© 2006 The McGraw

Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
Fourth
Edition
Beer
•
Johnston
•
DeWolf
6

1
Example 6.04
A square box beam is constructed from
four planks as shown. Knowing that the
spacing between nails is 44 mm. and the
beam is subjected to a vertical shear of
magnitude
V
= 2.7 kN, determine the
shearing force in each nail.
SOLUTION:
•
Determine the shear force per unit
length along each edge of the upper
plank.
•
Based on the spacing between nails,
determine the shear force in each
nail.
© 2006 The McGraw

Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
Fourth
Edition
Beer
•
Johnston
•
DeWolf
6

2
Example 6.04
For the upper plank,
3
64296
47
76
mm
18
mm
mm
mm
y
A
Q
For the overall beam cross

section,
4
4
12
1
4
12
1
10332
76
112
mm
mm
mm
I
© 2006 The McGraw

Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
Fourth
Edition
Beer
•
Johnston
•
DeWolf
Example 6.04
6

3
SOLUTION:
•
Determine the shear force per unit
length along each edge of the upper
plank.
length
unit
per
force
edge
4
.
8
2
8
.
16
0332
1
64296
7
.
2
4
3
mm
N
q
f
mm
N
mm
mm
kN
I
VQ
q
•
Based on the spacing between nails,
determine the shear force in each nail.
44mm
mm
N
4
.
8
f
F
N
369.6
F
© 2006 The McGraw

Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
Fourth
Edition
Beer
•
Johnston
•
DeWolf
6

4
Shearing Stresses in Thin

Walled Members
•
Consider a segment of a wide

flange
beam subjected to the vertical shear
V
.
•
The longitudinal shear force on the
element is
x
I
VQ
H
It
VQ
x
t
H
xz
zx
•
The corresponding shear stress is
•
NOTE:
0
xy
0
xz
in the flanges
in the web
•
Previously found a similar expression
for the shearing stress in the web
It
VQ
xy
© 2006 The McGraw

Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
Fourth
Edition
Beer
•
Johnston
•
DeWolf
6

5
Shearing Stresses in Thin

Walled Members
•
The variation of shear flow across the
section depends only on the variation of
the first moment.
I
VQ
t
q
•
For a box beam,
q
grows smoothly from
zero at A to a maximum at
C
and
C’
and
then decreases back to zero at
E
.
•
The sense of
q
in the horizontal portions
of the section may be deduced from the
sense in the vertical portions or the
sense of the shear
V
.
© 2006 The McGraw

Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
Fourth
Edition
Beer
•
Johnston
•
DeWolf
6

6
Shearing Stresses in Thin

Walled Members
•
For a wide

flange beam, the shear flow
increases symmetrically from zero at
A
and
A’
, reaches a maximum at
C
and
then decreases to zero at
E
and
E’
.
•
The continuity of the variation in
q
and
the merging of
q
from section branches
suggests an analogy to fluid flow.
© 2006 The McGraw

Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
Fourth
Edition
Beer
•
Johnston
•
DeWolf
6

7
•
The section becomes fully plastic (
y
Y
= 0) at
the wall when
p
Y
M
M
PL
2
3
•
For
PL
>
M
Y
, yield is initiated at
B
and
B’
.
For an elastoplastic material, the half

thickness
of the elastic core is found from
2
2
3
1
1
2
3
c
y
M
Px
Y
Y
Plastic Deformations
moment
elastic
maximum
Y
Y
c
I
M
•
Recall:
•
For
M = PL < M
Y
, the normal stress does
not exceed the yield stress anywhere along
the beam.
•
Maximum load which the beam can support is
L
M
P
p
max
© 2006 The McGraw

Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
Fourth
Edition
Beer
•
Johnston
•
DeWolf
6

8
Plastic Deformations
•
Preceding discussion was based on
normal stresses only
•
Consider horizontal shear force on an
element within the plastic zone,
0
dA
dA
H
Y
Y
D
C
Therefore, the shear stress is zero in the
plastic zone.
•
Shear load is carried by the elastic core,
A
P
by
A
y
y
A
P
Y
Y
xy
2
3
2
where
1
2
3
max
2
2
•
As
A’
decreases,
max
increases and
may exceed
Y
© 2006 The McGraw

Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
Fourth
Edition
Beer
•
Johnston
•
DeWolf
6

9
Sample Problem 6.3
Knowing that the vertical shear is 220
kN in a W250x101 rolled

steel beam,
determine the horizontal shearing
stress in the top flange at the point
a
located 108 mm from the edge of the
beam.
SOLUTION:
•
For the shaded area,
3
mm
03
3
.
259
2
.
122
6
.
19
108
E
Q
•
The shear stress at
a
,
mm
mm
E
It
VQ
6
.
19
06
164
mm
03
259.3E
220N
4
3
17.7MPa
© 2006 The McGraw

Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
Fourth
Edition
Beer
•
Johnston
•
DeWolf
6

10
•
When the force P is applied at a distance
e
to the
left of the web centerline, the member bends in a
vertical plane without twisting.
Unsymmetric Loading of Thin

Walled Members
•
If the shear load is applied such that the beam
does not twist, then the shear stress distribution
satisfies
F
ds
q
ds
q
F
ds
q
V
It
VQ
E
D
B
A
D
B
ave
•
F
and
F’
indicate a couple
Fh
and the need for
the application of a torque as well as the shear
load.
Ve
h
F
•
The point
O
is referred to as the
shear center
of
the beam section.
© 2006 The McGraw

Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
Fourth
Edition
Beer
•
Johnston
•
DeWolf
6

11
Unsymmetric Loading of Thin

Walled Members
•
Beam loaded in a vertical plane
of symmetry deforms in the
symmetry plane without
twisting.
It
VQ
I
My
ave
x
•
Beam without a vertical plane
of symmetry bends and twists
under loading.
It
VQ
I
My
ave
x
© 2006 The McGraw

Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
Fourth
Edition
Beer
•
Johnston
•
DeWolf
6

12
Example 6.05
•
Determine the location for the shear center of the
channel section with
b
= 100 mm.,
h
= 150 ., and
t
= 4
mm.
I
h
F
e
•
where
I
Vthb
ds
h
st
I
V
ds
I
VQ
ds
q
F
b
b
b
4
2
2
0
0
0
h
b
th
h
bt
bt
th
I
I
I
flange
web
6
2
12
1
2
12
1
2
2
12
1
2
3
3
•
Combining,
.
mm
100
3
150
2
100
3
2
mm
mm
b
h
b
e
mm
e
40
© 2006 The McGraw

Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
Fourth
Edition
Beer
•
Johnston
•
DeWolf
6

13
Example 6.06
•
Determine the shear stress distribution for
V
= 11 kN.
It
VQ
t
q
•
Shearing stresses in the flanges,
MPa
mm
mm
h
b
th
Vb
h
b
th
Vhb
s
I
Vh
h
st
It
V
It
VQ
B
6
.
14
150
100
6
150mm
4mm
100mm
11kN
6
6
6
6
2
2
2
2
12
1
•
Shearing stress in the web,
MPa
mm
mm
mm
mm
mm
mm
kN
h
b
th
h
b
V
t
h
b
th
h
b
ht
V
It
VQ
16
.
20
150
100
6
150
4
2
150
100
4
11
3
6
2
4
3
6
4
2
12
1
8
1
max
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