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Unformatted text preview: EM 319
FINAL EXAM Tot. NAME: SESSION: December 18th 2007, 24:30 pm, PHR 2.110 Closed books and notes exam — Formula sheet &
superposition tables provided Exercise 1 (30 points) In order to estimate the stresses that develop in the piston of a car engine solely
due to compressing of the airfuel mixture, the following model is used. The
piston is idealized as an assembly of a solid aluminum core sheathed by a steel
sleeve. That assembly is packed between two rigid plates, as in the figure. As the
piston is displaced by an amount at to the right under the action of a force F, it compresses the gas mixture inside the cylinder (see figure). The pressdre and
the volume of the mixture obey the law meconst. Neglecting the friction between the piston and the walls:
a) Find the pressure inside the cylinder as a function of the piston travel, x.
b) Find the stresses in the two members (core and sleeve) as a function of
x.(Note: both members are in compression.)
0) Plot the stresses from b) as functions of x, both on the same graph.
d) What is the Safety Factor against yielding of the two members of the
piston? (hint: use the stresses at x = s ).
Given: total piston travel (stroke): 5 = 10 cm, height of volume that remains at the end of the piston stroke: t=1cm, piston diameter: d=8cm, core diameter:
dC=7cm, material properties: EC =70 GPa, ES =210 GPa, yield stresses:
Um =320 MPa,0YIS =500 MPa, initial pressure in the cylinder (i.e, when x=Ocm) p0 =1a‘i.‘m=1.01><105 Pa. oyi'inoler Exercise 2 (35 points) Two beams ABC and DF are connected through the cable BF and carry no other
load. The cable is cooled by an amount AT = —1300 C and shontens, thus Causing the two beams to bend. Neglecting the weights ofthe three members: a) Find the centroid of the crosssection of beam ABC given in the figure in
the right, and the moment of inertia about the z—axis (usually that's the 3rd
step, but we need it here this time). b) Draw the Free Body Diagrams of the three members (ABC, DF and BF)
separately. Then, find the reactions at A, C, and D and the tension of the
cable BF (hint: use superposition. You will probably need a clear picture of
the deflected system, to write down the appropriate superposition). 0) Draw the shear force and the bending moment diagrams of beam ABC,
labeling all critical ordinates (include the sign convention). 0]) Calculate the normal stresses that develop at points a,b,c,d and e on the
crosssection at point B of the beam ABC. The tocation of these points is
indicated in the sketch of the crosssection (point b is 1 cm above the z—
axis, and point d is 3cm below that axis). e) Calculate the maximum normal stresses (tensile and compressive) that
develop in the beam ABC (magnitude and location along the beam). f) Calculate the maximum shear stresses that develop in the beam ABC (magnitude and location along the beam).
Given: L=1 m, E=70GPa, A=1cm2, 0:23x10'6/OC éy A tom tom Exercise 3 {35 gointsl A sign of dimensions 2.0mx1.2m is supported by a hollow circular pole having
outer diameter 220 mm and inner diameter180 mm, as in the figure. The sign is offset 0.5m from the centerline of the pole, and its lower edge is 6.0 m above the ground.
a) Find the stress resultants that act at the base of he pole (i.e. the opposites
of the reaction forces).
b) Draw the stress elements at points A and B, properly showing the specific
xyz coordinate system of the figure. c) Determine the principal stresses and the maximum shear stresses (either
inplane or outofplane) at points A and B. 0.5 m ...
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