Beer F.P., Johnston E.R., DeWolf J.T., Mazurek D.F. Mechanics of Materials
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REVIEW AND SUMMARY
This chapter was devoted to the analysis of beams and thin-walled
members under transverse loadings.
In Sec. 6.1 we considered a small element located in the vertical plane
of symmetry of a beam under a transverse loading (Fig. 6.61) and
found that normal stresses s
x
and shearing stresses t
xy
were exerted
on the transverse faces of that element, while shearing stresses t
yx
,
equal in magnitude to t
xy
, were exerted on its horizontal faces.
In Sec. 6.2 we considered a prismatic beam AB with a vertical
plane of symmetry supporting various concentrated and distributed
loads (Fig. 6.62). At a distance x from end A we detached from the
Stresses on a beam element
yx
xy
x
Fig. 6.61
B
P
1
P
2
w
A
x
C
y
z
Fig. 6.62
y
1
y
1
x
C
c
x
D
C'
N.A.
D'
y
z
Fig. 6.63
beam an element CDD9C9 of length Dx extending across the width
of the beam from the upper surface of the beam to a horizontal plane
located at a distance y
1
from the neutral axis (Fig. 6.63). We found
that the magnitude of the shearing force DH exerted on the lower
face of the beam element was
¢H 5
VQ
I
¢x (6.4)
where V 5 vertical shear in the given transverse section
Q 5 first moment with respect to the neutral axis of
the shaded portion A of the section
I 5 centroidal moment of inertia of the entire cross-
sectional area
Horizontal shear in a beam
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Shearing Stresses in Beams
and Thin-Walled Members
The horizontal shear per unit length, or shear flow, which was
denoted by the letter q, was obtained by dividing both members of
Eq. (6.4) by Dx:
q 5
¢
H
¢
x
5
V
Q
I
(6.5)
Dividing both members of Eq. (6.4) by the area DA of the horizontal
face of the element and observing that DA 5 t Dx, where t is the
width of the element at the cut, we obtained in Sec. 6.3 the following
expression for the average shearing stress on the horizontal face of
the element
t
ave
5
V
Q
It
(6.6)
We further noted that, since the shearing stresses t
xy
and t
yx
exerted,
respectively, on a transverse and a horizontal plane through D9 are
equal, the expression in (6.6) also represents the average value of t
xy
along the line D9
1
D9
2
(Fig. 6.64).
In Secs. 6.4 and 6.5 we analyzed the shearing stresses in a beam of
rectangular cross section. We found that the distribution of stresses
is parabolic and that the maximum stress, which occurs at the center
of the section, is
t
max
5
3
2
V
A
(6.10)
where A is the area of the rectangular section. For wide-flange
beams, we found that a good approximation of the maximum shear-
ing stress can be obtained by dividing the shear V by the cross-
sectional area of the web.
In Sec. 6.6 we showed that Eqs. (6.4) and (6.5) could still be used to
determine, respectively, the longitudinal shearing force DH and the
shear flow q exerted on a beam element if the element was bounded
by an arbitrary curved surface instead of a horizontal plane (Fig. 6.65).
Shear flow
Shearing stresses in a beam
yx
ave
ave
xy
D'
D'
D''
2
C''
1
D''
1
1
2
D'
Fig. 6.64
x
C
c
x
D
C'
D'
y
N.A.
z
Fig. 6.65
Shearing stresses in a beam
of rectangular cross section
Longitudinal shear on curved
surface
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This made it possible for us in Sec. 6.7 to extend the use of Eq. (6.6)
to the determination of the average shearing stress in thin-walled
members such as wide-flange beams and box beams, in the flanges of
such members, and in their webs (Fig. 6.66).
Review and Summary
N.A.
xz
y
t
z
(a)
N.A.
xy
y
t
z
(b)
Fig. 6.66
In Sec. 6.8 we considered the effect of plastic deformations on the
magnitude and distribution of shearing stresses. From Chap. 4 we
recalled that once plastic deformation has been initiated, additional
loading causes plastic zones to penetrate into the elastic core of a
beam. After demonstrating that shearing stresses can occur only in
the elastic core of a beam, we noted that both an increase in loading
and the resulting decrease in the size of the elastic core contribute
to an increase in shearing stresses.
In Sec. 6.9 we considered prismatic members that are not loaded
in their plane of symmetry and observed that, in general, both bend-
ing and twisting will occur. You learned to locate the point O of the
cross section, known as the shear center, where the loads should be
applied if the member is to bend without twisting (Fig. 6.67) and
found that if the loads are applied at that point, the following equa-
tions remain valid:
s
x
52
M
y
I
t
ave
5
V
Q
It
(4.16, 6.6)
Using the principle of superposition, you also learned to determine
the stresses in unsymmetric thin-walled members such as channels,
angles, and extruded beams [Example 6.07 and Sample Prob. 6.6]
Shearing stresses in thin-walled
members
e
O
P
Fig. 6.67
Plastic deformations
Unsymmetric loading
shear center
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REVIEW PROBLEMS
6.89 A square box beam is made of two 20 3 80-mm planks and two 20 3
120-mm planks nailed together as shown. Knowing that the spacing
between the nails is s 5 30 mm and that the vertical shear in the
beam is V 5 1200 N, determine (a) the shearing force in each nail,
(b) the maximum shearing stress in the beam.
s
s
s
120 mm
80 mm
20 mm
20 mm
Fig. P6.89
6.90 The beam shown is fabricated by connecting two channel shapes
and two plates, using bolts of
3
4
-in. diameter spaced longitudinally
every 7.5 in. Determine the average shearing stress in the bolts
caused by a shearing force of 25 kips parallel to the y axis.
6.91 For the beam and loading shown, consider section n-n and deter-
mine (a) the largest shearing stress in that section, (b) the shearing
stress at point a.
C12 20.7
16 in. in.
C
z
y
1
2
Fig. P6.90
90
120
1515 151530
20
20
20
40
20
72 kN
n
n
Dimensions in mm
1.5 m
0.5 m
0.8 m
a
Fig. P6.91
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Review Problems
6.93 For the beam and loading shown, consider section n-n and deter-
mine the shearing stress at (a) point a, (b) point b.
6.92 For the beam and loading shown, determine the minimum
required width b, knowing that for the grade of timber used,
s
all
5 12 MPa and t
all
5 825 kPa.
2.4 kN 4.8 kN
1 m
0.5 m
150 mm
AE
b
BC D
1 m1 m
7.2 kN
Fig. P6.92
Fig. P6.93 and P6.94
B
b
a
A
10 in.
20 in. 20 in.
25 kips
25 kips
n
7.25 in.
in.
1.5 in.
1.5 in.
3
4
8 in.
in.
3
4
in.
3
4
n
6.94 For the beam and loading shown, determine the largest shearing
stress in section n-n.
6.95 The composite beam shown is made by welding C200 3 17.1
rolled-steel channels to the flanges of a W250 3 80 wide-flange
rolled-steel shape. Knowing that the beam is subjected to a vertical
shear of 200 kN, determine (a) the horizontal shearing force per
meter at each weld, (b) the shearing stress at point a of the flange
of the wide-flange shape.
112 mm
a
Fig. P6.95
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Shearing Stresses in Beams
and Thin-Walled Members
6.96 An extruded beam has the cross section shown and a uniform wall
thickness of 3 mm. For a vertical shear of 10 kN, determine (a) the
shearing stress at point A, (b) the maximum shearing stress in the
beam. Also sketch the shear flow in the cross section.
60 mm
A
30 mm
16 mm16 mm
28 mm
Fig. P6.96
6.97 The design of a beam requires welding four horizontal plates to a
vertical 0.5 3 5-in. plate as shown. For a vertical shear V, deter-
mine the dimension h for which the shear flow through the welded
surfaces is maximum.
Fig. P6.97
2.5 in.
2.5 in.
h
h
0.5 in.
0.5 in.
4.5 in.4.5 in.
0.5 in.
6.98 Determine the location of the shear center O of a thin-walled
beam of uniform thickness having the cross section shown.
e
E
F
O
A
ab
D
B
h
G
Fig. P6.98
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Review Problems
6.99 Determine the location of the shear center O of a thin-walled
beam of uniform thickness having the cross section shown.
6.100 A thin-walled beam of uniform thickness has the cross section
shown. Determine the dimension b for which the shear center O
of the cross section is located at the point indicated.
E
60
60
F
e
O
A
D
B
in.
1
4
1.5 in.
1.5 in.
Fig. P6.99
Fig. P6.100
A
D
E
O
160 mm 200 mm
20 mm
20 mm
b
60 mm
GF
H
J
B
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COMPUTER PROBLEMS
The following problems are designed to be solved with a computer.
6.C1 A timber beam is to be designed to support a distributed load and
up to two concentrated loads as shown. One of the dimensions of its uniform
rectangular cross section has been specified and the other is to be determined
so that the maximum normal stress and the maximum shearing stress in the
beam will not exceed given allowable values s
all
and t
all
. Measuring x from
end A and using either SI or U.S. customary units, write a computer program
to calculate for successive cross sections, from x 5 0 to x 5 L and using given
increments Dx, the shear, the bending moment, and the smallest value of the
unknown dimension that satisfies in that section (1) the allowable normal
stress requirement, (2) the allowable shearing stress requirement. Use this
program to solve Prob. 5.65 assuming s
all
5 12 MPa and t
all
5 825 kPa, using
Dx 5 0.1 m.
6.C2 A cantilever timber beam AB of length L and of uniform rectangular
section shown supports a concentrated load P at its free end and a uniformly
distributed load w along its entire length. Write a computer program to deter-
mine the length L and the width b of the beam for which both the maximum
normal stress and the maximum shearing stress in the beam reach their largest
allowable values. Assuming s
all
5 1.8 ksi and t
all
5 120 psi, use this program
to determine the dimensions L and b when (a) P 5 1000 lb and w 5 0,
(b) P 5 0 and w 5 12.5 lb/in., (c) P 5 500 lb and w 5 12.5 lb/in.
B
t
h
A
x
1
x
3
x
2
x
4
a
b
L
P
1
w
P
2
Fig. P6.C1
w
P
B
b
8 b
L
A
Fig. P6.C2
b
n
b
2
V
b
1
h
n
h
2
h
1
Fig. P6.C3
6.C3 A beam having the cross section shown is subjected to a vertical
shear V. Write a computer program that, for loads and dimensions expressed
in either SI or U.S. customary units, can be used to calculate the shearing
stress along the line between any two adjacent rectangular areas forming
the cross section. Use this program to solve (a) Prob. 6.10, (b) Prob. 6.12,
(c) Prob. 6.21.
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Computer Problems
6.C4 A plate of uniform thickness t is bent as shown into a shape with
a vertical plane of symmetry and is then used as a beam. Write a computer
program that, for loads and dimensions expressed in either SI or U.S. cus-
tomary units, can be used to determine the distribution of shearing stresses
caused by a vertical shear V. Use this program (a) to solve Prob. 6.47, (b) to
find the shearing stress at a point E for the shape and load of Prob. 6.50,
assuming a thickness t 5
1
4
in.
6.C5 The cross section of an extruded beam is symmetric with respect
to the x axis and consists of several straight segments as shown. Write a
computer program that, for loads and dimensions expressed in either SI or
U.S. customary units, can be used to determine (a) the location of the shear
center O, (b) the distribution of shearing stresses caused by a vertical force
applied at O. Use this program to solve Probs. 6.66 and 6.70.
6.C6 A thin-walled beam has the cross section shown. Write a computer
program that, for loads and dimensions expressed in either SI or U.S. cus-
tomary units, can be used to determine the location of the shear center O
of the cross section. Use the program to solve Prob. 6.75.
y
1
y
x
y
2
x
2
x
1
x
n
Fig. P6.C4
x
2
x
1
y
1
t
2
t
1
y
n
t
n
e
y
2
O
V
y
x
Fig. P6.C5
O
b
2
t
0
t
1
t
2
t
i
t
n
a
n
a
i
a
i
a
n
a
1
a
1
a
2
a
2
e
b
i
b
n
Fig. P6.C6
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The aircraft shown is being tested to
determine how the forces due to lift
would be distributed over the wing.
This chapter deals with stresses and
strains in structures and machine
components.
436
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