1. Checking Out Stress States With Mohr’s Circle
(Credit for many illustrations is given to McGraw Hill publishers
and an array of internet search results)

2. Parallel Reading Chapter 8 Section 8.4 Section 8.5
(Do chapter 8 reading assignment problem set A)
Section 8.6
(Do chapter 8 reading assignment problem set B)

3. Lets Start by Considering Plane Stress
We take a 3D object and take a slice through it
Looking at the stresses in two dimensions.
There we will find 3
Stress components
Stresses in the 3rd Dimension
are zero

4. We Will Plot this State of Stress on Mohr’s Circle
Need to get our sign
Conventions right!
Negative =Compressive Stress
Positive = Tensional Stress σ
150 MPa
200 MPa

5. Now For Our Shear Stress Convention
Positive –
Shear rotates
Clockwise τ
Negative Shear
Rotates
Counter-Clockwise

6. So Which Shear Goes With Which Stress
(200,100) τ
100 MPa σ
Pick the ones
Acting on the
Same face.

7. Now We’ll Go After the Other Pairτ
(200,100) σ
Counter Clockwise
(-150,-100)

8. We Now Have Two Points on Mohr’s Circleτ
(200,100)
(25,0) σ
We could go after
The center graphically
Or calculate the center
(Obviously the shear values will cancel and leave
0 for the shear at the center)
(-150,-100)

9. We Can Also Get the Radiusτ
(200,100)
(25,0) σ
201.56 τ R
(-150,-100)

10. General Principles for 2D Mohr’s Circle
Add the tension and/or compression
Forces together. This will give you the
Center of Mohr’s Circle τ
For any two dimensional
Stress State σ

11. General Principle 2D Mohr’s Circle
The radius of Mohr’s Circle will always be
Given by the formula above
Interesting
To note this will
Also give the
Magnitude of
The maximum
Shear stress in
The plane

12. Well This is Really Cool But Why Are We Doing It?
We’ve already used Mohr’s Circle
To help us find shear or tensional
Distresses in other directions that
Have caused things to break in
Non-Obvious places or ways

13. We Can Also Use It to Find the Principle Stresses
These are the stresses when we find a direction
With no shear on any face.
These will be the minimum and maximum stresses
Acting in that plane. (We know that maximums
And minimums are commonly checked when
Making sure we have designed with enough of the
Right materials).

14. Getting Principle Stresses Analytically Can Be a Bear
Minimum
Stress
Maximum Stress
Admitting you can’t read them
Off Mohr’s Circle can be
Embarrassing

15. Describe the Stress in Any Orientation
Quickly checking our design
And material
For max
Shear
Tension
And
Compression
Is obvious A ϴ
Let the stars represent stress at some
Arbitrary orientation.

16. Obviously I can do Trig to figure out the State of stress at any angleϴ

17. Lets Take a Look at Our FE Exam Book
Here are the sign convention rules
For Mohr’s Circle which we hope you
Already know.
Hooke’s Law which you know without looking
Page 2

18. More of the FE Book on Mohr
Here are those very useful
Formulas for the center
And radius of Mohr’s circle
From any stress state.

19. Getting Principle Stresses
If you don’t have time to draw the
Circle to get the principle stresses
Or to get principle stresses from
Any stress state given.
While getting principle stresses is
Unlikely on the FE it may be darn likely
On in class quizes and tests.

20. Why the Excitement If we put something 400 psi Underground we
Expect the weight
From the ground above
To crunch down on us.
400 psi

21. What if the Squeeze Comes from Someplace Else?
400 psi
750 psi
In many places we (including Illinois) we have strong horizontal loads in the ground.
Design and orientation can be critical in building stable underground structures.

22. Finding and Using Principle Stress
Mining and Civil
Building tunnels to resist the actual maximum forces
Mechanical
Failure analysis of parts
We already know that we have a lot of different stress states in different orientations from a single type of load

23. Lets Do Some Principle Stress Finding!
One day Mindy Miner was looking for stress
Problems that kept making sections of roof
Fall in in her tunnels fall in even though she is
Only 300 ft deep.
North
-615 psi
234 psi N
She placed strain gages in a drill hole
Drilled to the east from a north south
tunnel
-345 psi
-345
East
234 psi
This is what she found
-615 psi

24. Mindy’s Problem
Because Mindy was a graduate of the Missouri Institute of Science and Technology
She had no idea what to do with the data and ask a smart student from SIU.
The Saluki suggested
Starting with Mohr’s
Circle.

25. Starting with Mohr’s Circle
(-615, 234)
North Facing stress
Center formula from FE book
(-480,0)
Plot the center
(-345, -234)
East facing stress
Plot the points
= 270 psi
Calculate the Radius

26. We Can Now Plot Our Maximums
Min and Max normal stress
Formulas from FE book.
(-210,0)
(-750,0)
= 750
480 – 271 = 210
Obviously the maximum shear
Is the radius of the circle
τmax = 270 psi

27. Now Where Are These Stresses
(-480,270)
Arctan(234/135)= 60 degrees
(-750,0)
(-210,0)
The minimum stress is 30
Degrees counter clockwise
From the east.
(-480,-270)

28. So What is Going On N -210 psi E -750 psi
(The killer stress >> 300 psi vertical stress)

29. That was So Fun Lets Try Another
Find the Principle Stresses τ
Help me plot
My points σ

30. Working Along (4,4) Someone find the center (10,-4) C =
Someone find the radius
R =
(C=7, R=5)

31. Closing in On Our Stress Maximums
(4,4) 5
(7,0)
Find the Minimum and
Maximum normal stresses
(10,-4)
Find the stress state at
Maximum shear.
Max =10, Min = 2, shear (7,5)

32. Lets Try for the Orientation
(7,5) 3
(2,0)
(12,0)
(7,0)
What is the angle? 4
(10,-4)
53.2

33. Drawing an Element Rotate Counter-Clockwise 26.6̊ 2 ksi
53.2̊
Rotate Counter-Clockwise 26.6̊
2 ksi
We of course could draw
A maximum shear force
Element in much the same
Way.
12 ksi

34. Assignment 10
Find the principle stresses using Mohr’s Circle for the following problems
8.4-7
8.4-8
8.4-10
Show and explain step by step how you are constructing the circle and then
How you read off the principle stresses
(Warning – if you fail to show and clearly explain your work and steps you can be marked wrong
Even if your result is correct)

35. Stress States Can be 3D Let us suppose we have the 3 principle
Stresses such that there is no shear stress
On any face of the cube.
Let us next put the principle stresses in order from the highest to the
least (remember tension positive – compression negative, negative
numbers will always be less than positive)
C < B < A

36. These Stresses Can be Used to Plot 3 Mohr’s Circles
Circle B-A represents the plane covering the axis of A and B
Circle C-B represents the plane covering the axis of B and C
Circle C-A represents the plane covering the axis of A and C

37. We Can Use These Circles to Pick Out the Maximum Shear Stress
We can also determine that we
Need to select which ever axis A and
C correspond to. A
Maximum stress will be in the AC at
45̊̊ from the A axis. C

38. Your FE Book Maximum shear in 3D is always
An average of the maximum and
Minimum principle stress.
But this does not tell you where the
Maximum shear stress is at!

39. So Why the Obsession with Maximum Shear?
Because it is the mode by which many of our materials
And corresponding structures fail.
Find it and design for it.

40. Finding the Maximum Shear on 2D Elements Wasn’t Hard If You Know All 3 Principle Axis – Is there Anything that can be done with 2D Planer Stress State Planes?
Yes!
If you have planer
stress there is no
shear stress across
it’s face. Therefore
that 3rd Axis is also
a principle axis!

41. Can We Find the Maximum Shear Stress from a 2D Plane Stress State?
Remember the order the stress from Highest, Middle, Lowest – A B C
For a Plane Stress State we know
The axis perpendicular to the plane
Has a 0 value for stress.

42. How Can We Use That? On Planer Stress States
If A and B fall on opposite sides of
The origin (one positive – one negative),
We know the Z axis is 0 so our plane
Has to be the big Mohr’s Circle.
We can read the maximum shear stress
Off of our circle!

43. But What If A and B are both Tension?
We know there is a zero stress
On the Z axis.
There is no way we have the big
Mohr’s Circle!
But we can easily draw the big Mohr’s
Circle and identify the maximum shear.

44. We Also Know Where That Maximum Shear Has to Be Located
The Maximum Shear Stress is at 45̊ to Plane
we are in.

45. And If They Are Both Compression?
The maximum shear will be on the other 45 degree plane
To the one we are in and will have a value of ½ our
Maximum compression.

46. So Lets Try One Here is our Planer Stress Element. We need the maximum
shear
We know the Z direction with 0 normal and 0 shear is a principle stress direction.
Is the cube we have now a principle stresses in x and y direction?
How can you tell?
Its not because there is shear on the faces

47. Standard Mohr’s Circle Procedure
Step 1 plot our stress τ σ
Now how do I
Plot 70 MPa Compression
And it’s shear?
Where does this
170 MPa Go
Is the shear that goes with it positive
Or negative?

48. Calculate the Center and Radius
(170,50)
(-70,-50)
(50,0) 130

49. Pick off the Maximum and Minimum Normal Stresses
130
130
(50,0)

50. Arrange the Principle Stresses In Order
0 , -80, 180
Put them in order
(0,0)
(-80,0)
(50,0)
(180,0)
Which two stars are on the
Edge of the largest Mohr’s
Circle?

51. So Which Mohr’s Circle Has the Maximum Shear Stress?
What is the Absolute Maximum Shear Stress?
(0,0)
(-80,0)
(50,0)
(180,0)
130

52. Which Plane Contains the Maximum Shear Stress?

53. Assignment 11
These problems all involve sketching Mohr’s Circle for a State of Plane
Stress and then finding the absolute maximum shear stress. You will
Sketch the relevant Mohr’s Circles and then show and explain how you
Get the absolute maximum shear stress. (Remember if you just sketch
A circle and write an answer without an explanation you will be marked
Wrong even if your result is correct).
Do , and 8.6-4