cee130-unknown2-mt2-Li-exam

cee130-unknown2-mt2-Li-exam - UNIVERSITY OF CALIFORNIA,...

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Unformatted text preview: UNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Mechanics of Materials (CE130) The Second Mid-term Examination Problem 1. Derive the differential equation for Bernoulli-Euler beam. Consider following infinitesimal beam element: (a) W E = (1(17); (1) (b) dM 71‘; = W55) (2) (1! points) Figure 1: A 2D infinitesimal element Problem 2. Draw shear & moment diagrams for the following beams (see: Fig. 2 (a)(b)) and label the peak values for the corresponding maximum shear and maximum moment. (25 points) U2 U2 (a) (b) Figure 2: Beams with external loads Problem 3. An I—beam shown in Fig. 3 is subjected to a shear force V at the given cross section. Find the maximum shear stress within the cross section. ' 1' VQ — Izt Q = / MA = Ag A I; = I;C + dgA parallel axis theorem 3 I M = % for rectangular cross section. (3) (20 points) Y Figure 3: The cross-section of an I—beam. Problem 4. A rectangular beam shown in Fig. 4 (a) is made of linear elastic-perfectly plastic material (shown in Fig 4 (b)). Suppose h = 400 mm, b = 200 mm, and the dimension of the elastic core is 2yo = 200 mm, and yield stress ayp = 100 AIPa. Find: (a) the stress distribution caused by a positive inelastic bending moment Alep; (b) the inelastic bending moment, 1116,, ? (c) the residual stress distribution after unloading. bh2 b '2 Al(unload) Hint: Ale}, : aypT _ pr_%, Aagunload) 2 ayes) _ Ugoad) = __z__I___y. z (20 points) (A) (3) Figure 4: An inelastic beam: (a) the geometry of the cross-section; (b) the stress-strain relation. Problem 5. Find the beam deflection 12(15) for the cantilever beam with a piecewise constant distributed load applied in 0 < .12 < a. The flexural rigidity, E1 = const.. (Recommend using singularity function method). (14 11 E1 (136‘? = we) (4) (Hint: q(:I:) = —q0(1- < :L' — a >0. ) (20 points) free end Figure 5: A cantilever beam with discontinous load distribution. BMW MA'OX hem I 2. OWL]; CEviac‘rL :3 if; igomml £8 E? 03 info 25 ‘3) V09+ WC%)-mA%—(v(x)+av):o E :> 0Vin EXT-AV E QM»): Aj AW E 60'» SM AV 2 a), m [w ow 0W :- (M Q’WL 0V ) b) imwmfo 9 4mm —V(>k) Aw~ CVUU-OK-Dlt +(mcu>+ am) ; 0 => am ; vcx) AM, + Wm {13392 2 W AXIS Smog) (Au) Z~90 (1990 m 153’) ; (if!) AX om VM). (5 Q. 3 1 \H/ Engineer‘s Computation Pad NO. 937 8'11E 9SIZIEDTLER® Tuna/x = V_® I21: V59 Show Qi'cliw waniva Vuflw 17,1: Engineer's Computation Pad E V 6? W “WW HS huge» main: cmé $6 pmw E L— ;k Wfiwr‘ C') E- ,A in m gave/yp/Lc/bmm G) is dampen a) Maw m W36 (,6 w on») among-«Q WV wand Mk N‘ {'3 C6 @5014 “E: - 5.1 Q: A3 5 ; (Letx t)x(lt+1:) + (int/Mt “I L E = low-rakimzk?’ G} W‘ I: 3 ‘3' V "S , 3 ¥ 1 3 W; 1;; PM? E Jr Ht gt)? + 1 t-(HH \2. L I; : axfflzseq Jr Let“ 3 3 : E §6’bt‘ “CM 2 V' '2153 3 a . 2 Sedtkl‘k, \q't Engineer's Computation Pad NO‘ 937 811E ‘QSDQEDTLER‘E’ b) {b w WMVB W Loom! 44M Momdmot '2. W M59: 073% ; $99.2») b M CM WWW MM'M SW9 wSWbUMOm Cu) 937 811E Engmeer's Computation Pad N0. 95MEDTLER® H923 (NC W6 dqflCM) ;~—q/o<i_ <%-q>0) olx‘i :) Eldgvm: V01): 'Wc (x—(u—a>‘)+C. c’Uu» V(0):o t.) Ct30 :9 Balm : - 0Wch -;<wa$‘) + c2 : mcu) out2 2 L m’Lo):0 3) (2:0 «:3 exam») 2 - W541? — l (wk-03) + C3 WOO-E1 6T“ 6 e 9(OLHD):O ' W ; bl 2 (a E17)UL)3— a 7;“ -_L “419), K (“bibs JrC “V 24 Zu< D + CV9; > ) L; ___Lm__J_ \' f. E11701) 2‘ “ CV; (“>65 (M- 0)“) + We.“ ((m)3_ b3) 2‘4 6 + we (WOYL b“) ~ (vowmmfi bi! 2’” E Engineer‘s Computation Pad 03) NO. 937 811E asmemER ...
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This note was uploaded on 12/01/2011 for the course CE 13972 taught by Professor Chow during the Spring '09 term at University of California, Berkeley.

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cee130-unknown2-mt2-Li-exam - UNIVERSITY OF CALIFORNIA,...

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