# Chapter 8: Stress & Strain

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18 Ιουλ 2012 (πριν από 5 χρόνια και 9 μήνες)

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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

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l

f

t

t

t

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
Cut-out 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 cross-section
The internal load P must pass through the centroid
of the cross-section 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
Uni-axial stress
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Average Shear Stress
Average shear stress:
Internalresultantshearforce
Internal

resultant

shear

force

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Single & Double Shear
Single-shear connection

1s桥慲獵牦慣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

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Single & Double Shear
Double-shear connection

2s桥慲

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

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䵡湵Mac瑵物rg⁅湧楮eer楮g
D

l

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

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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

a

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 cross-section 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 cross-section for which the depth and
thickness are 40 mm. If an axial force of 800 N is applied along the centroidalaxis
ofthebar
`
scross
of

the

bars

cross
-
sectional

area
,
determine

the

average

normal

stress

and

average shear stress acting on the material along a) section a-a and b) section b-b.
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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
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 mm-diameter 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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