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

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

Beam Subjected to Pure Bending

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

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

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

Strain variation along the depth Strength of Materials

Stress variation Strength of Materials

Moment of resistance Strength of Materials

Derivation continued…. Strength of Materials

Bending equation Strength of Materials

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

Loading Pattern Strength of Materials

Section Modulus Strength of Materials

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

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

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

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