Prof. Stephen M. Mutuli, Dept. of Mechanical & Manufacturing Engineering,
University of Nairobi (e-mail : mutulis1@gmail.com)
EXERCISE No. 2
Tutorials
FME 202 - Solid & Structural Mechanics Tutorial No 1
( Shear Forces And Bending Moment Diagrams)
2wN / m
2wN / m
Q1.
wN / m
6 wa 2 N  m
3a
6a
Fig Q1
Fig Q1 shows a beam of length 9a simply supported over a span length 6a and
loaded as indicated.
(i)
calculate the reactions at the supports
(ii)
draw to scale the Shear Force and the Bending Moment diagrams
3.6wN / m
Q2.
wN / m
8wa 2 N  m
6a
3a
Fig Q2
Fig Q2 shows a beam of length 9a meters and simply supported at the ends. The
beam supports a distributed load varying in magnitude linearly from zero at the left-hand
end to 3.6wN / m at a point distance 6a meters from the left-hand support; and another
uniformly distributed load of magnitude wN / m over the remaining 3a meters. A
clockwise couple of magnitude 8wa 2 N  m is applied at appoint 6a meters from the
left-hand support.
(i)
calculate the reactions at the supports
1
Prof. Stephen M. Mutuli, Dept. of Mechanical & Manufacturing Engineering,
University of Nairobi (e-mail : mutulis1@gmail.com)
(ii)
Draw to scale the Shear Force and the Bending Moment diagrams
2waN
Q3.
wN / m
4 wa 2 N  m
4 wa N  m
4a
2
4a
Fig Q3
Fig Q3 shows a cantilever beam of length 8a meters, fixed at the left hand end and
loaded as indicated by a distributed load, a point load and two clockwise couples.
(i)
calculate the reactions at the support
(ii)
draw to scale the Shear Force and the Bending Moment diagrams
Q4.
wN / m
4 wa 2 N  m
6a
3a
Fig Q4
Fig Q4 shows a beam of length 9a simply supported at the two ends and supporting
a distributed load varying in magnitude linearly from zero at the left-hand end to wN / m
and a clockwise couple of magnitude 4wa 2 N  m at a point distance 6a from the lefthand support.
(i)
Calculate the reactions at the supports
(ii)
Draw to scale the Shear Force and Bending Moment diagrams
2
Prof. Stephen M. Mutuli, Dept. of Mechanical & Manufacturing Engineering,
University of Nairobi (e-mail : mutulis1@gmail.com)
Q5.
80KN
120KN / m
120KN  m
2m
4m
Fig Q5
Fig Q5 shows a beam 6m long and simply supported on two supports 4m apart.
The beam carries a distributed load varying in magnitude linearly from zero at the lefthand end to 120KN / m at the right –hand end. A point load of magnitude 80KN and
an anti-clockwise couple of magnitude 120KN  m are applied at the left-hand end.
(i)
Calculate the reactions at the supports
(ii)
Draw to scale the Shear Force and the Bending moment diagrams.
Q6.
120KN
60KN / m
240KN  m
3m
3m
3m
Fig Q6
Fig Q6 shows a beam of length 9m and simply supported over a span of 6m . The
beam supports a distributed load varying in magnitude linearly from zero at the lefthand end to 60KN / m at the right-hand end, a point load of magnitude 120KN at the
center of the two supports and an anti-clockwise couple of magnitude 240KN  m at
the right-hand end of the beam.
(i)
Calculate the reactions at the supports
(ii)
Draw to scale the Shear Force and the Bending Moment diagrams
3
Prof. Stephen M. Mutuli, Dept. of Mechanical & Manufacturing Engineering,
University of Nairobi (e-mail : mutulis1@gmail.com)
Answers:
Q1.
.
(i) Y1  6.5wa , Y2  4wa
(ii)Values of Shear Force and Bending
Moments are given for different
values of the length of beam (x ) , measured
from the left-hand end
x
Shear
Bending
0
a
2a
3a
3a
4a
5a
6a
7a
8a
9a
Force
0
 1.1667wa
 2.667wa
 4.5wa
2wa
1.667wa
1.334wa
0.5wa
 0.667wa
 2.1667wa
 4wa
Moment
6wa 2
5.44 wa 2
3.556 wa 2
0
0
1.9445wa 2
3.556 wa 2
4.5wa 2
4.444 wa 2
3.056 wa 2
0
0  x  3a
3a  x  9a
Q2.
(i) Y1  5.61wa , Y2  8.19wa
(ii) Values of Shear Force and Bending Moments for different values of lengths
of the beam (x ) , measured from the left-hand end
x
Shear
Force
0
5.61wa
a
5.31wa
2a
4.41wa
3a
2.91wa
4a
0.81wa
5a
 1.89wa
Bending
Moment
0
5.51wa 2
10.42wa 2
14.13wa 2
16.04wa 2
15.55wa 2
x
Shear
Force
Bending
Moment
6a
 5.19wa
6a
 5.19wa
7a
 6.19wa
8a
 7.19wa
9a
 8.19wa
12.06wa 2
20.06 wa 2
14.37 wa 2
7.68wa 2
0
4
Prof. Stephen M. Mutuli, Dept. of Mechanical & Manufacturing Engineering,
University of Nairobi (e-mail : mutulis1@gmail.com)
Q3.
(i) V  4wa
, M  29.334wa 2
(ii) Values of Shear Force and Bending Moments for different values of lengths of
the beam (x ) measured from the left-hand end
0  x  4a
x
a
2a
3a
4a
0
4wa
3.875wa
3.5wa
2.875wa
2wa
Shear
Force
Bending
Moment
 29.334 wa 2
x
Shear
Force
Bending
Moment
 25.376 wa 2
 21.667 wa 2
 18.459wa 2
 16wa 2
4a  x  8a
4a
5a
2wa
2wa
6a
2wa
7a
2wa
8a
2wa
 12wa 2
 8wa 2
 6wa 2
 4wa 2
 10wa 2
Q4.
(i) Y1  1.222wa , Y2  1.778wa
(ii) Values of Shear Force and Bending Moment for different values of the lengths of
the beam (x ) measured from the left- hand end
x
Shear
Force
Bending
Moment
x
Shear
Force
Bending
Moment
0
1.222wa
a
1.139wa
2a
0.889wa
3a
 1.028wa
4a
 0.111wa
5a
 0.8613wa
0
1.1942wa 2
2.222 wa 2
2.196 wa 2
3.11wa 2
2.638wa 2
6a
 1.778wa
6a
 1.778wa
7a
 1.778wa
8a
 1.778wa
9a
 1.778wa
1.333wa 2
5.332 wa 2
3.554 wa 2
1.776wa 2
0
5
Prof. Stephen M. Mutuli, Dept. of Mechanical & Manufacturing Engineering,
University of Nairobi (e-mail : mutulis1@gmail.com)
Q5.
(i) Y1  330KN , Y2  110KN
(ii)Values of Shear Force and Bending Moment for different values of the lengths
of the beam (x ) measured from the left-hand end.
x
1m
2m
2m
0
 80KN
 90KN
 120KN
210KN
Shear
Force
Bending
Moment
x
Shear
Force
Bending
Moment
120KN  m
 203.33KN  m
 306.64KN  m
 306.64KN  m
3m
160KN
4m
90KN
5m
0
6m
 110KN
 119.91KN  m
6.88KN  m
53.75KN  m
0
Q6.
(i) Y1  100KN , Y2  290KN
(ii) Values of Shear Force and Bending Moment for different values of the lengths
of the beam (x ) measured from the left-hand end.
x (m)
3
0
1
2
Shear
100
96.67
86.67
70
Force(KN )
Bending
98.89
191.11
270
0
Moment ( KN  m)
x (m)
Shear
3
 49.97
4
 73.32
5
 103.33
6
 139.99
Force(KN )
Bending
Moment ( KN  m)
270
208.89
121.12
0
x (m)
Shear
6
150.01
7
106.68
8
56.69
9
0
Force(KN )
Bending
Moment ( KN  m)
0
128.93
211.17
240
6
7
8
9