MEE 320 Problems Chapter 6, Spring 2018
Lebanese American University · Strength of Material · 4 pages · Dr. Carine Habchi
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50 mm
s
s
50 mm
50 mm
100 mm
PROBLEM 6.1
Three full-size 50 100-mm boards are nailed together to form a
beam that is subjected to a vertical shear of 1500 N. Knowing that the
allowable shearing force in each nail is 400 N, determine the largest
longitudinal spacing s that can be used between each pair of nails.
SOLUTION
3
3
6
4
1
1 (100)(150)
28.125
10 mm
12
12
I
bh
6
4
28.125
10
m
2
(100)(50)
5000 mm
A
1
y 50 mm
3
3
6
3
1
250
10 mm
250
10
m
Q
Ay
6
3
6
(1500)(250
10
)
13.3333
10 N/m
28.125
10
VQ
q
I
qs 2 nail
F
3
nail
3
2
(2)(400)
60.0
10
m
13.3333
10
F
s
q
s 60.0 mm
1.5 m
100 mm
200 mm
40 mm
12 mm
12 mm
150 mm
0.3 m
10 kN
n
a
n
PROBLEM 6.10
For the beam and loading
shown, consider section n-n
and determine (a) the largest
shearing stress in that section,
(b) the shearing stress at
point a.
SOLUTION
At section n-n,
V 10 kN.
1
2
3
3
2
1 1
2 2
2
2
3
3
2
6
6
6
6
4
6
4
4
1
1
4
12
12
1
1
(100)(150)
4
(50)(12)
(50)(12)(69)
12
12
28.125
10
4 0.0072
10
2.8566
10
39.58
10 mm
39.58
10
m
I
I
I
b h
b h
A d
(a)
1 1
2 2
3
3
6
3
2
(100)(75)(37.5)
(2)(50)(12)(69)
364.05
10 mm
364.05
10
m
100 mm = 0.100 m
Q
A y
A y
t
3
6
3
max
6
(10
10 )(364.05
10
)
920
10 Pa
(39.58
10
)(0.100)
VQ
It
max
920 kPa
(b)
1 1
2
2
3
3
6
3
2
(100)(40)(55)
(2)(50)(12)(69)
302.8
10 mm
302.8
10
m
100 mm
0.100 m
Q
A y
A y
t
3
6
3
6
(10
10 )(302.8
10
)
765
10 Pa
(39.58
10
)(0.100)
a
VQ
It
765 kPa
a
2.4 kN
4.8 kN
7.2 kN
1 m
1 m
1 m
0.5 m
150 mm
A
E
b
B
C
D
PROBLEM 6.18
For the beam and loading shown, determine the
minimum required width b, knowing that for the
grade
of
timber
used,
all
12 MPa
and
all
825 kPa.
SOLUTION
0:
3
(2)(2.4)
(1)(4.8)
(0.5)(7.2)
0
2 kN
MD
A
A
Draw the shear and bending moment diagrams.
3
max
3
max
| |
7.2 kN
7.2 10 N
|
|
3.6 kN m
3.6 10 N m
V
M
Bending:
M
S
max
min
3
6
6
3
3
3
|
|
3.6 10
12 10
300 10
m
300 10 mm
M
S
For a rectangular section,
2
1
6
S
bh
3
2
2
6
(6)(300 10 )
80 mm
(150)
S
b
h
Shear: Maximum shearing stress occurs at the neutral axis of bending for a rectangular section.
2
3
2
1
8
3
1
12
3
3
3
3
1
1
1
,
,
2
4
8
1
12
(
)
3
2
(
)( )
3
(3)(7.2 10 )
87.3 10
m
2
(2)(150 10
)(825 10 )
A
bh
y
h
Q
Ay
bh
I
bh
t
b
V
bh
VQ
V
It
bh
bh
b
V
b
h
The required value of b is the larger one.
b 87.3 mm
PROBLEM 6.2
For the beam and loading shown,
consider section n-n and determine the
shearing stress at (a) point a, (b) the
shearing stress at point b.
SOLUTION
100 kN
A
B
R
= R
=
At section n-n,
V =100 kN
Locate centroid and calculate moment of inertia.
Part
2
A(mm )
y(mm)
3
Ay(mm )
d(mm)
2
4
Ad (mm )
4
I (mm )
3200
170
544000
55.4
9821312
106667
7200
90
648000
24.6
4357152
19440000
Σ
10400
1192000
14178464
19546667
2
6
1192000
114.6 mm
104.00
14178464
19546667
33.725 10
Ay
Y
A
I
Ad
I
= Σ
=
=
Σ
= Σ
+ Σ =
+
=
×
(a)
3
(20)(38)(114.6
19)
72656 mm
20 mm
Qa
Ay
t
=
=
−
=
=
6
(100000)(72656)
10.8 MPa
(33.725 10 )(20)
a
VQ
It
τ
=
=
=
×
(b)
9
(20)(76)(114.6
38)
1164.32 mm
20 mm
Qb
Ay
t
=
=
−
=
=
6
(100000)(116432)
17.3 MPa
(33.725 10 )(20)
a
VQ
It
τ
=
=
=
×
1