Module 34: Strains on any given plane using Mohr’s Circle

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来自 Georgia Institute of Technology 的课程

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

644 个评分

Georgia Institute of Technology

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

644 个评分

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.

从本节课中

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 32: Mohr’s Circle for Plane Strain2:28

Module 33: Determine Principal Strains, Principal Planes, and Maximum Shear Strain using Mohr’s Circle9:18

Module 34: Strains on any given plane using Mohr’s Circle6:04

与讲师见面

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

[MUSIC]

>> Welcome to module 34 Mechanics of Materials part one.

Today's learning outcome, if we're given a set of plane strain conditions we'll be

able to determine the strain on any given plane using Mohr's circle.

So here is the condition of plane strain and the strain transformation.

We had the equations that related to Mohr's circle, and

we also had Mohr's circle sign convention, as shown.

And last time we did part A,

which was to find the principal strains epsilon one and epsilon two,

and the maximum in plane shear strain and the orientation to the principal planes.

Now we want to do part two, which is the normal and shear strains on plane AA.

And so for plane AA we're rotating 50

degrees counter-clockwise on the stress block.

So theta sub A = 50 degrees on the stress block.

So, 2 theta sub A is going to be 100 degrees on Mohr's circle.

And so if I draw that,

we're going to go 100 degrees.

And so we're going to end up.

Somewhere up here.

And this is 2 theta sub A.

And so, remember from the last time we did this sort of an approach

on Mohr's circle for plane stress.

We'll call this angle phi.

And so, we find phi equals 2

theta sub A -2 theta sub p,

or 100- 24.8 degrees,

which is equal to 75.2 degrees.

Now we want to find, this is going to be point A up here, we

want to find what those stresses are.

And so, drop a vertical line down here.

So, for our triangle now,

sine of phi will be this opposite side over the hypotenuse, which is the radius.

And so we have sin of phi equals,

the opposite side is going to be the shear strain at A,

gamma sub A /2 over the radius, which was R, or 716 mu.

So we have gamma sub A /2 = R,

which is 716 times sin of 75.2

degrees, or gamma sub A over 2.

Shear strain /2 at point A will

end up being 692 mu radians.

And we could do a similar approach to find the normal strain at point A.

So we want this distance, and this will be the adjacent side.

So it's the cosine.

So we have Epsilon A equals 716

times cosine of 75.2 degrees.

But that's only this distance here.

We also need to add 300 to it to get out to point A.

So it's plus 300 mu.

And so epsilon A ends up equaling 483 mu, and

I'll put in millimeters per millimeter even though that's dimensionless.

So now I have my point A.

The strain conditions there, the normal strain,

is 483 mu, and the shear strain divided by 2 is 692 mu.

And the last thing we want to to do is show that, well,

let's call that our answer because that's the normal and shear strains on plane AA.

Let's show it on a properly oriented stress block.

So we are again going from a horizontal face.

And so I've got my horizontal face here.

I'm turning now to point A on the stress block.

I'm turning 50 degrees.

And so.

This angle is 50 degrees.

And we get to that 50 degrees,

the strains on that face are going to be 483.

It's positive, so it's in tension, mu.

So the shear strain is positive, and

this is the value of the shear strain divided by 2, 692.

So on my stress block it's going to be twice that for

the shear strain, and that's going to be 1384 mu.

And so here's a properly oriented block.

And it's got the strains showed on it, and the proper convention.

And you can do that for

any plane now at any angle using these Mohr's circle techniques.

And so that's where we'll leave off this time.

[MUSIC]

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