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Unformatted text preview: McGill University
Faculty of Engineering SOLID MECHANICS
CIVE207A01 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 CIVE207A 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 yaxis (see Fig. 1). The height of the post is 25 ft, and its section modulus is S : 10 ms. [Useful formulas: tan(29p)=21xy/(oxoy) & S = I/r, where I
is moment of inertia and r is outer radius] (a) Calculate the maximum tensile stress cm, and maximum inplane 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 inplane 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. GIVE207A 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 inplane shear stress is 78 MPa, and one of the two inplane 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 inplane principal stress. Show the principal stresses on a sketch of a properly oriented stress element. Fig. 4: CIVE207A 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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