# THEORIES OF FAILURE CLASS NOTES FOR MECHANICAL ENGINEERING

## ENGINEERING

###  Fig. MAXIMUM PRINCIPAL STRESS THEORY (RANKINE THEORY)

MAXIMUM PRINCIPAL STRESS THEORY (RANKINE THEORY)

#### Finally Maximum Principal Stress Actually present≤ σult/FOS

MAXIMUM SHEAR STRESS THEORY (Guest Theory or Tresca Theory)

####  Fig. Maximum Shear Stress Theory (Guest Theory)

#### Actually present maximum shear stress≤ allowable shear stress

MAXIMUM PRINCIPAL STRAIN THEORY

(SAINT VENANT’S THEORY)

####  Fig. Maximum Principal Strain Theory (st. Venant’s Theory)

#### If Actually present≤ Allowable strain, it is safe

MAXIMUM PRINCIPAL STRAIN ENERGY THEORY-HAIGH THEORY Fig. Maximum Principal Strain Energy Theory (Haigh Theory)

ALLOWABLE stress VALUES σallow, σallow, 0

Allowable strain energy

u=(1/2E)( σ2allow+ σ2allow–2µ σallow σallow)

Stresses actually present σ1, σ2 and σ3=0

Strain energy ACTUALLY present

u=(1/2E)( σ12+ σ22–2µ σ1 σ2)

FINAL IS

( σ12+ σ22–2µ σ1 σ2)= σ2allow

MAXIMUM SHEAR STRAIN ENERGY THEORY (VON MISES HENCKY THEORY)

####  Fig. Maximum Shear Strain Energy Theory (Von Mises Hencky Theory )

#### [( σ1—σ2)2 +(σ2–0)2 + (0- σ1)2 = 2 σ2allow

GRAPHICAL REPRESENTATION OF VARIOUS THEORIES OF FAILURE

#### Fig. Theories of Failure (Combined)

ABCD Max Principal Stress Theory Boundary

EBFGDHE is Max. Shear Stress Boundary

IJKLI is Max. Principal Strain Boundary

Dark Ellipse  Max. Shear Strain Energy Theory Boundary

Dotted Ellipse Max. Principal Strain Energy Theory Boundary