Stress is the ratio of applied load to the cross-sectional area of an
element in tension and isexpressed in pounds per square inch (psi) or
kg/mm2.
Load
L
Stress,
=
=
Area
A
Strain (normal)
A measure of the deformation of the material that
is dimensionless.
change in length
L
Strain,
=
=
original length
L
Modulus of elasticity
Metal deformation is proportional to the imposed
loads over a range of loads.
Since stress is proportional to load and strain
is proportional to deformation, this implies that
stress is proportional to strain. Hooke's Law is the
statement of that proportionality.
Stress
=
=
E
Strain
The constant, E, is the modulus of
elasticity, Young's modulus or the tensile modulus
and is the material's stiffness. Young's modulus is
in terms of 106 psi or 103
kg/mm2. If a material obeys Hooke's Law
it is elastic. The modulus is insensitive to a
material's temper. Normal force is directly
dependent upon the elastic modulus.
Proportional limit
The greatest stress at which a material is
capable of sustaining the applied load without
deviating from the proportionality of stress to
strain. Expressed in psi (kg/mm2).
Ultimate strength (tensile)
The maximum stress a material withstands when
subjected to an applied load. Dividing the load at
failure by the original cross sectional area
determines the value.
Elastic limit
The point on the stress-strain curve beyond which
the material permanently deforms after removing the
load .
Yield strength
Point at which material exceeds the elastic limit
and will not return to its origin shape or length if
the stress is removed. This value is determined by
evaluating a stress-strain diagram produced during
a tensile test.
Poisson's ratio
The ratio of the lateral to longitudinal strain
is Poisson's ratio.
lateral strain
=
longitudinal strain
Poisson's ratio is a dimensionless constant used
for stress and deflection analysis of structures
such as beams, plates, shells and rotating discs.
Bending stress
When bending a piece of metal, one surface of the
material stretches in tension while the opposite
surface compresses. It follows that there is a line
or region of zero stress between the two surfaces,
called the neutral axis. Make the following
assumptions in simple bending theory:
The beam is initially straight, unstressed
and symmetric
The material of the beam is linearly
elastic, homogeneous and isotropic.
The proportional limit is not exceeded.
Young's modulus for the material is the same
in tension and compression
All deflections are small, so that planar
cross-sections remain planar before and after
bending.
Using classical beam formulas and section
properties, the following relationship can be
derived:
3PL
Bending stress,b
=
2wt2
PL3
Bending or flexural modulus,E b
=
4wt3y
Where:
P
=
normal force
l
=
beam length
w
=
beam width
t
=
beam thickness
y
=
deflection at load point
The reported
flexural modulus is usually the initial modulus from
the stress-strain curve in tension.
The maximum
stress occurs at the surface of the beam farthest
from the neutral surface (axis) and is:
Mc
M
Max surface stress,max
=
=
I
Z
Where:
M
=
bending moment
c
=
distance from neutral axis to outer
surface where max stress occurs
I
=
moment of inertia
Z
=
I/c = section modulus
For a rectangular cantilever beam
with a concentrated load at one end, the maximum
surface stress is given by:
3dEt
max
=
2l2
the methods to reduce maximum stress is to keep the
strain energy in the beam constant while changing
the beam profile. Additional beam profiles are
trapezoidal, tapered and torsion.
Where:
d
=
deflection of the beam at the load
E
=
Modulus of Elasticity
t
=
beam thickness
l
=
beam length
Yielding
Yielding occurs when the design stress exceeds
the material yield strength. Design stress is
typically maximum surface stress (simple loading) or
Von Mises stress (complex loading conditions). The
Von Mises yield criterion states that yielding
occurs when the Von Mises stress,
exceeds the yield strength in tension. Often, Finite
Element Analysis stress results use Von Mises
stresses. Von Mises stress is:
(
1-
2
)2+ (
2-
3
)2+ (
1-
3
)2
=
2
where
1,
2,
3
are principal stresses.
Safety factor
is a function of design stress and yield strength.
The following equation denotes safety factor, fs.
YS
fs
=
DS
Where YS
is the Yield Strength and DS is the Design
Stress
L
