- [Instructor] So in this video, we're going to look
at the background to plane strain analysis.
The objective of plane strain analysis is to take
a solid 3D object and then slice it
in some plane to give us a 2D representation.
Now we have some theory which allows us
to mathematically deal with that situation.
So again to replace the full 3D analysis
with a 2D analysis with a typical cross section.
The motivation is that the analysis is much faster
and cheaper if we're using the 2D plane strain analysis.
We can really afford to saturate this 2D mesh
with for example the fillet radius around in here.
We can really put lots and lots of elements in there.
We get a very good value for this stress concentration.
If we try and do that with a 3D element,
then we're up against the cost of running 3D model
which gets very expensive.
So it's cheap, very effective.
Redesign and analysis is also much quicker.
It's much easier to take this sketch,
reestablish new configuration without worrying
about all the detail of the 3D object.
Analysis oversee again, much much faster and much cheaper.
So a little bit of background theory
behind the plane strain assumptions.
We assume that this little slice here is a slice
somewhere down a depth and basically the depth
just goes on forever.
So a representative section through here,
the idea is that it's infinitely deep
and the strain throughout that depth is zero.
That's why it's called plane strain analysis.
So strain is zero and the stresses which develop
through the depth are just secondary.
So we can focus on stresses in this 2D plane here.
So those will be the in-plane direct
and sheer stresses which we're going to be
representing the behavior.
Now we obviously lose detail of the runout
of a 3D object so there will be N features here,
N features here.
We can see our bolt here, bolts in here.
We kind of smear and smooth out the structure
so it looks like this mathematically,
rather than the actual detail of the 3D structure.
So we are saying this can be a representation
of this structure here.
One of the tricky things we have to watch out for
is the loading requirements
in the plane strain analysis,
it essentially becomes a load per unit depth of plane.
And now, conventionally we think of that as just being
a unit depth.
Actually, in solid work situation you can change that depth
to suit your purposes, so
we're actually working with a dept here of two inches.
So we can make that arbitrary depth two inches,
and then we have a load per unit depth
which would be the same as the load
being applied to the model here.
So, again, something to just watch out for.
So we must match the plane strain,
representational load per unit depth,
against the effective load per unit depth.
And a few simple hand calculations
will get you there quite easily.
Now, the 3D analysis results we're going to be looking at
are from this log which is loaded at the top here,
and that's going to disrupt the stress flow.
As soon as we have a bolt being loaded,
then the stress flow will run around here,
and we'll have a disturbance to the stress in here,
but we will have a very similar profile,
or stress gradient,
and this is what we're really interested in
in the region down in here.
So because of that local stress distribution influence,
all I've got to work with is
looking at the nominal stress or the average stress
in the upper section.
What would be the stress distribution
without the influence of the bolt?
Well, the load going in 12,000 pounds force,
going in vertically.
The area, cross section area,
the station here,
is one inch.
It's actually two inches by half inch section.
So the nominal stress is 12,000 psi.
So if we keep that in mind
as the theoretical nominal stress,
if there were no effector stress concentration around
the bolt here.
Now the 3D analysis, I've got two pictures here,
the first one is showing the stress,
and it's the maximum principal stress,
running down this line here.
Starting from the edge of the hole,
running just to the start of that route radius.
So you can see the stress is starting at zero
at the edge of the hole,
and then peaking up to the nominal
or the average stress, I'm interested in.
And this particular case using maximum principal stress
is the nominal stress in the upper region,
is about 14,200, if we look at that graph there.
So instead of the 12,000,
we get 14,200 because the influence of the bolt
in this 3D analysis in here.
If you look at the lower picture,
then we can see this is showing the stress
around this path here, so it's obviously
much finer scale.
So this value pick up from this value here,
and now we can see the peak stress concentration
peaking up to round about 80,000 here.
So that's what we're interested in.
Now if we look at the ratio of the peak stress here,
peak stress round about in here,
to the nominal average stress,
that's what we call a stress concentration factor,
or SCF, for about 5.6.
And that's really the metric we're going to use to
decide whether this method is effective or not.
So a stress concentration of 5.6,
so what is the peak stress here
compared to the average or nominal stress in here?
When I run the plane strain analysis,
I've shown both the regions on the same plot here.
So using maximum principal stresses again,
the nominal stress in this region through here,
which, again, we don't have a bolt,
so we just have that average stress there.
That's the average stress there,
and that matches our theoretical solution exactly
at 12,000 psi.
It's a good check to make sure we're putting
the right loading in.
When the stress gets to this point here,
that's the run out of that particular curve.
Now we're plotting the stresses around the curve here.
We can see, again, that point matches to that point.
And then we peak up the stress,
and the peak stress we're getting
is round about 66,000.
So we have a different nominal stress,
but we have kind of a kick up factor
or a magnification factor of the stress
as it flows around this feature here.
Again, that's what we call the stress concentration factor.
And in fact, here it's 5.5.
5.5 plays 5.6,
so the overall stress distribution,
the increase in stress in this local feature
in the plane strain analysis
is quite valid as a way of checking
compared to our 3D solid model.
So we're not going to be able to get
exactly the same number,
but the magnification effect,
the stress concentration factor, as it were,
of that route radius,
which is usually what we would be concerned with,
is something we can actually track.
And that's the basis of the approach
we're going to be taking.
So, in conclusion,
3D stress effects are important,
primarily with funneling the stresses underneath the bolt.
And we're using the stress concentrations factor
to calibrate between the 3D
and the 2D plane strain models.