Sunday, December 3, 2023

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 (σ) > failure stresses (Syt)

Condition for safe design,
Maximum principal stress  ≤  Permissible stress 
where, permissible stress = failure stress / Factor of Safety =SytN

σSytN


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( τmax=σ2)
  2. Biaxial loading when principal stresses are like in nature. ( τmax=σ2)
  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,

τmaxSysN=Syt2N

For tri-axial state of stress,
Max{|σ1σ22|,|σ2σ32|,|σ3σ12|} ≤  Syt2N

For bi-axial state of stress,
Max{|σ1σ22|,|σ22|,|σ12|} ≤  Syt2N




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,

 ε1SytEN

1E[σ1μ(σ2+σ3)]SytEN

for biaxial state of stress, σ3 = 0

σ1μ(σ2)SytN





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 = 12σ1ε1 + 12σ2ε2 + 12σ3ε3 

ε1=1E[σ1μ(σ2+σ3)]
ε2=1E[σ2μ(σ1+σ3)]
ε3=1E[σ3μ(σ2+σ1)]

we get,
TSEVol=12E[σ21+σ22+σ232μ(σ1σ2+σ2σ3+σ3σ1)]
TSEVol]Y.P.=12E(SytN)2
[σ21+σ22+σ232μ(σ1σ2+σ2σ3+σ3σ1)](SytN)2
for bi axial case σ3=0,

σ21+σ222μσ1σ2(SytN)2

Above Equation is an equation of ellipse whose semi major axis is Syt1μ and minor axis is Syt1+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 = 12 (average stress)(Volumetric strain)
VolSE/Vol=12σ1+σ2+σ33[12μE(σ1+σ2+σ3)]=12μ6E(σ1+σ2+σ3)2
DE/vol = TSE/vol - Vol SE/vol
DE/vol=1+μ6E[(σ1σ2)2+(σ2σ3)2+(σ3σ1)2]
DE/vol]YP=1+μ6E[2(SytN)2]
Condition for safe design,
DE/volDE/vol]YP
[(σ1σ2)2+(σ2σ3)2+(σ3σ1)2]2(SytN)2

for bi axial case σ3=0,

σ21+σ22σ1σ2(SytN)2

This Equation is an equation of ellipse whose semi major axis is 2Syt and minor axis is 2/3Syt


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

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