Theories of Failure

1. Maximum Principal Stress theory (RANKINE’S THEORY)
2. Maximum Shear Stress theory (GUEST AND TRESCA’S THEORY)
3. Maximum Principal Strain theory (St. VENANT’S THEORY)
4. Total Strain Energy theory (HAIGH’S THEORY)
5. Maximum Distortion Energy theory (VONMISES AND HENCKY’S THEORY)

Maximum Principal Stress Theory (MPST)

According to MPST, failure occurs when the value of maximum principal stress is equal to that of yield point stress. 

Condition for failure is,

Maximum principal stress ($\sigma$) > failure stresses (Syt)

Condition for safe design,

Maximum principal stress  ≤  Permissible stress 

where, permissible stress = failure stress / Factor of Safety =\( \frac{Syt}{N}\)

\[\boxed{\sigma ≤  \frac{Syt}{N}}\]

NOTE:

• This theory is suitable for brittle materials under all loading conditions (bi axial, tri axial etc.) because brittle materials are weak in tension.
• This theory is not suitable for ductile materials because ductile materials are weak in shear.
• This theory can be suitable for ductile materials when state of stress condition such that maximum shear stress is less than or equal to maximum principal stress i.e. 

1. Uniaxial state of stress( $τ_{max} = \frac{\sigma}{2}$)
2. Biaxial loading when principal stresses are like in nature. ( $τ_{max} = \frac{\sigma}{2}$)
3. Under hydrostatic stress condition (shear stress in all the planes is zero).

Maximum Shear Stress Theory (MSST)

According to this theory, failure occurs when maximum shear stress at any point reaches the yield strength. 

Condition for safe design,

\(\boxed{\tau_{max} ≤  \frac{Sys}{N} = \frac{Syt}{2N}}\)

For tri-axial state of stress,

Max{\(|\frac{σ_1 - σ_2}{2}|, |\frac{σ_2 - σ_3}{2}|, |\frac{σ_3 - σ_1}{2}|\)} ≤  $\frac{Syt}{2N}$

For bi-axial state of stress,

Max{\(|\frac{σ_1 - σ_2}{2}|, |\frac{σ_2}{2}|, |\frac{σ_1}{2}|\)} ≤  $\frac{Syt}{2N}$

NOTE:

• This theory is well suitable for ductile materials.
• MSST and MPST will give same results for ductile materials under uniaxial state of stress and biaxial state of stress when principal stresses are like in nature.
• MSST is not suitable for hydrostatic loading.

Maximum Principal Strain theory (M P St T)

According to this theory, failure occurs when maximum principal strain reaches strain at which yielding occurs in simple tension.

Condition for safe design,

 \(\boxed{ε_1 ≤  \frac{Syt}{EN}}\)

\(\frac{1}{E}[σ_1 - μ(σ_2 + σ_3)] ≤  \frac{Syt}{EN} \)

for biaxial state of stress, $σ_3$ = 0

\(σ_1 - μ(σ_2) ≤  \frac{Syt}{N} \)

Total Strain Energy theory (T St E T)

According to this theory, failure occurs when total strain energy per volume is equal to strain energy per volume at yield point in simple tension.

Condition for safe design,

 Total Strain Energy per unit volume  ≤  Strain energy per unit volume at yield point under tension test.

 Total Strain Energy per unit volume = $\frac{1}{2}σ_1ε_1$ + $\frac{1}{2}σ_2ε_2$ + $\frac{1}{2}σ_3ε_3$ 

$ε_1 = \frac{1}{E}[σ_1 - μ(σ_2 + σ_3)]$

$ε_2 = \frac{1}{E}[σ_2 - μ(σ_1 + σ_3)]$

$ε_3 = \frac{1}{E}[σ_3 - μ(σ_2 + σ_1)]$

we get,

\[\frac{TSE}{Vol} = \frac{1}{2E}[σ_1^2 + σ_2^2 + σ_3^2 - 2μ(σ_1σ_2 + σ_2σ_3 + σ_3σ_1)]\]

\[\frac{TSE}{Vol}\Bigg]_{Y.P.} = \frac{1}{2E}(\frac{Syt}{N})^2 \]

\[[σ_1^2 + σ_2^2 + σ_3^2 - 2μ(σ_1σ_2 + σ_2σ_3 + σ_3σ_1)] ≤ (\frac{Syt}{N})^2 \]

for bi axial case $σ_3 = 0$,

\(σ_1^2 + σ_2^2 - 2μσ_1σ_2  ≤ (\frac{Syt}{N})^2 \)

Above Equation is an equation of ellipse whose semi major axis is $\frac{Syt}{\sqrt{1-μ}}$ and minor axis is $\frac{Syt}{\sqrt{1+μ}}$

NOTE: This theory is suitable for hydrostatic stress condition.

Maximum Distortion Energy Theory (M D E T)

According to this theory, failure occurs when strain energy of distortion per volume is equal to strain energy of distortion per unit volume at yield point in simple tension.

Total strain energy/Vol = Volumetric strain energy/vol + distortion energy / volume

Volumetric Strain Eenrgy /vol = $\frac{1}{2}$ (average stress)(Volumetric strain)

\[Vol SE/Vol = \frac{1}{2}\frac{σ_1 + σ_2+ σ_3}{3}[\frac{1-2μ}{E}(σ_1 + σ_2+ σ_3)] = \frac{1-2μ}{6E}(σ_1 + σ_2+ σ_3)^2 \]

DE/vol = TSE/vol - Vol SE/vol

\[\boxed{DE/vol =  \frac{1 + μ}{6E}[(σ_1 - σ_2)^2 + (σ_2 - σ_3)^2 + (σ_3 - σ_1)^2]}\]

\[DE/vol]_{YP} =  \frac{1 + μ}{6E}[2(\frac{Syt}{N})^2]\]

Condition for safe design,

$DE/vol  ≤  DE/vol]_{YP}$

\[[(σ_1 - σ_2)^2 + (σ_2 - σ_3)^2 + (σ_3 - σ_1)^2] ≤  2(\frac{Syt}{N})^2\]

for bi axial case $σ_3 = 0$,

\(σ_1^2 + σ_2^2 - σ_1σ_2  ≤ (\frac{Syt}{N})^2 \)

This Equation is an equation of ellipse whose semi major axis is $\sqrt{2}Syt$ and minor axis is $\sqrt{2/3}Syt$

NOTE:

• This theory is best for ductile materials.
• It can not be applied materials under hydrostatic stress condition. 

Comparison among the different failure theories

Comparison of different failure theories