9 1 9 9.1
9 2 (1) 9.1 9.2 σ a = σ Y FS σ a : σ Y : σ b = M I c = M W FS : M : I : c : = σ b <σ a W :
9 3 (2) P/P Y l/l P Y =Σ 3 i=1σ Yi A i : l : : A 1 = A 2 = A 3 =100cm 2 : 1 3 (P A ) 3 P A = 235N/mm 2 30000mm 2 = 7050000N = 7.05 10 3 kn 1 3 2 (P Y ) 1 3 σ Y1 = 235MPa P Y = 235 20000 + 450 10000 = 9200000N = 9.20 10 3 kn P A /P Y =7.05 10 3 /9.20 10 3 =0.766 A ( l/l) A = σ/e = 235/(2.0 10 5 ) = 1175 10 6 2 ( l/l) B = σ/e = 450/(2.0 10 5 ) = 2250 10 6 σ Y1 = σ Y3 = 235MPa σ Y2 = 450MPa I II III A 1 A 2 A 3 σ Y 1 σ Y 2 σ Y 3 l σ σ y P/P Y 1.0 0.766 A B P l 0 ε y ε 0 1175 2250 l/l ( 10 6 )
9 4 M Y σ Y = M Y I z h/2 z0 σ Y z M =2 σ Y bzdz +2 bzdz z 0 0 z 0 h/2 M P =2 σ Y bzdz M P = σ Y Z Z : (cf. W: ) f = Z W : 0 h 2 9.3
9 5 9.1
9 6 9.2 : : M r = φ f M n M r : φ f : M n : M P (b f /2t f ) p : : F y 345 (MPa) 9.3 : M n = M P M P : = F Y Z F Y : Z : 9.4 : : M n = M r =(F Y F r )W
9 7 9.3 2 (x 1, y 1 ) I xy tan 2α = 1 2 (I x I y ) I 1 = I x cos 2 α + I y sin 2 α 2I xy sin α cos α 2 σ b = M 1 y + M 2 x = M 1 + M 2 I 1 I 2 W 1 W 2 x,y : I 2 = I x sin 2 α + I y cos 2 α +2I xy sin α cos α 9.5 9.6 z
9 8 9.4 9.8 9.7 τ zx = τ xz = VQ Ib V : I : 2 Q : 1 b : τ max = 3 V 2 bd = 3 2 τ ave τ ave = V bd : τ max = 4 V 3 πr = 4 2 3 τ ave τ ave = V πr : 2
9 9 9.5 9.9 H 9.10 9.11
9 10 x σ x x t + τ sx s t =0 s e s s τ sx t =[τ sx t] s=se t σ x s e x ds σ x =(M/I)z τ sx t =[τ sx t] s=se dm 1 s z(tds) dx I s e s dm/dx = V, z(tds) =Q s e τ sx t =[τ sx t] s=se VQ I τ sx = τ xs = 1 t(s) [τ xst] s=se VQ It(s) x 0 τ sx = τ xs =0 S e [τ xs t] s=se =0 τ xs = VQ It 9.12 2 = (τ xs3 t 3 τ xs1 t 1 τ xs2 t 1 )dx =0 τ xs3 t 3 = τ xs1 + τ xs2
9 11 9.1 (a) 0 s 1 b, s 1 = y + b V = dm/dx [τ xs1 ] s1 =0 =0 τ xs1 =[τ xs1 t f ] s1 =0 VQ I ( ) s1 t f ds 1 = V I = V I 0 y b d 2 ( d ) t f dy 2 d 2 t f(y + b) = V I s 1 = b (y =0) τ xs1 t f = V I d 2 t fb (b) 0 s 2 d, s 2 = z + d/2 9.13 τ sx = 1 t [τ sxt] s=se VQ It I = A z2 da 2bt f ( d 2 )2 }{{} + t wd 3 12 }{{} τ xs2 t w =[τ xs2 t w ] s1 =b VQ I [τ xs2 t w ] s2 =0 =[τ xs1 t f ] s1 =b = V d I 2 t fb τ xs2 t w = V d I 2 t fb V z I zt wdz d/2 = V I d 2 t fb V I t w 2 z 2 d2 4
9 12 9.6 σ x = M I z τ xs = 1 t : x z ( [τ xs t] se VQ I ) M xy = yσ xda = M A I yzda A b d M yz = 0 2 τ b d xs 1 t f ds 1 + 0 2 τ xs 3 t f ds 3 = V I y =0 0 b 2 d 2 t f 4 0 : F x,f z,m xz M xy =0 M yz =0 e = M V = b2 d 2 t f 4I 9.14 9.15
9 13 9.4 (z) : (y) : : σ cr = π2 EI z A f L 2 I z = t fb 3 12, A f = t f b ( 2 ) M 0 = ( π ) EIz GJ 1+ π2 EI w L L 2 GJ I w : (9.36) 9.16 GJ :
9 14 (9.36) x,y, z ζ,,η, ξ dw/dx ϕ ζ,,η, ξ M ζ = M cos ϕ M M η = M sin ϕ Mϕ M ξ = M sin(dw/dx) M(dw/dx) ζ d 2 v EI z = M (1) dx2 η d 2 w EI y = Mϕ (2) dx2 ξ GJ dϕ dx EI d 3 ϕ w dx = M dw 3 dx EI z : GJ : (3) EI y : EI w : (3) x (2) EI w d 4 ϕ dx 4 GJ d2 ϕ dx 2 M 2 EI y ϕ = 0 (4) ϕ = C 1 sinh α 1 x + C 2 cosh α 1 x + C 3 sin α 2 x + C 4 cos α 2 x α 1 = α 2 = λ 1 + λ 2 1 +4λ 2 2 λ 1 + λ 2 1 +4λ 2 2 λ 1 = GJ EI w λ 2 = M 2 (EI w )(EI y ) x =0,l ϕ =0( ) d 2 ϕ/dx 2 =0( ) C 1 0+C 2 1+C 3 0+C 4 1=0 C 1 0+C 2 α 2 1 + C 3 0 C 4 α 2 2 =0 C 1 sinh α 1 l + C 2 cosh α 1 l + C 3 sin α 2 l + C 4 cos α 2 l =0 C 1 α 2 1 sinh α 1 l + C 2 α 2 1 cosh α 1 l C 3 α 2 2 sin α 2 l C 4 α 2 2 cos α 2 l =0
9 15 0 1 0 1 0 α1 2 0 α2 2 sinh α 1 l cosh α 1 l sin α 2 l cos α 2 l α1 2 sinh α 1l α1 2 cosh α 1l α2 2 sin α 2l α2 2 cos α 2l 0 α1 2 + α2 2 0 sinh α 1 l cosh α 1 l cos α 2 l sin α 2 l α1 2 sinh α 1l α1 2 cosh α 1l + α2 2 cos α 2l α2 2 sin α 2l =0 =(α1 2 + α2 2 )α2 1 sinh α 1l sin α 2 l +(α1 2 + α2 2 )α2 2 sinh α 1l sin α 2 l =0 =(α1 2 + α2 2 ) [ α1 2 sinh α 1l sin α 2 l + α2 2 sinh α 1l sin α 2 l ] =(α1 2 + α2) 2 2 sinh α 1 l sin α 2 l =0 (α1 2 + α2) 2 0 sinh α 1 l 0 sin α 2 l =0 α 2 l = nπ (n =1, 2, 3, ) σ cr = M cr = π2 E ( W c 4 K l b )2 K =2 (A w /A c 2) = 3+A w /2A c (A w /A c > 2) (α 0.2) σ bu /σ y =1.0 (0.2 <α 2) σ bu /σ y =1.0 0.412(α 0.2) ( 2 <α) σ bu /σ y =1/α 2 M cr = ( ) π EIy GJ 1+ l π2 EI w l 2 GJ α = 2K π l σy b E
9 16 9.1 9.17 500cm I ( SM400) I M cr M u M a π 2 E σ cr = ( 4 2 5000 ) 2 = 308.4N/mm 2 250 M cr = W c σ cr = 6441 10 3 308.4 =1.986 10 9 N mm = 1.986 10 3 kn m α = σ y /σ cr = 235/308.4 =0.873 σ bu = {1.0 0.412(0.873 0.2)} 235 = 169.8 N/mm 2 M u = W c σ bu = 6441 10 3 169.8 =1.094 10 9 N mm = 1.094 10 3 kn m 9.17 I l/b = 5000/250 = 20,A w /A c =1.96 σ bag = 140 2.4(20 4.5) = 102.8 N/mm 2 9.5 I y = 328466cm 4 W y = I y /z c = 328466/51 = 6441cm 3 A w =98 1 = 98cm 2,A c =25 2 = 50cm 2 A w /A c =98/50 = 1.96 < 2 K =2 M a = W c σ bag = 6441 10 3 102.8 =0.662 10 9 N mm = 0.662 10 3 kn m M u /1.7 =0.644 10 3 kn m 9.1 9.1 I SM490Y M cr M u M a
9 17 9.2 9.18 I ( 500cm, SM400) I M u M a W c = I y /z c = 6408cm 3 W t = I y /z t = 5780cm 3 A w /A c = 100/50 = 2 K =2 σ cr = π2 E 4(2 20) 2 = 308.4N/mm2 α = 235/308.4 =0.873 9.17 I σ bu = {1.0 0.412(0.873 0.2)}235 = 169.8 N/mm 2 9.5 l/b = 500/25 = 20, A w /A c =2 σ bag = 140 2.4(20 4.5) = 102.8 N/mm 2 A(cm 2 ) z(cm) Az(cm 3 ) Az 2 (cm 4 )ori U-Flg.pl. 250 20 50 51 2550 130050 1-Web.pl. 1000 10 100 0 0 83333 L-Flg.pl. 200 20 40 51 2040 104040 A=190 510 I = 317423 e = 510/190 = 2.68 cm I y = 317423 190 2.68 2 = 316058 cm 4 z c =52.0 2.68 = 49.32 cm z c =52.0+2.68 = 54.68 cm M bu = W c σ bu =1.088 10 3 kn m M y = W c σ y =1.358 10 3 kn m M u =1.088 10 3 kn m M bag = W c σ bag =0.659 10 3 kn m M ta = W t σ ta = 5780 10 3 140 =0.809 10 3 kn m M a =0.659 10 3 kn m