1. Principle and Maximum Shearing Stresses (7.1-7.3)
MAE 314 – Solid Mechanics
Yun Jing
Principle and Maximum Shearing Stresses

2. Transformation of Stress
Recall the general state of stress at a point can be written in terms of 6 components: σx, σy, σz, τxy= τyx, τxz= τzx, τyz= τzy
This general “stress state” is independent of the coordinate system used.
The components of the stress state in the different directions do depend on the coordinate system.
Principle and Maximum Shearing Stresses

3. Transformation of Stress
Consider a state of plane stress: σz=τxz=τyz=0
Where does this occur?
Thin plate
Outer surface
Principle and Maximum Shearing Stresses

4. Transformation of Stress
What do we want to calculate?
Principle stresses (σ maximum and σ minimum)
Principle planes of stresses (orientation at which they occur)
Slice cube at an angle θ to the x axis (new coordinates x’, y’).
Define forces in terms of angle and stresses.
Principle and Maximum Shearing Stresses

5. Transformation of Stress
Sum forces in x’ direction.
Sum forces in y’ direction.
Principle and Maximum Shearing Stresses

6. Transformation of Stress
Trig identities
To get σy’, evaluate σx’ at θ + 90o.
Principle and Maximum Shearing Stresses

7. Transformation of Stress
Now, let’s perform some algebra:
Constants (we can find these stresses).
Variables
Principle and Maximum Shearing Stresses

8. Principle and Max Shearing Stress
Define
Plug into previous equation
Which is the equation of a circle with center at (σave,0) and radius R.
Principle and Maximum Shearing Stresses

9. Principle and Max Shearing Stress
We learned before that the principle stresses (maximum and minimum σ) occur when τx’y’ = 0.
Principle and Maximum Shearing Stresses

10. Example Problem
For the state of plane stress shown below, determine (a) the principal planes, (b) the principal stresses, (c ) the maximum shearing stress and the corresponding normal stress.
Mohr's Circle

11. MAE 314 – Solid Mechanics Yun Jing
Mohr’s Circle ( )
MAE 314 – Solid Mechanics
Yun Jing
Mohr's Circle

12. Constructing Mohr’s Circle
Given: σx, σy, τxy at a particular orientation
Draw axes
Plot Point O, center, at (σave,0)
Plot Point X (σx,τxy), this corresponds to θ = 0°
Plot Point Y (σy,-τxy), this corresponds to θ = 90°
Draw a line through XY (passes through O): This is the diameter of the circle
Draw circle t Y O s
2qP X
Mohr's Circle

13. Sign Convention
Mohr's Circle

14. Uses of Mohr’s Circle t s Find stresses on any inclined plane.
Rotate by 2θ1 to point X’.
This points gives the stress state (σx1, τxy1).
The point Y’ gives (σy1, -τxy1).
Find principle stresses.
Q1 is the point with the highest normal stress.
Q2 is the point with the smallest normal stress.
S1 is the highest shear stress = R.
S2 is the lowest shear stress = -R. S1
(save, tmax) t X’
(sx1, txy1)
2q1
2qQ1 s O Q2 Q1
(smin, 0)
(smax, 0)
2qP
(sy1, -txy1) Y’
2qQ2 X S2
(save, tmin)
Mohr's Circle

15. Example Problem
For the state of plane stress shown below, determine (a) the principal planes, (b) the principal stresses, (c ) the maximum shearing stress and the corresponding normal stress.
Mohr's Circle

16. Special Cases of Mohr’s Circle
Uniaxial tension
σmax = σx
σmin = 0
τmax = σx / 2
Hydrostatic pressure
σmin = σx
τmax = 0
Pure torsion
σmin = -σx
τmax = σx
Mohr's Circle

17. Example Problem
For the given state of stress, determine the normal and shearing
stresses after the element shown has been rotated (a) 25o clockwise
and (b) 10o counterclockwise.
Mohr's Circle

18. Example Problem
For the element shown, determine the range of values of xy for which the maximum tensile stress is equal to or less than 60 Mpa.
Mohr's Circle

19. 3D Stress States
Mohr’s circle can be drawn for rotation about any of the three principle axes.
These axes are in the direction of the three principle stresses.
The maximum shear stress is found from the largest diameter circle.
x-c plane
y-c plane
x-y plane
Mohr's Circle

20. Applied to Plane Stress
Recall σz = 0 (z-axis is one of the principle axes)
The third σ value is always at the origin.
Where does τmax occur?
2 possibilities
tmax is out of the x-y plane
tmax is in the x-y plane
Mohr's Circle

21. Example Problem
For the state of stress shown, determine two values of σy for which the
maximum shearing stress is 75 MPa.
Mohr's Circle

22. Thin-Walled Pressure Vessels (7.9)
MAE 314 – Solid Mechanics
Yun Jing
Thin-Walled Pressure Vessels

23. Thin-Walled Pressure Vessels
Cylindrical vessel with capped ends
Spherical vessel
Assumptions
Constant gage pressure, p = internal pressure – external pressure
Thickness much less than radius (t << r, t / r < 0.1)
Internal radius = r
Point of calculation far away from ends (St. Venant’s principle)
Thin-Walled Pressure Vessels

24. Cylindrical Pressure Vessel
Circumferential (Hoop) Stress: σ1
Sum forces in the vertical direction.
Longitudinal stress: σ2
Sum forces in the horizontal direction:
Thin-Walled Pressure Vessels

25. Cylindrical Pressure Vessel
There is also a radial component of stress because the gage pressure must be balanced by a stress perpendicular to the surface.
σr = p
However σr << σ1 and σ2 , so we assume that σr = 0 and consider this a case of plane stress.
Mohr’s circle for a cylindrical pressure vessel:
Maximum shear stress (in-plane)
Maximum shear stress (out-of-plane)
Thin-Walled Pressure Vessels

26. Spherical Pressure Vessel
Sum forces in the horizontal direction:
In-plane Mohr’s circle is just a point
Thin-Walled Pressure Vessels

27. Thin-Walled Pressure Vessels
Example Problem
A basketball has a 9.5-in. outer diameter and a in. wall thickness. Determine the normal stress in the wall when the basketball is inflated to a 9-psi gage pressure.
Thin-Walled Pressure Vessels

28. Thin-Walled Pressure Vessels
Example Problem
A cylindrical storage tank contains liquefied propane under a pressure of 1.5MPa at a temperature of 38C. Knowing that the tank has an outer diameter of 320mm and a wall thickness of 3mm, determine the maximum normal stress and the maximum shearing stress in the tank.
Thin-Walled Pressure Vessels

29. Thin-Walled Pressure Vessels
Example Problem
The cylindrical portion of the compressed air tank shown is fabricated of 8-mm-thick plate welded along a helix forming an angle β = 30o with the horizontal. Knowing that the allowable stress normal to the weld is 75 MPa, determine the largest gage pressure that can be used in the tank.
Thin-Walled Pressure Vessels