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)

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)

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

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

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

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

Now We’ll Go After the Other Pair τ (200,100) σ Counter Clockwise (-150,-100)

We Now Have Two Points on Mohr’s Circle τ (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)

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

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

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

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

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).

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

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.

Obviously I can do Trig to figure out the State of stress at any angle ϴ

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

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.

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.

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

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.

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

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

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.

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 615 345 234 Calculate the Radius

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

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)

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

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

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

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)

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

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

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)

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

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

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

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!

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.

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!

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.

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!

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.

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

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.

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

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?

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

Pick off the Maximum and Minimum Normal Stresses 130 130 (50,0) -80 180

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?

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

Which Plane Contains the Maximum Shear Stress?

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 8.6-1, 8.6-3 and 8.6-4