3 - Prob. 2.13- A block of linearly elastic material (E, v)...

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Unformatted text preview: Prob. 2.13- A block of linearly elastic material (E, v) is compressed between two rigid, perfectly smooth surfaces by an applied stress a, = —00, as depicted in Fig. P2.13-7. The only other nonzero stress is the stress 0, induced by the restraining surfaces at y = 0 and y = b. (a) Determine the value of the restraining stress 0,. (b) Determine Aa, the change in the x dimension of the block. (c) Determine the change-At in the thickness I in the z direction. U Sou/Hon: (a) rem-mining s+mas From 50,. 1.355) a: é, (<11, — 90;) =0 since val-mined (Y a VG’, = -'\?r0 Ty: "' .90", (b) Change in lea/39% From Eq. Z. 38 a, 6x= _'_(0”x - V0,) = AOL E T mug (01—002,) = 0L0} (12%|) E E No+c¥ since, 0094! , 179% l, Mentor/e, m is mega-Hm. (c) Change In mickncss From "Ear. 2.38 o, 62: "_’\_7_ (O'X*U:I) “At— ‘Prob. 2.1340. A block of linearly elastic material (E, v) is z placed under hydrostatic pressure: v. a a, = ti. = ‘p; I ‘17,, = -r" - 1,. = 0, as shown in Fig. P2.13-10i (a) Determine an expression for the extensional strain e, ( - a, III e‘). (1:) Determine an expression for the dilatation, ey. (c) The bulk modulus, k... of a material is defined as the ratio of the hydrostatic pressure, p, to the magnitude of the volume change per unit volume, Icy], that is, . .L "‘ I»: Determine an expression for the bulk modulus of this block of linearly elastic material. Express your answer in (Stresses onhiddcn faces n0t Shown) terms of and‘y. P113.” Soil/how NW ex= |_E[o’, wwwm] = (4;)(1194) e,< = (:Eflflzv-I) I b) an a+u+fon V: “be v“: (M éxa)(b+6~,b)(C-*€LC) =V(l+€x)(He\/)(|+€z) “ ‘3'V (“'EXf €-v= V*-V = (Hews—I ev: (1) Wk- Modulus Since 03‘:ng 0'; = - 1a is unfit:va compressive, ex (= e,= e2) is n69 a+Ive .ThCVOPOYG, (w \‘s m3afiv050- ‘4‘): i. = “.21”— lévl 6" WA column in a two-story building is fabricated from square structural steel tubing having a modulus of elas- ticity E = 210 GPa. The cross~sectional dimensions of the two segments are shown in Fig. P3.3-4b. Axial loads 1" = 200 kN and P, = 300 kN are applied to the column at levels —T A and B, as shown in Fig. Pas-4a. (:1) Determine the axial '1 = 3 m 150 m... stress al in segment AB of the column and the axial stress Q or; in segment BC of the column. (b) Determine the amount 8 by which the column is shortened. arc/@5294 .' M} Ific/a/J/flrrcr. EEu/'/'{r1‘am ,' :0 I. 400w ~F=o F,= 4001M ' I Mal/u 300"" 6 = {Oahu 530/: . D ' (ff/aria: F6 1’ F; r F, P334 -zao W ’300 éfil— F; ‘0 g — 2004M = -—-—-—————" = ‘4 , / I fl/ [/I‘YOMM)L-//HMAY'] ¢ 0 y é 5 —Jaoéu =—-— . —-———-—————; =.. ,. Ml? l4» - [(ZOUM)‘-(/76~~)] {$405 a @ [/4/ {AM/6,161] 9/ 60/401» . 57¢»: 64/ fi/C(/ a/cfl'rme [fiéfiflb/ I :1 = 2)" a5. _ (—4¢,mmn¢)/jm) 63= 5/ 2mm =-0,6z:em ,2 a 021, = {-rwoammm) “Mi/rm fl 2/05“ a fee mafl’z I / rm 4,5214 ,' /=—/4/f’zl = A450“ Prob. 3.3-9. A rigid beam AB of total length 3 m is supported by vertical rods at its ends, and it supports a downward load at C of P = 60 kN, as shown in Fig. P3.3-9. The diameters of the hanger rods are d1 = 25 mm and d, = 20 mm, and both are made of steel (E = 210 GPa). Neglect the weight of beam AB. (a) At what distance x from A must the load be placed such that u. = ml? (1:) How much is this displace- ment of the beam? (c) What are the corresponding axial stresses 0‘ and a: in the two hanger rods? fly/“5‘09: 7 M) [044/ A/fl(¢i.0/l K {26/ l/sz/ a6 = fl”. fiat-54' [’12va Wij, =0 : {a - P/d—x) =0 (I) mam/g =0; 52 - a :0 __ X *fffy-‘o! EIE'P‘O L—_ 57!»; én/ 24/4, - c/c KmaA-wv o/afiawa.’ (I. =74/‘T = ,5 m pcflrme/Tan (aorflar'vrfl twang/6;, w Z fll— I . ,_ ’ l ) 6 Fl [1.4. ) ’ 4,19 (0""44 {Ianxl/f’)! 6 zj ) d, - PX =0 "{4} 4 (owl/u (h) hue/[4)! _ ____ x ’ /f//1 9/ ) = [14’- {ZOMM)” m /5) fl/‘J‘fl/tczmzn/ gr) :ufi' lx: flow {/6} €= 1% = W = 57.3325 AM on, 4/3 — c; — ’{Z‘ = 6235’07mm 44. = #4” QJWM ’71. fr. I-i. [[7 fier/Ef. : -F,= MAM—5 = 2’0. wag/«U ¢e , a”: : [Z,776f/”/’(/ Zara/’2 -= 4554¢fl7flt ’ 27'4" Prob. 3.4-7. A steel pipe is filled with concrete, and the resulting column is subjected to a compressive load P - 80 hips. The pipe has an outer diameter of 12.75 in. and an inside diameter of 12.00 in. The elastic moduli of the steel and concrete are: E, '- 30 x 10’ ksi and E‘ a 3.6 x 10’ ksi. (a) Determine the stress in the steel and the stress in the concrete due to this loading. (b) If the initial length of the column is L = 12 ft, how much does the column shorten when the load is applied? (Ignore radial expansion of the concrete and steel due to Poisson’s ratio effect.) Solufion * (a) norm! S‘h’css swam dJSvI'Yl’DU‘HOYl '- 6 (>0 = e = Cons-mm s+ress w‘w'hm'on‘ rs = E56 r6 sEce rcsuiWhvcey repiace “3,930pr 95199 .___._—_—-—-—— Pg 0} Ag 315260200 in)" - _4 d5 Ail: 02 ‘7} => Po l’s Ps=<Fs As -' =E_6ES|Z\1.15I'n)"— 02.00 [109‘] EfiDj Column swh'on 7 +4“ zr-zlzoz-P—Pc—Ps =0 9; AI; [mHEc-Es) “115‘ an e: -4? 1m 7.1 (go -63) H139 5;] Y =>—Fvom S+re$$ dis‘rviwn’om Lx P° “'c ‘ APE/1r [121(26— 557+ males] WPS = _ 0.34:02 ksé = — P “3 4 ES/Trtnz‘cec—ESMuHS‘Es’J = — 2.84195 ksc 4 a. 96 ‘ es "‘ e ‘ 5L ‘ 1V[|21(E¢’Es)*|2.151551 = 0.0I5HI in. a: 4,354 no" in. New: column shelf-lays. Prob. 3.5-4. A steel pipe (E, = 200 Oh) surrounds a solid aluminum-alloy rod (& = 70 GPa), and together they are subjected to a compressive force of 200 kN acting on rigid end caps. Determine the shortening of this bimetallic com- pression member, and determine the normal stresses in the. pipe and in the rod. “Va/wan; .0557th {Ker éh/hj (0") dm/ 575532? (f, max/d; €= F = I jf’ / //z) 1/664? / at; 4=fl€ 1; 7:1: ALL: P=2oom ' L=0.5m P3.5-4 and P3.8-‘l3 F; *— 9 P /0.rm) //—[{0,0f0m) —/0, Mom) ‘J/Zoox/o ’5’, 5,:4/f/10“')-f’7 0, fl». [/aosnm )2 {701/0‘7») 5 A5560 / /0”)’:7 §€0mrfry a O/cfl/mdfilfl’} [ém C} = fl = —J-— (fifél/ffnl’ij‘) 6’) (In) 11;”: f?) -éno//f). {Fm/Le w /=/—;/5 6") (ml/o: (I) 4476/6”). ihffl — — - ,— 5, w 7—) ’ —/D —7 6-- fllfz — fr—fi—P = —g¢,53’w - = a uwmm 03—6, =—/,€'&,//?me~ If I F 455471” I: I ' =-— I ~ r 7’— /7[/aoromj-‘—/aa¢om)‘ i7 7/ $~4Z7flfl4él J = 5— ’ M =—/é.7¢7/m $5/a77mfl/d L ’91, 7 Mair/MP ...
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3 - Prob. 2.13- A block of linearly elastic material (E, v)...

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