homework_01 - 2007 Fall ME 572 Homework #1 HW#1 PARK, PHIL...

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Unformatted text preview: 2007 Fall ME 572 Homework #1 HW#1 PARK, PHIL /P Due Wednesday, September 19, 2007 1. Expand the following expressions and simplify where possible with the range of indices from 1 to 3 (a) 6,75% (b) @6116}, (c) dye)“ (d) 5.7%,: (e) em a3Tjk (f) eljk6j3§k2 2. Given 1 O {ai}= 2 , {bi}= 1 , 0 and [S17]: 2 0 0 1 1 0 1 3 0 O (a) Determine [T5] when T0. = flak (b) Determine {0,} when c,. = e1.ij jk (c) Determine {di} when dk = eijkaibj 3. If Ti]. =—TJ.,. and Si]. = S show that TijSiJ. =0 ii i 4. Constitutive laws can be usually simplified in the following normalized form as A9. = 6,73,“ +30. In order tp invert the relation, we usually follow the following procedure: (a) Deteninine Bkk in terms of AMC 7 (b) Use the results from (a) to show that 1 Bi] = Aij ‘Zél'jAkk 2007 Fall ME 572 Homework #1 PARK, PHILIP g 7. % EXfa/na/ tée affix/B273 Wa/ 3735/5041 / _—_____—————.\ 421e,? fess/:51? wit/z {‘42 rmoge of /;?d//2€S 740271-60 3 , C6) &'§‘k§k1 = cm 3,242 = 8‘ 9687207” 07” $1J‘I6‘k-=\é/c- §wk = 3H m + §m4k + $13 146. j” i=/) w : 5m +3: :4, +355 = m £:/)/<=2.‘§,,I/,2+ 0 + 0 =14; £=/) k=3 5 §HWB+ 0 + 0 : VB 1:2, kz/ : ESQ/l, + 3394/ IéEQI/gl = 144 2:24, #2: 0 +q¥uvn~+ 0 = LL 2:7.) 2'53: 0 +§E143+ 0 zlég X: 3, k=/ 'bgéj/w +§33 I/a/ :VBI I I i i=3] /<:)_~‘ 0 'I’ O +<§33V3L=3g I I I I I I g I (a) 6:3“ éjék. =(gn +§n+833>(é7;+§u+§33) = 3X3 =;/ I Aha/k3: a + 0 +535I/53 V33. w: $Pm7/ar’é ) 5;)” k : §x k/ PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN @>%©M:~ D 2007 Fall ME 572 Homework #1 PARK, PHIL/P 932‘ + gfak + gngzgfi gigem. + §fi3€3m -I fiffilk ~I— Sagas/i: a: 0 PHILIP PARK for eI/ergr k (SB-k =0. sz‘é’g‘k =0-/ CIVIL & ENVIROIUMEI‘ITAL ENGINEERING ~— UNIVERSITY OF MICHIGAN I E I I 2007 Fall ME 572 Homework #1 PARK, PHIL/P E 3‘ FM Qaé/C’m Z 56%;"? [fr GIVE/1 ) / 2 to 2 flax-TI 2 [551 (a) :Dez‘er/m/‘ne LTAji NAM = exjk at. 72:; reg-k4}; -‘= 667 Q/ + Ella). + a; I when 2:)“ “9 QQ‘k—Ta , A \~ 2H— 77; I 8/22; a”; A .— T2;— = E3 613 :O 71/ = Gama?» 2-513 =0 7/3 = 8/3.: a; =‘a3. 2—2 73/ ~ 831; an. 2 a; =2 13’813/al‘a4z/_ E1 :EBA/a4 :‘a/:’/ 1.713;} = 0 0 '3— 0 O / Z —/ 0 (A) :DetEW/We $Cc? (Me/’1 C; :4ij C2 =€gk 5)]t = 8,3125/2 ‘I‘ 81/333 PHELIP PARK 4’ 8321 51/ + 82.25 (823, skew (Symmetrfc ‘1?”507“ / “I” Elba/18w “Fe/652832., CiVlL & E!‘WlRONMEi\‘TAL Eil-JGH‘JEERING I § l i | 1 l 5 U l\!|\../ERS!TY OF P-iiiCHiG/QN 2007 Fall ME 572 Homework #1 PARK, PHILIP g, 54 w = c/ = em 3.23 mafia = a 3—0 =3, 2:2 “ CZ: 82/353 +3231)?» = “AT/5 1333/ = 0+/ =/ i A;3 ‘\ C3:€3/.>.IS/.z +932/53J z (SAL’SJ/ 2 ‘/—’Z 3 ’/ i _ W._,______W__w_____.mj (c) qDete/mMe f dz? Men 06C : 8:7A a; 06/6: 89‘): aibj : Euk 0H b; + 8131:6215; + €111<anb+el3k OLD-E; +€ale31m + 85.2): 3.7, /C=/ 3 d1 = amflbg +63“ EL k=2 3 dL figzayfij + €3|Z a; bl : “Q1133, + afi 13) 2Q: 3 3 0L»; :- 8113 4,53% 82-13 akb! : a* bi- 0L1 ID! at = CgbfdgbL=lxo~Oxz :0 ob. =dgbralbsr-ox0—Jxo =0 d3 =Ce‘bL—0Lzb.: Ix! —2><O :: Yak}: {YT 077 - ‘l 0 / / 2 l _ ___.__———————————————————————_——_—..———j PHiLlP PARK CWIL & ENWRONMENTAL ENGINEERENG Ui\J|‘_/EF;SITY OF WCHlG/W 2007 Fall ME 572 Homework #1 W Fkvéém 3. = *3; mm/ 55‘ = 5);: @Aad #5424 7315:;‘=OI I l = 77mg? 7L ESL-i. 7735/3 +354 +7EZ/SZLFI— 7:3 I v + T31183I'I‘ 733.233.:‘IL ~7—33 l 0) Men 779‘ “343 TI=TA=733=CU 697]“7] =9 271% => Two. \‘\ TISI/+TASLl+—EBE33 = O} \ (2) wéen 274:; ) 2‘73? 67446! 343‘: f»/’ Mar; the Sum/774.1570» 970 symmetric (pM/Dmentj Mei 9‘ ’55 I 731‘ 153* = 55+ (—713) 5:0; I 75%“743‘55 =0 . s H DH 733 UV + ‘FI Q} n ODD PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UI~J|VERSITY OF MICHIGAN 2007 Fall ME 572 Homework #1 PARK, PHIL/P g @nsfftm‘fle 144).? W (Se maria/4L! \S‘M/D/I’fl—é’a/ [a f/IZ, 79/00/7101 momma/7260i 7417271. 45 AA)“ = (99‘ Bkk+ By“ €3~ (I) tflé 74/90/74; frateaéme.‘ (a) fifter/mm 8% 7” fem: 075' Akk CA) W H12 VFW/465 740% (a) i?) 3/7010 fig} I I in order to hem: tée mat/2w , we mud/g 79/04) I Ba“ = A5 — 443:; m .mw- pg [2). ~Q/Mt7'0n (at) ‘Determme 51d: 7/» farm: 47/" 4/41. ° mu/flf/qcz 3);; f0 50% 370%, fig» 0) 3;; A)“ = 5]; $5 Bkk + CS) 89‘ 85c 53) a fire fro/kn? a7” Monaczéer 199/154». SLY/If}: = «42k. —— €314) “r00, 2“ §Jy4jk = 5m + 8;; LI: + élsAgk, x3:/ 3 $71 AIK 43;):le = AW 2:). ‘ 0 +§Lu41k 4— 0 =,4;_/<' 2:3 a + o +§>3243k 2143I<I WEV._._____._*.v—_W / ~_ r\7-~ \\ ’ZIJE/C — PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN ~ ' 2007 Fall ME 572 Homework #1 PARK, PHIL/P ,2 Tm_~’.—fl a fig ~%\(4) to 85:64 Wm 0% %\C3) ,45‘ = ' CS? 35 Bkk = 533% = 3 B/«/ (oj=&,+é‘a+§33 = 3) r M; = 83/ tAerL. 4;- = 3955+ J W i BKK = 3;)“ =<BH+BJJK+ B33 y :9 Wm 74a” I" Akk : :> Bkk = 'éA/(k / I/ (5) M68 f/Lz mac/2’; oyg (A) to 8/704) eg-(l). 6“ : Akk,/ efC?) o/b/Mé, 63(5) 57sz 33.0). / 245 = s:; m + 35 = 513(2- m) + 3‘; PHiLiP PARK CWlL & ENWROMVTENTAL ENGINEERiNG UNEVERSITY OF h’iiCHIGAN / "4 HM #/ , r f x? a Q /y, (/61) Q11” (Ii/(K, 1 / Gulch-31} )' 0'1/7“ 2122+ 033 ‘1 _ 3 r / r .r x K .-’ ,» (/5; .:.L~-:~.z€ = d/‘IO‘M = jinx/A /‘1 v" (c) dire/741 = é'fl :‘ pf J, ' . —\ (d) or; 67% :— é/‘le 1‘ C/ C / . _ = —,r- "7’ (e; 631 Q2 Tik = é3u< + 33w 4"? [>14 +/‘éiika / J 2 /3i& 2 " G / ,1, (r 1,, A ,i : (/31( a: in l '1» f/ 5‘3 pal {13 —/ /;g —” I"), Jr (6324 01342 +/é§/>>QZ by“? Vé/géj "3 T93 I - /’ ’i I '7' '- 11:514) Q; ~ ,n 4 . 2o { ,, . _ 0 6,}‘k0‘j3 1c; 3 Q'jl'CC‘EQGL + ejR—Jfi Cir; 4— QII‘C‘JJ‘Z ‘/‘ J‘ v" I H gm ‘1 Q:j \\ m «2 r, u \ "-. f, 4‘ = éiiiflc auz 5- éITS a3 (Ll/{i}; Q9- 3 .= J /’ 5" II Humjggim C) 'K" v, , :' ivxlaa (f I) ,3‘ : ,3 Mt! ~57Mwutrlc- 23:“ : A)? C WMMW/C fi:~T,, z) 3.!“ch a 77.20. 2 0 TIA 7:5 3 2 N 75‘” 25;; 16.75 - 0 ' ‘ 7’; 7:3 161/ NY}; A31; __ — O r Tu As . SE, A; 28;; 723‘ 93;“ : WEN 1.7742871). +773513-‘I g: Of PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN ~ 7gp { 3 '~‘ (3M3) =Cf “’2 C%'§k4 =€1~AL no ‘ 30 Ce“) fgkdafik : Q; @2127” 7L fiefl +§§fg T3 = 43 (<3me 63;, 7; J = as (77:71!) i :7 Embkm 1‘ , it: F0? 0 l O I §a£SIZQ§ )§\J>»72:g3 7 [5,92] :[2. ’3 03 "OK L z o t) 4 C00 713‘ : 8,54: 5‘11: =7 €314: +61); 0% +' 43 Ti}: TN u— 773’ f“ O 0 “vii? Ta ‘5, 7:5 ’ [ O i 75/ T» 73> 0 «:0 TEL: 01's :0 ...
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