CIVE_207_december2005

CIVE_207_december2005 - McGill University Faculty of...

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Unformatted text preview: McGill University Faculty of Engineering SOLID MECHANICS CIVE-207A01 Final Examination: 9:00 —— 12:00 AM, DECEMBER 15, 2005 Examiner: Prof. Y. Shao Co—examiner: Prof, McClure STUDENT NAME: ‘ W 7 MCGILL ID. NUMBER: INSTRUCTIONS: 1) This is a closed book examination. No crib sheet is allowed. 2) Only faculty standard calwlators are permitted. 3) This examination consists of five problems of a total of 5 pages, including cover page. 4) The examination will be marked out of 100. 5) Z‘Xfim ,‘5 imam amour- 6/060 CIVE-207A Final Examination December 15, 2005 p2 Problem 1 [20 marks]: A post having a hollow circular cross section supports a horizontal load P=3 60 lb acting at the end of an arm that is 4 it long. P is parallel to y-axis (see Fig. 1). The height of the post is 25 ft, and its section modulus is S : 10 ms. [Useful formulas: tan(29p)=21xy/(ox-oy) & S = I/r, where I is moment of inertia and r is outer radius] (a) Calculate the maximum tensile stress cm, and maximum in-plane shear stress tum at point A due to the load P. Point A is located on the “front” of the post, that is, at the point where the tensile stress due to bending alone is a maximum. (15 marks) (b) If the maximum tensile stress 0'...“ and maximum in-plane shear stress "cum at point A are limited to 16,000 psi and 6,000 psi, respectively, what is the largest permissible value of the load P? [Hintz The stresses at point A are linearly proportional to the load P] (5 marks) Fig. 1: Problem 2 [20 marks]: A Beam is loaded and Supported as shown in Fig. 2. Determine (a) The deflection at A. (15 marks) (b) The slope at D. (5 marks) The flexural rigidity of the beam is El. Express the results in terms of P, E1 and L. GIVE-207A Final Examination December 15, 2005 p3 Problem 3 {20 marks): A rigid beam of weight W = 800 lb hangs from three equally spaced wires, two of steel (S) and one of aluminum (A) (see Fig. 3). The diameter of the wires is 1/8 in. Before they were loaded, all three wires had the same length. (a) What temperature increase AT in all three wires would result in the entire load being carried by the steel wires? (Assume modulus of steel, ES : 30 x 106 psi, modulus of aluminum, Ea = 10 x 106 psi, coefficient of thermal expansion for steel, as = 6.5 x 106/°F, and coefficient of thermal expansion for aluminum, an 2 12 x 10'6/OF.) (14 marks) (b) Would aluminum wire be in compression or in tension if temperature increase is larger than AT? Briefly explain your answer without calculation. (3 marks) - (c) Would aluminum wire be in compression or in tension if temperature increase is less than AT? Briefly explain your answer without calculation. (3 marks) Problem 4: (20 marks) The state of plane stress at a point can be described by a known tensile stress 0, = 70 h/[Pa, an unknown tensile stress (5, and an unknown shear stress I, as indicated in Fig. 4. At this point the maximum in-plane shear stress is 78 MPa, and one of the two in-plane principal stresses is 22 MPa in tension. The directions of all stresses are shown on Fig. 4, Using Mohr’s circle method, (a) Determine the values of the two unknown stresses, labeled 0 and I on the figure. (b) Determine the second in-plane principal stress. Show the principal stresses on a sketch of a properly oriented stress element. Fig. 4: CIVE-207A Final Examination December 15, 2005 p.4 Problem 5: [20 marks) All metals in the cross section shown in Fig. 5 are 6 mm thick. The dimensions shown are centreline dimensions for the flanges and the webs. Assume that all of the section is effective in resisting flexural stresses and that only the web resists vertical shearing stresses. (Moment of inertia of the cross section, I = 12.5 x 106 mm4) (a) Locate the shear center of the cross section. (10 marks) (b) Determine the maximum shearing stress produced on the cross section by a vertical shear of8 kN. (10 marks) Useful formulas: Deflections and slo es of beams. (TY/W) - u = 3%“: — 3n: + x2) M Mr . X 0' = filmy - 6Lx + 3)?) A 4‘; _ r... _. M01. M01. GIVE—207A Final Examination December 15, 2005 p5 Useful formulas for section groaerties: Rectangle w E 7rd2 14:“? y'25 :7Tr2:—4-' —,_E ,Jfi _ wing" 2—2 "12 A‘_4k64 hh3 : 0 I 2 — I = 0 P 12 R 5'rrr4 S'ITd4 bh} h = z , IV. = T [F : r-(bz + if) 4 (’4 - 12 _ m4 _ 7T 4 ’ 2 _ 32 Semicircle A , wrz _, _ fl _ 2 Z 317 I g (93% - 64y“ } ‘ 7277 1.”,4 I, ; "é" I),Z * 0 1774 1 = M ...
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