Module 37: Find Principal Strains, Maximum Shear Strain, and Principal

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Do curso por Georgia Institute of Technology

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

513 classificações

Georgia Institute of Technology

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

513 classificações

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.

Na lição

Stress concentrations, Mohr’s Circle for Plane Strain, and measuring strains

In this section, we will learn about stress concentrations, and discuss plane strain, develop Mohr’s Circle for Plane Strain, and explore methods of measuring strain.

Module 35: Find Strains using Experimental Analysis Techniques5:34

Module 36: Find In-Plane Strains using Strain Gage Measurements7:09

Module 37: Find Principal Strains, Maximum Shear Strain, and Principal7:29

Conheça os instrutores

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

[MUSIC]

This is module 37 of Mechanics of Materials part 1.

Today's learning outcome is to calculate the principal strains,

the maximum in-plane shear strain, and the orientation of the principal planes based

on the strain gage Rosette measurements that we had.

And so, here's where we left off last time.

We had these three in plane strains, normal strains that were measured.

We went ahead, and found the in-plane strains,

both normal strains, and shear strain.

Now, today, we're going to go ahead, and use Mohr's Circle to find the principal

strains, max in-planes, shear strain, and the orientation of the principal planes.

So, here is our in-plane strains and what I'd like you to do now is

to draw a small element and show the strains on that element and comment back.

Okay? Here is the element with the strain shown

on it, and now, we're going to draw more circle for strain.

Let's start with a vertical face where I have 600 intention strain, normal strain,

so that's positive.

And then, I have a counterclockwise sheer strain of 150, but I plot gamma over two.

So, that's gonna be minus 75.

So, I go out 600, and then, down 75 for my horizontal face.

And for my vertical face, I've got 350 positive cuz it's in tension.

The shear strain is gonna cause counterclockwise rotation, so

it's positive by convention.

I use half of 150 again, so I'm gonna go 350.

Then up 75, and that's my vertical face.

So (350, 75), and I leave the mus off again for shorthand.

I can draw a line in between these two for my diameter.

And then, I can draw my Mohr's Circle.

Okay, the first thing I wanna do is find the center.

That's gonna be the average strain.

So, we've got (600 + 375) / 2 for

our average normal strain, which ends up being 475.

So, this is 475 and 0.

Let's now, do the radius.

So, I've got the radius is equal to, let's take this triangle here.

I've got 600 minus 475 squared

plus 75 squared.

And the square root of that will give me my radius, and the radius ends up equaling

if you calculate that out 145.8.

And so now, I can draw a line up from the center to the very peek of the circle.

That's going to be my point.

That will be 475 for

the normal strain, and then,

145.8, which means that

the gamma max over 2 is 145.8 or,

my max in-plane shear strain is going to equal

twice that, or 292 mu radians.

I still wanna find my principle strains.

If I add the radius to 475, that's gonna give me my first principal strain, so

epsilon sub one is going to be 475 plus 145.8.

And then, rounded to three significant figures.

That gives me 621 mu millimeters per millimeter.

And my other principal strain

epsilon 2 is going to be 475-radius of 145.8.

And so, that's going to equal 329

mu millimeters per millimeter.

That's our other principal strain.

And so, that is a good shear, excuse me, good more circle for strain.

Here I've gone ahead, and plotted it out so that it's much cleaner.

And easier to see.

The last thing I wanna do is find the orientation of

the principle plains from the XY axis.

So, let's take, from the horizontal face,

this angle to theta sub P on more circle.

So, sine of 2 theta sub P

is going to be equal to the opposite side which is 75 over

the hypotenuse which is the radius which is 145.8.

And so, if you calculate that out that gives you 2 theta sub P equals

30.96 degrees.

That's twice the angle on the small block.

Theta of sub P is equal to 15.5 degrees, and

we're rotating counter clockwise from the horizontal face.

So, I can now draw a properly oriented Stress Block, or Stress, or Strain Block.

So, we can now draw a properly oriented Block,

and so, I have my horizontal face, and

I'm gonna rotate 15 .5 degrees from the horizontal.

This is 15.5 degrees, and on that face we're

going to have epsilon sub 1 our principal strain,

principal normal strain is 621 mu.

Millimeters per millimeter.

It's positive so it's going to be in tension.

This is 621 mu millimeters per millimeters.

I turn 180 degrees on Mohr's Circle which is 90 degrees on my block.

And I get positive 329 mu millimeters a millimeter.

And we're gonna have equal on the other sides.

And so, that's a good oriented stress block.

And we've solved the problem.

And so, you should have a good handle now on finding strains experimentally,

converting them to the in-plane strains for two dimensions, and then,

using that data to find your max, or your in-plane principle strains,

and maximum shear strain, in-plane shear strain, and the orientation for them.

And so, we'll pick up again next time.

[MUSIC]

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