土木学会構造工学論文集(2014.3)

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1 Vol. 6A Numerical stud on ultimate strength of compressive flange considering statistical data for distributions of initial displacement and residual stress * ** *** **** Masato Komuro, Yoshiaki Okui, Katsuoshi Noaka, Takeshi Miashita, Kuniei Nogami, Masatsugu Nagai * komuro@news3.ce.muroran-it.ac.jp ** *** Ph. D **** The ultimate strengths of compressive steel plates simpl supported on three sides and one side free (outstanding plates) were numericall investigated b means of combining FE elasto-plastic analsis and Monte Carlo simulation considering the statistical data for variabilit of the initial displacement and residual stress. From this stud, the following results were obtained: (1) a standard deviation of the ultimate strength tends to increase in the region of the with-thickness ratio parameter ; (2) in the region of <.7, the effect of the residual stress on the ultimate strength is ver small; and (3) considering the statistical data for variabilit of the initial displacement and residual stress of the plate, a rational design formula can be proposed. Ke Words: ultimate strength, steel plate, FE analsis, initial displacement, residual stress 1. 1) 1 2) σ u σ >.7 = b σ 12(1 ν2 s ) t E s π 2 (1) k b t E s ν s k k k =.425 σu /σ.4 JSHB.2 McDermott (1969) Fukumoto et al. (1968) Nishino et al. (1968) )

2 mm 1 mm mm 28 SBHS5 SBHS7 JIS AASHTO LFD 2. Basler & Thurlimann 3) Komatsu & Kitada 4) 5) 6) 7) Basler & Thurlimann 3) σ u /σ = 1 (.45) = 1.53(.45) 1.36 (.45 < < 2) =(1/) 2 ( 2 ) 7) σ u /σ = 1 (.7) =(.7/) 4 (.7 < ) } (2) (3) σ rc w 5) σ u /σ = 1 ( ) = 1 ( β ) β (β 1) 2 ( < ) β = 1 +C( )+ ( ) σ rc C = C 1 C 2 ep C 3 σ C 1 = w b, C 2 = w b, C 3 = w = A Bln ( w b b ) A = 3.71ep ( 13.6 σ rc σ ( B = ep 13.2 σ ) rc σ σ rc σ.5, ) 1 89 w b 1 5 (4) 6) HT785 σ u /σ = 1 ( ) =(/) 3 ( < ) SM4 σ u /σ = 1 } ( 7) =(7/) 1 (7 < ) 2 σ rc σ rc /σ =.23 w w /b = 1/18 } (5) (6)

3 Usami Euler Curve Fukumoto Kitada (HT785) b b σu /σ Basler & Thurlimann a a w JSHB Euler Curve Basler & Thurlimann 3) Kitada (HT785) 6) Kitada (SM4) 6) Usami 5) Fukumoto 7) JSHB Kitada (SM4) α = a / b (a),, : fied = ϖ = w sin ( a ) (b) : free,, : fied : free,, : fied free, : free, : fied (c) 4 3 Euler < 1 Basler & Thurlimann > 1 Basler & Thurlimann < 1 > 1 SM4 > HT w b w /b σ rc σ u σ σ rc /σ σ u /σ σ u (1) (2) (1) (3) (2) σ u /σ (4) (3) (1) σ u /σ (5) (4) σ u /σ μ σ

4 1 ABAQUS (ver. 6.12) 6 12 S4 (6) 11 σ σ rc 6 frequenc N = 367 μ = 1/18 σ = 1/154 Weibull distribution (α =.1339, γ =, m = ) Gamma distribution (m = 2.164, λ = ) W = ma{w, AC ; W, BD } w /b 4) σ σ rc w σ ϖ = w sin ( a ) σ frequenc N = 248 μ =.23 σ =.145 normal distribution σ rc /σ 11) a b t 4 (b) w α(= a/b) 8) α = 2. ABAQUS 9) b 6 a 12 S4 11 ABAQUS 1 5 a σ σ σ rc 1) SBHS5 SBHS w /b 4) 6 μ = 1/18 σ = 1/154 Gamma Weibull Weibull σ rc 7 11) μ =.23 σ =.145

5 2 SM4 SM49 SM49Y SM57 SBHS5 SBHS7 ε/ε ε/ε ε/ε ε/ε ε/ε ε/ε E st /E s SM4 SM SM49Y SM57 SBHS5 SBHS7 5 1 ε/ε t (mm) 1 1 SM4, SM49, SM49Y SM57, SBHS5, SBHS7.4,.5,,.7,,.92, 4, 1.16, 8, 1.4 σ rc /σ, μ σ, μ, μ + σ, μ + 2σ, μ + 3σ w /b μ σ, μ, μ + σ, μ + 2σ, μ + 5σ σ (MPa) SM4 SM49 SM49Y SM57 SBHS5 SBHS7 3 2 Est 1 E s ε Mises 1 2) 2 σ ε E st /E s w/b μ+5σ μ+2σ μ+σ μ μ-σ FEM μ-σ μ μ+σ μ+2σ μ+3σ σ rc / σ 1 μ σ /18 1/ SM4 SM49 SM49Y SM57 SBHS5 SBHS t 1 mm w /b σ rc /σ 1

6 μ = 1/18 σ = 1/154.4 σ rc /σ =.23 w /b = μ-σ.2 w /b = μ SM4, =.4 w /b = μ+2σ δ/δ μ = 1/18 σ = 1/154.4 σ rc /σ =.23 w /b = μ-σ.2 w /b = μ SM4, = w /b = μ+2σ δ/δ μ = 1/18 σ = 1/154.4 σ rc /σ =.23 w /b = μ-σ.2 w /b = μ SM4, = 1.4 w /b = μ+2σ δ/δ (a) =.4 (b) = (c) = SM4 σ rc /σ =.23.4 w /b = 1/18 σ rc =.2 σ rc =.23σ SM4, =.4 σ rc =.52σ δ/δ.4 w /b = 1/18 σ rc =.2 σ rc =.23σ SM4, = σ rc =.52σ δ/δ.4 w /b = 1/18 σ rc =.2 σ rc =.23σ SM4, = 1.4 σ rc =.52σ δ/δ (a) =.4 (b) = (c) = SM4, w /b = 1/18 Mises stress (MPa) (σ) Mises stress (MPa) (σ) Mises stress (MPa) (σ) (a) σ rc /σ = (b) σ rc /σ = μ (c) σ rc /σ = μ + 2σ 13 Mises SM4 = SM4 SM57 3 (1) SM4 11 σ rc /σ =.23 w /b =.4, 1.4 σ σ δ δ w /b 12 w /b = 1/18 σ rc σ rc

7 .4 w /b = 1/18.4 w /b = 1/18.4 w /b = 1/18 σ rc = σ rc = σ rc =.2 σ rc =.23σ.2 σ rc =.23σ.2 σ rc =.23σ SM57, =.4 σ rc =.52σ SM57, = σ rc =.52σ SM57, = 1.4 σ rc =.52σ δ/δ δ/δ δ/δ (a) =.4 (b) = (c) = SM57, w /b = 1/18 μ = 1/18 σ = 1/154.4 σ rc /σ =.23 w /b = μ-σ.2 w /b = μ SM57, =.4 w /b = μ+2σ δ/δ μ = 1/18 σ = 1/154.4 σ rc /σ =.23 w /b = μ-σ.2 w /b = μ SM57, = w /b = μ+2σ δ/δ μ = 1/18 σ = 1/154.4 σ rc /σ =.23 w /b = μ-σ.2 w /b = μ SM57, = 1.4 w /b = μ+2σ δ/δ (a) =.4 (b) = (c) = SM57, σ rc /σ =.23 Strain (εst) Strain (εst) Strain (εst) (a) w /b = μ σ (b) w /b = μ (c) w /b = μ + 2σ 16 SM57 = (a) =.4 Mises σ rc σ = 215 (MPa) (2) SM57 14 w /b = 1/18 σ rc σ u /σ 1 15 σ rc /σ =.23 w /b

8 σu /σ FEM (All Grades) JSHB McDermott 12) Fukumoto et al. 13) Nishino et al. 14) σ rc /σ =.23 w /b = 1/ σ rc /σ =.23, w /b = 1/18 w /b 14 1 =.4 w /b = μ σ δ/δ SM57 SM (a) =.4 (ε st =.126, ε st /ε = 3) (3) 17 2) w /b = 1/18 σ rc /σ =.23 6 εpma / ε εpma / εst SM4 SM49 SM49Y SM57 SBHS5 SBHS ε pma /ε σ rc /σ =.23, w /b = 1/18 SM4 SM49 SM49Y SM57 SBHS5 SBHS ε pma /ε st σ rc /σ =.23, w /b = 1/ w.3 /b =, SM σ rc /σ 2 SM4, = σ rc /σ =.23 w o /b = 1/18 σ u /σ ε pma ε ε pma /ε ε pma /ε ε SM57 SBHS5 <.5 SBHS7 < ε pma /ε =.4 ε 12 SM57 SBHS5 19 ε pma ε st ε pma /ε st SM4, SM49

9 =.4 SM4 SM49 SM49Y SM57 SBHS5 SBHS = SM4 SM49 SM49Y SM57 SBHS5 SBHS = 1.4 SM4 SM49 SM49Y SM57 SBHS5 SBHS w/b (a) = w/b (b) = 21 2 σ rc /σ = w/b (c) = =.4 SM4 SM49 SM49Y SM57 SBHS5 SBHS σ rc/σ (a) = = 5 SM4 SM49 SM49Y SM57 SBHS5 SBHS σ rc/σ (b) = 22 2 w /b = 1/ = 1.4 SM4 SM49 SM49Y SM57 SBHS5 SBHS σ rc/σ (c) = 1.4 SM49Y ε pma /ε st 1 ε pma ε st SM57 SBHS5.5 ε pma /ε st 1 =.4 6 SBHS7 ε st ε ε pma /ε st 1 (.5) 5. n (n > 1) i (i = 1,,n) = f ( 1 n )+ε (7) ε σ rc /σ w /b σ u /σ σ u /σ = p + p 1 + p 1 + p p 11 + p p (8) p p p 3 3 σ rc /σ w /b p,, p FEM 1 w /b σ rc /σ SM4 = σ rc /σ =.23 w /b = 1/18 =.4,, 1.4 2

10 elative Frequenc Weibull dist. α =.1339 γ = m = μ =.52 σ = w /b 23 elative Frequenc Lognormal dist. μ =.23 σ = σ rc /σ 24 μ = μ σ N σ 25 = 4 σ u /σ σ u /σ σ rc /σ w /b =.4 μ = 329 σ = elative Frequenc (a) =.4 = 4.7 μ = 5493 σ = elative Frequenc 26 (c) = =.7.7 μ = σ = elative Frequenc (b) =.7 = 1.4 μ =.7721 σ = elative Frequenc (d) = 1.4 σ u /σ 6.1 σ u /σ σ rc /σ w /b σ rc /σ w /b ) 4) w /b 1/ σ rc /σ = 4 μ σ N σ u /σ μ σ = 4 1 = 4 N = 1 4 N = 1 5 1%

11 σ σu /σ All Grades σ μ μ-2σ.4 JSHB.2 Kitada (HT785) 6) Usami 5) Fukumoto 7) N 1 (N = 1 4 ) =.4,.7, 4, 1.4 =.4 σ u /σ > 27 σ σ 28 μ σ SM4 SM49 SM49Y SM57 SBHS5 SBHS (a) μ.15 SM4.1 SM49 SM49Y.5 SM57 SBHS5.4 SBHS7 1.4 (b) σ 29 >.7.5 < <.7.9 (μ 2σ) >.9 μ 29 σ u /σ μ σ (a) μ <.5 9

12 σ MCS Eq. (14) Eq. (17) esidual stress (σrc/σ) Initial displacement (w /b) σ σ σ (b).5 > σ u /σ Y w /b σ rc /σ X 1, X 2 Y X i Y = g(x 1, X 2 ) (9) FEM Y X i g g Talor Y 2 E(Y ) E(Y ) g(μ X1, μ X2 ) i=1 [ 2 ] g Xi 2 Var(X i ) (1) μ Xi X i Var(X i ) X i 1 Var(Y ) Var(Y ) 2 i=1 E 2 i Var(X i )+ 2 2 i=1 j=1 (11) E i E i E j Cov(X i, X j ) (11) E i = g(μ X 1, μ X2 ) X i (12) Cov(X i, X j ) X i, X j 1 Var(Y ) Var(Y ) 2 i=1 E 2 i Var(X i) (13) g f (g f ) σ Var(Y 2 ( ) g(μx1, μ X2 ) 2 ) Var(X i ) (14) X i i=1 g 2 E(Y ) E(Y )=g(μ X1, μ X2,, μ Xn )+ Y i ++ +Yi 2Y i (15) 8 σ Xi X i Y i Y i = g[ μ X1, μ X2,, μ Xi, ] Y i + = g[ μ X1, μ X2,, (μ Xi + σ Xi ), ] Yi = g[ μ X1, μ X2,, (μ Xi σ Xi ), ] Y i ++ = g[ μ X1, μ X2,, (μ Xi + 2σ Xi ), ] Yi = g[ μ X1, μ X2,, (μ Xi 2σ Xi ), ] (16)

13 4 f N γ μ σ f N γ(5%) γ(3%) γ(1%) Var(Y ) Var(Y ) n i=1 n i=1 ( Y + i ( Y + i Y i 2σ Xi Y i 2 ) 2 Var(X i ) ) 2 (17) (16) 5 Yi FEM μ σ (17) 3 σ MCS (14) (17) 3 31 (17) w /b 27 σ 1 w /b σ rc /σ <.7 σ rc /σ <.7 Nishino & Tall 15) 7. f N γ μ β T σ 32 β T σ μ f σ () β T 32 β T 26 β T f N γ μ β T σ = 1 γ f N (18) (18) μ σ 5%,3% 1 % β T 1.64, f N μ 4 = 4 5% γ = 6 1% γ = 9

14 ) (μ 2σ).9 >.9 2) 3) <.7 4) 5 % 6 1 % 9 1) II 212 2) No. 49, 28. 3) Basler, K., and Thurlimann, B. : Strength of Plate Girders in Bending, Journal of the Structural Division, ASCE, Vol. 87, No. 6, pp , ) Komatsu, S. and Kitada, T. : Statistical Stud on Compression Flange Plates, Journal of Structural Engineering, ASCE, Vol. 19, No. 2, pp , , Vol. 42A, pp , ), Vol. 46A, pp , 2. 7 (199) A , No.314, pp.15-27, ) ABAQUS/Standard user s manual, Ver. 6.12, Dassault Sstèms Simulia Corp., A(CD-OM), Vol. 53, pp , Fukumoto. Y. and Itoh, Y. : Basic Compressive Strength of Steel Plates from Test Data, Proc. of JSCE, No.344/I-1 (Structural Eng./Earthquake Eng.), pp , McDermott, J.F. : Local Buckling of A514 Steel Members, Proc. of ASCE, Paper 6761, , No. 16, pp , Nishino, F., Tall, L., and Okumura, T. : esidual Stress and Torsional Buckling Strength of H and Cruciform Columns, Transactions of the Japan Societ of Civil Engineers, No. 16, pp , Nishino, F., Tall, L. : esidual Stress and Local Buckling Strength of Steel Columns, Proceedings of the Japan Societ of Civil Engineers, No. 172, pp ,

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