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MECHANICS OF
MATERIALS
Third Edition
Ferdinand P. Beer
E. Russell Johnston, Jr.
John T. e!ol"
Le#ture Notes$
J. !alt Oler
Te%as Te#h &ni'ersit(
CHAPTER
© 2002 The McGraw-Hill Companies, Inc. All rights
8Principle StressesUnder a Given
Loading
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© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALST h i r d Beer ) Johnston ) e!ol"
Principle Stresses Under a Given Loading
Introduction
Principle Stresses in a Beam
Sample Problem 8.1
Sample Problem 8.2
Design of a Transmission Shaft
Sample Problem 8.3
Stresses nder !ombined "oadings
Sample Problem 8.#
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© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALST h i r d Beer ) Johnston ) e!ol"
Introduction
$ In !haps. 1 and 2% &ou learned ho' to determine the normal stress due
to centric loadsIn !hap. 3% &ou anal&(ed the distribution of shearing stresses in a
circular member due to a t'isting couple
In !hap. )% &ou determined the normal stresses caused b& bending
couples
In !haps. # and *% &ou e+aluated the shearing stresses due totrans+erse loads
In !hap. ,% &ou learned ho' the components of stress are transformed
b& a rotation of the coordinate aes and ho' to determine the
principal planes% principal stresses% and maimum shearing stress
at a point.
$ In !hapter 8% &ou 'ill learn ho' to determine the stress in a structural
member or machine element due to a combination of loads and
ho' to find the corresponding principal stresses and maimum
shearing stress
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MECHANICS OF MATERIA ST
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© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSTh i r d Beer ) Johnston ) e!ol"
Principle Stresses in a Beam
MECHANICS OF MATERIALST
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© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSTh i r d Beer ) Johnston ) e!ol"
Principle Stresses in a Beam
$ !ross-section shape results in large +alues of τ xy
near the surface 'here σ x is also large.
$ σ max ma& be greater than σ m
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MECHANICS OF MATERIALSTh i r d Beer ) Johnston ) e!ol"
Sample Problem !"
/ 1*0- force is applied at the end
of a 200#2 rolled-steel beam.
eglecting the effects of fillets and
of stress concentrations% determine
'hether the normal stresses satisf& a
design specification that the& be
e4ual to or less than 1#0 5Pa at
section A-A’.
S6"TI67$ Determine shear and bending
moment in Section A-A’
$ !alculate the normal stress at top
surface and at flange-'eb unction.
$ +aluate the shear stress at flange-
'eb unction.
$ !alculate the principal stress at
flange-'eb unction
MECHANICS OF MATERIALST
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MECHANICS OF MATERIALSTh i r d Beer ) Johnston ) e!ol"
Sample Problem !"
S6"TI67
$ Determine shear and bending moment inSection A-A’
( ) ( )
1*0
m-*0m3,#.01*0
===
A
A
V
M
$ !alculate the normal stress at top surface
and at flange-'eb unction.
( )
5Pa9.102
mm103
mm).905Pa2.11,
5Pa2.11,
m10#12
m*03*
=
===
×
⋅== −
c
yσ
S
M
bab
Aa
σ
σ
MECHANICS OF MATERIALST
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MECHANICS OF MATERIALSTh i r d Beer ) Johnston ) e!ol"
Sample Problem !"
$ +aluate shear stress at flange-'eb unction.
( )
( ) ( )( )( )
5Pa#.9#
m00,9.0m10,.#2
m10*.2)81*0
m10*.2)8
mm10*.2)8,.9**.1220)
)*
3*
3*
33
=
×
×==
×=×=×=
−
−
−
It
QV
Q
Abτ
$ !alculate the principal stress at
flange-'eb unction
( )
( )
( )5Pa1#05Pa9.1*9
#.9#2
9.102
2
9.102 22
22
21
21
ma
>=
+
+=
++= bbb τ σ σ σ
Design specification is not satisfied.
MECHANICS OF MATERIALST
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MECHANICS OF MATERIALSTh i r d Beer ) Johnston ) e!ol"
Sample Problem !#
The o+erhanging beam supports a
uniforml& distributed load and a
concentrated load. :no'ing that for
the grade of steel to used σ all ; 2) si
andτ all ; 1).# si% select the 'ide-
flange beam 'hich should be used.
S6"TI67
$ Determine reactions at A and D.
$
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MECHANICS OF MATERIALSTh i r d Beer ) Johnston ) e!ol"
Sample Problem !#
$ !alculate re4uired section modulusand select appropriate beam section.
section beam*2select 21
in,.119si2)
inip2) 3mamin
×
=⋅
==all
M S
σ
S6"TI67
$ Determine reactions at A and D.
ips)10
ips#90
=⇒=∑
=⇒=∑
A D
D A
R M
R M
$ Determine maimum shear and bending
moment from shear and bending moment
diagrams.
ips)3
ips2.12'ithinip).239
ma
ma
=
=⋅=
V
V M
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MECHANICS OF MATERIALSTh i r d Beer ) Johnston ) e!ol"
Sample Problem !#
$
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MECHANICS OF MATERIALSTh i r d Beer ) Johnston ) e!ol"
$esign o% a Transmission S&a%t
$ If po'er is transferred to and from the
shaft b& gears or sprocet 'heels% theshaft is subected to trans+erse loading
as 'ell as shear loading.
$ ormal stresses due to trans+erse loads
ma& be large and should be included indetermination of maimum shearing
stress.
$ Shearing stresses due to trans+erse
loads are usuall& small andcontribution to maimum shear stress
ma& be neglected.
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MECHANICS OF MATERIALSThi r d Beer ) Johnston ) e!ol"
$esign o% a Transmission S&a%t
$ /t an& section%
J
Tc
M M M I
Mc
m
z ym
=
+==
τ
σ
222
'here
$ 5aimum shearing stress%
( )
22ma
222
2
ma
2section%-crossannularorcircularafor
22
T M
J
c
J I
J
Tc
I
Mcm
m
+=
=
+
=+
=
τ
τ σ
τ
$ Shaft section re4uirement%
all
T M
c
J
τ
ma
22
min
+
=
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MECHANICS OF MATERIALSThi r d Beer ) Johnston ) e!ol"
Sample Problem !'
Solid shaft rotates at )80 rpm and
transmits 30 from the motor to
gears G and H = 20 is taen off at
gear G and 10 at gear H .:no'ing that σ all ; #0 5Pa% determine
the smallest permissible diameter for
the shaft.
S6"TI67
$ Determine the gear tor4ues and
corresponding tangential forces.
$
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MECHANICS OF MATERIALSThi r d Beer ) Johnston ) e!ol"
Sample Problem !'
S6"TI67
$ Determine the gear tor4ues and correspondingtangential forces.
( )
( )
( ))9.2m 199
>(802
10
*3.*m 398>(802
20
,3.3
m0.1*
m #9,
m #9,>(802
30
2
=⋅==
=⋅==
=⋅
==
⋅===
D D
C C
E
E E
E
T
T
!
T
"
# T
π
π
π π
$
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MECHANICS OF MATERIALShi r d Beer ) Johnston ) e!ol"
Sample Problem !'$ Identif& critical shaft section from tor4ue and
bending moment diagrams.
( )
m 13#,
#9,3,311*0 222
ma
22
⋅=
++=
+T M
MECHANICS OF MATERIALST h
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MECHANICS OF MATERIALShi r d Beer ) Johnston ) e!ol"
Sample Problem !'
$ !alculate minimum allo'able shaft diameter.
m2#.8#m02#8#.0
m101).2,2
shaft%circularsolida
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MECHANICS OF MATERIALShi r d Beer ) Johnston ) e!ol"
Stresses Under Combined Loadings
$ ish to determine stresses in slender
structural members subected toarbitrar& loadings.
$ Pass section through points of interest.
Determine force-couple s&stem at
centroid of section re4uired to maintain
e4uilibrium.
$ S&stem of internal forces consist of
three force components and three
couple +ectors.
$ Determine stress distribution b&
appl&ing the superposition principle.
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MECHANICS OF MATERIALShi r d Beer ) Johnston ) e!ol"
Sample Problem !(
Three forces are applied to a short
steel post as sho'n. Determine the principle stresses% principal planes and
maimum shearing stress at point H.
S6"TI67
$ Determine internal forces in Section
EG.
$ !alculate principal stresses and
maimum shearing stress.
Determine principal planes.
$ +aluate shearing stress at H .
$ +aluate normal stress at H .
MECHANICS OF MATERIALST h
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MECHANICS OF MATERIALShi r d Beer ) Johnston ) e!ol"
Sample Problem !(
S6"TI67
$ Determine internal forces in Section EG.
( ) ( ) ( ) ( )
( ) ( ) m3m100.0300
m#.8
m200.0,#m130.0#0
,##030
⋅===
⋅−=
−=
−==−=
z y
x
z x
M M
M
V # V
ote7 Section properties%
( ) ( )
( ) ( )
( ) ( ) )*3121
)*3121
23
m10,),.0m0)0.0m1)0.0
m101#.9m1)0.0m0)0.0
m10*.#m1)0.0m0)0.0
−
−
−
×==
×==
×==
z
x
I
I
A
MECHANICS OF MATERIALST h
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MECHANICS OF MATERIALSi r d Beer ) Johnston ) e!ol" Sample Problem !(
$ +aluate normal stress at H .
( ) ( )
( ) ( )
( ) 5Pa**.05Pa2.233.8093.8
m101#.9
m02#.0m#.8
m10,),.0
m020.0m3
m10#.*
#0
)*
)*23-
=−+=×
⋅−
×
⋅+
×=
−++=
−
−
x x
z z y
I
b M
I
a M
A
# σ
$ +aluate shearing stress at H .
( ) ( )[ ]( )
( ) ( )( )( )
5Pa#2.1,
m0)0.0m101#.9
m10#.8#,#
m10#.8#
m0),#.0m0)#.0m0)0.0
)*
3*
3*
11
=
×
×==
×=
==
−
−
−
t I
QV
y AQ
x
z yz τ
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