CN213 Mechanics of Materials 2
s
School of Environment & Technology
Semester 1 Examinations 2011-2012
CN213
MECHANICS OF MATERIALS 2
Instructions to Candidates
Time allowed: TWO hours
Answer ALL questions from Section A
Answer TWO questions from Section B
All questions carry equal marks
An Aide Memorie is provided
26 January 2012, 15:00hours
CN213 Strength of Materials 2 2011-12
SECTION A
Answer BOTH questions from this section
Question 1
(a) A hollow beam of rectangular cross section as shown in Figure 1 is subjected to
a bending moment of M = 5 kNm at an angle of 25 to the vertical principal axis.
(i) Find the moments My and Mz parallel to the principal axes of the beam.
(2 marks)
(ii) Find the neutral axis of the beam. Draw a sketch of the cross section of the
beam showing the neutral axis.
(5 marks)
(iii) Determine the stresses in points A and B. Indicate whether these stresses
are compressive or tensile.
(10 marks)
Figure 1
(b)
Using the strain gauge rosette shown in Figure 2derive an equation from the
strain transformation equation to determine the strains x, y and xy.
(8 marks)
y
C
B
45º
A
x
Figure 2
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CN213 Strength of Materials 2 2011-12
Question 2
(a)
A steel plate has a welded connection at an angle of 20˚ to the x-axis as shown
in Figure 3. It is subjected to a state of stress as shown in Figure 3.
(i)
From the state of stress as shown in Figure 3, construct the Mohr circle of
the stress situation in the steel plate.
(6 marks)
(ii)
Use Mohr’s circle to determine the in-plane shearing stress parallel to the
weld and the normal stress perpendicular to the weld.
(7 marks)
(iii)
Draw the state of stress at the weld in a properly oriented stress element.
(5 marks)
Figure 3
(b)
A steel plate as shown in Figure 4 is subjected to a force P = 75 kN. Determine
the minimum thickness t required if the allowable stress is 200 MPa.
(7 marks)
Figure 4
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CN213 Strength of Materials 2 2011-12
SECTION B
Answer TWO questions from this section
Question 3
(a)
The thin walled pipe is made up out of three steel plates that are welded
together as indicated in Figure 5 below. The cross-section of the pipe is an
equilateral triangle; the length of each side is 175 mm, the wall thickness is
t = 5 mm.
(i)
Determine the maximum torque T the pipe can resist, if the maximum
allowable angle of twist is  allow = 0.25°.
(G = 75 GPa, length L = 4 m)
(4 marks)
(ii)
Consider that one welded seam breaks due to poor workmanship along
the whole length of the beam.
Determine the angle of twist (in degrees) for a torque of T = 0.25 kNm.
(3 marks)
(iii)
Following on from (ii), assuming the thickness of one of the sides of the
triangles was changed to 6mm, state if you expect the maximum shear
stress in a side of thickness 5mm or 6mm and explain why.
(2 marks)
Figure 5
Question 3 continues on next page 
Page 4 of 12
CN213 Strength of Materials 2 2011-12
Question 3 (continued from previous page)
(b)
A cantilever beam made of channel section is subjected to a load P = 5 kN as
shown in Figure 6.
(i)
Determine the position of the shear centre of the channel section.
(4 marks)
(ii)
Determine the torque T applied to the section by the load P.
(2 marks)
(iii)
Assuming that the torque T = 350 Nm, determine the maximum shear
stress in the section.
(10 marks)
P
50 mm
P
8 mm
z
160 mm
12 mm
100 mm
Figure 6
Page 5 of 12
CN213 Strength of Materials 2 2011-12
Question 4
(a)
A wooden beam is made by gluing three 50mm × 100mm boards together to
form a solid beam as shown in Figure 7. The beam is loaded symmetrically
with a point load P = 1.5 kN.
Ignore the weight of the beam.
(i)
Determine if the shear stress in the glue is going to exceed the limiting
value of  = 0.25 MPa.
(8 marks)
(ii)
Determine if the tensile stress in the wood is going to exceed the limiting
value of  = 4.5 MPa.
(8 marks)
P
50 mm
50 mm
50 mm
L = 3.0 m
100 mm
Figure 7
(b)
A steel shaft with a diameter of 43 mm has a yield strength of Y = 300 MPa.
It is subjected to the axial load P = 350 kN and a torque T = 625 Nm as shown
in Figure 8.
Determine the factor of safety against yielding using the maximum principal
stress criterion.
(9 marks)
Figure 8
Page 6 of 12
CN213 Strength of Materials 2 2011-12
Question 5
(a)
A circular shaft as shown in Figure 9 is fixed at point C and subjected to a
torque at point A.
Part AB is made of steel with an allowable shear stress of steel = 100 MPa.
Part BC is made of aluminium with an allowable shear stress of alu = 70 MPa.
The diameter of part BC is 45 mm.
Ignore any stress concentrations in the shaft.
(i)
Determine the largest torque T that can be applied to the shaft without
exceeding the allowable stress in part BC.
(4 marks)
(ii)
Determine the diameter of part AB that is required to withstand a torque T
of 1300 Nm
(4 marks)
(iii)
Determine the angle of twist (in degrees) at point A for the same torque as
in (ii) assuming the diameter of section AB to be 40 mm. The shear
moduli for the materials are: Gsteel = 78 GPa and Galu = 26 GPa.
(4 marks)
Figure 9
Question 5 continues on next page 
Page 7 of 12
CN213 Strength of Materials 2 2011-12
Question 5 (continued from previous page)
(b)
A H-beam is made of wood by screwing together three wooden boards as
shown in Figure 8a. The beam is subjected to two point loads as shown in
Figure 8b.
The screws are 60mm apart along the length of the beam; the shear strength of
each screw is 4 kN.
Assuming that the beam will fail due to shear failure of the screws determine
the maximum load P the beam can safely resist.
(13 marks)
P/2
140 mm
P/2
20 mm
20 mm
100 mm
20 mm
1.2 m
Figure 10a
1.2 m
1.2 m
Figure 10b
Page 8 of 12
CN213 Aide-Memoire Mechanics of Materials 2
CN213 Mechanics of Materials AIDE-MEMOIRE 2011-2012
Page 9 of 12
CN213 Aide-Memoire Mechanics of Materials 2
STRESS AND STRAIN TRANSFORMATION
Stress transformation
 x1 
 y1 
x  y
2
x  y
 x1y1  

x  y
2
x  y

2
x  y
2
2
cos 2   xy sin 2
1,2 
tan 2p 
cos 2p 
sin 2p 
2
sin 2   xy cos 2
x  y 
max,min
2
 x   y 
2
Principal strains
2  xy
1,2 
1  2
2
x  y
2
tan 2p 
2
 x  y 
 
   xy 
 

 2 
 2 


2
 xy
( x   y )
2
 x  y 
   2xy
R  

2


cos 2p 
Strain transformation
 x1 
 xy
R
 x  y 
 
   xy 
R  

 2 
 2 


Mohr’s circle for plane stress
x  y
2R
2
 x  y 
  2xy
  
 2 
tan 2 s  
max 
 x  y 
   xy 2
 

2


2 xy
2
cos 2   xy sin 2
Principal stresses
x  y
Mohr’s circle for plane strain
Maximum shear stress
 y1 
 x1y1
2
x  y

2
x  y
2


x  y
2
x  y
x  y
2
2
sin 2p 
cos 2 
cos 2 
sin 2 
 xy
2
 xy
2
 xy
2
sin 2
sin 2
cos 2
x  y
2R
 xy
2R
Maximum shear strain
2
 x  y 
 
 max
   xy 
 

 2 
2
 2 


tan 2s  

x
2
 y 
 xy
Page 10 of 12
CN213 Aide-Memoire Mechanics of Materials 2
CENTROIDS AND MOMENTS OF AREA
TORSION
Centroid
Shear stress in a circular bar under torsion
x A
x
A
i
y A
y
A
i
i
i
i
i
i
i
i
i
z A
z
A
i
i
i
 zi A i
Qz 
i
i
 yi A i
bh3
Iz 
12
Parallel – axis therorem of moment of inertia
I y1  I y  A d2z
I z1  I z  A d2y
Polar moment of inertia of a circle (axis through centroid)
IP 
d 4
32
Angle of twist of a circular bar under torsion

TL
G Ip
i
Moment of inertia (axes through centroid) in
z-y plane
hb3
Iy 
12
TR
IP
i
First moment of area in z-y plane
Qy 
max 
Shear stress in thin walled closed section under torsion
aver 
T
2 t Am
Angle of twist of thin walled closed section under torsion

TL
2
4A m
G
 tii
s
i
Shear stress in thin walled open section under torsion
max 
T t max
J
with
J
1
bi t i3

3 i
Angle of twist of thin walled open section under torsion

TL
GJ
Page 11 of 12
CN213 Aide-Memoire Mechanics of Materials 2
HOOKE’S LAW
SHEAR STRESS IN BEAMS


x 
1
 x  ( y   z )
E
y 
1
 y  ( x   z )
E
z 

Shear stress in beams




1
 z  ( x   y )
E
V Qz
Iz b
Shear stress in beams with thin walled cross-sections

V Qz
Iz t
Shear flow in beams
BENDING
q
Normal stress due to bending
 x1  
Mz y
,
Iz
 x2 
My z
Iy
Normal stress due to unsymmetrical bending
x 
Px M z y M y z


A
Iz
Iy
Orientation of the neutral axis
tan  
I z My
V Qz
Iz
Shear centre of channel sections
V t f hb2
F
4 Iz
Fh
e
V
YIELD CRITERIA
The effective stress for the maximum distortion energy
criterion (von Mises)
I y Mz
e 

1
1   2 2   2   3 2   3  1 2
2

Page 12 of 12