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

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来自 Georgia Institute of Technology 的课程

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

599 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

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

与讲师见面

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

[MUSIC]

Hi, and welcome to module 19 of Mechanics and Materials I.

Today's learning outcome is to show that the principal planes are 90 degrees apart,

or what we'd say normal to each other, on mutually perpendicular planes.

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

We had our stress transformation equation for inclined planes.

We found the angle where max or minimum normal stresses occur.

And we said that the angle to those,

those are what we call the angle to the principal planes.

And that's where we see maximum or minimum normal stresses,

which are called principal stresses.

And we also made the note that shear stress is zero on these principal planes.

And so, here's again,

is our expression for the angle for the principal planes, it's a tangent function.

If I look at a tangent function, the tangent repeats every 180 degrees.

So, I go from like minus 90 degrees to plus 90 degrees minus, or

90 degrees to 270 degrees, etc.

And so for the values of, a given set of values for

sigma sub x, sigma sub y, and tao sub xy,

the values of 2 theta sub p are going to differ by 180 degrees.

Or theta sub p, the angles to the principal planes,

are going to be 90 degrees apart.

And so therefore, again, they're on mutually perpendicular planes or

normal to each other.

And so here's our stress block in the xy frame.

We can rotate to our principal planes.

We have normal stresses only.

No shear stresses on those principal planes and

we see that these principal planes are 90 degrees apart.

And so, recalling back to earlier parts of the class,

I said that we're using three-dimensional,

we can use a three-dimensional state of stress at a point.

We've reduced it down to two dimensions for plane stress.

But I mentioned earlier that I was going to show that an infinite number of

planes can be passed through a point but we really only need three mutually

perpendicular planes to completely describe this state of stress.

And we're starting to see that now because I can arbitrarily

turn my stress block and I have two planes for my perpendicular,

excuse me, for my principal stresses, in the xy plane.

And I'd have a third plane in the z direction.

And so, it's all starting to come together.

And we'll continue on with this next module.

[MUSIC]

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