Module 18: Principal Stresses/Principal Planes

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From the course by Georgia Institute of Technology

Mechanics of Materials I: Fundamentals of Stress & Strain and Axial Loading

478 ratings

Georgia Institute of Technology

Mechanics of Materials I: Fundamentals of Stress & Strain and Axial Loading

478 ratings

This course explores the topic of solid objects subjected to stress and strain. The methods taught in the course are used to predict the response of engineering structures to various types of loading, and to analyze the vulnerability of these structures to various failure modes. Axial loading with be the focus in this course. ------------------------------ The copyright of all content and materials in this course are owned by either the Georgia Tech Research Corporation or Dr. Wayne Whiteman. By participating in the course or using the content or materials, whether in whole or in part, you agree that you may download and use any content and/or material in this course for your own personal, non-commercial use only in a manner consistent with a student of any academic course. Any other use of the content and materials, including use by other academic universities or entities, is prohibited without express written permission of the Georgia Tech Research Corporation. Interested parties may contact Dr. Wayne Whiteman directly for information regarding the procedure to obtain a non-exclusive license.

From the lesson

Stresses on Inclined Planes

In this section, we will develop the stress transformation equations for inclined planes and introduce Mohr’s Circle for Plane Stress

Module 16:Stresses on Inclined Planes – Sign Convention5:31

Module 17: Transformation Equations for Plane Stress8:01

Module 18: Principal Stresses/Principal Planes5:18

Module 19: Principal Stresses/Principal Planes (cont.)2:48

Module 20: Maximum and Minimum In-Plane Principal Stresses7:58

Module 21: Maximum In-Plane Shear Stress5:57

Meet the Instructors

Dr. Wayne Whiteman, PE
Senior Academic Professional
Woodruff School of Mechanical Engineering

[MUSIC]

Well can you believe it?

We're already on module 18 of Mechanics and Materials Part one.

Today's learning outcomes are to derive the angles to the Principal Planes

where maximum and minimum normal stresses are going to occur and

we're going to define those as principal stresses.

And we're going to show that the shear stress is zero on these principal planes.

So here's where we left off last time.

I applied equilibrium to our stress block.

We came up with our stress transformation equations.

So if we knew the stresses, the normal stresses and

shear stresses on any two planes form the external loading condition,

we could find sheer stress and normal stress on an arbitrary inclined plane.

And again, these were the transformation equations, and

you'll note that these equations, we've derived these solely from equilibrium.

And so they're applicable to stresses for any material.

Whether it's linear or non linear, elastic or inelastic, it doesn't matter.

And so since we're talking about stresses we use the forces on the stress block.

But in the future I'm just going to refer from now on to the stresses themselves.

So this is the stress block I'll show in the future.

And so for any plane at an angle theta we find sigma sub n, sigma sub t.

As I mentioned before, I showed some parts and any engineering part,

any engineering structure.

All structural machine design, we want to know what planes and

at what angles the maximum values are for the sheer stress and the normal stress.

Because that's going to tell us where our material is or

our structure is likely to fail.

And we've already done tensions tests to find out

what material properties the type of material has.

And so, this all starts to come together.

And so, let's first find the angle at where the max and

minimum normal stresses are going to occur.

And my question to you is, how can we do that?

How can we find that angle?

And so what you should say is well, to find the angle to the maximum,

minimum, normal stress, we want to find where sigma sub n,

the derivative of sigma sub n with respect to theta goes to zero,

because that's the mathematical definition of how to find a max or a min.

And so we're going to take d theta sub n with respect to theta and

this term doesn't have a theta so that goes to 0.

This term we've got a cosine 2 theta.

So the derivative of that is going to be

minus since we're going to go from cosine theta to sin theta.

The 2 comes out, so it cancels with this 2 so

we get minus sigma sub x minus sigma sub y sin 2 theta.

And on this one we're going to take the derivative

of sin 2 theta so the 2's going to come out so

we get plus 2 tau xy cosine to theta.

And so that we want to set equal to 0 for the maximum or the min.

What we get is, if I work this out,

is I now have sin of two theta over cosine of two

theta which is the same as tangent of 2 theta.

And this is going to be the tangent to the principal plane.

So I'm going to label that now as theta sub p because this is

a very special theta.

That's equal to, we get 2

Tau xy

over sigma sub x minus sigma sub y.

And so that is the result.

That's how we can find the angle to the principal planes.

And so theta sub p is the angles to what are defined as the principal planes and

these are the planes where the maximum and minimum normal stresses are going to

occur, and we're going to define those as principal stresses.

Okay, so here we go with those results.

You'll note that for

the principal planes when I took the derivative this is what I found.

If I now compare that with my shear equation,

you can see if I divide by 2 here, this cancels and

I divide by 2, that means that the value on the right hand

side here of my shear stress equation is equal to 0.

So what that means is on the principal planes, a very important

note is that the shear stress is zero on the principal planes.

So that completes module 18 and we'll see you next time.

[MUSIC]

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