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7/29/2019 shearing stresses
1/26
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.
6Shearing Stresses in
Beams and Thin-
Walled Members
7/29/2019 shearing stresses
2/26
2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALShird
Edition
Beer Johnston DeWolf
6 - 2
Shearing Stresses in Beams and
Thin-Walled Members
IntroductionShear 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 DeformationsSample Problem 6.3
Unsymmetric Loading of Thin-Walled Members
Example 6.05
Example 6.06
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2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALShird
Edition
Beer Johnston DeWolf
6 - 3
Introduction
00
0
00
xzxzz
xyxyy
xyxzxxx
yMdAF
dAzMVdAF
dAzyMdAF
t
t
tt
Distribution of normal and shearing
stresses satisfies
Transverse loading applied to a beam
results in normal and shearing stresses intransverse sections.
When shearing stresses are exerted on the
vertical faces of an element, equal stressesmust be exerted on the horizontal faces
Longitudinal shearing stresses must exist
in any member subjected to transverse
loading.
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2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALShird
Edition
Beer Johnston DeWolf
6 - 4
Shear on the Horizontal Face of a Beam Element
Consider prismatic beam
For equilibrium of beam element
A
CD
ADDx
dAyI
MMH
dAHF 0
xVxdx
dMMM
dAyQ
CD
A
Note,
flowshearI
VQ
x
Hq
xI
VQH
Substituting,
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2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALShird
Edition
Beer Johnston DeWolf
6 - 5
Shear on the Horizontal Face of a Beam Element
flowshearI
VQ
x
H
q
Shear flow,
where
sectioncrossfullofmomentsecond
aboveareaofmomentfirst
'
2
1
AA
A
dAyI
y
dAyQ
Same result found for lower area
HH
qI
QV
x
H
q
axisneutralto
respecthmoment witfirst
0
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2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALShird
Edition
Beer Johnston DeWolf
6 - 6
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 isV= 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.
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7/26 2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALShird
Edition
Beer Johnston DeWolf
6 - 7
Example 6.01
46
2
3
121
3
12
1
36
m1020.16
]m060.0m100.0m020.0
m020.0m100.0[2
m100.0m020.0
m10120
m060.0m100.0m020.0
I
yAQ
SOLUTION:
Determine the horizontal force per
unit length or shear flow q on the
lower surface of the upper plank.
mN3704
m1016.20
)m10120)(N500(
46-
36
I
VQq
Calculate the corresponding shear
force in each nail for a nail spacing of25 mm.
mNqF 3704)(m025.0()m025.0(
N6.92F
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MECHANICS OF MATERIALShird
Edition
Beer Johnston DeWolf
6 - 8
Determination of the Shearing Stress in a Beam
The average shearing stress on the horizontal
face of the element is obtained by dividing theshearing force on the element by the area of
the face.
ItVQ
xt
x
I
VQ
A
xq
A
Have
t
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 atD1
andD2 are significantly higher than atD.
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9/26 2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALShird
Edition
Beer Johnston DeWolf
6 - 9
Shearing Stresses txyin Common Types of Beams For a narrow rectangular beam,
A
V
c
y
A
V
Ib
VQxy
2
3
12
3
max
2
2
t
t
For American Standard (S-beam)
and wide-flange (W-beam) beams
web
ave
A
VIt
VQ
maxt
t
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10/26 2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALShird
Edition
Beer Johnston DeWolf
6 - 10
Further Discussion of the Distribution of
Stresses in a Narrow Rectangular Beam
2
2
12
3
c
y
A
Pxyt
I
Pxyx
Consider a narrow rectangular cantilever beam
subjected to load P at its free end:
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-Venants 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.
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MECHANICS OF MATERIALShird
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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,
psi120psi1800 allallt
determine the minimum required depth
dof 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.
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12/26 2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALShird
Edition
Beer Johnston DeWolf
6 - 12
Sample Problem 6.2
SOLUTION:
Develop shear and bending momentdiagrams. Identify the maximums.
inkip90ftkip5.7
kips3
max
max
M
V
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MECHANICS OF MATERIALShird
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Beer Johnston DeWolf
6 - 13
Sample Problem 6.2
2
2
61
2
61
3
121
in.5833.0
in.5.3
d
d
dbc
IS
dbI
Determine the beam depth based on allowable
normal stress.
in.26.9
in.5833.0
in.lb1090psi1800
2
3
max
d
d
S
Mall
Determine the beam depth based on allowable
shear stress.
in.71.10
in.3.5
lb3000
2
3psi120
2
3 max
d
d
A
Vallt
Required beam depth is equal to the larger of the two.
in.71.10d
MECHANICS OF MATERIALS
E
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14/26 2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALShird
Edition
Beer Johnston DeWolf
6 - 14
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 anelement defined by the curved surface
CDDC.
a
dAHF CDx 0
Except for the differences inintegration areas, this is the same
result obtained before which led to
I
VQ
x
Hqx
I
VQH
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E
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MECHANICS OF MATERIALShird
Edition
Beer Johnston DeWolf
6 - 15
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.
MECHANICS OF MATERIALSh
Ed
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16/26 2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALShird
Edition
Beer Johnston DeWolf
6 - 16
Example 6.04
For the upper plank,
3in22.4
.in875.1.in3in.75.0
yAQ
For the overall beam cross-section,
4
3
1213
121
in42.27
in3in5.4
I
SOLUTION:
Determine the shear force per unitlength along each edge of the upper
plank.
lengthunitperforceedge
in
lb15.46
2
in
lb3.92
in27.42
in22.4lb6004
3
qf
I
VQq
Based on the spacing between nails,
determine the shear force in eachnail.
in75.1in
lb15.46
fF
lb8.80F
MECHANICS OF MATERIALSh
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MECHANICS OF MATERIALShird
Edition
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6 - 17
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
xI
VQH
It
VQ
xt
Hxzzx
tt
The corresponding shear stress is
NOTE: 0xyt
0xzt
in the flanges
in the web
Previously found a similar expression
for the shearing stress in the web
It
VQxy t
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MECHANICS OF MATERIALShird
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6 - 18
Shearing Stresses in Thin-Walled Members
The variation of shear flow across the
section depends only on the variation ofthe first moment.
I
VQtq t
For a box beam, q grows smoothly fromzero at A to a maximum at Cand Cand
then decreases back to zero atE.
The sense ofq in the horizontal portions
of the section may be deduced from thesense in the vertical portions or the
sense of the shear V.
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MECHANICS OF MATERIALShird
dition
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6 - 19
Shearing Stresses in Thin-Walled Members
For a wide-flange beam, the shear flow
increases symmetrically from zero atA
and A, reaches a maximum at Cand the
decreases to zero atEand E.
The continuity of the variation in q andthe merging ofq from section branches
suggests an analogy to fluid flow.
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MECHANICS OF MATERIALShird
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6 - 20
The section becomes fully plastic (yY= 0) at
the wall when
pY MMPL 2
3
For PL >MY, yield is initiated atB and B.
For an elastoplastic material, the half-thickness
of the elastic core is found from
2
2
3
11
2
3
c
yMPx YY
Plastic Deformations
momentelasticmaximum YYc
IM Recall:
For M = PL < MY, the normal stress doesnot exceed the yield stress anywhere along
the beam.
Maximum load which the beam can support is
L
MP
pmax
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MECHANICS OF MATERIALShird
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6 - 21
Plastic Deformations
Preceding discussion was based on
normal stresses only
Consider horizontal shear force on an
element within the plastic zone,
0 dAdAH YYDC
Therefore, the shear stress is zero in theplastic zone.
Shear load is carried by the elastic core,
A
P
byA
y
y
A
PY
Y
xy
2
3
2where1
2
3
max
2
2
t
t
As Adecreases, tmax increases and
may exceed tY
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MECHANICS OF MATERIALShird
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6 - 22
Sample Problem 6.3
Knowing that the vertical shear is 50
kips in a W10x68 rolled-steel beam,
determine the horizontal shearing
stress in the top flange at the point a.
SOLUTION:
For the shaded area,
3in98.15
in815.4in770.0in31.4
Q
The shear stress at a,
in770.0in394in98.15kips50
4
3
It
VQt
ksi63.2t
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6 - 23
Unsymmetric Loading of Thin-Walled Members
Beam loaded in a vertical plane
of symmetry deforms in thesymmetry plane without
twisting.
It
VQ
I
Myavex t
Beam without a vertical plane
of symmetry bends and twists
under loading.
It
VQ
I
Myavex t
MECHANICS OF MATERIALSh
Ed
B J h D W lf
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MECHANICS OF MATERIALSirddition
Beer Johnston DeWolf
6 - 24
When the force P is applied at a distance e to theleft 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 distributionsatisfies
FdsqdsqFdsqVIt
VQ E
D
B
A
D
Bave t
Fand F indicate a couple Fh and the need forthe application of a torque as well as the shear
load.
VehF
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Ed
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MECHANICS OF MATERIALSirdition
Beer Johnston DeWolf
6 - 25
Example 6.05
Determine the location for the shear center of the
channel section with b = 4 in., h = 6 in., and t= 0.15 in.
I
hFe
where
I
Vthb
dsh
st
I
Vds
I
VQdsqF
b bb
4
22
0 00
hbth
hbtbtthIII flangeweb
6
212
12
12
12
2121
233
Combining,
.in43
.in62
in.4
3
2
b
h
be .in6.1e
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MECHANICS OF MATERIALSirdition
Beer Johnston DeWolf
Example 6.06
Determine the shear stress distribution for
V= 2.5 kips.
ItVQ
tq t
Shearing stresses in the flanges,
ksi22.2
in6in46in6in15.0
in4kips5.26
66
62
22
2121
hbthVb
hbth
Vhb
sI
Vhhst
It
V
It
VQ
Bt
t
Shearing stress in the web,
ksi06.3in6in66in6in15.02
in6in44kips5.23
62
43
6
4
2121
81
max
hbth
hbV
thbth
hbhtV
It
VQt
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