CIVE_207_april2007

CIVE_207_april2007 - CIVE 207 Solid Mechanics Winter 2007...

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Unformatted text preview: CIVE 207 Solid Mechanics - Winter 2007 McGill University Faculty of Engineering Department of Civil Engineering and Applied Mechanics CIVE 207 Solid Mechanics FINAL EXAMINATION , Wednesday, 18 April 2007 2-5 PM Examiner: Prof. G. McClure 1 fi . Mr (fly/LL *' , ‘ / Associate Examiner: Prof. S.C. Shrivastava \Jr 3710‘ wLajgér INSTRUCTIONS: 1. 4. 5. 6. This is a CLOSED BOOK examination. A useful sketch is given on page 2 and additional reference material is appended as indicated in Note 6. FACULTY STANDARD CALCULATOR permitted ONLY. This examination consists of SIX parts. Attempt ALL parts. The questionnaire comprises 11 pages. The examination will be marked out of 100. Note that the questionnaire contains the following appendices: A) Properties of plane areas B) Deflections and slopes of uniform beams ONE 207 Solid Mechanics ' I Winter 2007 Angles that locate principal planes in plane stress was. ‘L' .2 1”“; 'l .. in} " ‘3 CIVE 207 Solid Mechanics ' 3 Winter 2007 Part 1 (15 marks total) A circular steel rod AB (diameter d1 = 15 mm and length L1 = 1100 mm) has a bronze sleeve (outer diameter d2 = 21 mm and length L2 = 400 mm) shrunk onto it so that the two parts are securely bonded together (see Figure P1). Calculate the total elongation of the steel bar due to a temperature rise of 350°C. Material properties: Steel: ES = 210 GPa; GS = 77 GPa; as =12 x10'6/°C Bronze: Eb = 110 GPa; Gb = 42 GPa; ab = 20 x10'6/°C Figure P1 CIVE 207 Solid Mechanics Winter 2007 Part 2 (15 marks) A vertical force is applied to the pipe wrench as shown in Figure P2. Determine the largest combined shear stress in the pipe at the wall B. Show this stress state clearly on a sketch of a differential element with proper notation, orientation and sign. The pipe has an outer diameter of 25 mm and a wall thickness of 3 mm. Figure P2 ONE 207 Solid Mechanics ' I Winter 2007 Part 3 (15 marks) A stainless steel water tank (Figure P3) is filled to the top as shown. The tank has a wall thickness of 20 mm throughout and an outside diameter of 10 m. Determine the stress state at the base of the tank and represent it on a differential element. Use clear notation and indicate proper signs. What is the maximum shear stress in the tank? Density of stainless steel = 7920 kg/m3 Density of water = 1000 kg/m3 Figure P3 CIVE 207 Solid Mechanics . ' Winter 2007 Part 4 (15 marks) A beam ABCD rests on simple supports at B and C as shown in Figure P4. The beam has a slight initial curvature so that end A is 15 mm above the elevation of the supports and end D is 10 mm above. Find the loads P and Q, acting at points A and D, respectively, that will move points A and D downward to the level of the supports. The beam has a uniform flexural rigidity El = 5.0 x 106 N-mz. Figure P4 ONE 207 Solid Mechanics . ' Winter 2007 Part 5 (20 marks) Determine the location of the shear centre (SC) of the thin-walled cross section shown in Figure P5. Neglect terms in t3 when evaluating the second moment of area of the cross section. Figure P5 ONE 207 Solid Mechanics - ' Winter 2007 Part 6 (20 marks) A sign is supported by a steel pipe (see Figure P6) having outer diameter 100 mm and inner diameter 80 mm. The dimensions of the sign are 2.0 m x 0.75 m, and its lower edge is 3.2 m above the base. The horizontal wind pressure against the sign is 1.8 kPa. The self weight of the structure is neglected. Determine the maximum in-plane shear stresses due to the wind pressure on the sign at points A, B and C, located on the outer surface at the base of the pipe. Represent the stress state at each point on a differential element with proper signs and indices. Sketch (approximately) the Mohr’s circle for plane stress corresponding to each case and indicate clearly the position of the diameter representing the stress state calculated for each of the three differential elements. Figure P6 Section X-X ONE 207 Solid Mechanics APPENDIX A Geometric properties of common planar shapes Moments of Inertia of Common Geometric Shapes 1'31)! 1 ‘ ' 'Il'jb'lh 1th Ebjh - l'._,£nh(bl + h") Rectangle I 3 36b}? 12-bit“ _--. II Triangle JI Circie / f} 2 _\‘ _ $7135] C I 4 ;r J0 .—_ 5177" Semicirclc if Quarter circle Ellipse J0 = ivabfizfi + b?) Winter 2007 CIVE 207 Solid Mechanics - ' Winter 2007 Centroids of Common Shapes of Areas and Lines Shape Triangular area Quarter-circular area Semicircular area Scmiparabolic area Parabolic area Parabolic spandrel Circular sector Quartepcircuim arc Semicircular arc Y’:_..»" n: , rsina Arc of circle 5,-1.1 ----- -—- — _ (3 2m 10 ONE 207 Solid Mechanics Winter 2007 APPENDIX B Formulas for beam deflections and slopes 'l‘abku 8-19 Beam Defiections anti Skopes Load and Support (Length L) _ Slope at End Maximum Deflection I Equaiion of Eiastic Curve (+ A) (+ upward) ~ _ (+ upward) PL: .P!.-‘ a = ""2???“ "max 3 _fi PVZ ‘ I 2 —~‘— '31,. —- .' atx = L atx = L ‘ 65H ‘) Vm'm : Hex? ) 1 ' z: _ —4{.. + 6!: -. m=L ‘ 2451“ r } .I "ntnx : WK: t N ) - 1 3 ' \f:— w (IOIH— .l()1,"x+3Lx‘—.r). atx =1. IEOHL MI.2 me; = + ”E! ‘ _ 3915' atx = L 2‘“ mu} — b2)“ .3, ._ ._.___ “ Nina! = div—£13m Pbx 2 2 3 (L ”b —.1) « v = ~6Efl. ‘ -— “Pb(31f _ 4b2)| O < \1 4: a m _ 485! ' ‘ Peg/)2 i 1’ 3 W atx = a 3m E '“J? ' ‘ Px ' P1” == ~~—-. 3L” — 4x“ ”“3“ : _4SEI 1 435” L atx=I./2 055x51;- I .. SWI.J WV . 2 __H_ V a . 1 ‘““"‘ 384E! . v = -- Emu-3 — 2m +49) a1 1: = {4'2 ML: ’ = w.— lfll‘ .gfiglf atleJvi-j ‘__J'fo 2— 2) 6EIL _ ML! (cum _ _ 16E!- INJI I11“ 1] ...
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