38
MECHANICS OF MATERIALS
UNIAXIAL STRESS-STRAIN
Stress-Strain Curve for Mild Steel
♦
The slope of the linear portion of the curve equals the
modulus of elasticity.
DEFINITIONS
Engineering Strain
ε
=
∆
L
/
L
0
, where
ε
=
engineering strain (units per unit),
∆
L
=
change in length (units) of member,
L
0
=
original length (units) of member.
Percent Elongation
% Elongation
=
100
o
L
L
⎛⎞
∆
×
⎜⎟
⎝⎠
Percent Reduction in Area (RA)
The % reduction in area from initial area,
A
i
, to final area,
A
f
, is:
%RA
=
100
if
i
AA
A
−
×
True Stress is load divided by actual cross-sectional area.
Shear Stress-Strain
γ
=
τ
/
G
, where
= shear strain,
τ
=
shear stress, and
G
=
shear modulus
(constant in linear force-deformation
relationship).
()
ν
+
=
1
2
E
G
, where
E = modulus of elasticity
v
=
Poisson's ratio
, and
=
– (lateral strain)/(longitudinal strain).
Uniaxial Loading and Deformation
σ
=
P/A
, where
σ
= stress on the cross section,
P
= loading, and
A
= cross-sectional area.
ε
=
δ
/
L
, where
δ
= elastic longitudinal deformation and
L
= length of member.
AE
PL
L
A
P
E
=
δ
δ
=
ε
σ
=
THERMAL DEFORMATIONS
δ
t
=
α
L
(
Τ
–
o
), where
δ
t
=
deformation caused by a change in temperature,
α
=
temperature coefficient of expansion,
L
=
length of member,
=
final temperature, and
o
= initial temperature.
CYLINDRICAL PRESSURE VESSEL
Cylindrical Pressure Vessel
For internal pressure only, the stresses at the inside wall are:
i
r
i
o
i
o
i
t
P
r
r
r
r
P
−
>
σ
>
−
+
=
σ
0
and
2
2
2
2
For external pressure only, the stresses at the outside wall
are:
o
r
i
o
i
o
o
t
P
r
r
r
r
P
−
>
σ
>
−
+
−
=
σ
0
and
2
2
2
2
, where
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:
2
2
2
i
o
i
i
a
r
r
r
P
−
=
σ
These are principal stresses.
♦
Flinn, Richard A. & Paul K. Trojan,
Engineering Materials & Their Applications,
4th ed., Houghton Mifflin Co., 1990.