Short Notes: Failure Theories | Short Notes for Mechanical Engineering PDF Download

 Page 1


Failure Theories 
When a machine element is subjected to a system of complex stress system, it is 
important to predict the mode of failure so that the design methodology may be 
based on a particular failure criterion.
Theories of failure are essentially a set of failure criteria developed for the ease of 
design.
In machine design an element is said to have failed if it ceases to perform its 
function.
There are basically two types of mechanical failure:
1. Yielding-This is due to excessive inelastic deformation rendering the machine 
part unsuitable to perform its function. This mostly occurs in ductile 
materials.
2. Fracture- in this case the component tears apart in two or more parts. This 
mostly occurs in brittle materials.
Failure: In order to design the cottor joint and find out the dimensions, failure, in 
different parts and different cross-sections are considered.
There are many instances when a ductile material may fail by fracture. This may 
occur if a material is subjected to
(a) Cyclic loading
(b) Long term static loading at elevated temperature
(c) Impact loading
(d) Work hardening
(e) Severe quenching
Page 2


Failure Theories 
When a machine element is subjected to a system of complex stress system, it is 
important to predict the mode of failure so that the design methodology may be 
based on a particular failure criterion.
Theories of failure are essentially a set of failure criteria developed for the ease of 
design.
In machine design an element is said to have failed if it ceases to perform its 
function.
There are basically two types of mechanical failure:
1. Yielding-This is due to excessive inelastic deformation rendering the machine 
part unsuitable to perform its function. This mostly occurs in ductile 
materials.
2. Fracture- in this case the component tears apart in two or more parts. This 
mostly occurs in brittle materials.
Failure: In order to design the cottor joint and find out the dimensions, failure, in 
different parts and different cross-sections are considered.
There are many instances when a ductile material may fail by fracture. This may 
occur if a material is subjected to
(a) Cyclic loading
(b) Long term static loading at elevated temperature
(c) Impact loading
(d) Work hardening
(e) Severe quenching
Factor of safety (f.o.s): The ratio of ultimate to allowable load or stress is known all 
factor of safety i.e. The factor of safety can be defined as the ratio of the material 
strength or failure stress to the allowable or working stress.
The factor of safety must be always greater than unity. It is easier to refer to the 
ratio of stresses since this applies to material properties, 
f.o.s = failure stress / working or allowable stress;
Ductile and Brittle Materials
• A ductile material deforms significantly before fracturing.
• Ductility is measured by % elongation at the fracture point.
• Materials with 5% or more elongation are considered ductile
• The limiting strength of ductile materials is the stress at yield point.
• A brittle material yields very little before fracturing, the yield strength is 
approximately the same as the ultimate strength in tension.
• The ultimate strength in compression is much larger than the ultimate 
strength in tension.
• The limiting strength of brittle materials is the ultimate stress
Static Failure Theories Predicting failure in members subjected to uni-axial stress 
is both simple and straight-forward. But, predicating the failure stresses for 
members subjected to bi-axial or tri-axial stresses is much more complicated.A 
large numbers of different theories have been formulated. The principal theories of 
failure for a member subjected to biaxial stress are as follows:
• Maximum Principal/Normal Stress Theory (Rankine’s Theory)
• Maximum Shear Stress Theory (Guest’s Theory)
• Maximum Principal /Normal Strain Theory (Saint’s Theory)
• Maximum Strain Energy Theory (Haigh’s Theory)
• Maximum Distortion Energy Theory (Hencky & Von Mises Theory)
Maximum Principal Stress Theory (Rankine Theory):
• If one of the principal stresses a l (maximum principal stress), o2 (minimum 
principal stress) or o3 exceeds the yield stress, yielding would occur.
• In a two dimensional loading situation for a ductile material where tensile and 
compressive yield stress are nearly of same magnitude:
c r -i = ± Oy 
C T 2 = ±C T y
Page 3


Failure Theories 
When a machine element is subjected to a system of complex stress system, it is 
important to predict the mode of failure so that the design methodology may be 
based on a particular failure criterion.
Theories of failure are essentially a set of failure criteria developed for the ease of 
design.
In machine design an element is said to have failed if it ceases to perform its 
function.
There are basically two types of mechanical failure:
1. Yielding-This is due to excessive inelastic deformation rendering the machine 
part unsuitable to perform its function. This mostly occurs in ductile 
materials.
2. Fracture- in this case the component tears apart in two or more parts. This 
mostly occurs in brittle materials.
Failure: In order to design the cottor joint and find out the dimensions, failure, in 
different parts and different cross-sections are considered.
There are many instances when a ductile material may fail by fracture. This may 
occur if a material is subjected to
(a) Cyclic loading
(b) Long term static loading at elevated temperature
(c) Impact loading
(d) Work hardening
(e) Severe quenching
Factor of safety (f.o.s): The ratio of ultimate to allowable load or stress is known all 
factor of safety i.e. The factor of safety can be defined as the ratio of the material 
strength or failure stress to the allowable or working stress.
The factor of safety must be always greater than unity. It is easier to refer to the 
ratio of stresses since this applies to material properties, 
f.o.s = failure stress / working or allowable stress;
Ductile and Brittle Materials
• A ductile material deforms significantly before fracturing.
• Ductility is measured by % elongation at the fracture point.
• Materials with 5% or more elongation are considered ductile
• The limiting strength of ductile materials is the stress at yield point.
• A brittle material yields very little before fracturing, the yield strength is 
approximately the same as the ultimate strength in tension.
• The ultimate strength in compression is much larger than the ultimate 
strength in tension.
• The limiting strength of brittle materials is the ultimate stress
Static Failure Theories Predicting failure in members subjected to uni-axial stress 
is both simple and straight-forward. But, predicating the failure stresses for 
members subjected to bi-axial or tri-axial stresses is much more complicated.A 
large numbers of different theories have been formulated. The principal theories of 
failure for a member subjected to biaxial stress are as follows:
• Maximum Principal/Normal Stress Theory (Rankine’s Theory)
• Maximum Shear Stress Theory (Guest’s Theory)
• Maximum Principal /Normal Strain Theory (Saint’s Theory)
• Maximum Strain Energy Theory (Haigh’s Theory)
• Maximum Distortion Energy Theory (Hencky & Von Mises Theory)
Maximum Principal Stress Theory (Rankine Theory):
• If one of the principal stresses a l (maximum principal stress), o2 (minimum 
principal stress) or o3 exceeds the yield stress, yielding would occur.
• In a two dimensional loading situation for a ductile material where tensile and 
compressive yield stress are nearly of same magnitude:
c r -i = ± Oy 
C T 2 = ±C T y
• Yielding occurs when the state of stress is at the boundary of the rectangle.
• At a point a, the stresses are still within the elastic limit but at b, ol reaches 
ay although a 2 is still less than ay.
• Yielding will then begin at point b
Maximum Principal Strain Theory (St. Venant’s theory):
• Yielding will occur when the maximum principal strain just exceeds the strain 
at the tensile yield point in either simple tension or compression. If el and e2 
are maximum and minimum principal strains corresponding to ol and o2, in 
the limiting case:
% = ^ ( a i - V 02)
% = p K .- V C T ,)
N - N
|a2|>|o.
It implies:
Ee, = a, - va2 - ±a0 
E£t = a , - va, = ± a 0 •
• Boundary of a yield surface in Maximum Strain Energy Theory is given below
C T i =CT0+ V O 2
Page 4


Failure Theories 
When a machine element is subjected to a system of complex stress system, it is 
important to predict the mode of failure so that the design methodology may be 
based on a particular failure criterion.
Theories of failure are essentially a set of failure criteria developed for the ease of 
design.
In machine design an element is said to have failed if it ceases to perform its 
function.
There are basically two types of mechanical failure:
1. Yielding-This is due to excessive inelastic deformation rendering the machine 
part unsuitable to perform its function. This mostly occurs in ductile 
materials.
2. Fracture- in this case the component tears apart in two or more parts. This 
mostly occurs in brittle materials.
Failure: In order to design the cottor joint and find out the dimensions, failure, in 
different parts and different cross-sections are considered.
There are many instances when a ductile material may fail by fracture. This may 
occur if a material is subjected to
(a) Cyclic loading
(b) Long term static loading at elevated temperature
(c) Impact loading
(d) Work hardening
(e) Severe quenching
Factor of safety (f.o.s): The ratio of ultimate to allowable load or stress is known all 
factor of safety i.e. The factor of safety can be defined as the ratio of the material 
strength or failure stress to the allowable or working stress.
The factor of safety must be always greater than unity. It is easier to refer to the 
ratio of stresses since this applies to material properties, 
f.o.s = failure stress / working or allowable stress;
Ductile and Brittle Materials
• A ductile material deforms significantly before fracturing.
• Ductility is measured by % elongation at the fracture point.
• Materials with 5% or more elongation are considered ductile
• The limiting strength of ductile materials is the stress at yield point.
• A brittle material yields very little before fracturing, the yield strength is 
approximately the same as the ultimate strength in tension.
• The ultimate strength in compression is much larger than the ultimate 
strength in tension.
• The limiting strength of brittle materials is the ultimate stress
Static Failure Theories Predicting failure in members subjected to uni-axial stress 
is both simple and straight-forward. But, predicating the failure stresses for 
members subjected to bi-axial or tri-axial stresses is much more complicated.A 
large numbers of different theories have been formulated. The principal theories of 
failure for a member subjected to biaxial stress are as follows:
• Maximum Principal/Normal Stress Theory (Rankine’s Theory)
• Maximum Shear Stress Theory (Guest’s Theory)
• Maximum Principal /Normal Strain Theory (Saint’s Theory)
• Maximum Strain Energy Theory (Haigh’s Theory)
• Maximum Distortion Energy Theory (Hencky & Von Mises Theory)
Maximum Principal Stress Theory (Rankine Theory):
• If one of the principal stresses a l (maximum principal stress), o2 (minimum 
principal stress) or o3 exceeds the yield stress, yielding would occur.
• In a two dimensional loading situation for a ductile material where tensile and 
compressive yield stress are nearly of same magnitude:
c r -i = ± Oy 
C T 2 = ±C T y
• Yielding occurs when the state of stress is at the boundary of the rectangle.
• At a point a, the stresses are still within the elastic limit but at b, ol reaches 
ay although a 2 is still less than ay.
• Yielding will then begin at point b
Maximum Principal Strain Theory (St. Venant’s theory):
• Yielding will occur when the maximum principal strain just exceeds the strain 
at the tensile yield point in either simple tension or compression. If el and e2 
are maximum and minimum principal strains corresponding to ol and o2, in 
the limiting case:
% = ^ ( a i - V 02)
% = p K .- V C T ,)
N - N
|a2|>|o.
It implies:
Ee, = a, - va2 - ±a0 
E£t = a , - va, = ± a 0 •
• Boundary of a yield surface in Maximum Strain Energy Theory is given below
C T i =CT0+ V O 2
Maximum Shear Stress Theory (Tresca Theory):
• Yielding would occur when the maximum shear stress just exceeds the shear 
stress at the tensile yield point. At the tensile yield point a2= o3 = 0 and thus 
maximum shear stress is oy/2.
This gives us six conditions for a three-dimensional stress situation:
°3 °1
= ±Oy
= ± O y
= - ° y
• Yield surface corresponding to maximum shear stress theory in biaxial stress 
situation is given below :
^ 2
Maximum strain energy theory ( Beltrami’s theory):
• Failure would occur when the total strain energy absorbed at a point per unit 
volume exceeds the strain energy absorbed per unit volume at the tensile 
yield point. This may be given:
1
2
Substituting, el, e2 , e3 and ey in terms of stresses we have
(5j2 + a J +<x,2 - 2 u ( a fa ? + a 2a 3 + c s i a ]) = c v2
( \
2
f \
2
( \
+
a 2
- 2 v
U J
l C T yJ
2
L a y J
Page 5


Failure Theories 
When a machine element is subjected to a system of complex stress system, it is 
important to predict the mode of failure so that the design methodology may be 
based on a particular failure criterion.
Theories of failure are essentially a set of failure criteria developed for the ease of 
design.
In machine design an element is said to have failed if it ceases to perform its 
function.
There are basically two types of mechanical failure:
1. Yielding-This is due to excessive inelastic deformation rendering the machine 
part unsuitable to perform its function. This mostly occurs in ductile 
materials.
2. Fracture- in this case the component tears apart in two or more parts. This 
mostly occurs in brittle materials.
Failure: In order to design the cottor joint and find out the dimensions, failure, in 
different parts and different cross-sections are considered.
There are many instances when a ductile material may fail by fracture. This may 
occur if a material is subjected to
(a) Cyclic loading
(b) Long term static loading at elevated temperature
(c) Impact loading
(d) Work hardening
(e) Severe quenching
Factor of safety (f.o.s): The ratio of ultimate to allowable load or stress is known all 
factor of safety i.e. The factor of safety can be defined as the ratio of the material 
strength or failure stress to the allowable or working stress.
The factor of safety must be always greater than unity. It is easier to refer to the 
ratio of stresses since this applies to material properties, 
f.o.s = failure stress / working or allowable stress;
Ductile and Brittle Materials
• A ductile material deforms significantly before fracturing.
• Ductility is measured by % elongation at the fracture point.
• Materials with 5% or more elongation are considered ductile
• The limiting strength of ductile materials is the stress at yield point.
• A brittle material yields very little before fracturing, the yield strength is 
approximately the same as the ultimate strength in tension.
• The ultimate strength in compression is much larger than the ultimate 
strength in tension.
• The limiting strength of brittle materials is the ultimate stress
Static Failure Theories Predicting failure in members subjected to uni-axial stress 
is both simple and straight-forward. But, predicating the failure stresses for 
members subjected to bi-axial or tri-axial stresses is much more complicated.A 
large numbers of different theories have been formulated. The principal theories of 
failure for a member subjected to biaxial stress are as follows:
• Maximum Principal/Normal Stress Theory (Rankine’s Theory)
• Maximum Shear Stress Theory (Guest’s Theory)
• Maximum Principal /Normal Strain Theory (Saint’s Theory)
• Maximum Strain Energy Theory (Haigh’s Theory)
• Maximum Distortion Energy Theory (Hencky & Von Mises Theory)
Maximum Principal Stress Theory (Rankine Theory):
• If one of the principal stresses a l (maximum principal stress), o2 (minimum 
principal stress) or o3 exceeds the yield stress, yielding would occur.
• In a two dimensional loading situation for a ductile material where tensile and 
compressive yield stress are nearly of same magnitude:
c r -i = ± Oy 
C T 2 = ±C T y
• Yielding occurs when the state of stress is at the boundary of the rectangle.
• At a point a, the stresses are still within the elastic limit but at b, ol reaches 
ay although a 2 is still less than ay.
• Yielding will then begin at point b
Maximum Principal Strain Theory (St. Venant’s theory):
• Yielding will occur when the maximum principal strain just exceeds the strain 
at the tensile yield point in either simple tension or compression. If el and e2 
are maximum and minimum principal strains corresponding to ol and o2, in 
the limiting case:
% = ^ ( a i - V 02)
% = p K .- V C T ,)
N - N
|a2|>|o.
It implies:
Ee, = a, - va2 - ±a0 
E£t = a , - va, = ± a 0 •
• Boundary of a yield surface in Maximum Strain Energy Theory is given below
C T i =CT0+ V O 2
Maximum Shear Stress Theory (Tresca Theory):
• Yielding would occur when the maximum shear stress just exceeds the shear 
stress at the tensile yield point. At the tensile yield point a2= o3 = 0 and thus 
maximum shear stress is oy/2.
This gives us six conditions for a three-dimensional stress situation:
°3 °1
= ±Oy
= ± O y
= - ° y
• Yield surface corresponding to maximum shear stress theory in biaxial stress 
situation is given below :
^ 2
Maximum strain energy theory ( Beltrami’s theory):
• Failure would occur when the total strain energy absorbed at a point per unit 
volume exceeds the strain energy absorbed per unit volume at the tensile 
yield point. This may be given:
1
2
Substituting, el, e2 , e3 and ey in terms of stresses we have
(5j2 + a J +<x,2 - 2 u ( a fa ? + a 2a 3 + c s i a ]) = c v2
( \
2
f \
2
( \
+
a 2
- 2 v
U J
l C T yJ
2
L a y J
Above equation results in Elliptical yield surface which can be viewed as:
Distortion energy theory (Von Mises yield criterion):
• Yielding would occur when total distortion energy absorbed per unit volume 
due to applied loads exceeds the distortion energy absorbed per unit volume 
at the tensile yield point. Total strain energy ET and strain energy for volume 
change Ev can be given as:
e t = ^ ( c t i£i + a 2E2 +O3E3) and Ev = | a av£a
Substituting strains in terms of stresses the distortion energy can be given as:
( o | + a 2 + 03 - o , a 2 - a 2a 3 - a . o , )
r: _ r: + 2 , 2
- ty- tv -
6E
At the tensile yield point, al = a y , a2 = a3 = 0 which gives,
p _ 2( 1+ y ) , 2
d > 6E y
The failure criterion is thus obtained by equating Ed and Edy, which gives
In a 2-D situation if a3 = 0, so the equation reduces to, •
(
2
( 2 ( \ ( \
° 1
j \ S i .
CT2
(7 C7 1 a .
V y j V y y \ y / V y /
• This is an equation of ellipse and yield equation is an ellipse.
• This theory is widely accepted for ductile material.