Fourth Edition
CHAPTER
6
MECHANICS OF
MATERIALS
Ferdinand P. Beer
E. Russell Johnston, Jr.
John T. DeWolf
Lecture Notes:
J. Walt Oler
Texas Tech University
Shearing Stresses in
Beams and ThinWalled Members
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
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 txy 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
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6-2
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Introduction
• Transverse loading applied to a beam
results in normal and shearing stresses in
transverse sections.
• Distribution of normal and shearing
stresses satisfies
F x    x dA  0
F y   t xy dA   V
F z   t xz dA  0
  y t xz  z t xy dA  0
M y   z  x dA  0
M z    y  x   M
M
x

• When shearing stresses are exerted on the
vertical faces of an element, equal stresses
must be exerted on the horizontal faces
• Longitudinal shearing stresses must exist
in any member subjected to transverse
loading.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6-3
Fourth
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Shear on the Horizontal Face of a Beam Element
• Consider prismatic beam
• For equilibrium of beam element
 F x  0   H    D   C dA
A
M
H 
D
 MC
I
 y dA
A
• Note,
Q   y dA
A
dM
MD MC 
x  V x
dx
• Substituting,
H 
VQ
x
I
q 
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
H
x

VQ
 shear
flow
I
6-4
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Shear on the Horizontal Face of a Beam Element
• Shear flow,
q 
H
x

VQ
 shear flow
I
• where
Q   y dA
A
 first moment of area above y1
I 
2
 y dA
A A'
 second moment of full cross section
• Same result found for lower area
q 
H 

x
Q  Q  0
VQ
  q
I
 first moment wit h respect
to neutral axis
H   H
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6-5
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Example 6.01
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.
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.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6-6
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Example 6.01
SOLUTION:
• Determine the horizontal force per
unit length or shear flow q on the
lower surface of the upper plank.
q 
VQ

( 500 N )(120  10
16.20  10
I
Q  Ay
 3704 N
  0 . 020 m  0 . 100 m  0 . 060 m 
 120  10
I 
1
12
6
 0 . 020
m
4
m
• Calculate the corresponding shear
force in each nail for a nail spacing of
25 mm.
m  0 . 100 m 
3
 2 [ 1  0 . 100 m  0 . 020 m 
12
  0 . 020 m  0 . 100 m  0 . 060 m  ]
2
 16 . 20  10
m
3
m )
3
3
6
-6
6
m
4
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
F  ( 0 . 025 m ) q  ( 0 . 025 m )( 3704 N m
F  92 . 6 N
6-7
Fourth
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
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.
t ave 

H
A

q x
A

VQ  x
I t x
VQ
It
• On the upper and lower surfaces of the beam,
tyx= 0. It follows that txy= 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 D1
and D2 are significantly higher than at D.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6-8
Fourth
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Shearing Stresses txy in Common Types of Beams
• For a narrow rectangular beam,
2
3V 
y 
1 

t xy 

2
Ib
2 A 
c 
VQ
t max 
3V
2 A
• For American Standard (S-beam)
and wide-flange (W-beam) beams
t ave 
VQ
t max 
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
It
V
A web
6-9
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Further Discussion of the Distribution of
Stresses in a Narrow Rectangular Beam
• Consider a narrow rectangular cantilever beam
subjected to load P at its free end:
2
3P
y 
1 

t xy 
2
2 A 
c 
x 
Pxy
I
• Shearing stresses are independent of the distance
from the point of application of the load.
• Normal strains and normal stresses are unaffected by
the shearing stresses.
• From Saint-Venant’s principle, effects of the load
application mode are negligible except in immediate
vicinity of load application points.
• Stress/strain deviations for distributed loads are
negligible for typical beam sections of interest.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6 - 10
Fourth
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 6.2
SOLUTION:
• Develop shear and bending moment
diagrams. Identify the maximums.
• Determine the beam depth based on
allowable normal stress.
A timber beam is to support the three
concentrated loads shown. Knowing
that for the grade of timber used,
 all  1800 psi
t all  120 psi
• Determine the beam depth based on
allowable shear stress.
• Required beam depth is equal to the
larger of the two depths found.
determine the minimum required depth
d of the beam.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6 - 11
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 6.2
SOLUTION:
Develop shear and bending moment
diagrams. Identify the maximums.
V max  3 kips
M max  7 . 5 kip  ft  90 kip  in
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6 - 12
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 6.2
• Determine the beam depth based on allowable
normal stress.
 all 
M max
S
3
90  10 lb  in.
1800 psi 
 0 . 5833
in.  d
2
d  9 . 26 in.
3
I  1 bd
• Determine the beam depth based on allowable
shear stress.
12
S 
I
c
2
 1bd
6
t all 
2
 1 3 . 5 in. d
6
  0 . 5833 in. d
2
3 V max
2
120 psi 
A
3 3000 lb
2 3.5 in.  d
d  10 . 71 in.
• Required beam depth is equal to the larger of the two.
d  10 . 71 in.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6 - 13
Fourth
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Longitudinal Shear on a Beam Element
of Arbitrary Shape
• We have examined the distribution of
the vertical components txy on a
transverse section of a beam. We now
wish to consider the horizontal
components txz of the stresses.
• Consider prismatic beam with an
element defined by the curved surface
CDD’C’.
 F x  0   H    D   C dA
a
• Except for the differences in
integration areas, this is the same
result obtained before which led to
H 
VQ
I
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
x
q 
H
x

VQ
I
6 - 14
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Example 6.04
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.
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.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
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