0:48

made some assumptions about the flow of stress through an object.

We've assumed that it's a uniform stress distribution and

that we're dealing with homogenous and isotropic materials.

And this is a pretty good assumption if you have a uniform cross-sectional area or

uniform geometry throughout the object and if you're calculating the stresses

at a sufficient distance between the point load and the support.

So what I mean by that is if F right here is my point load,

I need to be calculating the stresses a certain distance back from that

point load in order for this uniform stress distribution to be accurate.

And that's called Saint-Venant's principle, and Dr.

Whiteman talks about it in more detail in his lecture.

1:40

So unfortunately in engineering, or

fortunately, most components don't have uniform geometry.

Because we have cabling, we have holes, we have different diameters,

we have all sorts of different types of components.

And so it's common for components to have what we call discontinuities, or

notches, or holes, or changes in radius in their geometry.

And what happens is if you think of stress as a pressure flowing through the object,

the stress is going to concentrate around these discontinuities,

around the notches, around the holes, around the changes in area.

And you can see that in this example right here where we have bar with a hole in it,

and you can see the stress concentrating around the hole.

So the stress is going to be a bit higher right up next to the hole.

So, how do we determine what the actual stress is in

a component when we have these discontinuities in geometry?

Well, we use something called a stress concentration factor.

And a stress concentration factor, in this class, we'll refer to it as capital K,

subscript t, and what it does is it helps you increase the stress at

a discontinuity to reflect what the stress on the object actually is.

So your actual stress would be this sigma maximum right here or

your tau maximum right here, and then you can see the Kt is

multiplying times sigma nom, which is your nominal stress.

So this is the stress that would occur if there was a uniform distribution of

stress through the parts.

So if the stress was uniformly distributed across this rod, you would get sigma nom.

And since it concentrates at this hole when you calculate the stress at this

hole, you have to multiply it by Kt to get what's actually occurring in the rod,

this increase in stress due to the distribution.

4:02

So where do we get these stress concentration factors?

Well, most stress analysts will buy a book called

Peterson's Stress Concentration factors.

And it's probably the most famous book, and it's full of every stress

concentration factor that you can think of based off of loading and geometry.

And so here's a chart that's showing some stress concentration factors.

And you can see up here, they show the geometry and

the type of loading it's in.

So this is a shaft.

It has a change in diameter, and it's axially loaded at the ends.

And what we would expect, so you can see I have a shaft over here with the axial

loads, and we'd expect to see the stress concentrating in these areas.

And so if you look at the geometry of the specific shaft that you're calculating,

you'll be able to figure out the stress concentration factor.

So we have geometric ratios on the x-axis and

on these lines running through the chart.

And those help you determine the stress concentration factor on the y-axis,

which you can then plug in to this equation and

figure out the actual stress at this point.

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So a really important thing to understand when

doing stress analysis is when to use Kt.

And let's start out in design and think of loading in two different ways.

So we have fatigue loading which we'll talk about in Unit 3.

And that's when we have dynamic loads, they're changing over time

like a car engine that's revving from zero to 600 rpm, it's changing.,

What we're talking about in this unit are static loads.

So when fatigue loads, you'll always use a stress concentration factor.

The stress concentration factor will be capital K, lower case or

subscript f, so Kf, and that's your stress concentration factor for fatigue.

For static loading,

you need to look at how is the material behaving in your operating conditions.

And if your material is behaving in a ductile manner, so

your strain at fracture is greater than 0.05.

And you've taken in consideration your temperature, if it's being

exposed to high levels of radiation, if there's been any hydrogen embrittlement or

any residual stresses that could make this material behave in a brittle manner.

Just like last lecture when we talked about the ductile to

brittle transition temperature.

So if you've taken into account all of these things,

you're sure that your material is behaving in a ductile manner,

then you don't need to use a Kt, you don't need to use a stress concentration factor.

And that's because at that place of the discontinuity,

the geometry discontinuity, you'll get localized plastic yielding.

So this localized yielding will cause strain hardening to occur at that point,

which will increase the yield strength, and that makes it unnecessary for

you to use the stress concentration factor.

7:16

You have to use a stress concentration factor in a brittle material every time,

and it's very important.

So in a brittle material, you use the static stress concentration factor Kt.

Okay, so continuing on, let's take a look at an example.

So if we have this shaft or rod, and it has a change in diameter, so

it starts out with a diameter of 24 millimeters, and then there's a fillet

radius of 4 millimeters, and it goes down to a diameter of 16 millimeters.

And they're saying it's aluminum, normally, it behaves in a ductile manner.

However, here it's operating below its transition temperature, so

it's going to behave in a brittle manner.

What's the max stress in the rod at a load of 40 newtons,

and we can see that the rod is loaded in tension?

And then here, we have the stress concentration factor chart.

There's a lot of these charts all over the Internet, in the back of textbooks, and

again, the most prominent resource is Peterson's Stress Concentration Factors.

And you can see,

the first thing is to make sure this is the correct chart to use.

So we see here, we have a rod, two different diameters and

an axial load with a fillet radius in between.

The second thing over here is it's going to tell you

what is it defining as your sigma nom.

And this is really important to understand what area are they calculating

your nominal stress from.

And here you see this little d,

that means they're calculating the nominal stress at the smaller diameter.

So that's exactly what we're going to do.

We're going to say stress, axial stress equals F/A.

And we need to use the smaller diameter to get my A,

so 4F divided by pi d squared is going to be, let's see.

40 newtons times 4 divided

by pi times 0.016 meters squared,

and I get about 0.2 megapascals.

Okay, so the next thing we need to do is we need to figure out

our stress concentration factor based off of this geometry.

And looking at the chart, we need two ratios to do that.

We need the R/D ratio, which is going to be 4 divided by 16, which is 0.25.

And then, we need a D/d ratio,

which is going to be 24 divided by 16, which is 1.5.

So here's my 0.25 place on the chart.

This line right here is 1.5, so I can see that I cross

it right about here, which is right about here.

So my Kt is going to be 1.52.

Okay, so now I figured out my sigma nom and my Kt, and I can plug in to

figure out the maximum stress or the stress that's actually occurring.

So my sigma max right here, which is going to be sigma nom times Kt,

so my 0.2 megapascals times 1.52, and

I get a stress of 0.3 megapascals in this example.

So you can see,

a stress concentration factor has a significant increase in the stress.

It's definitely 150% higher in this case, and therefore, it's

very important to consider if you need to use a stress concentration factor or not.