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

d

Sut = 300 MPa

Directly we can get (d) without using any failure

theory because only uniaxial load (P)

𝜎1 ≤ Syt

4Pπd2 ≤ Syt

P

2.

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

P

T

d

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.

Condition for failure is,

Maximum principal stress ( 1) failure stresses (Syt or Sut )

and Factor of safety (F.O.S) = 1

If 1 is +ve then Syt or Sut

1 is –ve then Syc or Suc

Condition for safe design,

Factor of safety (F.O.S) > 1

Maximum principal stress ( 1) ≤ Permissible stress ( per)

where permissible stress = Failure stress

Factor of safety = SytN or

SutN

1 ≤ SytN

or SutN

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 = 12 )

(ii) Biaxial state of stress when principal stresses are like in nature (Absolute max = 12 )

(iii) Under hydrostatic stress condition (shear stress in all the planes is zero).

2. Maximum Shear Stress theory (M.S.S.T)

Condition for failure,

Maximum shear stress induced at a critical Yield strength in shear under tensile point under triaxial combined stress test

Absolute max (Sys)T.T or Syt2

unknown therefore use Syt

Condition for safe design,

Maximum shear stress induced at a critical ≤ Permissible shear stress (τper)

tensile point under triaxial combined stress

where,

Permissible shear stress = Yield strength in shear under tension test

Factor of safety = (Sys)T.T

N = Syt2N

Absolute max ≤ (Sys)T.T

N or Syt2N

For tri-axial state of stress,

larger of [| σ1 - σ22 |, |σ2 - σ3

2 |, |σ3 - σ1

2 |] ≤ Syt2N

larger of [ |σ1 – σ2|, | σ2 – σ3|, | σ3 – σ1|] ≤ SytN

For Biaxial state of stress, σ3 = 0

|σ12 | or |

σ1 - σ22 | ≤

Syt2N

|σ1| ≤ SytN when σ1, σ2 are like in nature Eqn (2)

|σ1 – σ2| ≤ SytN when σ1, σ2 are unlike in nature 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) Yielding strain under tensile test (ε Y.P.)T.T

ε1 (ε Y.P.)T.T or SytE

where E is Young’s Modulus of Elasticity

Condition for safe design,

Maximum Principal strain ≤ Permissible strain

where Permissible strain = Yielding strain under tensile test

Factor of safety = (ε Y.P.)T.T

N = SytEN

ε1 ≤ SytEN

1E [σ1 - µ(σ2 + σ3)] ≤

SytEN

σ1 - µ(σ2 + σ3) ≤ SytN

for biaxial state of stress, σ3 = 0

σ1 - µ(σ2) ≤ SytN Eqn (4)

4. Total Strain Energy theory (T.St.E.T)

Condition for failure,

Total Strain Energy per unit volume Strain energy per unit volume at yield point (T.S.E. /vol) 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

εE.L

Total Strain Energy per unit volume = 12 σ1 ε1 +

12 σ2 ε2 +

12 σ3 ε3 Eqn (6)

(triaxial)

Strain energy per unit volume up

to Elastic limit (E.L) = 12 σE.L εE.L

ε1 = 1E [σ1 - µ(σ2 + σ3)]

ε2 = 1E [σ2 - µ(σ1 + σ3)] Eqn (7)

ε3 = 1E [σ3 - µ(σ1 + σ2)]

By substituting equations (6) in equations (5)

T.S.E. /vol = 1

2E [σ12 + σ22 + σ32 - 2µ (σ1 σ2 + σ2 σ3 +σ3 σ1)] (8)

To get [(S.E /vol) Y.P.] T.T ,

Substitute σ1 = σ = SytN , σ2 = σ3 = 0 in equation (8)

[(S.E /vol) Y.P.] T.T = 1

2E ( SytN )

^2 (9)

By Substituting equations (8) and (9) in equation (5), the following equation is obtained

σ12 + σ22 + σ32 - 2µ (σ1 σ2 + σ2 σ3 +σ3 σ1) ≤ (SytN )

^2

for biaxial state of stress, σ3 = 0

σ12 + σ22 - 2µ σ1 σ2 ≤ (SytN )

^2 (10)

Note:

1. Eqn (10) is an equation of ellipse (x2 + y2 - xy = a2).

2. Semi major axis of the ellipse =

√ =

√ = 1.2 Syt

Semi minor axis of the ellipse =

√ =

√ = 0.87 Syt

3. Total strain energy theory is suitable under hydrostatic stress condition.

5. Maximum Distortion Energy Theory (M.D.E.T)

For

µ = 0.3

Condition for failure,

Maximum Distortion Energy/volume Distortion energy/volume at yield point (M.D.E/vol) 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 (12)

Under hydrostatic stress condition, D.E/vol = 0

and

Under pure shear stress condition, Volumetric S.E/vol = 0

From equation (8)

T.S.E/vol = 1

2E [σ12 + σ22 + σ32 - 2µ (σ1 σ2 + σ2 σ3 +σ3 σ1)]

Volumetric S.E/vol = 12 (Average stress) (Volumetric strain)

= 12 (

σ1 + σ2 + σ33 ) [(

1-2µE ) (σ1 + σ2 + σ3) ]

Vol S.E/vol = 1-2µ6E (σ1 + σ2 + σ3)

2 (13)

From equation (12) and (13)

D.E/vol = 1+µ6E [(σ1 - σ2)

2 + (σ2 - σ3)2 + (σ3 - σ1)2] (14)

To get [(D.E/vol) Y.P.] T.T ,

Substitute σ1 = σ = SytN , σ2 = σ3 = 0 in equation (14)

[(D.E/vol) Y.P.] T.T = 1+µ3E (

SytN )

^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] ≤ 2 (SytN

)^2

For biaxial state of stress, σ3 = 0

σ1 2 + σ22 – σ1 σ2 ≤ (SytN

) ^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.

Ration of SYSSYt

by using theories of failure

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

σ1 = τ , σ2 = - τ and τ = 16T

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.

(a) SYSSYt

in Maximum Principal stress theory

According to M.P.S.T,

Considering Factor of safety (N) = 1

σ1 ≤ Syt or

σ1 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

SYSSYt

= 1

(b) SYSSYt

in Maximum shear stress theory

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

SYSSYt

= 12

(c) SYSSYt

in Maximum principal strain theory

According to M.P.St.T,

σ1 - µ(σ2) Syt

τ - µ(-τ) Syt

τ(1+ µ) = Syt

Sys = Syt

1+ µ

for µ = 0.3

SYSSYt

= 0.77

(d) SYSSYt

in Total strain energy theory

According to T.St.E.T,

σ12 + σ22 - 2µ σ1 σ2 Syt2

τ2 + τ2 + 2 τ2 = Syt2

τ =

√

Sys =

√

for µ = 0.3

SYSSYt

= 0.62

(d) SYSSYt

in Maximum distortion energy theory

According to M.D.E.T,

σ1 2 + σ22 – σ1 σ2 Syt2

τ2 + τ2 + τ2 = Syt2

τ =

√

Sys =

√

SYSSYt

= 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.O.F Me and Te Equations

M.P.S.T Me = 1

2 [ M + √ ] = 32 d

3 σper

M.S.S.T Te = √ = 16 d3 τper

M.D.E.T Me = √

=

32 d3 σper

M

T T

M

d

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.

T.O.F σt equations

M.P.S.T σt = 12

[σx + √ ] = SytN

M.S.S.T σt = √ = SytN

M.D.E.T σt = √ = SytN

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

σx σx

τxy

τxy

(a) M.P.S.T :- Square

Syt

Syc Syt

Syc = -Syt

(b) M.S.S.T :- Hexagon

σ1 -σ1

σ2

-σ2

σ1

σ2

-σ1

-σ2

σ1 -σ2 = Syt

σ1 -σ2 = -Syt

(c) M.P.St.T :- Rhombus

(c) M.D.E.T :- Ellipse

Syt -Syt

Syt

-Syt

σ2

-σ2

σ1 - σ1

σ1 -σ1

M.D.E.T

M.S.S.T

σ2

- σ2

Syt

-Syt

-Syt

Syt

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

Syt

Syc Syt

Syc = -Syt

σ1 -σ1

σ2

- σ2

Syt

-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