MECHANICS OF
MATERIALS
Third Edition
Ferdinand P. Beer
E. Russell Johnston, Jr.
John T. DeWolf
Lecture Notes:
J. Walt Oler
Texas Tech University
CHAPTER
©
2002
The McGraw

Hill Companies, Inc. All rights reserved.
6
Shearing Stresses in
Beams and Thin

Walled Members
©
2002
The McGraw

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

2
Shearing Stresses in Beams and
Thin

Walled Members
Introduction
Shear on the Horizontal Face of a Beam Element
Example
6
.
01
Determination of the Shearing Stress in a Beam
Shearing Stresses
t
xy
in Common Types of Beams
Further Discussion of the Distribution of Stresses in a ...
Sample Problem
6
.
2
Longitudinal Shear on a Beam Element of Arbitrary Shape
Example
6
.
04
Shearing Stresses in Thin

Walled Members
Plastic Deformations
Sample Problem
6
.
3
Unsymmetric Loading of Thin

Walled Members
Example
6
.
05
Example
6
.
06
©
2002
The McGraw

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

3
Introduction
©
2002
The McGraw

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

4
Introduction
©
2002
The McGraw

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

5
Shear on the Horizontal Face of a Beam Element
•
Consider prismatic beam
•
For equilibrium of beam element
A
C
D
A
D
D
x
dA
y
I
M
M
H
dA
H
F
0
x
V
x
dx
dM
M
M
dA
y
Q
C
D
A
•
Note,
flow
shear
I
VQ
x
H
q
x
I
VQ
H
•
Substituting,
©
2002
The McGraw

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

6
Shear on the Horizontal Face of a Beam Element
flow
shear
I
VQ
x
H
q
•
Shear flow,
•
where
section
cross
full
of
moment
second
above
area
of
moment
first
'
2
1
A
A
A
dA
y
I
y
dA
y
Q
•
Same result found for lower area
H
H
Q
Q
q
I
Q
V
x
H
q
axis
neutral
to
respect
h
moment wit
first
0
©
2002
The McGraw

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

7
Example
6
.
01
A beam is made of three planks,
nailed together. Knowing that the
spacing between nails is
25
mm and
that the vertical shear in the beam is
V
=
500
N, determine the shear force
in each nail.
SOLUTION:
•
Determine the horizontal force per
unit length or shear flow
q
on the
lower surface of the upper plank.
•
Calculate the corresponding shear
force in each nail.
©
2002
The McGraw

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

8
Example
6
.
01
4
6
2
3
12
1
3
12
1
3
6
m
10
20
.
16
]
m
060
.
0
m
100
.
0
m
020
.
0
m
020
.
0
m
100
.
0
[
2
m
100
.
0
m
020
.
0
m
10
120
m
060
.
0
m
100
.
0
m
020
.
0
I
y
A
Q
SOLUTION:
•
Determine the horizontal force per
unit length or shear flow
q
on the
lower surface of the upper plank.
m
N
3704
m
10
16.20
)
m
10
120
)(
N
500
(
4
6

3
6
I
VQ
q
•
Calculate the corresponding shear
force in each nail for a nail spacing of
25
mm.
m
N
q
F
3704
)(
m
025
.
0
(
)
m
025
.
0
(
N
6
.
92
F
©
2002
The McGraw

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

9
Determination of the Shearing Stress in a Beam
•
The
average
shearing stress on the horizontal
face of the element is obtained by dividing the
shearing force on the element by the area of
the face.
It
VQ
x
t
x
I
VQ
A
x
q
A
H
ave
t
•
On the upper and lower surfaces of the beam,
t
yx
=
0
. It follows that
t
xy
=
0
on the upper and
lower edges of the transverse sections.
•
If the width of the beam is comparable or large
relative to its depth, the shearing stresses at
D
1
and
D
2
are significantly higher than at
D.
©
2002
The McGraw

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

10
Shearing Stresses
t
xy
in Common Types of Beams
•
For a narrow rectangular beam,
A
V
c
y
A
V
Ib
VQ
xy
2
3
1
2
3
max
2
2
t
t
•
For American Standard (S

beam)
and wide

flange (W

beam) beams
web
ave
A
V
It
VQ
max
t
t
©
2002
The McGraw

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

11
Sample Problem
6
.
2
A timber beam is to support the three
concentrated loads shown. Knowing
that for the grade of timber used,
psi
120
psi
1800
all
all
t
determine the minimum required depth
d
of the beam.
SOLUTION:
•
Develop shear and bending moment
diagrams. Identify the maximums.
•
Determine the beam depth based on
allowable normal stress.
•
Determine the beam depth based on
allowable shear stress.
•
Required beam depth is equal to the
larger of the two depths found.
©
2002
The McGraw

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

12
Sample Problem
6
.
2
SOLUTION:
Develop shear and bending moment
diagrams. Identify the maximums.
in
kip
90
ft
kip
5
.
7
kips
3
max
max
M
V
©
2002
The McGraw

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

13
Sample Problem
6
.
2
2
2
6
1
2
6
1
3
12
1
in.
5833
.
0
in.
5
.
3
d
d
d
b
c
I
S
d
b
I
•
Determine the beam depth based on allowable
normal stress.
in.
26
.
9
in.
5833
.
0
in.
lb
10
90
psi
1800
2
3
max
d
d
S
M
all
•
Determine the beam depth based on allowable
shear stress.
in.
71
.
10
in.
3.5
lb
3000
2
3
psi
120
2
3
max
d
d
A
V
all
t
•
Required beam depth is equal to the larger of the two.
in.
71
.
10
d
©
2002
The McGraw

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

14
Example
6
.
04
A square box beam is constructed from
four planks as shown. Knowing that the
spacing between nails is
1
.
5
in. and the
beam is subjected to a vertical shear of
magnitude
V
=
600
lb, 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.
©
2002
The McGraw

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

15
Example
6
.
04
For the upper plank,
3
in
22
.
4
.
in
875
.
1
.
in
3
in.
75
.
0
y
A
Q
For the overall beam cross

section,
4
3
12
1
3
12
1
in
42
.
27
in
3
in
5
.
4
I
SOLUTION:
•
Determine the shear force per unit
length along each edge of the upper
plank.
length
unit
per
force
edge
in
lb
15
.
46
2
in
lb
3
.
92
in
27.42
in
22
.
4
lb
600
4
3
q
f
I
VQ
q
•
Based on the spacing between nails,
determine the shear force in each
nail.
in
75
.
1
in
lb
15
.
46
f
F
lb
8
.
80
F
©
2002
The McGraw

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

16
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
t
t
•
The corresponding shear stress is
•
NOTE:
0
xy
t
0
xz
t
in the flanges
in the web
•
Previously found a similar expression
for the shearing stress in the web
It
VQ
xy
t
©
2002
The McGraw

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

17
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
t
•
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
.
©
2002
The McGraw

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

18
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 the
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.
©
2002
The McGraw

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

19
Sample Problem
6
.
3
Knowing that the vertical shear is
50
kips in a W
10
x
68
rolled

steel beam,
determine the horizontal shearing
stress in the top flange at the point
a
.
SOLUTION:
•
For the shaded area,
3
in
98
.
15
in
815
.
4
in
770
.
0
in
31
.
4
Q
•
The shear stress at
a
,
in
770
.
0
in
394
in
98
.
15
kips
50
4
3
It
VQ
t
ksi
63
.
2
t
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