CIVE_207_december2007

# CIVE_207_december2007 - McGill University Faculty of...

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Unformatted text preview: McGill University Faculty of Engineering SOLID MECHANICS CIVE—207A Final Examination: 2:00 - 5:00 PM, DECEMBER 20, 2007 Examiner: Prof. Shao Associate examiner: Prof. McClure INSTRUCTIONS: 1) This is a closed book and closed notes examination. 2) Only faculty standard calculators are permitted. 3) This examination consists of FIVE problems of a total of 7 pages, including cover page and formula sheets. 4) The examination will be marked out of 100. CIVE-207A' Final Examination December 20, 2007 p.2 Problem 1 (20 marks}: A wooden I beam is made up with a narrow lower ﬂange because of space limitations, as shown in Fig. l. The lower ﬂange is fastened to the web with nails spaced longitudinally 6 in. apart, and the vertical boards in the lower ﬂange are glued in place. The beam is subjected to a vertically downward shear force of 400 lb. The moment of inertia for the whole section around the neutral axis is 2640 in4. (1) Sketch the shear ﬂow diagram; (2) Determine the shear stress in the glued joints and the shear force carried by each nail in the lower ﬂange. (3) Calculate the maximum shear stress. Fig. 1: Problem 2: {20 marks) In the beam AB with both ends ﬁxed as shown in Fig. 2, the left-hand support, A, has settled a distance 50 below the right-hand support, B. Modulus of elasticity E, moment of inertia I, beam length L, and settled distance 80 are given. Determine: (1) Reactions at supports A and B; (2) Deﬂection at mid-span cf the beam. (3) Draw the shear force and bending moment diagrams of the beam. Express the results in terms of E, I, L and 80. The effect of axial force is negligible. CIVE-207A Final Examination December 20, 2007 p.3 Problem 3 120 marks): The state of stress at three different points in a loaded structure is shown in Fig. 3. For each case, (1) Draw the Mohr’s circle; (2) Use the Mohr’s circle to determine the principal stresses and show them on properly oriented stress element; (3) Use the Mohr’s circle to ﬁnd the maximum shear stresses with the associated normal stresses and show them on properly oriented stress element. Fig. 3: '4ksi _ I 4ksi (A) (13) Problem 4: (20 marks! A 100-mm-diameter shaft is loaded on left-hand side free end and ﬁxed to the wall section ABCD on right-hand side, as shown in Fig. 4. Determine the maximum normal stress and maximum shearing stress at section ABCD. Fig. 4: The cross sectional area A, the ﬁrst moment of semi-circular section Q, the moment of inertia I, and the polar moment of inertia J are calculated as follows: A = 301? =§(100)Z = 7854mm: = 7854(10‘6)m2 _”/2(4’ 232 3 13 3 q~—63 150m Q——U‘)( =50) =§(50) =83N3(10)mm =83.3:(10 )m 2 F, 30)“ = gen)“ = 4.909(106)mm4 = 490900451114 J = gcdf = Q50)4 = 9.817(10‘)mm4 = 9‘817(10‘6)m4 I CIVE-207A Final Examination December 20, 2007 p.4 Problem 5 (20 marks): The temperature in a furnace is measured by means of a stainless steel wire placed in it (Fig. 5). The stainless steel wire is rigid enough to be able to carry either tensile or compressive force. The wire is fastened to the end of a cantilever beam outside the furnace. The strain at point A is measured by a strain gage glued to the outside of the beam and is a measure of the temperature. At reference temperature, the strain reading is set to zero and the wire carries no stress before the temperature is changed. Assuming that the full length of the wire is heated to the furnace temperature, what is the change in furnace temperature if the strain reading at point A is —100x10'6 in./in.? The mechanical properties of the materials are as follows: For stainless steel wire: coefﬁcient of thermal expansion ass = 9.5 x 10'6 per 0F, modulus ESS = 30 x 106 psi, area Awire = 5 X 10'4 inz; For aluminium bar as a beam (rectangular cross section): coefﬁcient of thermal expansion oral = 12 x 10'6 per 0F, modulus E31 = 10 x 106 psi, moment of inertia Ibeam = 6.5 x 10'4 m4, the depth of cross section of beam = 0.25 in. Fig. 5: Furnace Aluminum bar Strain gage\ |:l—.‘i: 0.93? in. Hymi 02053 35.:qu 0‘»? 85?? M CIVE-207A Final Examination December 20, 2007 p.5 Fundamental Equations of Mechanics of Materials Axial Load Normal Stress a _ 5 A Displacement 6 _ IL P(x)dx 0 A(x)E PL 3 — 2 E 5T = at ATL Torsion Shear stress in circular shaft _ TP T _ 1 where J = gc“ solid cross section I = gm,“ — cf) tubular Cross section Power P = Ta) = 27rfT Angle of twist ' _ I,“ T(x)dx 0 we TL 2 2 — d) J G Average shear stress in a thin-walled tube 7 _ L avg Shear Flow I q = Tavgt = m Bending Normal stress _ My 0 _ 1 Shear Average direct shear stress V Tavg = X Transverse shear stress VQ 1' = — t Shear ﬂow I VQ q = 7 : —- I Stress in Thin-Walled Pressure Vessel pr pr 0'1 = 7 0'2 = E Sphere — — pr Stress Transformation Equations ax+oy {Fax—cry 2 2 O'xr= 03,—0'y ' 7,3,: — 2' sm29+7x,00520 Principal Stress rxy tan 20 = —— " (a'x — a'y)/2 a")[ + a", a" - a", 2 (71.2 = 2 i ( 2 ‘i‘ 7i}. Maximum in-plane shear stress (0 — o- )/2 tan 26: = _ __"—y 1-,}, or - a" 2 Tmax = < x 2 y) + Ti)" 01 + 0y 0'an = Absolute maximum shear stress T __ omax — 0min bs _ 3m 2 _ Umax + 0min Uavg _ 2 cos 20 + rxy sin 20 CIVE—207A Final Examination December 20, 2007 p.6 Material Property Relations Geometric Properties of Area Elements Poisson’s ratio 6 lat V = - 6long Generalized Hooke’s Law 1 ex = E [03: _ V(0'y + 02)] 1 6y = Elk-y _ V(Ux + where E "5 RelaﬁOns Between w, V, M dV (1M 3: ‘WW 7‘ V Elastic Curve El: 1 p Bee.“ng . Crztzcal anal load “ZEI Pa, = (m2 9 ICL‘ Le Critical Stress ' 711E c, = , = V I A ‘7 (KL/r)2 ' / (a) One ﬁxed end, (5) Both ends (9) One ﬁxed end, (d) Both ends one free end pinned one pinned end ﬁxed Exparabolic area GIVE-207A Final Examination December 20, 2007 p.7 Appendix D. Beam Deflections and Slopes Maximum I Beam and Loading Elastic Curve Deflection Slope at End Equation of Elastic Curve PL2 P 2E1 y _ 6731063 _ 31"? Fora>bz Forx<a Pb L2 — (72 3/2 Pb L2 — .92 ——(—1) aA=——{——) y= P” [xJ—(Lz—bzm 9w’iEIL GEIL 6EIL LZ—b3 Fag—£2 22 atxm= 3 BB=-i~——(———2 Forx=a y=—Pab ...
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## This note was uploaded on 12/01/2009 for the course CIVE 207 taught by Professor Shao during the Winter '09 term at McGill.

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CIVE_207_december2007 - McGill University Faculty of...

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