Chapter 8: Stress & Strain
Mechanics of Materials : relationship between the external loads
and the
intensity of the internal forces
acting within body.
By using the principles of statics, determine the forces
and deformations
acting both on and within the member.
Size of the members, their deflection and their stability depend also on the
type of material
from which the members are made.
Material behaviour is of vital im
p
ortance for develo
p
in
g
the necessar
y
ppgy
equations used in mechanics of materials.
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Normal stress
Normal stress
䥦瑨lftll
瑩lt
䥦
瑨
攠湯牭e
l
f
潲捥爠o
t
牥獳⁰r
汬
猠
t
敮e
楬
攠e
t
牥獳
If the force or stress pushes compressive stress
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Shear stress
Shear stress : force per unit area, acting tangent
to differential area
Orientation of the area
Direction
line
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General state of stress
Cutout a cube which represents the
state of stressacting around the chosen
point in the body.
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Average Normal Stress in an Axially Loaded
Bar
P is applied along the centroidal axis of the cross
section.
Homogeneous and isotropic material
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Average Normal Stress in an Axially Loaded
Bar
We will not consider the regions of the bar near its
end. A
pp
lied force causes localized distortions.
pp
Consider the stress distribution within the bar’s
midsection.
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Average Normal Stress Distribution
Due to
e
q
uilibrium:
q
Average normal stress on the crosssection
The internal load P must pass through the centroid
of the crosssection since the uniform stress
distribution will produce zero moments about any x
andyaxespassingthroughthispoint
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and
y
axes
passing
through
this
point
.
Equilibrium
Uniaxial stress
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Average Shear Stress
Average shear stress:
Internalresultantshearforce
Internal
resultant
shear
force
周楳楳慮
慰灲潸業慴a敱畡瑩潮
扵b
䍲潳猭獥捴楯湡氠慲敡
周楳
楳
慮
慰灲潸業慴a
敱畡瑩潮
扵b
獵sf楣楥湴汹捣畲慴攠景爠浡湹湧楮敥物湧m
慰灬楣慴楯湳a
䵡湵Mac瑵物rg⁅湧楮eer楮g
Single & Double Shear
Singleshear connection
1s桥慲獵牦慣s
却敥S
W潯o
For the steel connection
, the nut is not
tightened to a great extent. Therefore the
friction between the plates is neglected.
1
獨敡s
獵牦慣s
䵡湵Mac瑵物rg⁅湧楮eer楮g
卩湧汥慰S
橯楮j
Single & Double Shear
Doubleshear connection
2s桥慲
却敥S
W潯o
For the steel connection
, the nut is not
tightened to a great extent. Therefore the
friction between the plates is neglected.
2
獨敡s
獵牦慣s
Dbll橩j
䵡湵Mac瑵物rg⁅湧楮eer楮g
D
潵
扬
攠
l
慰a
j
o
i
n
t
Equilibrium
Consider a volume element taken at a point located on the surface of any
sectioned area on which the average shear stress acts:
P
u
r
e
ue
Shear
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Deformation
Applied force tend to change the shape and the size of the body.
Deformation
p
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In
general
,
the
deformation
of
a
body
will
not
be
uniform
throughout
its volume.
Strain
Normal strain : Elongation or contraction of a line
segment per unit length
Strain is a dimensionless quantity.
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Strain
Shear strain : The change in angle that occurs between
twolinesegmentsthatwereoriginallyperpendicularto
two
line
segments
that
were
originally
perpendicular
to
each other
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Cartesian Strain Components
Consider a small volumetric
element
乯牭慬s瑲慩湳捡畳ca
捨慮来楮癯汵浥
潦瑨t
乯牭慬
獴牡楮s
捡畳c
a
捨慮来
楮
癯汵浥
潦
瑨t
牥捴慮杵污爠敬敭敮r
Shear strains cause a change in its shape
The state of strain at a point in a body is defined
by:
by:
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Example Problem
Determine the average normal stress in each segment of the bar shown on the
figure below. The structure has a rectangular crosssection of 35 mm x 10 mm.
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Example Problem
The 80 kg lamp is supported by two rods AB and BC as shown in the figure. If
AB has a diameter of 10 mm and BC has a diameter of 8 mm, determine the
averagenormalstressineachrod
average
normal
stress
in
each
rod
.
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Example Problem
The bar shown in the figure has a square crosssection for which the depth and
thickness are 40 mm. If an axial force of 800 N is applied along the centroidalaxis
ofthebar
`
scross
sectionalareadeterminetheaveragenormalstressand
of
the
bars
cross

sectional
area
,
determine
the
average
normal
stress
and
average shear stress acting on the material along a) section aa and b) section bb.
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Example Problem
Theinclinedmemberinthefigureissubjectedtoacompressiveforceof3000N.
Determinetheaveragecompressivestressalongthesmoothareasofcontact
defined
by
AB
and
BC
and
the
average
shear
stress
along
the
horizontal
plane
defined
by
AB
and
BC
,
and
the
average
shear
stress
along
the
horizontal
plane
definedbyEDB.
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Example Problem
Theplateisdeformedintothedashedshapeshowninthefigure.Ifinthis
deformedshapehorizontallinesontheplateremainhorizontalanddonotchange
their
length
determine
a)
the
average
normal
strain
along
the
side
AB
b)
the
their
length
,
determine
a)
the
average
normal
strain
along
the
side
AB
,
b)
the
averageshearstrainintheplaterelativetothexandyaxes.
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Example Problem
Thesquaredeformsintothepositionshownbythedashedlines.Determinethe
a)averagenormalstrainalongeachdiagonal(ABandCD)andb)averageshear
strain
at
each
of
its
corners
(AC)
Side
D
and
B
remains
horizontal
strain
at
each
of
its
corners
(A
,
C)
.
Side
D
and
B
remains
horizontal
.
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Example Problem
Determine the intensity w of the maximum distributed load that can be
supported by the hanger assembly so that an allowable shear stress of 100 MPa
itddith12
ditblttAdBdllbltil
i
s no
t
excee
d
e
d
i
n
th
e
12
mm
di
ame
t
er
b
o
lt
s a
t
A
an
d
B
an
d
an a
ll
owa
bl
e
t
ens
il
e
stress of 150 MPais not exceeded in the 15 mmdiameter rod AB.
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Example Problem
The pedestal supports a load P at its center. If the material has a mass density ρ,
determninethe theradial dimension r as a function of z so that the average normal
stressinthepedestalremainsconstantThecross
sectionsarecircular
stress
in
the
pedestal
remains
constant
.
The
cross

sections
are
circular
.
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