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Torsional Shear Stress. Shafts in Series and Parallel. Bending Stress in Straight Beams. Bending Stress in Cur ved Beams. Pr incipal Stresses and Principal Planes. Determination of Principal Stresses for a Member Subjected to Biaxial Stress. Application of Principal Stresses in Designing Machine Members. Theories of Failure under Static Load. Maximum Principal or Nor mal Stress Theor y (Rankine’s Theory). Maximum Shear Stress Theory (Guest’s or Tresca’s Theory). Maximum Principal Strain Theor y (Saint Venant’s Theory). Maximum Strain Energy Theory (Haigh’s Theory). Maximum Distortion Energy Theory (Hencky and Von Mises Theory). Eccentric Loading—Direct and Bending Stresses Combined. Shear Stresses. | CONTENTS c H A P T E R 5 Torsional and Bending Stresses in Machine Parts 1. Introduction. 2. Torsional Shear Stress. 3. Shafts in Series and Parallel. 4. Bending Stress in Straight Beams. 5. Bending Stress in Curved Beams. 6. Principal Stresses and Principal Planes. 7. Determination of Principal Stresses for a Member Subjected to Biaxial Stress. 8. Application of Principal Stresses in Designing Machine Members. 9. Theories of Failure under Static Load. 10. Maximum Principal or Normal Stress Theory Rankine s Theory . 11. Maximum Shear Stress Theory Guest s or Tresca s Theory . 12. Maximum Principal Strain Theory Saint Venant s Theory . 13. Maximum Strain Energy Theory Haigh s Theory . 14. Maximum Distortion Energy Theory Hencky and Von Mises Theory . 15. Eccentric Loading Direct and Bending Stresses Combined. 16. Shear Stresses in Beams. 5.1 Introduction Sometimes machine parts are subjected to pure torsion or bending or combination of both torsion and bending stresses. We shall now discuss these stresses in detail in the following pages. 5.2 Torsional Shear Stress When a machine member is subjected to the action of two equal and opposite couples acting in parallel planes or torque or twisting moment then the machine member is said to be subjected to torsion. The stress set up by torsion is known as torsional shear stress. It is zero at the centroidal axis and maximum at the outer surface. Consider a shaft fixed at one end and subjected to a torque T at the other end as shown in Fig. 5.1. As a result of this torque every cross-section of the shaft is subjected to torsional shear stress. We have discussed above that the 120 CONTENTS Torsional and Bending Stresses in Machine Parts 121 torsional shear stress is zero at the centroidal axis and maximum at the outer surface. The maximum torsional shear stress at the outer surface of the shaft may be obtained from the following equation t T C . 0 - - . i r J I where t Torsional shear stress induced at the outer surface of