CIVE_207_april2006

CIVE_207_april2006 - CIVE 207 Solid Mechanics Winter 2006 2...

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Unformatted text preview: CIVE 207 Solid Mechanics Winter 2006 2;. _ 9;. McGill University s Faculty of Engineering Department of Civil Engineering and Applied Mechanics ONE 207 Solid Mechanics FINAL EXAMINATION Friday, April 28, 2006 2-5 PM Examiner: Prof. G. McClure Associate Examiner: Prof. S.C. Shrivastava INSTRUCTIONS: 1. This is a CLOSED BOOK examination. Useful information on the Mohr’s circle for plane stress is given on page 2 and additional reference material is appended. 2. FACULTY STANDARD CALCULATOR permitted ONLY. 3. This examination consists of SIX parts. Attempt ALL parts. 4. The questionnaire contains 12 pages. 5. The examination will be marked out of 100. 6. The questionnaire contains the following appendices: A) Properties of plane areas B) Deflections and slopes of uniform beams C) Properties of wide-flange beams in SI units CIVE 207 Solid Mechanics Winter 2006 Angles that locate the principal planes in the Mohr’s circle for plane stress ONE 207 Solid Mechanics Winter 2006 Part 1 (15 marks total) After a load P = 150 kN is applied to the pin-connected structure shown in Fig. P1, the temperature increases 100°C. The thermal coefficients of expansion are 22 x 10'6/°C for the aluminum alloy rod A and 12 x 10'6/°C for the steel rod B. The moduli of elasticity of the aluminum alloy and steel are 75 GPa and 200 GPa, respectively. The cross- sectional area of members A and B are 1000 mm2 and 500 mm2, respectively. If member CD is rigid, determine the normal stresses in bars A and B and the vertical displacement of point D. You must sketch the deflected shape of the system and indicate the rod deformations (mechanical and thermal). Figure P1 ClVE 207 Solid Mechanics Winter 2006 Part 2 (20 marks total) a) 15 marks Sketch a typical steel column curve, Le. a graph giving the compressive axial stress vs. the slenderness ratio of the column, covering the full range of possible slenderness ratios, for an ideal pin-ended column under centric load. Identify and label the regions corresponding to the different column failure modes. Sketch an example ofa column failure mode corresponding to each region. Write the equation for the axial stress in each region of the curve, using the following symbols for the material properties: fy is the yield stress fu is the ultimate stress E is the modulus of elasticity And Eris the reduced modulus. b) 5 marks List five parameters that affect the capacity of a column in a structure. ONE 207 Solid Mechanics Winter 2006 Part 3 (10 marks total) Select the lightest wide-flange steel beam that can be used to support the load shown in Fig. P3. The allowable flexural stresses in tension and compression are 160 MPa and the allowable average shearing stress in the web is 82 MPa. Verify that your design is adequate after consideration of the self-weight of the beam. You must sketch the shear force and bending moment diagrams of the beam. Figure P3 2.0 ltN-“In ONE 207 Solid Mechanics Winter 2006 Part 4 (15 marks total) A beam is loaded and supported as shown in Fig. P4. Determine the reactions at supports A and B in terms of the variables P, a, L and the flexural rigidity of the beam, EI. Figure P4 ONE 207 Solid Mechanics Winter 2006 Part 5 (20 marks total) All metal in the cross section shown in Fig. P5 is 5 mm thick. The dimensions shown are centerline dimensions for the flanges and the webs. Assume that all of the section is effective in resisting flexural stresses and that only the web portions resist vertical shearing stresses. a) 15 marks Locate the shear center of the cross section with respect to the center of the right web. b) 5 marks Determine the maximum shearing stress produced on the cross section by a vertical shear force P of 15 kN. Figure P5 ONE 207 Solid Mechanics Winter 2006 Part 6 (20 marks total) A 100-mm-diameter shaft is loaded and supported as shown in Fig. P6 : the cross section with points A to D is perfectly fixed. a) 10 marks Determine the normal and shearing stresses at points A, B, C and D on a section at the wall. Neglect stress concentrations. b) 10 marks Determine the principal stresses and maximum shearing stresses at points A, B, C and D. Which is (are) the most severely stressed point(s)? NB: Use the system of coordinates shown in the figure and be rigorous in your notation and sign for stresses. Figure P6 150 kN " F50 nun___ I'll-{)0 mm ONE 207 Solid Mechanics Winter 2006 APPENDIX A Properties of plane areas Centroids of Common Shapes of Areas and Lines Shape Area Trianguiar area i); Quarter-circular area if; ‘ r2 Semicmzular area 32.. Semiparabolic area 4 Parabolic area %}g ah i’arabolic spandrcl T Circular secror a ,1 Quarter-circular are fit Semicircular are 1" Arc of circle 2M ONE 207 Solid Mechanics Winter 2006 Moments of Inertia of Common Geometric Shapes L z .13an “1-. II n" ilk-h!“ Rectangle 1’. = .93"! l_,. = libs}: - n [IE-£311“): + h") Triangle Semicirclc Quarter circle 10 ONE 207 Solid Mechanics APPENDIX B Table [1‘19 Beam Deflections and Slopes Deflections and slopes of uniform beams Maximum Deflection Winter 2006 Pb(L2 — b?) m1} m b2)”2 1 6L5] m w 9J5LEI' x at X = 1 2 a L91! ; [/92 ‘ 2 fl atx=v/(L~~b 1/3 ,1; 62 — + P“: ‘1” ‘ _ m3!) — 4121) “m ‘ LE 599*“ “ _ 48151 b < u at .X = 1 m) max '1 V - ~Pacb2 afx — a ““ 31511, ‘ ‘ 5 :r U2 P 6 _ N PL2 ' 1651 x atx=0 v ‘_WPL3 {.91 J l rye: PL? mm L E; 0 # ' atx = ‘ ()3 = < I ‘inuu + ' atx = L 7 Ir W at = .- wLJ i 24E] I x I atx = 0 v _ H Swif‘ gt] 1 {a} “ 38415! 0 . 6 + M-‘L3 flix : L]? 5 : -—-—-- "um: ‘ 24E] atx a L __ _ __ *__ F 2 _ 3 I" a!” 9 2 4% i , I: _ M]. X l Lmax l at x = 0 _ r :9; I i [/91 azx — Lm 0 -' ML W rm” 2 _ 3—13? v = _.L.M at x = L Eifl'fin 165’ L Load and Support Siope at End Equation of Elastic Curve (Length L) (+ A) (+ upfvard) (+ upwaré)_ 6 PL2 } PL3 " __ ‘mnx = “m ,2 EEI 3E1 1IE_£(3L*M guzl. aix=L 6127 WL3 wL‘ : —— = .— 1 613] SE! “5“” 2 2 _ ~ v -- 4Lx - 6.12 at x' a L m = L 24.51“ i ) WI.1 WL" : _ Vnmx : - WY: 1 a a 3- ' 'm—— ' ,'—-EOL‘ ’5L.'— -, msz atx=L ” IZQEJLUOI H“ T x} 9fl+fl£ __«E_ML1 E, anx— I v : J’.‘ atx = L atx = L Pbx \: = _.._._.. (SEIL O f x f a 11 ONE 207 Solid Mechanics Winter 2006 APPENDIX C Properties of wide-flange beams in SI units I Tahie 11—2 Wide-Flange [Seams {SI Units) Web “AXEX'”X _ __.. __. _ Mix—Y __.T. Area Depth Width Thickness Thickness I (10" S (It)3 r I (10‘J 5 (1113 r Designation ‘ (mmz) (mm) (mm) (mm) [mm] mm“) mm}; (mm) mm‘] mmE) (mm) 0/914 x 342 43610 912 418 32.0 19.3 6245 13715 373 391 1870 94.7 X 2.38 30325 915 305 25.9 16.5 4060 8880 366 123 805 63.5 “"338 X 299 38130 855 400 29.2 18.3 4735 1 1210 356 312 1560 90.4 X 226 28850 851 294 26.3 16.1 3395 7980 343 114 775 62.7 X 193 24710 840 292 21.7 14.7 2795 6655 335 90.? 620 60.7 “"762 X 196 25100 770 268 25.4 15.6 2400 6225 310 81.6 610 57.2 x 161 20450 758 266 19.3 13.8 1860 4900 302 60.8 457 541.6 VVG86 X 217 27675 695 355 24.8 15.4 2.345 6735 290 184 1040 81.5 X 140 17370 684 254 18.9 12.4 1360 3980 277 5.1.6 406 53.8 \‘1’610X 155 19740 011 324 19.1 12.7 1290 4230 257 108 667 73.9 x 125 15935 612 229 19.6 11.9 985 3210 249 39.3 342 49.5 X 92 11750 603 179 15.0 10.9 645 2145 234 14.4 161 35.1 “7533 x 130 19225 543 312 20.3 12.7 1005 3720 229 103 660 73.4 x 124 15675 544 212 21.2 13.1 762 2800 220 33.9 320 46.5 X 92 11305 533 209 15.6 10.2 554 2080 217 23.9 228 45.0 “7457 X 144 18365 472 283 22.1 13.6 728 3030 199 33.7 592 67.3 X 113 E4385 463 280 17.3 10.8 554 2395 196 63.3 452 66.3 X 89 11355 463 192 17.? 10.5 410 1770 190 20.9 213 42.9 W406 X 149 18970 431 265 25.0 14.9 620 2870 180 77.4 585 64.0 X 100 12?10 415 260 16.9 10.0 397 1915 177 49.5 380 62.5 X 60 7615 407 178 12.8 7.7 216 1060 168 12.0 135 39.9 X 39 4950 399 140 8.8 6.4 125 62.9 159 3.99 57.2 28.4 W356 X 179 32775 368 373 23.9 15.0 574 31 15 158 206 1105 95.0 X 122 15550 363 257 21.7 13.0 367 2055 154 (31.6 430 63.0 X 64 8130 347 203 13.5 7.7 178 1025 148 18.5 185 48.0 X 45 5710 352 171 9.8 6.9 121 688 146 3.16 95.4 37.8 “"305 x 143 18195 323 309 22.9 14.0 347 2145 138 112 728 73.5 X 97 12325 308 305 15.4 9.9 223 1440 134 72.4 477 76.7 X 74 9435 310 205 16.3 9.4 164 1060 132 23.4 228 49.8 X 45 5670 313 166 11.2 6.6 99.1 633 13?. 3.45 102 38.6 “1254 X 89 11355 260 56 1?.3 10.7 142 1095 112 48.3 377 65.3 X ()7 8580 257 204 15.7 3.9 103 805 110 22.2 218 51.1 X 45 5.705 266 148 13.0 7.6 70.8 531 1 11 6.95 94.2 34.8 X 33 4185 258 146 9.1 6.1 49.1 380 108 4.75 65.1 33.8 W203 X 60 7550 210 205 34.2 9.1 60.8 582 89.7 20.4 200 51.8 x 46 5890 203 203 11.0 7.2 45.8 451 88.1 15.4 152 51.3 X 36 4570 201 165 10.2 6.2 34.5 342 86.7 7.61 92.3 40.9 X 22 2865. 206 102 8.0 6.2 20.0 193 83. 1.42 27.9 32.3 \V152 X 37 4735 162 154 11.6 3.1 22.2 274 68.6 7.12 91.9 33.6 X 24 3060 160 102 10.3 6.6 13.4 16? 66.0 1.84 36.1 24.6 “"127 X 24 3020 127 1.27 9.1 6.1 8.87 139 54.1 3.13 49.2 32,3 W002 X 19 2470 106 103 8.8 7.1 4.70 89.5 43.7 1.61 31.1 23.4 ‘W means widoflange beam, followed by the umnina? depth in mm, then [110 mass in kg per meter or length. 12 ...
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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_april2006 - CIVE 207 Solid Mechanics Winter 2006 2...

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