Module 36: Find In-Plane Strains using Strain Gage Measurements

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

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Georgia Institute of Technology

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

467 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.

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

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

[MUSIC]

Welcome to Module 36 of Mechanics and Materials Part I.

Today's learning outcome is to calculate the in-plane strains based on strain gage

rosette measurements, and we talked about strain gage rosettes last module.

So, here's a strain guage rosette generically.

We are able to measure the normal strain in three different directions at

three angles and we can come up with three equations then.

The normal strains in the a, b and the c direction are measured and

the three unknowns are on the right hand side are two normal strains

epsilons of x and epsilons of y and the shear strain epsilons of xy.

Once we know those, we can then solve for in-plane principal strains,

the principal planes and the max in-plane shear strain.

So here's an example.

I have a strain rosette.

It's going to be a 45-degree strain rosette

placed on the surface of a critical point of an engineering part and

we measured the following Ea, Ev, and Ec normal strains.

And we want to determine the in-plane stresses in this module and

then next module we'll go ahead and use more circle

to determine the principal planes and maximum shear in-plane shear strength.

Okay, so to start off with, for a 45 degree rosette,

theta sub a is going to be 45, I am sorry theta sub a is 0 degrees,

because it's aligned with the x-axis.

Theta sub b is going to be 45 degrees and

theta sub c is going to be 90 degrees.

Similarly, if you had a 60 degree strain rosette,

you'd have 0 degrees, 60 degrees and 120 degrees.

Okay, so let's now go ahead and calculate,

we know epsilon a, epsilon b and epsilon c.

Let's go ahead and use the equation.

So let's start with epsilon a.

Epsilon a is measured to be 350 and its mu,

millimeters per millimeters if its dimensionless.

I'm going to leave the mu's off just as I calculate just a shorten

my calculations but put it back on in my final answer.

So epsilon sub a is 350

mu equals epsilon sub x which is unknown.

Cosine squared epsilons of a we said was 0 degrees

+ epsilons of y sine squared

0 degrees + gamma

xy sine 0 degrees cosine 0 degrees.

And we know that cosine of 0 degrees is 1, so that's squared is 1.

Sine squared is 0.

Sine squared of 0 degrees is 0.

And this should be 0 degrees, not theta.

And sine of 0 is 0 degrees.

Or sine of 0 degrees is 0.

And so what we end up with is epsilon sub x.

The in-plane normal strain in the x direction is 350 mu

millimeters per millimeter, and so that's our one answer.

Let's do the same thing now for epsilon sub c and

so epsilon sub c is measured as 600 mu.

That's equal to epsilon sub x times

cosine squared 90 degrees now for

epsilon sub c + epsilon sub y times

sine squared 90 degrees + gamma

xy sine of 90 degrees cosine of 90 degrees.

And with that, we know that cosine squared or cosine of 90 degrees is 0.

So cosine squared of 90 degrees is 0.

Even though epsilon sub x has a value that term zeros out.

Sine of 90 degrees is 1.

So that squared is 1.

And again, cosine of 90 degrees is 0.

So this term zeros out.

And we end with now epsilons of y equals 600 mu millimeters per millimeter.

And that's our normal strain in the y direction in-plane.

Finally, we'll use the epsilon b, the third equation,

and so we have epsilon sub b is measured to be 400 mu,

and 400 mu is equal to, I guess I was going to leave the mus off so

let's erase that off and we'll put it in the answers.

But we have 400 times epsilon x cosine squared 45 degrees and

+ epsilon y sine squared 45 degrees + gamma

xy sine of 45 degrees cosine of 45 degrees.

And we have epsilon sub x was 350.

We've already measured that above or calculated that above.

Sine squared of 45 degrees is 0.5.

Epsilon sub y is 600.

And sine squared of 45.

You know what, this should be and I'll just correct this here.

This should actually be a cosine, right?

Cosine squared, 45 degrees for, we're using the b equation.

But that still comes out to be 0.5, so there's no change there.

Sine squared of 45 is also 0.5.

And then sine of 45 times cosine of 45 is also 0.5.

And now if you calculate the only unknown left in this equation is gamma xy and

so gamma xy ends up equaling -150 mu and

that's a radians because it's an angle.

Dimension is radians and so that's our other answer.

So we've gone ahead and calculated the end plane and this is not stresses,

this should be strains.

And that's it.

We'll carry on with the second part, part B, next time.

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