Third Edition MECHANICS OF MATERIALS

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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