2/- /2`Bp2/ 7`QK i?2 i?`22 +QHH Tb2 KQ/2b /2b+`B#2/ T`2pBQmbHvX h?2 bbm;h2 M/ i?2 /Qm#H2 btbhhqp2` ivt2b? p2 #Qi? r p2h2m;i? Q7 9 7 +2ib #mi i?2v? p2 T? b2 /Bz2`2M+2 Q7? H7 7 +2i QM i?2 +B`+mK72`2M+2X at2+b}+ HHv- i?2 /Qm#H2 btbhhqp2` KQ/2-? b +B`+mK72`2MiB H MQ/2b BM i?2 KB//H2 Q7 7 +2ib- bbkbh ` iq i?2 T ii2`m /2b+`B#2/ #v "mhbqm (Rj)X h?2b2 irq KQ/2b + M HbQ #2 +QMbB/2`2/ b `?QK#B+ T ii2`m- bbkbh ` iq i?2 /B KQM/@ b? T2/ +QHH Tb2 Q7 +vhbm/`b+ H b?2hhb- (jj)x n+ = np 4 UjXR9V 2π n+ UjXR8V w+,r (θ) = bbm (n+ θ) UjXReV l+ = lk = w+,k (θ) = bbm n+ θ + π n+ UjXRdV h?2 `2bmHi2/ T ii2`mb `2 BHHmbi` i2/ BM };bx jxk M/ jxj 7Q` i?2 bbm;h2 M/ i?2 /Qm#H2 btbhhqp2` + b2b `2bT2+iBp2HvX

jk AKT2`72+iBQMb a b a b a - imperfection wave b - windowing l U V ~ i T ii2`mx U#V j. K2b?X 6B;m`2 jxk, abm;h2 btbhhqp2` BKT2`72+iBQMb QM /2+ ;QM H bt2+bk2mx a b a b a - imperfection wave b - windowing l U V ~ i T ii2`m 7Q` a>ax U#V j. K2b? Q7 /2+ ;QM H bt2+bk2mx 6B;m`2 jxj,.qm#h2 btbhhqp2` BKT2`72+iBQMb QM /2+ ;QM H bt2+bk2mx

n =2 l = l = 2π n w (θ) = (n θ) a b a b a - imperfection wave b - windowing l

n = n /2 n =2 b /200 b /100 U 0.006 U 0.010 U 0.016 U = Δw l

l =4 rt l =25t

l 150

n p

n p 46 12 2 3 4 = 13248 46 12 2 3 6 = 19872 n p = b εt n p 46 12 = 552 n p E = ε

50 30 0 1200 1000 800 0 10 0 10 20 30 10 1200 1000 800 0 σ 50 30 10 20 30 0.7 0.2 0.3 10 20 30 0.7 0.4 0.3 0.2 σ /σ, n p n

n,p σ σ 50 30 10 1200 800 10 20 30 0 σ 3000 2500 2000 1500 1000 500

9X8X :2QK2i`B+ MQM@HBM2 ` 9j U V np = 10, p+ = 57 U#V np = 24, p+ = 57 U+V np = 10, p+ = 30, U/V np = 24, p+ = 30 6B;m`2 9X9, :L /BbTH +2K2Mi }2H/b i mhibk i2 HQ /X *QHQm` K T p Hm2b BM JS X U V np = 5, p+ = 48, NK t = jj 9yy FL- U#V np = 6, p+ = 51, NK t = jk 8yy FL 6B;m`2 9X8, hrq 2/;2 + b2b Q7 bb;mb}+ MiHv 2H2p i2/ `2bBbi M+2X *QHQm` K T p Hm2b BM JS X

2.4 50 30 2.0 1.6 1.6 1.6 10 20 30 3.0 2.5 2.0 1.5 σ /σ σ σ x, y, z σ

Plate imp. 50 2.7 1.8 2.1 2.1 2.4 1.8 1.5 1.8 Single spillover imp. 2.5 30 1.5 2.0 10 20 30 Double spillover imp. 10 20 30 Multiple spillover imp. 1.5 50 30 10 20 30 10 20 30 σ /σ

1.81.8 1.8 50 30 50 30 2.7 2.1 2.1 2.4 1.5 1.8 Plate imp. 10 20 30 Double spillover imp. 0.4 10 20 30 0.4 1.6 Single spillover imp. 0.4 10 20 30 Multiple spillover imp. 10 20 30 0.4 2.5 2.0 1.5 σ σ 50 30 50 30 1.8 2.42.4 2.12.1 1.5 1.5 1.5 Plate imp. 10 20 30 Double spillover imp. 2.0 1.8 1.5 1.5 10 20 30 1.8 1.5 1.6 Single spillover imp. 10 20 30 Multiple spillover imp. 0.4 10 20 30 2.5 2.0 1.5 σ /σ

n < 7 σ σ /f

93 LmK2`B+ H bbkmh ibqmb U V np = 10, p+ = 30- U#V np = 10, p+ = 57 U+V np = 24, p+ = 30- U/V np = 24, p+ = 57 6B;m`2 9XRy, 1t KTH2b Q7 K tbkmk HQ / bi i2- :JL rbi? aj88x U V np = 10, p+ = 30 U#V np = 10, p+ = 57 U+V np = 24, p+ = 30 U/V np = 24, p+ = 57 6B;m`2 9XRR, 1t KTH2b Q7 K tbkmk HQ / bi i2- :JL rbi? adyyx

50 30 00 S355 75 10 20 30 S700 00 80 10 20 30 0.720 1.1 0.7 σ /f ε λ =0.15

6 50 8 4 0.76 Plate imp. 8 0.72 Single spillover imp. 0.7 0.4 30 50 30 2 Double spillover imp. 8 6 4 0.4 0.48 0.72 8 Multiple spillover imp. 8 4 0.72 0.7 0.4 σ f 50 30 50 30 0.75 5 5 Plate imp. Double spillover imp. 0.7 5 0.7 0.4 Single spillover imp. 0.7 0.4 Multiple spillover imp. 6 8 4 0.72 6 0.3 0.7 0.4 0.3 σ f

n p n p p n n > 30 f

50 0.72 S355, fca 6 0.48 0.4 S700, fca 0.4 30 50 30 50 30 8 4 10 20 30 S355, fcb 4 8 0.72 4 0.4 0.48 10 20 30 S355, fcc σ, f 0.72 4 6 0.4 10 20 30 6 0.48 0.7 10 20 30 S700, fcb 0.7 0.4 0.3 10 20 30 S700, fcc 0.7 0.3 0.4 10 20 30 n =18 0.7 0.4 0.3 n p n p f n p

10 <n < 28 n =18 n =18 n =18 n =18 N,, γ =

1.4 1.61.61.8 N,, γ =1.1 50 30 1.1 1.4 1.4 S355, fca 1.4 10 20 30 S355, fcb 50 1.61.4 1.6 1.6 1.6 1.4 1.1 1.1 1.1 1.4 1.1 S700, fca 1.3 10 20 30 S700, fcb 1.5 1.4 1.1 1.1 1.1 1.6 1.4 30 50 30 2.1 1.5 10 20 30 S355, fcc 2.1 1.8 1.5 10 20 30 1.4 1.4 10 20 30 S700, fcc 1.4 10 20 30 N, N,

00 50 30 50 30 50 30 80 00 1.120 S355, fca 1.120 00 80 10 20 30 S355, fcb 1.120 00 1.120 10 20 30 S355, fcc 00 N, N,, 00 10 20 30 00 1.120 00 80 S700, fca 10 20 30 S700, fcb 1.120 00 10 20 30 S700, fcc 00 00 10 20 30 00 1.120 80 1.800 00 1.3 2.0 1.8 1.6 1.4 S355, fca S700, fca 50 30 50 30 50 30 N, N,, 80 80 00 0.720 10 20 30 S355, fcb 80 00 0.720 0.480 10 20 30 S355, fcc 00 0.720 0.480 10 20 30 80 80 00 0.720 0.480 10 20 30 S700, fcb 80 0.720 0.480 0.0 10 20 30 S700, fcc 0.720 00 0.480 10 20 30 0.0 0.7 0.4

n =20 p =42 σ, f n > 20

100% (double spillover) 20% 80% 0.7 0.4 6 4 0.48 8 0.72 % % 6 4 % % 0.72 0.78 6 0.7 8 0.72 4 4 80% 20% 0.76 0.72 8 0.72 8 4 100 (plate) 0.76 8 0.72 0.4 2 2 σ, f

1.1 1.1 0% 100% 0.7 1.1 1.1 20% 80% 1.3 1.1 2.0 1.8 5 1.1 % % 1.1 1.4 1.4 1.3 80% 20% 5 1.35 1.65 1.5 5 % % 1.1 1.4 1.5 1.3 100% 0% 5 1.35 1.5 1.4 1.5 1.65 1.8 1.6 1.4 N, N,

50 30 10 20 30 fab. class A fab. class B fab. class C eq. (4.1) 10 20 30 n = ξ + p 27, 27 p ζ ζ = (364U 2 5U +0.021) 10 3 ξ =24 0.02f [ Δw U =, Δw ] l l f f n p b /t p ε n p

n 20

n 12 p > n =16 n

n =12 n =25 n p n =[16, 20, 24] p =[30,, 50]

50 30 10 20 30 fca fcb fcc n p t r

750 500 250 0 750 500 Stress, σeng [MPa] 250 0 750 500 250 0 cp3 0.0 2.5 5.0 7.5 10.0 12.5 15.0 Strain, ε eng [%]

f f f

1.5 Perimeter, [mm] 0.0 0 100 200 300 0 500 0 700 Height, z coordinate [mm]

700 0 500 0 z 300 200 100 0 100 50 0 50 x 100 0 50 100 D Fitted circle 100 50 y B2 Scanned points C Fitted plane B1 A

Deviation, w [mm] 0 100 200 300 0 500 0 700 Height, z coordinate [mm]

200 EN 1993-1-5 150 bf w 100 EN 1990-2 50 0 SP1 SP2 SP3 SP4 SP5 SP6 SP7 SP8 SP9 b /Δw b /200 b /100 l =2b 2b 2b l /750

1000 800 Bow imp. limit 0 l f w 0 200 0 SP1 SP2 SP3 SP4 SP5 SP6 SP7 SP8 SP9 Fab. class A Fab. class B Fab. class C l =2b Δw l /2 l /

0 200 0 0 0 200 0 0 original windowed Amplitude, Δw [mm] 0.0 0.4 0.2 original windowed 0.0 0.003 Δw lm 0.002 0.001 0.000 2 10 20 30 2 10 20 30 l Total length to half-wavelength ratio, l m

1500 Original 1250 l f w 1000 750 Bow imp. limit 500 250 0 00 Windowed Fab. classes A, B, C 3000 l f w 2000 1000 0 SP1 SP2 SP3 SP4 SP5 SP6 SP7 SP8 SP9 Bow imp. limit Fab. classes A, B, C l /Δw

welds Specimens 1, 2, 3 02 04 06 08 10 12 14 16 Specimens 4, 5, 6 02 07 12 17 Specimens 7, 8, 9 02 05 08 11 14 17 20 23

3y 1tT2`BK2Mib 6B;m`2 8XR8,.Bb bb2k#h2/ bt?2`b+ H #2 `BM;X 6B;m`2 8XRe, JQmMiBM; bt2+bk2m QM i?2 i2bibm; 7` K2 rbi? i?2 7Q`F HB7i2`X

N f r γ = γ =1.1 σ

f n =16 n =20 n =24 f n =16

0.75 SP3 =16 0 p c =47.5 0.25 0.00 SP6 =20 p c =51.4 SP9 =24 p c =51.8 σ fy Average stress, 0.75 0 0.25 0.00 SP2 =16 p c =37.5 SP5 =20 p c =38.0 SP8 =24 p c =41.1 0.75 SP1 =16 0 p c =27.5 0.25 SP4 =20 p c =27.9 SP7 =24 p c =28.2 0.00 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 Average strain, ε [%] 0.0 0.1 0.2 0.3 0.4

σ p, r, b, N, f n N, N N, N, N, N, σ p, r, b, N, f, N N, N, N, N,

Load, F [KN] 2000 1000 0 12C 12F 0.0 0.1 0.2 0.3 0.4 Strain, ε [%] 02C 02F 0.0 0.1 0.2 0.3 0.4 Strain, ε [%] 1000 500 0 1.6 1.4 N exp N b,rk,plt 10 20 30 N exp N b,rk,shl 10 20 30 110 100 90 80 70 p c 50 30 Bulson Aoki Harraq Godat Anas Manoleas

p 110 n > 20 n 10 n 10

n p f U n κ, n >κ, κ = ξ + p 27, ζ 27 p 4.1, ζ = 7.4, 34.2, ξ =24 0.02f f f

n p f U

50 4 0.72 Plate imp. 8 0.76 Single spillover imp. 0.4 30 8 4 Double spillover imp. 0.7 Multiple spillover imp. 0.7 50 30 0.7 0.4 0.7 0.4 0.4 0.3 σ GMNIA f y

50 30 50 30 0.76 8 Plate imp. 4 0.72 4 8 Double spillover imp. 0.7 0.4 0.3 0.7 Single spillover imp. 0.4 Multiple spillover imp. 0.7 0.3 0.4 0.7 0.4 0.3 σ GMNIA f y 50 30 50 30 5 5 Plate imp. 0.75 5 5 0.7 Double spillover imp. 0.7 0.3 0.4 Single spillover imp. 0.7 0.4 0.3 Multiple spillover imp. 0.7 0.4 0.7 0.4 0.3 σ GMNIA f y

50 30 50 30 0.7 5 Plate imp. 5 0.75 Double spillover imp. 0.7 0.3 0.4 0.75 Single spillover imp. 0.7 0.3 0.4 Multiple spillover imp. 0.75 0.45 0.3 0.7 0.4 0.3 σ GMNIA f y

100% (double spillover) 20% 80% 0.72 0.7 0.4 % % 6 0.4 0.48 0.72 0.7 0.4 % % 4 6 0.7 8 4 8 0.7 5 80% 20% 0.75 5 4 8 0.72 100 (plate) 8 0.76 4 0.4 0.3 σ, f

0.7 100% (double spillover) 0.7 20% 80% 0.4 % % 0.4 0.4 % % 0.7 4 6 0.48 0.7 0.72 8 0.7 5 80% 20% 5 0.75 8 0.72 100 (plate) 4 0.76 4 8 0.4 0.3 σ, f

100% (double spillover) 0.3 20% 80% 0.7 0.4 % % 0.7 0.4 0.72 % % 6 0.4 0.7 0.48 0.7 0.72 80% 20% 4 6 6 0.75 5 8 4 100 (plate) 0.7 5 0.4 6 8 5 0.3 σ, f

100% (double spillover) 0.7 0.4 % % 0.3 20% 80% 0.7 0.3 0.4 % % 0.7 0.4 0.4 0.7 0.7 0.72 80% 20% 6 5 100 (plate) 0.75 5 5 0.7 0.4 8 4 5 0.3 σ, f

100% (double spillover) 0.3 0.7 20% 80% 0.3 0.7 0.4 0.4 0.7 % % 0.3 % % 0.7 0.4 0.7 0.4 8 80% 20% 6 0.72 0.48 4 0.7 100 (plate) 5 5 0.75 0.4 0.3 σ, f

50 30 50 30 50 30 00 50 0% 100% 0.750 00 50 10 20 30 % % 50 00 50 10 20 30 80% 20% 00 00 00 00 1.350 1.800 10 20 30 00 50 00 50 20% 80% 0.750 00 00 50 10 20 30 % % 00 1.350 50 1.500 10 20 30 100% 0% 00 00 10 20 30 2.200 2.000 1.800 1.650 2.25 2.00 1.75 1.50 5 0 0.75 N, N,

00 00 1.350 50 30 50 30 50 30 1.350 00 50 0% 100% 00 0.0 00 0.750 50 10 20 30 % % 50 0.750 1.500 00 50 10 20 30 80% 20% 00 00 1.500 50 00 1.750 2.000 10 20 30 1.350 00 00 50 20% 80% 0.750 10 20 30 % % 50 00 00 1.350 50 00 50 00 1.500 10 20 30 100% 0% 00 00 1.500 50 1.750 2.000 10 20 30 N, N, 2.2502.500 1.650 2.5 2.0 1.5

00 1.350 50 30 1.120 0% 100% 20% 80% 0.720 00 80 10 20 30 % % 50 1.120 00 80 80 80 1.120 80 1.120 00 80 1.120 10 20 30 % % 00 00 1.100 00 00 2.2 2.0 1.8 1.6 1.4 30 50 30 50 10 20 30 80% 20% 50 00 1.650 1.500 10 20 30 1.300 10 20 30 100% 0% 00 00 00 2.000 1.800 10 20 30 N, N,

1.65 1.1 1.3 0% 100% 1.1 1.1 0.7 1.1 1.1 % % 1.1 1.3 1.3 1.3 1.3 1.1 1.1 20% 80% 0.7 1.1 % % 1.3 1.1 1.4 1.6 1.5 2.75 2.50 2.25 2.00 1.75 1.50 80% 20% 1.1 1.1 100% 0% 5 5 1.35 1.35 5 1.5 1.8 5 1.5 1.75 2.02.25 2.5 2.5 0 0.75 N, N,

5 2.0 0% 100% 5 0.75 5 5 20% 80% 0.75 5 5 1.35 3.0 5 % % 0.75 5 1.35 1.35 1.35 5 5 % % 1.35 5 1.5 1.35 1.5 1.65 2.5 2.0 80% 20% 100% 0% 2.8 1.5 1.6 5 1.5 1.5 1.75 2.0 2.0 2.4 N, N,

SP1 SP2 SP3 t =

SP4 SP5 SP7 t =

SP6 SP8 SP9 t =