1. 3. Stresses in Machine Elements Lecture Number – 3.1 Prof. Dr. C. S. Pathak Department of Mechanical Engineering Sinhgad College of Engineering, Pune Strength of Materials

2. Agenda Theory of Simple Bending Assumptions Derivation 1 Illustrative Numerical 1 Workout Example Strength of Materials

3. Beam Subjected to Pure Bending

4. Assumptions 1. The beam is initially straight and unstressed 2. The material is homogeneous and isotropic 3. The beam loaded within elastic limit 4. Young’s modulus is same in tension and compression 5. Plane cross–section remains plane before and after bending 6. Each layer of the beam is free to expand or contract 7. Radius of curvature is large compared with dimension of cross sections Strength of Materials

5. Bending Deformations Strength of Materials bends uniformly to form a circular arc cross-sectional plane passes through arc center and remains planar length of top decreases and length of bottom increases a neutral surface must exist that is parallel to the upper and lower surfaces and for which the length does not change stresses and strains are negative (compressive) above the neutral plane and positive (tension) below it member remains symmetric

6. Derivation Strength of Materials Constant BM is applied Beam will bend @ O, R tension, compression Ref – Strength of Materials by Dr. R. K. Bansal, Laxmi Publications

7. Strain variation along the depth Strength of Materials

8. Stress variation Strength of Materials

9. Moment of resistance Strength of Materials

10. Derivation continued…. Strength of Materials

11. Bending equation Strength of Materials

12. Moment of Inertia Moment of inertia is a measure of the resistance of the section to – applied moment or – load that tends to bend it. Moment of inertia depends on shape and not material It is a derived property Strength of Materials

13. Loading Pattern Strength of Materials

15. Section Modulus Strength of Materials

18. Illustrative Example A cast-iron machine part is acted upon by a 3 kN-m couple. Knowing E = 165 GPa and neglecting the effects of fillets, determine (a)the maximum tensile and compressive stresses, (b)the radius of curvature. Ref:- Mechanics of Materials by Beer and Johnston

19. SOLUTION Strength of Materials Based on the cross section geometry, calculate the location of the section centroid and moment of inertia. Apply the elastic flexural formula to find the maximum tensile and compressive stresses. Calculate the curvature

20. Strength of Materials SOLUTION Based on the cross section geometry, calculate the location of the section centroid and moment of inertia.

21. Strength of Materials SOLUTION Calculate the curvature Apply the elastic flexural formula to find the maximum tensile and compressive stresses.