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SCHOOL OF COMPUTING, ENGINEERING AND MATHEMATICS
SEMESTER 2 EXAMINATIONS 2011/2012
MATERIALS ENGINEERING
ME334
DR. E.M. SAZHINA
Time allowed:
TWO hours
Answer:
Any FOUR questions out of SIX
Each question carries 25 marks
Items permitted:
Any approved calculator
Items supplied:
Formulae sheet (attached: pages 9 - 11)
Marks for part of the questions are indicated in brackets ( )
May/June 2012
Page 1 of 11
Question 1
A steel vessel is under cyclic loading of high internal pressure. A fluctuating tensile
stress varies in a cyclic manner from zero to 290 MPa. The vessel is made of steel
alloy with a value of yield stress equal to 360 MPa.
The results of fatigue testing for the steel alloy, with a zero static component, are
shown below in Figure Q1:
S-N curve
Stress amplitude (MPa)
450
400
350
300
250
200
150
100
2
3
4
5
6
7
8
9
Log N
Figure Q1
(a) Describe the setup of the rotating-bending beam experiment for fatigue testing,
and the concept of an S - N plot.
(5 marks)
(b) Explain the influence of corrosion on the fatigue limit for a ferrous specimen.
(5 marks)
(c) Estimate the fatigue limit of the pressurised vessel using the Soderberg criterion
for cyclic stress with a non-zero static component.
(5 marks)
(d) Describe the concept of auto-frettage, and its benefits for increasing fatigue life
of a pressurised vessel.
(10 marks)
ME334 (2011/2012)
Page 2 of 11
Question 2
A pressurised vessel is designed as a cylinder under internal pressure of 50 MPa
with inner radius R1 = 250 mm and outer radius R2 = 330 mm. The external pressure
is normal atmospheric.
The yield stress (elastic limit) of the material is 300 MPa.
500 mm
660 mm
Figure Q2
(a) Calculate the factor of safety at the critical point under the thick-walled cylinder
approximation. Briefly explain your selection of the critical point location under
Tresca failure theory.
(10 marks)
(b) The pressure vessel is redesigned as a cylinder of the same dimensions under
internal pressure of 50 MPa, whilst an external pressure of 20 MPa is applied by
shrinking a compound cylinder onto it. The inner radius is R1 = 250 mm and the
outer radius R2 = 330 mm. Calculate the factor of safety at the critical point
under Tresca failure theory.
(10 marks)
(c) Explain the benefits of the compound-cylinder technique for enhancing the
safety of the working cylinder.
(5 marks)
ME334 (2011/2012)
Page 3 of 11
Question 3
Stress analysis at a point in the exhaust pipe of a micro-light aircraft gave the
following system of stresses:
s x = 28 MPa
s y =150 MPa
s z = 62 MPa
 xy   yz   zx  0
(a) Describe the concept of octahedral planes and stresses and calculate the
octahedral normal and shear stresses for this stress system.
(5 marks)
(b) Explain the derivation of the von Mises failure theory using the concept of
‘octahedral shear stress’ and calculate the factor of safety by von Mises failure
criteria. Take the yield stress to be equal to 240 MPa.
(10 marks)
(c) Describe the concepts of creep and creep testing and how the Larson-Miller
method is used for predicting the long-term creep behaviour from data obtained
over relatively short periods of time. Calculate the time to rupture (in days) for
steel at a temperature of 1000 K if the Larson-Miller parameter is equal to
23 x 103 under applied stress. Take C = 20.
(10 marks)
ME334 (2011/2012)
Page 4 of 11
Question 4
(a) A torque T = 75 Nm is applied to a thin-walled steel tube having a rectangular
cross-section 12 mm by 72 mm. The tube is of constant wall thickness of 4 mm
as shown in Figure Q4a. All dimensions are given relative to the median line in
the wall of the tube. Explain the concept of shear flow and calculate the shear
stress in the tube wall.
12 mm
72 mm
Figure Q4a: Cross-section of a thin-walled tube under torsion
(5 marks)
(b) Sketch Mohr’s circle of stress for an element in the tube wall assuming a pure
shear state of stress. Calculate the factor of safety by Tresca failure theory
assuming that the yield stress for the tube material is equal to 180 MPa.
(10 marks)
(c) A torque T = 75 Nm is applied to a solid bar having a rectangular cross-section
as shown in Figure Q4b. Indicate the location of maximal stress and calculate
its value.
t = 12 mm
b = 72 mm
Figure Q4b: Rectangular section of a solid bar under torsion
(10 marks)
ME334 (2011/2012)
Page 5 of 11
Question 5
An aircraft component is made of an alloy with plane strain fracture toughness
K Ic = 50 MPa m . A non-destructive test has established that the maximal internal
crack half-length is a = 3.0 mm.
(a) Explain the concepts of stress intensity factor and the plane strain fracture
toughness in Mode I as a material property.
(10 marks)
(b) Calculate the stress intensity factor under an applied tensile stress of 200 MPa
assuming that Y = 1.2. Will the component fracture?
(5 marks)
(c) A large steel bar under tension is drilled with a transverse circular hole of
diameter 20 mm as shown in Figure Q5a. The nominal tensile stress is 30 MPa.
Two smaller relief holes are drilled in close proximity to the original hole as
shown in Figure Q5b. Calculate the maximal stress in both cases, by assuming
that the stress distribution can be approximated by an elliptical hole with
a = 20 mm and b = 30 mm following the notations of Figure Q5b. Explain this
method of stress relief by using the following expression:
a
b
 m   o (1  2 )
2a
2a
2b
Fa
tig
ue
lim
it
30 MPa
Figure Q5a
30 MPa
Figure Q5b
(10 marks)
ME334 (2011/2012)
Page 6 of 11
Question 6
(a) The data from a series of Charpy impact tests on ductile cast iron are shown in
Figure Q6a. Explain the setup of the Charpy test and estimate the ductile-tobrittle transition temperature from the plot.
Figure Q6a
(10 marks)
(b) A large sheet of glass is under a tensile stress of 6 MPa. It is known that there
is a central slit in the middle of the glass sheet as shown in Figure Q6b.
Determine the maximum crack length that is possible without fracture. Assume
that the modulus of elasticity E is 60 GPa and that the specific surface energy
GC = 0.5 Jm-2 for the glass.
2a
Figure 6b
(5 marks)
Question 6 continues on the next page
ME334 (2011/2012)
Page 7 of 11
Question 6 (continued)
(c) A cylindrical vessel has an internal diameter D1 = 180 mm. The outer diameter
is D2 = 204 mm as shown in Figure Q6c:
D2 = 204 mm
D1 = 180 mm
Fig. Q6c
The internal pressure in the vessel is 3 MPa. The vessel is made of a steel alloy
with a yield stress of 210 MPa and a plane strain fracture toughness given by:
K Ic = 50 MPa m
Calculate the upper limit for tensile stress in the vessel walls for the leak-beforebreak (LBB) condition and compare it with the yield stress. Assume that Y = 1.1
for this geometry.
(10 marks)
ME334 (2011/2012)
Page 8 of 11
ME334 MATERIALS ENGINEERING – LIST OF EQUATIONS
All the symbols have their usual meaning
Axisymmetric stress systems
Stresses in thick-walled cylinders under pressure:
H  A
B
r2
L  A
;
;
r  A
B
r2
The constants A, B are given by
P1 R12  P2 R22
A
( R22  R12 )
B
P1  P2
R12 R22
2
2
( R2  R1 )
Thin-walled cylinders under internal pressure:
H 
pR1
t
Bending:
L 

y

pR1
2t
where R1 is the inner radius
M E

I
R
I   y 2 dA
Second moment of area:
A
For a rectangle about N.A.:
For a circle about diameter:
bd 3
12
 r4
I
4
I
Torsion of circular sections:
T 

Ip
r
; 0 < r < R where τ is the shear stress produced by torque T,
I p is the polar second moment of area
Solid circular shaft:
Ip 
Hollow circular shaft: I p 
ME334 (2011/2012)
 R4
2

2
R
4
o
 Ri4

Page 9 of 11
Torsion of non-circular sections


Thin-walled closed sections:
T  2At
Thin-walled open sections:
 max 
T
k1bt 2
1.5
1.75 2.0
2.5
3.0
4.0
6.0
8.0
10.0
b/t 1.0

k1 0.208 0.231 0.239 0.246 0.258 0.267 0.282 0.299 0.307 0.313 0.333
Fracture mechanics.
Griffith’s equation:  a  2 EGc
2
Stress intensity factor K  Y a
Creep
Larson-Miller parameter (K-h): P1  T (C  log t r )
Fatigue
n
n
n1 n2

 3  ...  i  ..  1
N1 N 2 N 3
Ni

Soderberg criteria  a   N (1  m )
 yield
Miner’s law
Failure theories
Factor of safety SF=  yield /  E ;
Tresca failure criterion:
 E   1   3 where  1   2   3

Von Mises failure criterion:  E  0.5 ( 1   2 ) 2  ( 2   3 ) 2  ( 3   1 ) 2

Principal stresses in plane stress (2D case):
 1 ,2 
x  y
2

ME334 (2011/2012)
1
2
( 
x
  y )2  4 xy2 
Page 10 of 11
Three-dimensional stress and strain analysis
Normal and shear stresses on the octahedral planes
 oct 
1   2   3
3
ME334 (2011/2012)
 oct 
1
3
 1   2  2   2   3  2   3   1  2
Page 11 of 11