2. Strain
2. Strain
CHAPTER OBJECTIVES
z Define concept of
normal strain
z Define concept of
shear strain
z Determine normal
and shear strain in
engineering
applications
Chapter 3: Strain
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2. Strain
2. Strain
CHAPTER OUTLINE
2.1 DEFORMATION
Deformation
z Occurs when a force is applied to a body
z Can be highly visible or practically unnoticeable
z Can also occur when temperature of a body is
changed
z Is not uniform throughout a body’s volume, thus
change in geometry of any line segment within
body may vary along its length
1. Deformation
2. Strain
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2. Strain
2. Strain
2.1 DEFORMATION
2.2 STRAIN
Normal strain
z Defined as the elongation or contraction of a line
segment per unit of length
z Consider line AB in figure below
z After deformation, ∆s changes to ∆s’
To simplify study of deformation
z Assume lines to be very short and located in
neighborhood of a point, and
z Take into account the orientation of the line
segment at the point
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2. Strain
2. Strain
2.2 STRAIN
2.2 STRAIN
Normal strain
z Defining average normal strain using εavg (epsilon)
εavg =
∆s − ∆s’
∆s
Normal strain
z If normal strain ε is known, use the equation to
obtain approx. final length of a short line segment
in direction of n after deformation.
∆s’ ≈ (1 + ε) ∆s
z As ∆s → 0, ∆s’ → 0
ε=
lim
∆s − ∆s’
B→A along n ∆s
z Hence, when ε is positive, initial line will elongate,
if ε is negative, the line contracts
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2. Strain
2. Strain
2.2 STRAIN
Normal Strain
Units
z Normal strain is a dimensionless quantity, as
it’s a ratio of two lengths
z But common practice to state it in terms of
meters/meter (m/m)
z ε is small for most engineering applications, so
is normally expressed as micrometers per
meter (µm/m) where 1 µm = 10−6
z Also expressed as a percentage,
e.g., 0.001 m/m = 0.1 %
σ=
ε=
P
= stress
A
δ
text, p. 48
L
= normal strain
σ=
ε=
2P P
=
2A A
δ
L
Stress & Strain: Axial Loading
P
A
2δ δ
ε=
=
2L L
σ=
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2. Strain
2. Strain
2.2 STRAIN
2.2 STRAIN
Shear strain
z Defined as the change in angle that occurs
between two line segments that were originally
perpendicular to one another
z This angle is denoted by γ (gamma) and
measured in radians (rad).
Shear strain
z Consider line segments AB and AC originating
from same point A in a body, and directed along
the perpendicular n and t axes
z After deformation, lines become curves, such that
angle between them at A is θ’
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2. Strain
2. Strain
2.2 STRAIN
EXAMPLE 2.3
Plate is deformed as shown in figure. In this
deformed shape, horizontal lines on the on
plate remain horizontal and do not change
their length.
Shear strain
z Hence, shear strain at point A associated with n
and t axes is
γnt =
π
− θ’
2
Determine
(a) average normal strain
along side AB,
(b) average shear strain
in the plate relative to
x and y axes
z If θ’ is smaller than π/2, shear strain is positive,
otherwise, shear strain is negative
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2. Strain
2. Strain
EXAMPLE 2.3 (SOLN)
EXAMPLE 2.3 (SOLN)
(a) Line AB, coincident with y axis, becomes
line AB’ after deformation. Length of line
AB’ is
(a) Therefore, average normal strain for AB is,
(εAB)avg =
AB’ = √ (250 − 2)2 + (3)2 = 248.018 mm
AB’ − AB
248.018 mm − 250 mm
=
250 mm
AB
= −7.93(10−3) mm/mm
Negative sign means
strain causes a
contraction of AB.
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2. Strain
2. Strain
EXAMPLE 2.3 (SOLN)
EXAMPLE 2
(b) Due to displacement of B to B’, angle
BAC referenced from x, y axes changes
to θ’.
Since γxy = π/2 − θ’, thus
γxy = tan−1
= 0.0121 rad
( 250 mm3 mm
− 2 mm )
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2. Strain
2. Strain
EXAMPLE 3
EXAMPLE 4
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2. Strain
2. Strain
CHAPTER REVIEW
CHAPTER REVIEW
z Loads cause bodies to deform, thus
points in the body will undergo
displacements or changes in position
z Normal strain is a measure of
elongation or contraction of small line
segment in the body
z Shear strain is a measure of the change
in angle that occurs between two small
line segments that are originally
perpendicular to each other
z State of strain at a point is described
by six strain components:
a) Three normal strains: εx, εy, εz
b) Three shear strains: γxy, γxz, γyz
c) These components depend upon the orientation
of the line segments and their location in the
body
z Strain is a geometrical quantity
measured by experimental techniques.
Stress in body is then determined from
material property relations
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