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MT1_Solutions

MT1_Solutions - 3U tu'i‘(to ﬁt y ME 423 Intermediate...

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Unformatted text preview: 3U tu'i‘ (to. ﬁt y ME 423 Intermediate Mechanics of Deformable Solids Fall, 2005 Midterm 1 Note: This is a closed-book and closed-note examination. A formula sheet is provided. All questions must be answered in the space provided under each problem (use the back of the sheet, if necessary). Pledge: l have neither given nor received any unauthorized assistance on this exam. Signature: (20) 1. Due to applied loads, the square plate shown assumes the conﬁguration denoted by the dotted lines (not drawn to scale). (:1) Compute the strains 5,, 5),, 1/1}, in the deformed body, assuming inﬁnitesimal strain theory and a plane strain condition. (b) [)0 the strains in the body satisfy the small strain assumptions? Explain. a: ﬂu + QIX 4-day + 62?)”, “(010)20 => 80:0 u(1,o)=a.00+ ~r——> 0,:0.004 u(0,r)=‘0.oo.¥ =—_—> 02:9,”; 0mm): 0.0r2 : 0.004+0.oo£+03 V(‘10): 0-025 @éf:0v02£ V(0,I): 0.1-0} @151": 0.40.? 1 I L51 ELL/— ' :2 9%, M +6)! P 914%,, 0.034 (5) N0. {y?0.403’ IJ‘ MC’T I'MALL COMPM-ED TD OWE. (20) 2. At a point in a loaded structure, the principal stresses are found to be: 0'I =121MPa 0'2 : 20MPa 0'3 = JilMPa With respect to a Cartesian (xyz) coordinate system that does not correspond to the principal directions, the stress state is: 100 I” O [0']: I” —50 0 Mind 0 0 0': Determine a: and 1”. (ﬁg {Mt/MIANTJ‘.‘ I ._J/X+G’y+0/:=l :014’0/11—0/3 f /00— 5‘0 +52 H [7(+20_7/ riff—f 7. 1 1 :7: —: 0’)? ‘1" 03‘: 0/2 +QQ’TX/tfl’2’2 -,[:‘f2 2 ca (WW) + (ml/20) + arc/(2»— 73,: «a —— a : (r21)(20) 4— [20)[-7/) + Fwy/W) :3 [217: 5?, 9 mm (20) 3. A body in equilibrium has the state of stress shown below. If no body forces are acting, determine the three constants k] , k2 , k3 and the functions f, , f2 , f3. (£21392) (4)122) [a]: (kzya) (19332) (6)222) (f3) (7332) Pym/um 0F [CY]: X 0 #- lbw—PH)“; :0 'L'T? K3j-7 (20) 4. For the case of plane ﬁrms, write (a) the constitutive relations (Hooke’s law) for a linearly thermo-elastic isotropic material and (b) the stress equilibrium equations. (c) In addition to the strain-displacement relations, what other sets of equations must be satisﬁed to solve a lane elastici rohlern'? p. M _. D W p 6'2 e 7x 2 - ’5} 2 F SN -: 95-2 P y g} + 0(A7— 0/ a/ £7 , 4/7} +7511— +oz47' _. (X7 6,. "E yX/y f 6 h 2(l‘f‘f/j -' é'z/Xy : 0 (H ax + (32’ + F" 51:29” 4— + Fyfa (C) COMPAﬂErnTy coypnvw/J‘ Bo (AA/p W 66%} {TWA/f (6) 5. In any analysis: (a) what is the first step? DE FIVE- THE 6”qu / OgijTZI/EJ‘ (b) what is the ﬁnal step? A F J I ? VAL rMTr //< Ech IF‘ magmas}, (6) 6. Are the following statements true or false? (a) In an isotropic body, an applied shear strain does not give rise to any normal stresses. 7- (b) In an unconstrained isotropic body, an applied temperature change does not give rise to any shear stresses. (3) 7. Where are the equilibrium equations deﬁned? AT A Porr/‘F M 77h? WW (5) 3. Write Saint Venant’s Principle and give a brief example of its application to a mechanics problem. IF A «1‘7me 0F“ Fang/w {4677/1/61 m A f/hﬂzc RE 63mm 0F Aw ELAJ‘WC J’oan If NEPd/IML 3y WOW J‘ﬂﬂmq Ha/VA’L-En/T Paws” rpm, W IWJ‘J’EJ” cam/6:5 Wﬂxxc/m may m/ 7H: Mao/ﬁnerer aF 7711” W» 8567044 {9 I‘D {Sbt %> zpét p 1b -:l:im FQRmLfLA SHEET 77-; R 55*— JD IMENS‘JUNAL “mm SFOKMA “rm/u: : .S'TRE'S‘J (NVHRIANTS I I, :: o; + a”), +6”; :2 = ego; +0262 +®oa-’r;; wig-'21: Is :- 01: 7x)» 2;?- Z/r'r 5r 77’? 753a: 77/2 ‘72 STREJ‘J‘ C(A’F/C’.’ @3‘12‘53 +1.2 of!” #13:0 ...
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MT1_Solutions - 3U tu'i‘(to ﬁt y ME 423 Intermediate...

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