THEORIES OF FAILURE

In this article we discuss the theories of failure :-

1) MAXIMUM PRINCIPAL STRESS THEROY (RANKINES THEROY)

2)MAXIMUM PRINCIPAL STRAIN THEROY (ST.VENANT’S THEORY

3)MAXIMUM SHEAR STRESS THEORY

(GUEST & TRESCA’S THEORY)

4) MAXIMUM STRAIN ENERGY THEORY (HAIGH’S THEORY)

5) MAXIMUM SHEAR STRAIN ENERGY/DISTORTION ENERGY THEORY/MISES-HENKY THEROY

MAXIMUM PRINCIPAL STRESS THEROY (RANKINES THEROY)

According to Rankine theory, permanent set takes place under a state of complex stress,

when the value of maximum principal stress is equal to that of yield point stress as found in a simple tensile test. 

𝜎1,2 ≤ 𝜎𝑌  for no failure.

𝜎1,2 ≤ σY /FOS  for design.

Note: For no shear failure 𝜏 ≤ 0.57 𝜎𝑌

Graphical representation

For brittle materials which do not fail by yielding but fail by brittle fracture, this theory 

gives satisfactory result. The graph is always square even for different of values of 𝜎1 , 𝜎2.

MAXIMUM PRINCIPAL STRAIN THEROY (ST.VENANT’S THEORY

According to Venants theory, a ductile material begins to yield when the maximum principal strain reaches the strain at which yielding occurs in simple tension. 

𝜖1,2 ≤ σY/ E 

 for no failure in uni-axial loading.

σ1 /E– 𝜇 σ2/ E - 𝜇 σ2/ E ≤ σY /E

For no failure in tri-axial loading.

𝜎1 – 𝜇𝜎2 – 𝜇𝜎3 ≤ 𝜎𝑌/ 𝐹𝑂𝑆 

for design. Here, 𝜖 = principal strain

𝜎1, 𝜎2 and 𝜎3 = Principal stresses

MAXIMUM SHEAR STRESS THEORY

(GUEST & TRESCA’S THEORY)

According to this theory, failure of a specimen subjected to any combination of loads

when the maximum shearing stress at any point reaches the failure value equal to that

developed at the yielding in an axial tensile or compressive test of the same material.

Graphical Representation

𝜏𝑚𝑎𝑥 ≤𝜎𝑦/2

 for no failure

𝜎1, 𝜎2 are maximum and minimum principal stresses respectively. Here,

 𝜏𝑚𝑎𝑥 = Maximum shear stress

𝜎𝑦 = permissible stress

This theory is well justified for ductile materials.

MAXIMUM STRAIN ENERGY THEORY (HAIGH’S THEORY)

According to this theory, a body under complex stress fails when the total strain energy on the body is equal to the strain energy at elastic limit in simple tension.

{𝜎1/2 + 𝜎2/2 + 𝜎3/2)- 2𝜇 (𝜎1𝜎2 + 𝜎2𝜎3 + 𝜎3𝜎1)} ≤ 𝜎𝑦/2  for no failure

MAXIMUM SHEAR STRAIN ENERGY/DISTORTION ENERGY THEORY/MISES-

HENKY THEROY

It states that inelastic action at any point in a body, under any combination of stress begins, when the strain energy of distortion per unit volume absorbed at the point is  equal to the strain energy of distortion absorbed per unit volume at any pint in a bar  stressed to the elastic limit under the state of uniaxial stress as occurs in a simple  tension/compression test.

• It can not be applied for material under hydrostatic pressure.

• All theories will give same result if loading is uniaxial.