2. THEORIES OF FAILURE
Maximum Normal Stress Theory
Maximum Normal Strain Theory
Maximum Shear Stress Theory (Tresca Yield
Criterion)
Strain Energy Theory
Maximum Distortion Energy Theory
Out of these five theories of failure, the maximum normal
stress theory and maximum normal strain theory are only
applicable for brittle materials, and the remaining three
theories are applicable for ductile materials.
3. Maximum Normal Stress Theory (Rankine Theory)
Maximum Normal Strain Theory (St-Venant
Theory)
Maximum Shear Stress Theory (Tresca Yield
Criterion)
Strain Energy Theory (Haigh Theory)
Maximum Distortion Energy Theory (Von Misses)
4. INTRODUCTION ( CONTD..)
Stress-Analysis is performed on a component to
determine
The required “size or geometry” (design)
an allowable load (service)
cause of failure (forensic)
For all of these, a limit stress or allowable stress value for
the component material is required.
Hence, a Failure-Theory is needed to define the onset or
criterion of failure
5. INTRODUCTION (CONTD..)
FAILURE
•Occurs if a component can no longer function as intended.
•Failure Modes:
• yielding: a process of global permanent plastic deformation.
Change in the geometry of the object.
• low stiffness: excessive elastic deflection.
• fracture: a process in which cracks grow to the extent that the
component breaks apart.
•buckling: the loss of stable equilibrium. Compressive loading
can lead to bucking in columns.
6. INTRODUCTION (CONTD..)
FAILURE PREDICTION
• The failure of a statically loaded member in uni-axial
tension or compression is relatively easy to predict.
• One can simply compare the stress incurred with the
strength of the material.
• When the loading conditions are Complex (i.e. biaxial
loading, sheer stresses) then we must use some method
to compare multiple stresses to a single strength value.
• These methods are known failure theories
7. INTRODUCTION (CONTD..)
NEED FOR FAILURE THEORIES
To design structural components and calculate
margin of safety.
To guide in materials development.
To determine weak and strong directions.
8. MAXIMUM NORMAL STRESS THEORY
• this theory postulates, that failure will occur in the
structural component if the maximum normal stress in
that component reaches the ultimate strength, u
obtained from the tensile test of a specimen of the same
material.
• Thus, the structural component will be safe as long as the
absolute values of the principle stresses 1 and 2 are
both less than u:
1 = U and 2 = U
• This theory deals with brittle materials only.
• The maximum normal stress theory can be expresses
graphically as shown in the figure. If the point obtained by
plotting the values 1 and 2 of the principle stress fall
within the square area shown in the figure, the structural
component is safe.
• If it falls outside that area, the component will fail.
1
2
u
u
-u
-u
9. This theory alson known as Saint-Venant’s Theory
According to this theory, a given structural component is safe
as long as the maximum value of the normal strain in that
component remains smaller than the value u of the strain at
which a tensile test specimen of the same material will fail.
As shown in the figure, the strain is maximum along one of the
principle axes of stress if the deformation is elastic and the
material homogenous and isotropic.
Thus denoting by 1 and 2 the values of the normal strain
along the principle axes in plane of stress, we write
1 = u and 2 = u
MAXIMUM NORMAL STRAIN THEORY
10. MAXIMUM NORMAL STRAIN THEORY (CONT)
Making use of the generalized Hooke’s Law, we could express
these relations in term of the principle stresses 1 and 2 and
the ultimate strength U of the material.
We would find that, according to the maximum normal strain
theory, the structural component is safe as long as the point
obtained by plotting 1 and 2 falls within the area shown in
the figure where is Poisson’s ration for the given material.
1
2
U
U
-U
-U
1
U
1
U
11. MAXIMUM SHEARING STRESS THEORY
• This theory is based on the observation that yield
in ductile materials is caused by slippage of the
material along oblique surfaces and is due
primarily to shearing stress.
• A given structural component is safe as long as
the maximum value max of the shearing stress in
that component remains smaller than the
corresponding value of the shearing stress in a
tensile test specimen of the same material as the
specimen starts to yield.
• For a 3D complex stress system, the max shear
stress is given by:
max = ½ (1-2)
• On the other hand, in the 1D stress system as
obtained in the tensile test, at the yield limit, 1=
Y and 2=0, therefore:-
• max= ½ Y
12. MAXIMUM SHEARING STRESS THEORY (CONT.)
Thus,
Graphically, the maximum shear stress criterion
requires that the two principal stresses be within the
green zone as shown in the figure.
signs
same
have
and
signs
opposite
have
and
-
-
2
1
2
1
-
2
1
and
2
1
2
1
Y
2
Y
1
2
1
Y
2
1
2
1
Y
2
1
max
Y
max
13. MAXIMUM DISTORTION ENERGY THEORY
This theory is based on the determination of the distortion
energy in a given material, i.e. of the energy associated with
changes in shape in that material (as opposed to the energy
associated with changes in volume in the same material).
A given structural component is safe as long as the maximum
value of the distortion energy per unit volume in that material
remains smaller than the distortion energy per unit volume
required to cause yield in a tensile test specimen of the same
material.
The distortion energy per unit volume in an isotropic material
under plane stress is:
-
G
6
1
U
2
2
2
1
2
1
d
14. MAXIMUM DISTORTION ENERGY THEORY (CONT)
In the particular case of a tensile test specimen that is starting
to yield, we have:-
This equation represents a principal stress ellipse as
illustrated in the figure
Von Mises criterion also gives a reasonable estimation of
fatigue failure, especially in cases of repeated tensile and
tensile-shear loading
2
2
2
1
2
1
2
Y
2
2
2
1
2
1
2
Y
2
Y
Y
d
2
Y
1
G
6
1
G
6
,
Thus
G
6
U
and
0
,
15. PROBLEM 1
The solid shaft shown in Figure has a radius of 0.5 cm and
is made of steel having a yield stress of 360 MPa.
Determine if the loadings cause the shaft to fail according
to Tresca and von mises theories.
32.5 Nm
15 kN
1 cm
18. PROBLEM 2
The state of plane stress shown
occurs at a critical point of a steel
machine component. As a result of
several tensile tests, it has been
found that the tensile yield strength
is Y=250 MPa for the grade of
steel used. Determine the factor of
safety with respect to yield, using:
(a) the maximum shearing stress
theory
(b) the maximum distortion energy
theory
80 MPa
25 MPa
40 MPa
19. SOLUTION:
92
.
1
MPa
65
MPa
125
F.S
-
:
is
yield
respect to
h
safety wit
of
factor
tha
,
Therefore
MPa
65
-
:
circle
s
mohr'
or
formula
using
by
determined
be
can
stress
shearing
maximum
The
MPa
125
250MPa
2
1
2
1
is
yield
at
stress
shearing
ing
correspond
the
,
MPa
250
Since
theory
stress
shearing
Maximum
.
)
a
(
MPa
45
and
MPa
85
circle,
s
mohr'
or
formula
using
by
determined
be
can
and
stresses
principle
The
25MPa
and
,
MPa
40
80MPa,
Given
max
Y
max
Y
Y
Y
2
1
2
1
xy
y
x
20. SOLUTION (CONT)
19
.
2
S
.
F
S
.
F
250
3
.
114
S
.
F
250
45
45
85
85
S
.
F
Criterion
Energy
Distortion
Maximum
.
)
b
(
2
2
2
2
Y
2
2
2
1
2
1
