Jan 2018 Mech4
Jan 2018 Mech4
Jan 2018 Mech4
TALLAGHT
Bachelor of Engineering (Honours)
Bachelor of Engineering (Ab Initio)
Mechanical Engineering
Full Time
Mechanics 4
Internal Examiners
Mr S. Tiernan
External Examiners
Mr P. McCormick
Day: Tuesday
Date: 09/01/2018
Time: 9.30 to 11.30am
Instructions to Candidates
Question 1 25 marks
(a) A simply supported wooden beam is loaded centrally with weights and the
central deflection measured (figure Q1(a)). The results are shown in figure
Q1(b). The beam has a rectangular cross section 25mm x 5mm. Calculate
the Modulus of Elasticity (E) of the wood.
FL3
Deflection
48EI
Deflection Measured
0.25m
Loads
Reaction Reaction
A 0.5m
B
(6 marks)
(b) Two plastic strips (2mm x 25mm) are now bonded onto each side of the
wood as shown in Figure Q1(c). Using the method of transformation
calculate the deflection of the composite beam when the central load is
40N.
2mm
Plastic
5mm Wood
2mm 25mm
Figure Q1(c)
(8 marks)
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Bachelor of Engineering in Mechanical Engineering Academic Year 2017/2018
Semester 5 Jan 2018 Mechanics 4
(c) Calculate the maximum stress in the wood and in the plastic, of the
composite beam, when the 40N load is applied.
(8 marks)
(d) Calculate the factor of safety in the wood and in the plastic given that the
yield stress of the wood is 10 MPa and the yield stress of the plastic is 30
MPa. If the beam is loaded excessively how will it fail?
(3 marks)
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Bachelor of Engineering in Mechanical Engineering Academic Year 2017/2018
Semester 5 Jan 2018 Mechanics 4
Question 2 25 marks
(a) A simply supported beam is as shown below (figure Q2(a)). It has a central
load W.
L
X /2
W
W
x
L
L
Figure Q2(a)
Derive the equation below for the displacement of the beam, at a distance x:
1 Wx3 WL2 x
y
EI 12 16
(8 marks)
WL3
(b) Hence show that the maximum deflection is y
48EI
(2 marks)
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Bachelor of Engineering in Mechanical Engineering Academic Year 2017/2018
Semester 5 Jan 2018 Mechanics 4
(i) Write down an equation in terms of x for the bending moment about
‘XX’. ‘x’ is the distance measured for the left hand side of the beam.
(3 marks)
(ii) Using Macaulay’s method derive an equation for the deflection of the
beam at a distance x from the left hand side.
(8 marks)
(iii) Calculate the deflection of the beam, shown in figure Q2(b), 2m from
the left hand end.
(4 marks)
Given that:
d2y
BM xx EI
. dx 2
X
30kN 30kN
2m
4m
6m
RB
RA x
X
Figure Q2(b)
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Bachelor of Engineering in Mechanical Engineering Academic Year 2017/2018
Semester 5 Jan 2018 Mechanics 4
(a) Draw a sketch of a cylinder showing the directions of the hoop, longitudinal
and radial stresses.
(3 marks)
(b) A pressurised thin boiler cylinder with a diameter of 0.5m and wall
thickness of 30mm. The internal pressure is 2 MPa.
(i) Calculate the hoop and longitudinal stresses in the cylinder.
(ii) If the cylinder is manufactured from a material with a yield strength of
160MPa, calculate the factor of safety.
(iii) Is this an appropriate factor of safety if it contains an inflammable gas?
(6 marks)
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Bachelor of Engineering in Mechanical Engineering Academic Year 2017/2018
Semester 5 Jan 2018 Mechanics 4
Question 4 25 marks
(i) The principal stresses and the angles at which they act.
(ii) The max shear stress and the angle of the plane on which it acts.
(iii) Von Mises Stress
(iv) The normal and shear stresses on a plane at 150 to the horizontal.
(12 marks)
2
30MN/m
2
40MN/m
2
20MN/m
Figure Q4
(b) Given that the shaft is manufactured from aluminium with a yield stress of
270MPa. Will the component fail and determine the factor of safety based
on the following theories:
(i ) Max shear stress theory.
(ii ) Von Mises theory.
(iii ) Max principal stress theory.
(7 marks)
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Bachelor of Engineering in Mechanical Engineering Academic Year 2017/2018
Semester 5 Jan 2018 Mechanics 4
Question 5 25 marks
(a) Describe what is meant by buckling and when it should be considered in
mechanical designs.
(4 marks)
(b) A steel framed building is to use I beams as vertical columns. Each column
is to support 100kN and is 8m high. Both ends of the beam may be
regarded as fixed. A factor of safety of 6 is to be used.
(i) If the beam will fail by buckling, calculate the minimum second moment
of area (I) required.
(5 marks)
(ii) Select a suitable steel column from the table below.
(3 marks)
(iii) What load will cause this column to yield in direct compression?
(4 marks)
(iv) Calculate the buckling loads if the column is 4, 6 and 8m long.
(4 marks)
(v) Sketch a graph of the load versus column length. Mark on this graph the
failure lines due to buckling and pure compression. Explain what this
graph means in terms of column length and failure load.
(5 marks)
Note: For steel E= 200 GPa and the yield stress = 420 MPa
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Bachelor of Engineering in Mechanical Engineering Academic Year 2017/2018
Semester 5 Jan 2018 Mechanics 4
M E T G
Bending Torsion
I y R J r L
𝒅𝟐 𝒚
Slope & Deflection 𝑀 = 𝐸𝐼
𝒅𝒙𝟐
Hoop Strain H
1
H L Longitudinal strain L
1
L H
E E
H
Hoop Stress H E H 2 L Longitudinal Stress L E L
1 1
2
Thin Cylinders
PD PD
Hoop Stress H Longitudinal stress L
2t 4t
Pd 2
Change in Length L
Pd
1 2 L Change in Diameter d 2
4tE 4tE
Change in Volume V
Pd
5 4 V Bulk Modulus K
PV
4tE V
B B
Radial Stress r A Hoop stress H A
r2 r2
Pr P r
2 2
Longitudinal stress L 1 1 2 22 2
r2 r1
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Bachelor of Engineering in Mechanical Engineering Academic Year 2017/2018
Semester 5 Jan 2018 Mechanics 4
Stresses (2D)
x y x y
2
Principle Stresses 1 , 2 xy2
2 2
2 xy
tan 2 P
Angle of one Principle Plane
x y
y x y
n Cos 2 xy Sin 2
x
2 2
x y
n Sin 2 xy Cos 2
2
Buckling
L pin pin
0.7 L fixed pin
Le
0.5L fixed fixed
2 L fixed free
2 EI
Critical Load PCrit
L2e
𝐼 = 𝐴𝑟 2 , r = radius of gyration
𝐿
𝑆𝑙𝑒𝑛𝑑𝑒𝑟𝑛𝑒𝑠𝑠 𝑅𝑎𝑡𝑖𝑜 =
𝑟
𝑏𝑑 3
Rectangle about mid line: 𝐼=
12
𝜋𝑑 4
Circle about diameter: 𝐼=
64
𝜋𝑑 4
Circle about axis: 𝐼=
32
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