1. Purpose of Mohr’s Circle Visual tool used to determine the stresses that exist at a given point in relation to the angle of orientation of the stress element. There are 4 possible variations in Mohr’s Circle depending on the positive directions are defined.

2. Sample Problem  x = 6 ksi  y = -2 ksi  xy = 3 ksi Some Part A particular point on the part x y x & y  orientation

3. Mohr’s Circle  (CW)  x-axis y-axis  x = 6 ksi  y = -2 ksi  xy = 3 ksi (6 ksi, 3 ksi) 6 3 (-2 ksi, -3 ksi) 2 3 of Mohr’s Circle Center of Mohr’s Circle

4. Mohr’s Circle  (CW)    avg  = 2  ksi x-face y-face (6 ksi, 3ksi) (-2 ksi, -3ksi)  (  avg,  max )  x = 6 ksi  y = -2 ksi  xy = 3 ksi (  avg,  min )

5. Mohr’s Circle  (CW)   x-face y-face (6 ksi, 3ksi)   x = 6 ksi  y = -2 ksi  xy = 3 ksi 4 ksi    avg + R  7 ksi    avg – R  ksi (  avg,  max ) (2 ksi, 5 ksi) (  avg,  min ) (2 ksi, -5 ksi) 3 ksi R

6. Mohr’s Circle  (CW)   x-face y-face (6 ksi, 3ksi)   x = 6 ksi  y = -2 ksi  xy = 3 ksi 22 4 ksi (  avg,  max ) (2 ksi, 5 ksi) (  avg,  min ) (2 ksi, -5 ksi) 3 ksi

7. Principle Stress  (CW)   x-face (6 ksi, 3ksi)   1 = 7 ksi  2 = -3 ksi 22 4 ksi (  avg,  max ) (2 ksi, 5 ksi) (  avg,  min ) (2 ksi, -5 ksi) 3 ksi  = 18.435° Principle Stress Element Rotation on element is half of the rotation from the circle in same direction from x-axis

8. Shear Stress  (CW)   x-face y-face (6 ksi, 3ksi)   avg = 2 ksi  max = 5 ksi 22 4 ksi (  avg,  max ) (2 ksi, 5 ksi) (  avg,  min ) (2 ksi, -5 ksi) 3 ksi 22 Maximum Shear Stress Element  = 26.565°

9. Relationship Between Elements  avg = 2 ksi  max = 5 ksi  = 26.565°  1 = 7 ksi  2 = -3 ksi  x = 6 ksi  y = -2 ksi  xy = 3 ksi  = 18.435°  +  = 18.435 ° + 26.565 ° = 45 °

10. What’s the stress at angle of 15° CCW from the x-axis?  = ? ksi  = ? ksi  = ? ksi Some Part A particular point on the part x y U & V  new axes @ 15 ° from x-axis 15° U x V

11. Rotation on Mohr’s Circle  (CW)    avg  = 2  ksi x-face y-face  (  avg,  max ) (  avg,  min ) 30° 15° on part and element is 30° on Mohr’s Circle (  U,  U ) (  V,  V )

12.  U =  avg + R*cos(66.869°)  U = 3.96 ksi  V =  avg – R*cos(66.869°)  V = 0.036 ksi  UV = R*sin(66.869°)  UV = 4.60 ksi Rotation on Mohr’s Circle  (CW)    avg  = 2  ksi x-face y-face  (  avg,  max ) (  avg,  min ) 66.869° R (  U,  U ) (  V,  V )

13. What’s the stress at angle of 15° CCW from the x-axis?  U = 3.96 ksi  V  =.036 ksi  = 4.60 ksi Some Part A particular point on the part x y 15° U x V

14. Questions? Next: Special Cases

15. Special Case – Both Principle Stresses Have the Same Sign Cylindrical Pressure Vessel X Y Z

16. Mohr’s Circle xx  (CW)  yy xx yy Mohr’s Circle for X-Y Planes This isn’t the whole story however…  x =  1 and  y =  2

17. Mohr’s Circle 11  (CW)  yy xx zz  z = 0 since it is perpendicular to the free face of the element.  z =  3 and  x =  1 33 Mohr’s Circle for X-Z Planes xx  z = 0  max  xz

18. Mohr’s Circle 11  (CW)  22 yy xx zz  z = 0 since it is perpendicular to the free face of the element. 33  max  xz  1 >  2 >  3

19. Pure Uniaxial Tension  y = 0  x = P/A Ductile Materials Tend to Fail in SHEAR  1 =  x  2 = 0 Note when  x = S y, S ys = S y /2

20. Pure Uniaxial Compression  y = 0  x = P/A  1 = 0  2 =  x

21. Pure Torsion T T  1 =  xy  2 = -  xy CHALK 11 Brittle materials tend to fail in TENSION.

22. Uniaxial Tension & Torsional Shear Stresses Rotating shaft with axial load. Basis for design of shafts.  x = P/A  xy = Tc/J  1 =  x /2+R  x /2  x,  xy )  2 =  x /2-R  0,  yx )