Subject: Machine Design -I

Topic: Machine Design consideration

Difficulty: High

Theories of failure

• Maximum principal stress theory (Rankine’s theory)
• Maximum shear stress theory (Coulomb, Tresca and Guest’s theory)
• Distortion energy theory (Huber von Mises and Hencky’s theory)
• Maximum strain theory (St. Venant’s theory)
• Maximum total strain energy theory (Haigh’s theory)

Maximum Principal Stress Theory (Rankine’s theory):

• The theory states that the failure of the mechanical component subjected to bi-axial or tri-axial stresses occurs when the maximum principal stress reaches the yield or ultimate Strength of the material.
• If σ1 , σ2 and σ3 are the three principal stresses at a point on the component and σ1 > σ2 > σ3. then according to this theory, the failure occurs whenever…… σ1 =Syt or σ1 =Sut whichever is applicable.
• The theory considers only the maximum of principal stresses and disregards the influence of the other principal stresses. The dimensions of the component are determined by using a factor of safety.
• For Tensile Stresses

• For Compressive Stresses

Maximum shear stress theory (Coulomb, Tresca and Guest’s theory):

• The theory states that the failure of a mechanical component subjected to bi-axial or tri-axial stresses occurs when the maximum shear stress at any point in the component becomes equal to the maximum shear stress in the standard specimen of the tension test, when yielding starts.

• In the tension test, the specimen is subjected to uni-axial stress (σ1) and (σ2 = 0). The stress in the specimen of tension test and the corresponding Mohr’s circle diagram are shown in Fig. From the figure,

• When the specimen starts yielding (σ1 = Syt), the above equation is written as

• Therefore, the maximum shear stress theory predicts that the yield strength in shear is half of the yield strength in tension, i.e.,