Module 10 – Elastic flexural formula

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From the course by Georgia Institute of Technology

Mechanics of Materials III: Beam Bending

189 оценки

Georgia Institute of Technology

Mechanics of Materials III: Beam Bending

189 оценки

This course explores the analysis and design of beam bending problems. Prerequisite Knowledge: You will need to have successfully completed my earlier course “Mechanics of Materials I: Fundamentals of Stress and Strain and Axial Loading” in order to be successful 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

Elastic Beam Bending

In this section, we will learn how to analyze and design for elastic beam bending.

Module 7 – Strain-Curvature relationship7:56

Module 8 – Locate Neutral Axis/Surface6:02

Module 9 – Moment-Curvature relationship5:53

Module 10 – Elastic flexural formula3:12

Module 11 – Area moment of inertia4:21

Module 12 – Section modulus4:11

Meet the Instructors

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

[MUSIC]

Welcome back to Mechanics of Materials part III.

We're moving right along in the course.

We did a review of shear force and bending moment diagrams.

We did the elastic flexural stresses and strains and

now today we're going to come up with the all important Elastic Flexural Formula.

And so our learning outcome is to derive the elastic flexural formula.

And so here's where we left off.

Well we found sigma sub x for linear elastic material was

proportional to curvature and the distance y form the neutral axis.

And we derived last module the moment-curvature relationship,

where curvature was proportional to moment.

Okay, so here are those relationships again.

Let's go ahead and substitute in for kappa, which is now m over ei.

And that will give us the elastic flexural formula,

which is sigma x e= the e's cancel, so

I get -My over I, very important relationship.

There it is again, the Elastic Flexural Formula.

And it says that stress now is directly proportional to the bending moment, M.

It's inversely proportional to the area moment of inertia I, and

it's also varies linearly with the distance y from the neutral axis.

And this is all predicated on the fact

that we're working with a linear elastic material.

And if you recall back if you did my mechanics materials part two course

on torsion, this has a direct analogy here to the shear stress that we found for

a circular bar and torsion.

Shear stress is analogous to, in the case of flexural formula,

normal stress T is analogous to M, the moment.

The distance y from the neutral axis is analogous to the distance rho from

the center of our circular bar.

And I, the area moment of inertia, is analogous to the the polar moment inertia.

And so you can see that it's a different application but

there's a lot of similarities between those two problems.

And so here is the flexural formula once again.

The maximum stress occurs where y

is the greatest distance from the neutral axis, and you can see in my picture here,

that would be down at this bottom fiber here on my cross section.

And so we call distance c, and

that's the furthest distance on the cross section from the neutral axis.

That's where the maximum stress will occur, is down here and

it's shown there as c.

And so we now have the Elastic Flexural Formula,

we know how to find the maximal stress on the cross section.

And that's a good point to take a break and come back next module.

[MUSIC]

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