UNIVERSITY OF KWAZULU-NATAL
MECHANICAL ENGINEERING
EXAMINATIONS: JUNE 2015
SUBJECT, COURSE AND CODE: STRENGTH OF MATERIALS 2: ENME3ST
DURATION: THREE HOURS
TOTAL MARKS: 100
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1.
Internal Examiners:
S. Adali
S. Davrajh
Independent moderator:
J. Padayachee
There are six questions. ANSWER ONLY FIVE QUESTIONS. You may choose which
of the five questions you would like to answer. If in the event you choose to answer all
six questions, only the first five answered will be marked. Each question carries 20
marks.
2.
Marks will be assigned only where candidates express themselves clearly and legibly.
3.
If you consider that any information is unclear or missing, state clearly any
assumptions necessary and complete the question.
4.
Refer to the formulas on pages 7-11 where necessary.
NOTE: This exam paper consists of 11 pages. Please see that you have all of them.
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UNIVERSITY OF KWAZULU-NATAL, EXAMINATIONS: MAY-JUNE 2015
SUBJECT, COURSE AND CODE: STRENGTH OF MATERIALS 2: ENME3ST
QUESTION 1
(20 marks, sections a, b = 10 marks )
a) The simply supported beam is subjected to moments M  40 kNm at the supports A
and B as shown in the figure. The uniformly distributed load on the beam is
q0  8 kN/m.
Determine the deflection of the mid-point C using the method of integration.
Use the following data: E  200 GPa, I  129106 mm4
b) The statically indeterminate beam ACB shown in the figure has a moment M 0 acting
on it at the mid-point C as shown in the figure.
Use the method of superposition to determine the reaction forces R A and RB at
supports A and B, and the reaction moment M A at support A.
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UNIVERSITY OF KWAZULU-NATAL, EXAMINATIONS: MAY-JUNE 2015
SUBJECT, COURSE AND CODE: STRENGTH OF MATERIALS 2: ENME3ST
QUESTION 2
(20 marks)
The frame shown in the figure below has a constant bending stiffness of EI . The frame
is subjected to a distributed load of q0 and a concentrated load P at point B as shown in
the figure. Using the unit load method:
a) Determine the vertical displacement of point C.
b) Determine the slope at the point C.
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QUESTION 3
(20 marks)
The bent rod has a diameter of 15 mm and is subjected to a force of 600 N as shown in the
figure. Determine the principal stresses and the maximum in-plane shear stress that are
developed at point A and point B. Also, determine the principal stresses and ABSOLUTE
maximum shear stress at point C shown in the diagram on the right. Show the results on
properly oriented elements located at these points.
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UNIVERSITY OF KWAZULU-NATAL, EXAMINATIONS: MAY-JUNE 2015
SUBJECT, COURSE AND CODE: STRENGTH OF MATERIALS 2: ENME3ST
QUESTION 4
(20 marks)
A pressurized cylindrical tank with flat ends is loaded by torques T and tensile forces P
(see the figure below). The tank has an inner radius of r  125 mm and wall thickness
t  6.5 mm. The internal pressure p  7.25 MPa and the torque T  850 Nm.
(a)
What is the maximum permissible value of the forces P if the allowable tensile
stress in the wall of the cylinder is 80 MPa?
(b)
[10]
If force P  114 kN, what is the maximum acceptable internal pressure in the
tank if the allowable tensile stress in the wall of the cylinder is 80 MPa?.
[10]
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UNIVERSITY OF KWAZULU-NATAL, EXAMINATIONS: MAY-JUNE 2015
SUBJECT, COURSE AND CODE: STRENGTH OF MATERIALS 2: ENME3ST
QUESTION 5
(20 marks)
a) The horizontal beam ABC shown in the figure is supported by column BD and CE.
The beam is prevented from moving horizontally by the pin support at end A. Each
column is pinned at its upper end to the beam, but at the lower ends, support D is a
guided support and support E is pinned. Both columns are solid steel bars (E = 200
GPa) of square cross section with width equal to 16 mm. A load Q acts at distance a
from column BD. Assume failure only by buckling.
1. If the distance a =0.5 m, what is the critical value Qcr of the load? [8]
2. If the distance a can be varied between 0 and 1 m, what is the maximum possible
value of Qcr? What is the corresponding value of the distance a? [6]
Question 5 b overleaf
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UNIVERSITY OF KWAZULU-NATAL, EXAMINATIONS: MAY-JUNE 2015
SUBJECT, COURSE AND CODE: STRENGTH OF MATERIALS 2: ENME3ST
b. Beam ACB has a pin support at A and is supported at C by a steel column with square
cross section ( E  190 GPa, b  42 mm ) and height L  5.25 m. The column is pinned
at C and fixed at D. The column must resist a load Q at B with a factor of safety 2.0
with respect to the critical load. What is the maximum permissible value of Q ?
[6]
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QUESTION 6
(20 marks, sections a, b = 10 marks each)
A lever subjected to a downward force of F  1800 N. The lever is keyed to a round bar with
a diameter of d  25 mm as shown in the figure. Other dimensions are shown in the figure.
Yield stress of the material is  ys  900 MPa.
a) Determine the safety factor of the shaft with respect to Tresca failure criterion.
b) Determine the safety factor of the shaft with respect to Von Mises failure criterion.
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UNIVERSITY OF KWAZULU-NATAL, EXAMINATIONS: MAY-JUNE 2015
SUBJECT, COURSE AND CODE: STRENGTH OF MATERIALS 2: ENME3ST
APPENDIX - FORMULAS
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DEFLECTION OF BEAMS
d 4v
 q ( x)
dx 4
Load equation:
EI
Shear-force equation:
d 3v
EI 3  V ( x)
dx
Bending-moment equation:
EI
d 2v
 M ( x)
dx2
BEAMS/SHAFTS
Normal stress:

Bending stress:
 
Shear stress due to torsion:

P
A
My
I
Tr
J
Maximum shear stress in a rectangular cross-section due to shear load V :

3V
2A
Maximum shear stress in a circular cross-section due to shear load V :

4V
3A
Circular cross-sections:
I
 r4
J
4
 r4
A   r2
2
PRESSURE VESSELS
pr
2t
Spherical Pressure Vessels:

Cylindrical Pressure Vessels:
Circumferential stress:
c 
pr
t
Longitudinal (axial) stress:
a 
pr
2t
BUCKLING
Pcr 
 2 EI
(KL ) 2
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UNIVERSITY OF KWAZULU-NATAL, EXAMINATIONS: MAY-JUNE 2015
SUBJECT, COURSE AND CODE: STRENGTH OF MATERIALS 2: ENME3ST
ENERGY METHOD
Strain energy for bending
M2
U 
dx
2 EI
Castigliano’s theorem
 M 2 dx
i 
Fi  2 EI
Unit load method

 M 2 dx
i 
M i  2 EI
Mm
dx
EI
STRESS/STRAIN EQUATIONS
Principal Stresses:
 1 , 2 
Maximum shear stress:
 max 
 x  y
2
 x  y
 
2

2

2
   xy


 max   min
2
FAILURE THEORIES
Maximum stress criterion:
Tresca criterion:
Von Mises criterion:
max  
 yield
Sf
  yield

2
max   
Sf




 VM   12   1 2   22

1/ 2

 yield
Sf
1/ 2
 VM
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 (   2 ) 2  ( 2   3 ) 2  ( 3   1 ) 2 

  1

2



 yield
Sf
UNIVERSITY OF KWAZULU-NATAL, EXAMINATIONS: MAY-JUNE 2015
SUBJECT, COURSE AND CODE: STRENGTH OF MATERIALS 2: ENME3ST
SUPERPOSITION TABLES
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UNIVERSITY OF KWAZULU-NATAL, EXAMINATIONS: MAY-JUNE 2015
SUBJECT, COURSE AND CODE: STRENGTH OF MATERIALS 2: ENME3ST
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UNIVERSITY OF KWAZULU-NATAL, EXAMINATIONS: MAY-JUNE 2015
SUBJECT, COURSE AND CODE: STRENGTH OF MATERIALS 2: ENME3ST
STRESSES ON AN INCLINED SECTION
 x1 
 y1 
 x  y
2
 x  y
2
 x1y1  


 x  y
2
 x  y
2
 x  y
2
cos 2   xy sin 2 ,
cos 2   xy sin 2
sin 2   xy cos 2
PRINCIPAL STRESSES AND MAXIMUM IN-PLANE SHEAR STRESS
 1 , 2 
 x  y
 x  y
 
2

2
2
2

2
   xy
,


 x  y 
2
   xy

2


 max(in  plane)  
 xy
 x  y 



 x  y


2
, tan 2 s   
ABSOLUTE MAXIMUM SHEAR STRESS:
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tan 2 p 
 xy
 xy 
1   2
2
;



2



 ,
x 
 average 
1
2 ;
y 
2
2
 x  y
2