Theories of failure
Introduction
Theories of failure are those theories which help us to determine the safe dimensions of
a machine component when it is subjected to combined stresses due to various loads
acting on it during its functionality.
Some examples of such components are as follows:
1. I.C. engine crankshaft
2. Shaft used in power transmission
3. Spindle of a screw jaw
4. Bolted and welded joints used under eccentric loading
5. Ceiling fan rod
Theories of failure are employed in the design of a machine component due to the
unavailability of failure stresses under combined loading conditions.
Theories of failure play a key role in establishing the relationship between stresses
induced under combined loading conditions and properties obtained from tension test
like ultimate tensile strength (Sut) and yield strength (Syt).
Examples:
1.
Syt = 200 MPa
Sut = 300 MPa
d
Directly we can get (d) without using any failure
theory because only uniaxial load (P)
𝜎1 ≤ Syt
4P
πd2 ≤ Syt
P
2.
Member is subjected to both Twisting moment and
uniaxial load, hence combined loading conditions.
We cannot determine (d) directly in this case
because failure stresses under combined loading
conditions are unknown.
d
T
P
So, different scientists give relationships between
Stresses induced under combined loading conditions and (Syt and Sut) obtained using
tension test which are called theories of failure.
Various Theories of Failure
1.
Maximum Principal Stress theory also known as RANKINE’S THEORY
2.
Maximum Shear Stress theory or GUEST AND TRESCA’S THEORY
3.
Maximum Principal Strain theory also known as St. VENANT’S THEORY
4.
Total Strain Energy theory or HAIGH’S THEORY
5.
Maximum Distortion Energy theory or VONMISES AND HENCKY’S THEORY
1.
Maximum Principal Stress theory (M.P.S.T)
According to M.P.S. T
Condition for failure is,
Maximum principal stress (
1)
failure stresses (Syt or Sut )
and Factor of safety (F.O.S) = 1
If
1 is
1
+ve then Syt or Sut
is –ve then Syc or Suc
Condition for safe design,
Factor of safety (F.O.S) > 1
Maximum principal stress (
1)
≤ Permissible stress (
per)
Syt Sut
Failure stress
where permissible stress = Factor of safety = N or N
1
Syt
Sut
≤ N or N
Eqn (1)
Note:
1. This theory is suitable for the safe design of machine components made of brittle
materials under all loading conditions (tri-axial, biaxial etc.) because brittle materials
are weak in tension.
2. This theory is not suitable for the safe design of machine components made of ductile
materials because ductile materials are weak in shear.
3. This theory can be suitable for the safe design of machine components made of
ductile materials under following state of stress conditions.
(i) Uniaxial state of stress (Absolute
max =
1
2 )
(ii) Biaxial state of stress when principal stresses are like in nature (Absolute
(iii) Under hydrostatic stress condition (shear stress in all the planes is zero).
max =
1
2 )
2. Maximum Shear Stress theory (M.S.S.T)
Condition for failure,
Maximum shear stress induced at a critical
point under triaxial combined stress
Yield strength in shear under tensile
test
Syt
(Sys)T.T or 2
Absolute max
unknown therefore use Syt
Condition for safe design,
Maximum shear stress induced at a critical
tensile point under triaxial combined stress
≤ Permissible shear stress (τper)
where,
Permissible shear stress =
Absolute
max ≤
(Sys)T.T
Syt
Yield strength in shear under tension test
=
=
N
Factor of safety
2N
(Sys)T.T
N
or
Syt
2N
For tri-axial state of stress,
σ3 - σ1
larger of
[| σ1 2- σ2 |, |σ2 2- σ3 |, |
larger of
[ |σ1 – σ2|, | σ2 – σ3|, | σ3 – σ1|]
For Biaxial state of stress, σ3 = 0
σ
σ1 - σ2
| 21 | or |
2
|≤
Syt
2N
2
|]
≤
≤
Syt
2N
Syt
N
|σ1| ≤
|σ1 – σ2| ≤
Syt
when σ1, σ2 are like in nature
N
Syt
when σ1, σ2 are unlike in nature
N
Eqn (2)
Eqn (3)
Note:
1. M.S.S.T and M.P.S.T will give same results for ductile materials under uniaxial state
of stress and biaxial state of stress when principal stresses are like in nature.
2. M.S.S.T is not suitable under hydrostatic stress condition.
3. This theory is suitable for ductile materials and gives oversafe design i.e. safe and
uneconomic design.
3.
Maximum Principal Strain theory (M.P.St.T)
Condition for failure,
Maximum Principal strain (ε1)
ε1
(ε Y.P.)T.T
Yielding strain under tensile test (ε
Y.P.)T.T
Syt
or E
where E is Young’s Modulus of Elasticity
Condition for safe design,
Maximum Principal strain ≤ Permissible strain
where Permissible strain =
ε1
Syt
≤ EN
Syt
1
E [σ1 - µ(σ2 + σ3)] ≤ EN
Yielding strain under tensile test (ε Y.P.)T.T Syt
=
= EN
Factor of safety
N
Syt
σ1 - µ(σ2 + σ3) ≤ N
for biaxial state of stress, σ3 = 0
Syt
σ1 - µ(σ2) ≤ N
4.
Eqn (4)
Total Strain Energy theory (T.St.E.T)
Condition for failure,
Total Strain Energy per unit volume
(T.S.E. /vol)
Strain energy per unit volume at yield point
under tension test (S.E /vol) Y.P.] T.T
Condition for safe design,
Total Strain Energy per unit volume ≤ Strain energy per unit volume at yield point
under tension test.
Eqn (5)
σE.L
Strain energy per unit volume up
1
to Elastic limit (E.L) = 2 σE.L εE.L
εE.L
1
1
1
Total Strain Energy per unit volume = 2 σ1 ε1 + 2 σ2 ε2 + 2 σ3 ε3
(triaxial)
Eqn (6)
ε1
1
= E [σ1 - µ(σ2 + σ3)]
ε2
1
= E [σ2 - µ(σ1 + σ3)]
ε3
1
= E [σ3 - µ(σ1 + σ2)]
Eqn (7)
By substituting equations (6) in equations (5)
1
T.S.E. /vol = 2E [σ12 + σ22 + σ32 - 2µ (σ1 σ2 + σ2 σ3 +σ3 σ1)]
(8)
To get [(S.E /vol) Y.P.] T.T ,
Syt
Substitute σ1 = σ = N , σ2 = σ3 = 0 in equation (8)
1 Syt
[(S.E /vol) Y.P.] T.T = 2E ( N )^2
(9)
By Substituting equations (8) and (9) in equation (5), the following equation is obtained
Syt
σ12 + σ22 + σ32 - 2µ (σ1 σ2 + σ2 σ3 +σ3 σ1) ≤ ( N )^2
for biaxial state of stress, σ3 = 0
Syt
σ12 + σ22 - 2µ σ1 σ2 ≤ ( N )^2
(10)
Note:
1. Eqn (10) is an equation of ellipse (x2 + y2 - xy = a2).
2. Semi major axis of the ellipse =
Semi minor axis of the ellipse =
√
√
=
=
√
√
= 1.2 Syt
= 0.87 Syt
For
µ = 0.3
3. Total strain energy theory is suitable under hydrostatic stress condition.
5.
Maximum Distortion Energy Theory (M.D.E.T)
Condition for failure,
Maximum Distortion Energy/volume
(M.D.E/vol)
Distortion energy/volume at yield point
under tension test (D.E/vol) Y.P.] T.T
Condition for safe design,
Maximum Distortion Energy/volume ≤ Distortion energy/volume at yield point
under tension test
(11)
T.S.E/vol = Volumetric S.E/vol + D.E/vol
D.E/vol = T.S.E/vol - Volumetric S.E /vol
Under hydrostatic stress condition,
and
Under pure shear stress condition,
(12)
D.E/vol = 0
Volumetric S.E/vol = 0
From equation (8)
1
T.S.E/vol = 2E [σ12 + σ22 + σ32 - 2µ (σ1 σ2 + σ2 σ3 +σ3 σ1)]
1
Volumetric S.E/vol = 2 (Average stress) (Volumetric strain)
1 σ1 + σ2 + σ3
= 2(
3
) [(
1-2µ
E )
(σ1 + σ2 + σ3) ]
1-2µ
Vol S.E/vol = 6E (σ1 + σ2 + σ3)2
(13)
From equation (12) and (13)
1+µ
D.E/vol = 6E
[(σ1 - σ2)2 + (σ2 - σ3)2 + (σ3 - σ1)2]
To get [(D.E/vol) Y.P.] T.T ,
Syt
Substitute σ1 = σ = N , σ2 = σ3 = 0 in equation (14)
(14)
1+µ Syt
[(D.E/vol) Y.P.] T.T = 3E ( N )^2
(15)
Substituting equation (14) and (15) in the condition for safe design , the following
equation is obtained
[(σ1 - σ2)2 + (σ2 - σ3)2 + (σ3 - σ1)2]
Syt
≤ 2 ( N )^2
For biaxial state of stress, σ3 = 0
S
σ1 2 + σ22 – σ1 σ2 ≤ ( Nyt ) ^2
(16)
Note:
1. Equation (16) is an equation of ellipse.
2. Semi major axis of the ellipse =
√
Syt
Semi minor axis of the ellipse = √
Syt
3. This theory is best theory of failure for ductile material. It gives safe and economic
design.
4. This theory is not suitable under hydrostatic stress condition.
SYS
Ration of S by using theories of failure
Yt
1. Sys (Yield strength in shear) is obtained from torsion test.
2. Torsion test is conducted under pure torsion i.e. pure shear state of stress (σx = σy= 0;
τxy = τ ).
3. Under pure shear state of stress
16T
σ1 = τ , σ2 = - τ and τ = d3
4. Sys can also be obtained by applying theories of failure for pure shear state of stress
condition.
5. When yielding in shear occurs under pure shear state of stress, τ = Sys.
SYS
(a) S in Maximum Principal stress theory
Yt
According to M.P.S.T,
Considering Factor of safety (N) = 1
σ1 ≤ Syt
σ1
or
Syt
But in pure shear state of stress, σ1 = τ
τ = Syt
When yielding occurs in shear under pure shear state of stress, τ = Sys
Sys = Syt
SYS
SYt = 1
SYS
(b) S in Maximum shear stress theory
Yt
According to M.S.S.T,
|σ1 – σ2| ≤ Syt
But in pure shear state of stress, σ1 = τ and σ2 = -τ
τ – (-τ) = Syt
2 τ = Syt
When yielding occurs in shear under pure shear state of stress, τ = Sys
SYS 1
SYt = 2
SYS
(c) S in Maximum principal strain theory
Yt
According to M.P.St.T,
σ1 - µ(σ2)
τ - µ(-τ)
Syt
Syt
τ(1+ µ) = Syt
Sys =
Syt
1+ µ
for µ = 0.3
SYS
SYt = 0.77
SYS
(d) S in Total strain energy theory
Yt
According to T.St.E.T,
σ12 + σ22 - 2µ σ1 σ2
Syt2
τ2 + τ2 + 2 τ2 = Syt2
τ=
√
Sys =
√
for µ = 0.3
SYS
= 0.62
SYt
SYS
(d) S in Maximum distortion energy theory
Yt
According to M.D.E.T,
σ1 2 + σ22 – σ1 σ2
Syt2
τ2 + τ2 + τ2 = Syt2
τ=
√
Sys =
√
SYS
SYt = 0.577
Equivalent Bending Moment (Me) and Twisting Moment (Te) equations
These equations should be used when the component is subjected to both Bending
Moment and Twisting Moment simultaneously.
T
T
M
M
d
T.O.F
M.P.S.T
M.S.S.T
M.D.E.T
Me and Te Equations
Me = 1 [ M + √
2
Te = √
Me = √
] = 32 d3 σper
= 16 d3 τper
=
32
d3 σper
Normal Stress Equations (σt equations)
Normal stress equations should be used when a point in a component is subjected to
normal stress in one direction only and a shear stress.
τxy
σx
σx
τxy
σt equations
T.O.F
M.P.S.T
σt = 1 [σx + √
2
Syt
] = N
Syt
M.S.S.T
σt = √
= N
M.D.E.T
σt = √
= N
Syt
Shape of safe boundaries for theories of failure
Graphical representation or safe boundaries are used to check whether the given
dimensions of a component are safe or not under given loading conditions.
As per theories of failure for ductile material, Syc = - Syt
(a) M.P.S.T :- Square
σ2
Syt
-σ1
Syc
Syt
σ1
Syc = -Syt
-σ2
(b) M.S.S.T :- Hexagon
σ2
σ1 -σ2 = -Syt
-σ1
σ1
σ1 -σ2 = Syt
-σ2
(c) M.P.St.T :- Rhombus
σ2
Syt
-
σ1
σ1
-Syt
Syt
-Syt
-σ2
(c) M.D.E.T :- Ellipse
σ2
M.D.E.T
Syt
M.S.S.T
-Syt
-σ1
σ1
Syt
-Syt
-
σ2
Note :1. Semi major axis of the ellipse =
Syt
√
Semi minor axis of the ellipse = √
Syt
2. As the area bounded by the curve increases, failure stresses increases thereby
decreases dimensions and hence cost of safety.
In all the quadrants
Area bounded by the MDET curve
Aread bounded by MSST curve
Hence
(Dimensions)MDET
(Dimensions)MSST
(c) T.St.E.T :- Ellipse
σ2
Syt
-Syt
-σ1
Syc
Syt
Syc = -Syt
-
σ2
σ1
Syt
Note:
Semi- major axis of the ellipse =
Semi- minor axis of the ellipse =
√
√
For Objective Questions
1. All the theories of the failure will give the same result when uniaxial state of stress
Examples –
1. Bar subjected to uniaxial load
2. Beam subjected to pure bending
2. All the theories of the failure will give the same result when one of the principal
stresses is very large as compared to the other principal stresses.
3. For pure shear state of stress, all the theories of failure will give the different result.
(a) MDET and MSST will be used under pure shear state of stress.
(b) MDET will be preferred over MSST.
4. MSST and MDET are not valid for hydrostatic state of stress condition.
5. TSET and MPST will be used for hydrostatic state of stress condition. TSET will be
preferred over MPST.
References
1. Introduction to Machine Design by V.B Bhandari
2. NPTEL content and Videos