0:00

[MUSIC]

Welcome to Module 13 of Mechanics of Materials part III.

We've actually come to the exciting part of the course where we're going to start

working on some real world engineering problems.

And so today's learning outcome is to solve an elastic beam bending problem for

the maximum flexural stress in both tension and compression.

And so as a review, we're looking at beam bending in the elastic range.

The stresses all remain in the elastic range.

This was the elastic flexural formula where y was the distance from the neutral

axis to the point where we wanted to find the stress in the x direction.

Where the flexural stress.

0:43

And we said that the maximum stress was as shown where C

is the furthest distance from the neutral axis.

And so that allows us to find where the maximum stress occurs.

And so let's go ahead and apply this to a real world problem.

This is a typical bridge.

I'm going to simplify it some, we'll do some simplifications of the system, but

the process is going to be remaining the same, regardless, and you'll know how,

even for a more complicated system, to go through and solve one of these problems.

So we're going to look at a strip of one beam, and

how much load it carries for a bridge.

And it's carrying this truck And so the beam itself,

and the structure, and this is true for almost all bridges,

is that the beam and the structure itself carries a lot of weight.

And I'm going to estimate it for this one beam span to be 250 pounds per foot.

1:44

So the constant load in a bridge structure is not insignificant,

it is a big part of the design.

But in addition to that where the truck is located

we'll include an additional 350 pound per foot.

Now it's actually point loads at the wheels but I've modeled its distributed

load actually at a track vehicle would be a more distributed type load.

But again the procedure is the same, regardless.

And so let's go ahead and proceed.

2:16

And so here is my structure.

I'm going to use a rectangular cross-section.

And so to start with.

We'll look at other cross-sections later on, and most beams are what we

call in bridges girders, are I-beams and we'll work with I-beams later on but

to start let's just look at this simple rectangular cross section.

And so, two parts to the worksheet.

We want to draw the shear and moment diagrams and determine where in this

beam does the maximum moment occur because that's what we want to design for.

And then once we find where that maximum moment occurs

we're going to determine the maximum flexural stress and

the maximum compressor stress in the cross section at that location.

What I would like you to do first is I would like you to draw the shear and

moment diagram to determine where the maximum moment occurs and

what that value is.

And so, you should be able.

A real pro at doing shear moment diagrams.

If you need some refresher, go back to my applications in engineering mechanics

course and I have several modules on how to proceed.

And so, do that on your own.

And when you do that, these are the results you should find and and for

the moment diagram we find that the maximum moment occurs at the center of

the beam or 60 foot from the left or the right edge and

it's value is 865 625 foot paths and so

now we have our M max the maximum moment that our beams going to have to carry.

We know that it's at the center of the bridge structures beam, or girder,

and so we know also that sigma max

4:08

from our first slide is equal to Mc over I.

Well we know what M is, we just found that.

C now is the distance from the neutral axis to the outermost fiber.

And in this case because of symmetry the neutral axis

occurs right in the center and c is equal distance of

12.5 inches up from the neutral axis.

Or 12.5 inches down from the neutral axis.

And which one will be in tension.

Which outer surface will be in tension and which will be in compression based on

the loading that's shown.

5:19

Okay so I for rectangular cross section is

one-twelfth the base times the height cubed or

1/12, the base in this case is 10 inches and

the height is 25 inches, so cube that.

And if you run those numbers, you get 13,020 inches to the fourth.

5:51

So there's our equation for the maximum flexural stress.

We saw that it was going to be, the c is the same for tension and

compression so we're going to have the maximum at the top and the bottom,

just one's in compression, one's in tension.

We found out what the maximum moment is we have our c we have our I.

And so I can go ahead and substitute in, so I get sigma max = M 865,625 ft- lb.

And I'm

6:36

per foot times c is 12.5 inches and that's divided

by I which is 13,020 inches to the fourth.

And again, if you do that calculation you get

9,972 pounds per inches squared.

7:49

And sigma yield for steel.

And so 9.972 is less than 36, so we are okay.

The beam remains in the elastic region.

Okay, beam

remains in the elastic region.

8:22

So it's going to be 9.972 ksi compression at the top and

it's going to be 9.972 tension at the bottom.

Those are the maximum flexural stress and

tension hand compression, and so here's a And that little graph,

a three dimensional graph of that where we have maximum tension at the top or

excuse me, maximum compression at the top and maximum tension at the bottom.

And so that's a good standard elastic flexural problem for beams.

And we'll see you next time.

[MUSIC]