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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. CIVE207A' 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 lefthand support, A, has settled a
distance 50 below the righthand 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 midspan 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. CIVE207A 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 100mmdiameter shaft is loaded on lefthand side free end and ﬁxed to the wall section ABCD on righthand 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
semicircular 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 CIVE207A 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 CIVE207A 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 thinwalled 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 ThinWalled 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 inplane 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 = Elky _ 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 GIVE207A 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.
 Winter '09
 Shao

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