Principal Stresses and Strain and Theories of Failure

Principal Stresses and Strain and Theories of Failure Strength of Materials Prof. A. S. PATIL Department of Mechanical Engineering Sinhgad Academy of Engineering, Pune

6. 3 MOHR’S CIRCLE From the stress-transformation equations

Mohr Circles ( x + y)/2 R

Mohr Circles u. A point on the Mohr circle represents the x’ and xy’ values on a specific plane. u is measured counterclockwise from the original x-axis. u. Same sign convention for stresses as before. i. e. , on positive planes, pointing positive directions positive, and ….

Mohr Circles 2 …. . when we rotate the plane by °, we go 2 ° on the Mohr circle. x’ xy’

Mohr Circles x’ 2 max xy’ 1

A Mohr Circle Problem 200 k. Pa 1. The stresses at a point A are shown on right. 60 k. Pa A Find the following: 40 k. Pa Ø major and minor principal stresses, Ø orientations of principal planes, Ø maximum shear stress, and Ø orientations of maximum shear stress planes. 06 -12 -2014 Strength of Materials 7

200 k. Pa Drawing Mohr Circle 60 k. Pa A 40 k. Pa F 120 O D B C R = 100 E (k. Pa) G (k. Pa) 06 -12 -2014 Strength of Materials 8

Principal Stresses 1= 220 2= 20 (k. Pa) 06 -12 -2014 (k. Pa) R = 100 Strength of Materials 9

Maximum Shear Stresses max = 100 (k. Pa) 06 -12 -2014 Strength of Materials (k. Pa) 10

200 k. Pa Positions of x & y Planes 60 k. Pa on Mohr Circle A 40 k. Pa R = 100 60 120 40 (k. Pa) 60 (k. Pa) 06 -12 -2014 tan = 60/80 = 36. 87° Strength of Materials 11

200 k. Pa Orientations of Max. Shear 26. 6° Stress Planes 60 k. Pa A 53. 1° 36. 9 ° (k. Pa) 06 -12 -2014 40 k. Pa (k. Pa) 116. 6° Strength of Materials 12