homework_09

# homework_09 - J HW#9 1 Phi{o[Dark Due Date December 3 2007...

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Unformatted text preview: J HW#9 1. Phi/{o [Dark Due Date: December 3, 2007 crack for one cross section as shown. The structural member is made of A572 structural steel with K ,6 = 38 MPax/m for operation at —30°C. If the safety factor for the structural member against fracture. is assigned to be 1.5, determine the maximum allowable bending moment about the x axis as shown. The moment of inertia with respect to the x axis is 1.29 x104‘m4 . "at: tgmm I3,\ mm we; 1 / > f v 1 CVQOK X, 303mm 7-5 mm—" (Lag Mm 2. Consider a double cantilever beam specimen with a thickness of B, a height of h, and a crack length of a as shown. The specimen is subject to the bending moment M as shown. Derive the stress intensity factor K , as a function of M and the given geometric parameters. Assume ~h is much smaller than a. The loading and the geometry are symmetric with respect to the crack line. The material is assumed to be - linear elastic. ‘ 3. The J integral can be expressed as l 3U J=—— — B[aa)A where B is the thickness of the specimen, a is the crack length, U is the strain energy and A is the displacement. Consider a three-point bend specimen as shown. The load-displacement curves are idealized to be elastic for three neighboring crack lengths as shown. The Young’s modulus E for the material is 200 GPa. Here; P represents the load and w is the width of the specimen. For the specimen, w ~= 50 mm and B = 20 mm. ‘ ' (a) Estimate the value of J for a/ w = 0.6 at A = 1 mm (b) Estimate the stress intensity factor K at A = 1 mm. Assume plane stress conditions. ' ' MEMZ HM? PA" M ./, 7 ,.%é/ém f l— Lem (under Fare LemoIan, . 3 Inc: /.Z?X/0_¢m 4“ Mater/’44 AMZ. K15: 33> MPa/Z— m: ~30°C Assume w 5mg? 743213;» /I§‘. ' 2991297271122, fie max/771m az/ékmé/e 5975/7/1qu Wa/zE/rt. ° Stress M W Wafer 7, Pure Maﬁa Ud't—‘g ==> DEM=531WM 5W“ : 3:3 mm =/s—/,;Mm= a./5”/fm , . waﬁd WmX/Mf/E” F61): /, A2. ' f S 4 /3‘ - Afpm/Wwbn 74;” W crack bugLé/L. ' I ' “prowl/503 wows/L :=> R00 Vaﬂués mammrmd ﬁeuﬁaI/owma I” , PHILIP PARK CIVIL 8: ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN Fa) =a,2«5§(/—DZ)‘L+ meatfon o4) {HOD Mtk a/b FQ)=H2 acceFﬁLUe (EM—L. C(0%9¥’Y~m”>’ Cmaé {mod/L dz/fmm 5:203 mm. % 4/5 =20792’ yam?) = (,/¢§f)//z,(w 3420421 r/) we AMJK‘. (More ansermtﬁ/e D 0 231779;? ﬁfe/257% 741%)". I KI = F53) 07446 Ed”: wézre J ” HM t/az d/oMaA/e bend/ff)? moment. 54‘? X K1 S KIC 7 /9 117(4) me/T’La < 300M315» wwwyr mm x/Zxoo/I < 38 Mai»? EI£MM \$1? 0.1% MEL 0/57!- y M— < WDML X—-L 7—5472» %X “74—9; X/0‘%.\X 51mm 0943/7 N—m. = 9543/? k/V—m. <M)a//aw4£fg = q“ {45/7 MAM. / PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN M15502 HIM? W?“ Bark. ﬂ 3: B 3 ‘Ml’c'éﬂes‘g _ kﬁn, 293N742 z‘ée stregs {My}? 7%660)’ K1, 3/? M (J::EM . I , a Su'wnwre I °Pofenﬂa€ @gg_ ¢. GIT/en M ‘9 [m1 (britm/ ¢=M<9f= ¢=_._M_av- El h? ° Eneri; @5458 @1714 Q ~ /a<é t a R M“ Q ’ 2.13%" ° Stress ﬁfe/4:77 @dz)’ ) Afr. KZI=EGI VKP/a/ﬂest‘ress) 7gp .swaﬂ B) L KI z/f-LGr (fW 5374/77) 760’” W 8) 1% /C’L” /2W _ D—M; ‘-£XEEP ’ RM M __3 KI =25—k 2 (PM mg) 5 . ——‘—_ff PHILIP PARK CIVIL 81 ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN M5372 HM? P/r’ Burk. .3. @é/em 3, 722 J, @ﬁgmfé am .49 375/6er 4.97 =—— <33) , ,,,B t tIIFcknagg 7 a? crack bath. ' M: strain energg ' AI aITslbéLce mmt .52200 6104 GIUZSIWMML Bngmm.., £le wogx MOI (AI—ML). "j = MK. M333; M (4/4} :0‘60) A— : ~I~WX20X/@3Xa00/3 (MPM) 2 =3 /0 N—m . 'I VMrM. “(4/0 :ﬂ,o’/> IA=IMM ; r 6/; ﬂ/ng AIAZ*&I§— Npm~ PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN (I?) Est/777m? d2, gifts: sz‘ensFtéL 7&5?”- I16 42‘ A =/mm 745mm 2/4412. «stress a)? ftfajys, , , 7 j: GI: g2. (f/d/me. SW85; KIA = EQJ’. PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN .- ‘ , ,v_ f5 ‘ 7:01’ (Ed/33f, Cir—(1:15. (f< 015) t} /<:/_/2‘0’\q7:a/ = An.» «Lg-7:5: 7 .L. I 7...!) ’43:}— 50.“![6/ 711564;, (5‘) I! [Lt/(LLCLO/ /l40. r—“ r“ *7 M: <. .,_ 33m ‘ [IlfélU/ﬁur /‘/2_></.r’x z" - 2 (ﬁx. .zuL” a.“ (3...?! a 0. '70} \i 7bXD. 0/1" _ l .-' . , ' ‘s V!“ L'- Le. 1-14 on«.c=.;_ Eff-j a. -—- c-l ’—-- — f .305 U 10/973,149 M9-—:M¢9_ _ 1:14 E1313 /' L ;( cow: c/ 4/7. = i :2/(' 7:15.01 L. 2' ' ~ 1 (AMA :le0.01Qxlo— oogi’EOxlo-O)(lx\$ 03) ~:-g,5 ”-m, a . AOL = AWN 'w :(0. {31-0 soucsomoa) = 3x15“ m. é 3= -4 , (-05) =_ 5x16“ N/m. ﬂoxla‘3 S 200‘“ Z (1;) Fay piomé s’mm (anti/khan. ’ KI = ES. . . a . ~. _- ._. § Iyr, ‘ -- ‘ bmq Ca 3 5310 NI . _ aim E:.m30 69:.- 2 q‘ A .. K: = «9510 K“) ASXK) 15 7 L =_ 1x10 B; --m, .K.r : 108 P0152“ = {00 HPav'Yﬁ ', ' HW#7 Prbb’gmi L—A @— / ,, ,. /" , ~anf'te, width . I i PHHJP PARK CW”. 8 ENVIRONMENTAL ENGINEERING UNIVERSETY OF MICHIGAN ...
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