Module 2: Differential Equation for the elastic curve of a beam

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來自 Georgia Institute of Technology 的課程

Mechanics of Materials IV: Deflections, Buckling, Combined Loading & Failure Theories

150 個評分

Georgia Institute of Technology

Mechanics of Materials IV: Deflections, Buckling, Combined Loading & Failure Theories

150 個評分

This course explores the analysis and design of engineering structures considering factors of deflection, buckling, combined loading, & failure theories ------------------------------------------------- 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.

從本節課中

Deflections and Statically Indeterminate Beam Structures

In this section, we will learn how calculate the deflections, or deformations, of engineering structures subjected to loads. We will then use those techniques is solving statically indeterminate beam problems

Module 1: General Analysis Approach1:53

Module 2: Differential Equation for the elastic curve of a beam4:18

Module 3: Deflection of a simply supported beam with a concentrated load at the center6:33

與講師見面

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

[MUSIC]

Welcome back to Mechanics of Materials Part IV.

Today's learning outcome is to derive the differential equation for

the elastic curve of a beam.

Again, Mechanics of Materials is the foundation for all structural and

machine design.

We start off with an engineering structure, we apply some external loads.

This generates internal forces in moments, stresses and strains and

we look at the structural performance.

As far as structural performance considerations,

we look at the fact that we don't want to have normal stress failure,

we don't want to have shear stress failure.

In this course, we're also going to look at no excessive deflections or

no buckling.

And so there are numerous examples of requirements for

engineering structures to stay within a specified deflection under a given load.

And I'd like you to go ahead and research some of those on your own and

then come on back So

we're going to look at beam deflection as start in this course.

And we'll start with a simply supported beam, which has a pin and

a roller at the ends.

This is a model of that, and we apply moments to each end,

and we get some deflection or bending.

And this is exaggerated, it's an exaggerated shape when loaded.

Here is a model of the beam with a pin on one side and

a roller on the other and if we applied moments we'd get some deflection.

I can't bend this enough to show you the deflection with this model.

So I have my pool noodle model, where if I put moments on each side,

you can see how we get the beam deflection.

We came up in the last course, my part three course,

with a moment curvature relationship and this is what it was.

Kappa was called the curvature and we found it was proportional to the moment.

It was also equal to 1 over the radius of curvature, which is one over rho.

We said that EI was the flexural rigidity or

the resistance of the beam to bending for a given curvature.

So we made some assumptions.

We were operating in the linear elastic region.

We were working with pure bending or

flexure under constant bending moment and we have no shear force.

We actually find that the beam deflection due to shearing can be negligible.

So, here again is our Moment-curvature relationship where one

over the radius of curvature is equal to M over EI.

The Curvature equation, which you can find in any standard calculus

textbook is shown here where y is in the transverse direction or

the direction of the deflection and x is in the direction along the beam.

And we're looking at small deformations and so

dy/dx is a small value and if we square that like we do in the numerator, that's

much much less than 1 since it's a small number squared and we can Neglect it.

And so our curvature, 1 over the radius of curvature becomes d squared y dx squared.

And then we can substitute that into the top equation for

the moment curvature relationship.

And we get a differential equation for the elastic curve of the beam shown here.

So we have this differential equation for the elastic curve of a beam.

If we have an equation for the moment along the beam now,

we can find the deflections by integrating twice and

using boundary conditions to find the constants of integration.

And so what we're looking for is why.

Our sign convention will be that the bending moment is positive,

if we have a smiley face here.

This is the same as we've used in my previous courses where negative is

shown on the right.

So, if it's a positive bending moment, d squared y dx squared is positive.

If it's a negative bending moment, then d squared y dx squared is negative.

And so for horizontal beams, I will always select

deflection y as being positive upward for beam deflection problems.

And we'll get started with looking at that in the next module.

[SOUND]

[MUSIC]

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