33
MECHANICS OF MATERIALS
UNIAXIAL STRESSSTRAIN
StressStrain Curve for Mild Steel
♦
The slope of the linear portion of the curve equals the
modulus of elasticity.
DEFINITIONS
Engineering Strain
ε
=
∆
L
/
L
o
, where
ε
=
engineering strain (units per unit),
∆
L
=
change in length (units) of member,
L
o
=
original length (units) of member.
Percent Elongation
% Elongation
=
L
L
100
o
#
D
c
m
Percent Reduction in Area (RA)
The % reduction in area from initial area,
A
i
, to final area,
A
f
, is:
%
RA =
A
A
A
100
i
i
f
#

e
o
Shear StressStrain
γ
=
τ
/G
,
where
γ
=
shear strain,
τ
=
shear stress, and
G
=
shear modulus
(constant in linear torsionrotation
relationship).
,
G
v
E
2 1
where
=
+
^
h
E
=
modulus of elasticity
v
=
Poisson’s ratio
, and
=
– (lateral strain)/(longitudinal strain).
STRESS, PSI
STRESS, MPa
MECHANICS OF MATERIALS
Uniaxial Loading and Deformation
σ
= P/A
, where
σ
=
stress on the cross section,
P
=
loading, and
A
=
crosssectional area.
ε
=
δ
/L
,
where
δ
=
elastic longitudinal deformation and
L
=
length of member.
E
L
P
A
AE
PL
=
=
=
v f
d
d
True stress is load divided by actual crosssectional area
whereas engineering stress is load divided by the initial area.
THERMAL DEFORMATIONS
δ
t
= α
L
(
T
–
T
o
)
,
where
δ
t
=
deformation caused by a change in temperature,
α
=
temperature coefficient of expansion,
L
=
length of member,
T
=
final temperature, and
T
o
=
initial temperature.
CYLINDRICAL PRESSURE VESSEL
Cylindrical Pressure Vessel
For internal pressure only, the stresses at the inside wall are:
P
r
r
r
r
P
and
t
i
o
i
o
i
r
i
2
2
2
2
=

+
=
v
v
For external pressure only, the stresses at the outside wall are:
,
P
r
r
r
r
P
and
where
t
o
o
i
o
i
r
o
2
2
2
2
=

+
=
v
v
σ
t
=
tangential (hoop) stress,
σ
r
=
radial stress,
P
i
=
internal pressure,
P
o
=
external pressure,
r
i
=
inside radius, and
r
o
=
outside radius.
For vessels with end caps, the axial stress is:
P
r
r
r
a
i
o
i
i
2
2
2
=

v
σ
t
,
σ
r
, and
σ
a
are principal stresses.
♦
Flinn, Richard A. & Paul K. Trojan,
Engineering Materials & Their Applications,
4th ed., Houghton Mifflin Co., Boston, 1990.
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3±
MECHANICS OF MATERIALS
When the thickness of the cylinder wall is about onetenth or
less of inside radius, the cylinder can be considered as thin
walled. In which case, the internal pressure is resisted by the
hoop stress and the axial stress.
t
Pr
t
P r
2
and
t
i
a
i
=
=
v
v
where
t
= wall thickness.
STRESS AND STRAIN
Principal Stresses
For the special case of a
twodimensional
stress state, the
equations for principal stress reduce to
,
2
2
0
a
b
x
y
x
y
xy
c
2
2
!
=
+

+
=
v
v
v
v
v
v
x
v
d
n
The two nonzero values calculated from this equation are
temporarily labeled
σ
a
and
σ
b
and the third value
σ
c
is
always
zero in this case. Depending on their values, the three roots are
then labeled according to the convention:
algebraically largest
=
σ
1
,
algebraically smallest
=
σ
3
,
other
=
σ
2
. A typical 2D stress element is shown below with
all indicated components shown in their positive sense.
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 Shear Stress, maximum shear stress, Sut Sy

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