MECHANICS OF MATERIALS

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© 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