CN243 Mechanics of Materials
s
School of the Environment
Semester 2 Examinations 2006-2007
CN243
MECHANICS OF MATERIALS
Instructions to candidates
Time allowed: TWO hours
Answer FOUR questions in total as follows:
Attempt BOTH questions from Section A, and
Attempt any TWO questions from Section B.
All questions carry equal marks
The necessary formulae are provided with this examination paper.
Friday 1st June 2007 12:00 – 14:00 hours
Page 1 of 10
CN243 Mechanics of Materials
SECTION A
Answer BOTH questions from this section
Question 1
(a) A wooden beam of rectangular cross section (b = 100 mm, h = 150 mm)
is supported by two adjacent roof trusses Figure 1. The beam supports
the roof material; this imposes a maximum bending moment of M = 12
kNm onto the beam as shown in Figure 2.
(i) Find the moments Myand 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.
(3 marks)
(iii) Find the maximum compressive stress and the maximum tensile
stress in the beam.
(8 marks)
b = 100 mm
h = 150 mm
1
M = 12 kNm
2
 = 26.5
Figure 1
Figure 2
Question 1 continues on next page 
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CN243 Mechanics of Materials
Page 3 of 10
CN243 Mechanics of Materials
Question 1 (continued from previous page)
(a) A link of non-uniform cross-section is subjected to a tensile load P of
magnitude 150 kN as shown in Figure 3. Part AB is parallel to A′B′, part
BC is tapered and part CE is parallel to C’E’. The link is symmetric about
EE′.
A
B
60 mm
CE
P
15 mm
A’
B’
90 mm
30 mm
15 mm
P
30 mm
C’ E’
End elevation
Side elevation
Figure 3
Using only the given geometry, calculate and indicate where the
maximum normal stresses (tensile and compressive) will occur in the
link away from the loaded points.
(12 marks)
Page 4 of 10
CN243 Mechanics of Materials
Question 2
(a) The principal strains and stresses at a point on the surface of a tank are
1 = 1500*10-6 and 2 = 1000*10-6 , 1 = 396 MPa and 2 = 317 MPa,
respectively. Assuming a plane stress case, determine the corresponding
modulus of elasticity and Poisson’s ratio at a point in the same plane.
(6 marks)
(b) Use the state of stress shown in Figure 4 to calculate and sketch the
corresponding principal stresses and their orientation angle.
(5 marks)
y = 30 MPa
x = 60 MPa
 = -20 MPa
Figure 4
(c) A short concrete cylinder with a diameter of d = 200 mm is subjected to a
torque of T= 5000 Nm and an axial force of F as shown in Figure 5.
Based on the maximum principal stresses, determine whether the
cylinder fails if the axial force is F = 400 kN (compressive) or F = 100 kN
(tensile).
The ultimate compressive strength of the concrete is ult = 25 MPa and
the concrete can withstand a tensile stress of 2 MPa.
(14 marks)
T
F
F
x
T
Figure 5
Page 5 of 10
CN243 Mechanics of Materials
SECTION B
Answer ANY TWO questions from this section
Question 3
(a)
The wooden beam shown in Figure 6 is made by nailing together two
boards of dimensions 30 mm x 200 mm and 30 mm x 180 mm
respectively. The beam is subjected to a uniformly distributed load of
1.0 kN/m over BC and a concentrated load of 2.0 kN at C.
(i)
Calculate the vertical reactions and draw the shear force diagram
for the beam.
(3 marks)
(ii)
Calculate the position of the neutral axis and the second moment
of area of the beam cross-section.
(4 marks)
(iii)
Determine the maximum spacing of the nails in each of the
portions BC and CD if each nail can safely resist 1.2 kN in shear.
mm
(8 marks)
2.0 kN
200 mm
30
180 mm
1.0 kN/m
B
D
C
3.0 m
3.0 m
30
mm
Figure 6
Question 3 continues on next page 
Page 6 of 10
CN243 Mechanics of Materials
Question 3 (continued from previous page)
(b)
A rectangular rubber bearing ABCD in Figure 7 is subjected to a vertical
load Py and a horizontal load Px through a rigid steel plate in the x-y
plane. After load application it assumes the deformed shape shown by
the dashed lines.
y
Py
400 mm
A
B
20 mm
500 mm
B′
Px
C
x
C′
8 mm
D
Figure 7
(i)
Calculate the normal strain along the edges AB′ and DC′ and
diagonals B′D and AC′ of the deformed rubber bearing.
(4 marks)
(ii)
(c)
Calculate the maximum shear strain in the bearing.
(3 marks)
For the applied shear force V, draw shear flow patterns along the
segments of the thin-walled members shown in Figure 8.
(3 marks)
V
V
V
Figure 8
Page 7 of 10
CN243 Mechanics of Materials
Question 4
The tie CD in Figure 9 has a rectangular cross-section of 80 mm x 100 mm
and is pinned at C whilst being free at D. The tie is loaded in the x-y plane by
point loads Px = 40 kN and Py = 50 kN which are applied at the centroid of
the cross-section at D.
50 kN
y
D
40 kN
A
A
2.5 m
80 mm
100 mm
Section A-A
C
x
2m
Figure 9
(a)
Calculate the member internal force components with respect to the x-y
co-ordinate axis system.
(2 marks)
(b)
Calculate the average normal and shear stress components with
respect to the x-y co-ordinate axis system for a differential element
located away from the loaded points.
(8 marks)
(c)
Calculate the average normal stress in the longitudinal direction of the
tie using the resultant of the applied loads.
(2 marks)
(d)
Use Mohr’s circle to show that the normal and shear stress components
obtained for a co-ordinate axis system x1 – y1 with the x1-axis oriented
along CD agree with the direct calculation in (c) above.
(13 marks)
Page 8 of 10
CN243 Mechanics of Materials
Question 5
(a)
A tensile test is carried out on a specimen as shown in Figure 10. The
applied load and measured elongation of the specimen are given in
Table 1. Assume that the material reacts in linearly elastic – perfectly
plastic manner,
Table 1
L = 80 mm
P
Load
(kN)
P
d = 12 mm
Figure 10
0
11.30973
33.9292
39.58407
39.58407
Elongation
(mm)
0
0.048
0.144
0.192
0.560
(i) Calculate the Young’s modulus of the material.
(2 marks)
(ii) Determine the yield stress and the yield strain of the material
(3 marks)
(iii) After applying the maximum load, the load is released. Determine
the final length of the specimen.
(4 marks)
Question 5 continues on next page 
Page 9 of 10
CN243 Mechanics of Materials
Question 5 (continued from previous page)
(b) The stepped aluminium rod shown in Figure 11 has a circular cross
section and is fixed to a wall at point C. The rod is subjected to axial
loads P1=25 kN and P2=12 kN as shown id Figure 11. Determine the
strain energy density in the rod if E = 70 GPa.
(5 marks)
A
B
C
P2 = 12 kN
P1 = 25 kN
25 mm
400 mm
10 mm
600 mm
Figure 11
(c) The thin pipe shown in Figure 12 has a length L = 4.0 m. It has a
triangular cross-section (equilateral triangle, length of sides s = 200 mm,
wall thickness t = 5 mm).
Determine the maximum torque T which the pipe can resist, if the
maximum allowable shear stress is max = 100 MPa and the tube is
restricted to twist no more than  = 0.0025 radians.
The shear modulus is G = 75 GPa.
(11 marks)
60
t = 5 mm
L = 4.0 m
T
s = 200 mm
Figure 12
Page 10 of 10