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PROFESSOR: Ladies and gentlemen,
welcome to this

lecture on nonlinear finite
element analysis of solids and

structures.

In this lecture I'd like
to discuss with

you structural elements.

Structural elements are, of
course, employed to model

beam, plate and shell structures
and are most

important elements.

Because of their importance in
engineering practice much

research and development effort
has been focused on the

development of efficient
structural elements.

In this and the next lecture I'd
like to discuss with you

some modern effective
elements.

We will first discuss shell
elements and then beam

elements and then we go back to
use the shell elements, a

concept that we discussed in
some actual applications.

When we do structural analysis
we should keep one method in

mind, namely that in a
geometrically nonlinear

analysis, a flat shell, referred
to as a plate, goes

very rapidly over into the
behavior of a shell because of

the curvature that develops
as the plate deforms.

Therefore, to analyze
geometrically nonlinear plates

we really are quite well off
using general shell elements.

And I'd like to now focus our
attention on the development

of general shell elements that
we actually then employ to

analyze plates as well
as general shells.

There are various solution
approaches that one can follow

for the development of
efficient elements.

And one such approach is
the use of general

beam and shell theories.

I'm thinking of beams as well
as shells although I want to

focus our attention on shells,
really, in this first lecture.

One starts with beam or shell
theory as general as possible

and develops the governing
differential equations.

From the governing differential
equation one

develops variational
formulations and one

discretizes those variation
formulations using finite

element interpolations.

This is the approach
that is taken.

The disadvantage of that
approach is, in general, lack

of generality of the approach.

You're starting off with
a particular shell

theory, beam theory.

You of course would like to
have that shell theory as

general as possible but you
will find that really, the

shell theory that you're
starting of will mostly only

be applicable to a certain
class of shells.

Therefore your finite elements
that you're developing would

also be only applicable to that
same class of shells.

And that means in engineering
practice, in actual usage, the

user has to be very familiar
for which the particular

element developed,
as shown here.

I'll discuss now for what
kinds of shells this

particular shell element
is really applicable.

We would rather like to have
shell elements that are

applicable to any shell.

Of course this is a
very big aim, very

difficult to achieve.

But this is what we would
like to have ideally

in engineering practice.

The other difficulty with this
approach is that frequently a

large number of nodal degrees of
freedom have to be carried

along in the development
of the shell elements.

And what I'm thinking of there
is that you don't just have

translations and rotations at
the nodal points, which has

engineering degrees of freedom
that we would like to have and

see in for the element, but
you also have to have

additional degrees of freedoms
relating to the curvatures in

the elements.

And so on.

So this of course would mean, or
does mean, difficult use of

such elements and these reasons,
these two reasons

here have really driven the
research and development

efforts in different
directions.

Another approach is to
use simple elements.

Simple elements, but a large
number of elements, then to

model very complex beam
and shell structures.

As an example I'd like to just
refer you to a 3-node

triangle, a flat element in
which the plate-bending

behavior is modeled in a
particular way and the plane

stress behavior is modeled as
a constant strain, constant

stress element.

These two are superimposed,
these two behaviors are

superimposed and you have a very
simple shell element that

in certain analyses can
be quite effective.

Of course we have to recognize
in this approach that the

coupling between the membrane
and bending action is only

introduced at the
element nodes.

That is a major disadvantage.

And that is the reason why you
need so many elements to model

a shell and that if we use a
triangular 3-noded element

with just a constant strain,
constant stress element to

model the membrane behavior, the
membrane action, of course

the membrane action is quite
poorly approximated.

And that is another reason why
we need so many elements to

model a complex shell
behavior.

Here on this viewgraph I'm
showing a picture of the

element that I just
referred to.

Here you have a triangular
element in the three

dimensional space.

Notice we introduce a local
coordinate system, x bar 2, x

bar 3, x bar 1.

Local coordinate system, and
in this local coordinate

system we measure the
displacements and rotations.

And we superimpose for this
element the bending behavior,

the bending behavior
corresponding to this degree

of freedom, that degree of
freedom and this degree of

freedom at every node,
of course.

We add, or superimpose onto
this bending behavior the

membrane behavior which
corresponds to these two

degrees of freedom.

And we immediately notice that
corresponding to this degree

of freedom we don't have a real
physical stiffness and we

introduce a little artificial
stiffness as shown down here.

Now this artificial stiffness
has to be selected and you

want to select it such as to
take out the singularity out

of the system but yet make the
stiffness not too big so as to

destroy the behavior
of the element.

Because it's an artificial
stiffness you really want to

make it as small as possible
just to take out the

singularity out of this element
when you apply this

element in the modeling of
general shell structures.

This artificial stiffness
actually is quite bothersome

in nonlinear analysis.

It can provide problems and
we really don't like it.

But if we use this approach we
have to introduce it and well,

we have to, so to say, live with
it, with the difficulties

that we encounter.

But the other approach that
I want to discuss with you

really quite extensively in
these two lectures, namely

using the isoparametric
elements, having curved

elements there as well, we
don't introduce this

artificial stiffness anymore
because our experiences of

introducing it in nonlinear
analysis show us that there

are many difficulties that
come into the analysis

procedure if you have this
artificial stiffness.

So we got rid of it.

We don't need to use it any
longer for the more modern

elements that I will be talking
about just now.

The approach for these modern
elements is to use

isoparametric interpolations
and we talk then about the

isoparametric (degenerate)
beams and shell elements.

"Degenerate" because we
degenerate these elements, or

we obtain these elements I
should say, by degeneration

from three dimensional
behavior.

We will talk much more
about it just now.

But in essence we are saying
we take the 3-D continuum

equations and we degenerate
those equations to the

particular shell behavior and
beam behavior for beams that

we would like to capture.

The resulting elements can be
used to model quite general

beam and shell structures.

And that is, of course, a very
large advantage in engineering

practice if you can use the
same elements to model a

variety of structures.

The basic approach of this
isoparametric interpolation is

to use the total and updated
Lagrangian formulations that

we developed earlier.

We talked in the earlier
lectures quite extensively

about the total updated
Lagrangian formulation, the

continuum mechanics equations,
as well as the finite element

discretization of the continuum
mechanics equations.

But we applied the final
discretization only to 2-D and

3-D solid elements.

Now we want to do the same
for shell elements.

We recall that in the T.L.
formulation the governing

equation is this one here, which
is nothing else than the

principal of virtual work
operating on the second

Piola-Kirchhoff stress and
the variation on the

Green-Lagrange strain.

This integral has taken over
the original volume of the

structure, of the
element, when we

develop a finite element.

And this of course, the internal
virtual work, and

here we have the external
virtual work.

Notice the linearization of
this left integral here

resulted into these
three integrals.

We went through that
linearization in quite some

detail and we talked about
the individual terms.

I don't think it's now necessity
to review that

material anymore.

Please refer to the
earlier lectures.

The same approach of course
we used also for the U.L.

formulation.

Here is the general starting
point, the principle of

virtual work now, using second
Piola-Kirchhoff stresses refer

to the configuration
at time t.

Variations in the Green-Lagrange
strains refer

to the time configuration
at time t.

This gives the internal virtual
work corresponding to

time t plus delta t,
t plus delta t.

And this is the external virtual
work, same external

virtual work of course that
we are having in the total

Lagrangian formulation.

The linearization of the left
hand side integral here

results into these
three integrals.

Once again we talked about
the linearization quiet

extensively and if you refer to
the earlier lectures surely

you recognize individual terms
that you're seeing here now.

We use these governing equations
for the total

Lagrangian and updated
Lagrangian formulation to

develop the general
shell elements.

And what we have to do now is
to impose on these equations

the basic assumptions of
beam and shell action.

And let us go now through these
basic assumptions one by

one carefully.

The first assumption is that
the material particles

originally on a straight line
normal to the mid-surface of

the beam or the shell remain
on that straight line

throughout the response
history.

This is one most important
assumption.

You must have encountered this
assumption or at least some

form of it already earlier in
your discussion of beam

theories and possibly
shell theories.

Let's look at this assumption
more closely, so.

For beams we would say plane
sections initially normal to

the mid-surface remain
plane sections during

the response history.

This is basically saying
the same thing

what I just said earlier.

And if you look at this closely
you recognize that we

do not say that the plane
sections initially normal to

the mid-surface remain plane
sections during the response

history and remain normal
to the mid-surface.

We don't say that.

That's of course being
said when you use the

Euler-Bernoulli beam theory.

We don't say this, that the
plane sections remain normal

to the mid-surface throughout
the response history.

We don't say that and because we
don't say that we in effect

include in an approximate
way shear deformations.

In other words, we look here
at the effect of transverse

shear deformations is included,
and hence the lines

initially normal to the
mid-surface do not remain

normal to the mid-surface
during the deformations.

Let's look at what this
means pictorially.

Here we have a section of a beam
at time 0 and we draw a

line normal, that is, at 90
degrees to the mid-surface,

shown as a dashed line.

And we identify particles
on that line.

Here we have four
such [? rad ?]

material particles.

Now the beam will move, deform,
go through large

displacements, large
rotations.

But actually we assume
small strains.

And we see that these material
particles which were

originally up here are
now down here.

We identify that these material
particles are still

on a straight line.

But this straight line is not
anymore normal, that it is at

90 degrees to the mid-surface.

And because it is not any more
normal to the mid-surface we

do include shear deformations
approximately because we

assume that the shear
deformations are constant

throughout the thickness
of the beam.

This is a most important
assumption.

We're looking here what looks
like a beam but actually, if

you think of another dimension
here, you can directly see

that the same picture is
also applicable to

the motion of a shell.

The second important assumption
is that the stress

in the direction normal to the
beam or shell mid-surface is

zero throughout the
response history.

In other words, there is no
stress developed normal to the

mid-surface.

But notice that here is a stress
along the material

fiber that is initially normal
to the mid-surface is

considered.

Now this material fiber which
is initially normal to the

mid-surface will not remain
normal to the

mid-surface as I just said.

And in the motion we will
consider always this stress in

the direction of that material
fiber which was originally

normal to the mid-surface.

So after motion has taken
place we are not really

talking anymore exactly--

you want to look in great detail
at what's happening--

we don't talk anymore exactly
about the stress, that is

normal to the current
mid-surface.

But we always talk about the
stress in the direction of the

fiber that was initially normal
to the mid-surface.

That's being said here.

And the third assumption, also
most important assumption, is

that the thickness of the beam
or shell remains constant.

Here then we clearly identify
that we are using really a

small-- we are assuming small
strain conditions, but we

allow for very large
displacements and rotations.

Well, with these three
kinematic and static

assumptions clearly identified
we are now ready to actually

develop the shell element
interpolations.

And let's go at that.

The first point is that we
incorporate the geometric

assumptions.

Straight lines normal to the
mid-surface remain straight,

put here in quotes, that
geometric assumption and the

geometric assumption that the
shell thickness remains

constant throughout
the whole motion.

These were two assumptions
that we just discussed.

We incorporate that into our
shell element formulation by

using the appropriate geometric
and displacement

interpolations.

We incorporate the condition of
zero stress normal to the

mid-surface, I put it in quotes
here, because remember,

we are talking about the stress
in the direction of the

fiber that was originally normal
to the mid-surface.

We incorporate this condition
by using the appropriate

stress-strain law.

Let's talk first about this
assumption and then about this

assumption, how we're using
these to actually develop our

shell elements.

To focus our attention
I'd like to talk

about a 9-node element.

However.

we will later on see that in
practice actually we don't use

the 9-node element very much.

We actually recommend the use
of a 16-node element and a

4-node element but this
9-node element in some

analysis is also used.

And it certainly is an element
with which I can discuss with

you, share with you all the
experiences regarding the

formulation of the elements
because what we're talking

about now really is applicable
to any of the elements.

In fact, we're talking about
variable number nodes

elements, where the number of
nodes can be selected by the

analyst and the geometric
assumptions that we're now

talking about are the same for
any one of these elements.

How do we go about
the formulation?

Well, one important point is
that we introduce at each node

lying on the mid-surface, and
here we see such 9-nodes, a

director vector, a director
vector t v n k, t referring

already to the geometry
at time t.

Of course this director vector
is actually input by the

analyst for time 0 and then
it evolves with the

motion of the shell.

Also we're introducing for the
analysis, and that's being

done automatically in the
computer program, these two

vectors here, t v 1 k and t v
2 k, which are normal to the

director vector.

These vectors are calculated
automatically in

the computer program.

We talk more about it
a little later.

Notice that the thickness here
at this node is a k, and

notice that v n k at node
k-- k of course

stands for that node--

acts into the direction off
the thickness here.

Notice that such [? triad ?]

of vectors is of course
being worked with at

each of these nodes.

And notice that the thickness
at the nodes can change.

The element is defined
as follows.

As far as the analyst is
concerned the initial nodal

point coordinates of all
the nodal points on the

mid-surface must be input.

Also the initial director vector
must be input, here now

you see the zero, and the
thickness at every

node must be input.

Notice if these director vectors
at all of the nodes

are known, with the thicknesses
at the nodes of

course, then we can interpolate
the thickness at

any point of the mid-surface
of the shell and we can

interpolate the director vector
corresponding to any

point on the mid-surface.

Here such point on the
mid-surface we get the

thickness at that point, from
the thicknesses that we have

here, and from the
director vectors.

And of course we are also
getting the director vector at

this point from these
director vectors.

So the analyst must put in the
nodal point coordinates of the

mid-surface nodes and the
direction cosines of these

director vectors.

Much of it of course
can be generated

in a practical analysis.

We use an isoparametric
coordinate system with

coordinates r, s, and t.

The coordinates r and s
correspond to a measure in the

mid-surface.

The coordinate t is measured
in the direction of the

director vectors.

The geometry at time 0 is
interpolated as shown here in

this equation.

0 x i gives us the coordinates,
three, i goes

from one to three, of any
material particle in the

stationary coordinate frame.

I should point out once more
very strongly that we use a

stationary Cartesian coordinate
frame, x1, x2, x3,

to describe the geometry of the
element and to work with

our element.

This coordinate frame, x1,
x2, x3, is stationary.

In that stationary coordinate
frame of course we are

measuring the coordinates
of any material particle

corresponding to time
0, corresponding

to time t, et cetera.

The same way as we discussed it
in earlier lectures when we

talked about the analysis of
solids, 2-D and 3-D solids,

and when we talked about the
continuum mechanics equations.

So here these coordinates of the
material particles, as the

material particles are moving
through the stationary

coordinate frame, are given
by the right hand side.

And what do we see here?

k is 1 to n, n is the number of
nodes for the element that

I've shown you and
would be nine.

h k are the interpolation
functions corresponding to the

two-dimensional surface
of the element.

In other words, these h
k's are really the 2-D

interpolation functions as we
are used to see them for plane

stress, plane strain, and
axisymmetric analysis.

Same interpolation functions.

These are the nodal point
coordinates at time 0.

Here we have t, that is the
third isoparametric

coordinate.

We talked about it just now.

k, going from 1 to n again.

a k are the thicknesses
at the nodal points.

h k here is exactly the same
h k that you see here.

And these are the direction
cosines of

the director vectors.

Director vector n means director
vector k, or normal.

n really stands for normal, but
it's really the director

vector referring to the
director vector k,

of course, a node.

0 means time 0 and i means
the components, the three

components of the
director vector.

That's what we're
looking at here.

Now if you leave this term out
then you would have simply the

interpolation of the mid-surface
as for a membrane

element, of course curved
mid-surface.

This term here is added in to
take into account the effect

of the shell thickness.

Similarly at time t we have,
applying the same

interpolation t,
x, i here now.

Here we have t, x, i, k, same
h k that we talked about

earlier, same summing that
we talked about just now.

Here now t, v, n, i, k, the
direction cosines, also

director vectors.

Director vector k
at time t now.

Notice that all we have done in
this term and in that term

is to substitute for the 0 that
we had here and that we

had there, the t now.

Of course this t means
time t, this t here

means coordinate t.

That's why we put a circle
around it and wrote it out

there once more.

This is the coordinate, the
isoparametric coordinate t.

r s coordinates go in here, t
coordinate goes in there.

What has happened here is that
originally our director vector

might look like shown in
this picture here.

The node coordinates are given
here, 0 x i k, and in the time

from time 0 to time t this
director vector moves to look

as shown now here.

And of course the nodal point
itself has moved as well.

So it's these quantities here,
these [? rad ?] quantities

that we're using here, which
carries a curve.

Well, to obtain the displacement
of any material

particle within the shell we
proceed now exactly the same

way as we have proceeded
in the development

of 3-D solid elements.

We take the interpolation for
the geometry at time t,

subtract the interpolation for
the geometry at time 0 and we

get the displacements
corresponding to time t.

The result is this
equation here.

Here we have the displacements
of any material

particle in the shell.

Here we have the nodal point
displacements from

time 0 to time t.

Here we have the director
vectors at time t, so to say,

minus the director vectors
at time 0.

Really these are the direction
cosines corresponding to the

director vector at time t, minus
the direction cosines of

the director vector
corresponding at time 0.

Of course this quantity here is
obtained by this equation

and that quantity here is simply
obtained by subtracting

the right hand sides
corresponding to t x i and 0 x

i the way you just have seen
them on the previous

viewgraphs.

The incremental displacement
from time t to time t plus

delta t is similarly obtained
from this relationship here.

And the result is shown here
where now we have here the

increments in the nodal
point displacements.

And here we have the increments
in the direction

cosines of the director vectors
from time t to time t

plus delta t.

The equation is given
right there.

Well, with the equations that
we have developed so far we

are almost ready to establish
the strain displacement

matrices for the T.L. and U.L.
formulations of the element.

We have the coordinate
interpolations for the

material particles.

We discussed those.

And we have at the moment also
the interpolation of the

incremental displacements of the
material particles in the

shell element in terms of
nodal point incremental

displacements and the increments
of the direction

cosines of the director vectors
at the nodal points.

What we, however, want is
to have the incremental

displacements in terms of the
nodal point displacements and

nodal point rotations.

The nodal point rotations
because the incremental nodal

point displacements and nodal
point rotations are the

engineering type quantities that
we can nicely deal with

in a computer program when we
analyze shell structures.

So what we want is to express v
n i k, the increments in the

direction cosines of the
director vectors, from time t

to time t plus delta
t in terms of

the nodal point rotations.

And that is achieved
as follows.

Here we have the stationary
coordinate frame x1, x2, x3.

e 1 is a vector into the x1
direction, e 2 the vector into

the x2 direction, e 3 the vector
into the x3 direction.

Here we show v n k for
nodal point k.

In other words, a director
vector, at time 0.

This one is input
by the analyst.

These two are calculated
in the program.

And of course there are such two
for every nodal point k.

For every nodal point k we also
put in a director vector.

Now the convention that is used,
that can be used for the

v 1 and v 2 calculations, that
convention is given down here.

Notice that if v n points into
the e 3 direction then v 1

points into the 1 direction
and v 2

points into the 2 direction.

By that I mean into the e 1 and
into the e 2 directions.

Notice, this is a detail, that
when v n points into the e 2

direction this formula breaks
down and you need to use some

other convention.

But that is a detail.

We don't really need
to discuss that

very much here now.

Anyways, let us say then at
every nodal point k, v n has

been input by the user, v
1, v 2, calculated by

the computer program.

We notice that these
two are once again

perpendicular to v n.

Then we can directly say that
the increment in the direction

cosines of the vector, the
director vector, is given via

this relationship here.

Now I've written this down
already for time t.

Of course it also holds for
time 0, all you do is

substitute for t, 0.

It holds in fact for any time
of the motion that we are

considering.

Let's see why this holds.

Well, this picture here shows
what's happening.

Here we have v n, here we have
v 2, here we have v 1.

Notice alpha is the rotation
about v 1, beta is the

rotation about v 2.

Notice that with the rotation
beta about v 2 here, we of

course have an increment in
this vector shown by this

component because this here is
now v n at time t plus delta t

when alpha k is 0.

When alpha k is 0.

Now of course you would also
have the alpha k component

coming in and that means you
have to add also this term.

Notice that once you have
obtained this vector you want

to normalize its
lengths again.

But this picture here shows
why this term is a correct

term to use in this formula
and you can extend this

picture to also include
this term.

With this relationship we can
substitute for v n and that's

now done in this equation.

We have substituted for v n and
have now our increments in

the displacements for the
material particle within the

shell in terms of nodal point
incremental displacements and

rotations about these v 1 and
v 2 axes that we defined.

Well, having established this
interpolation for the

incremental displacements and,
of course, the interpolations

for the geometries of the
element at time 0, at time t,

we can directly establish the
strain displacement matrices.

And we will see we can then
set up the k matrix, the f

vector, the elements that go
into the equilibrium equations

the way that we discussed it
in the earlier lectures.

And from the solution of k u
equals r or k delta u equals

delta r in nonlinear analysis,
we of course get our nodal

point rotations, alpha
k and beta k.

And once we have calculated
these nodal point rotations we

obtain by this relationship
here the v n at time

t plus delta t.

In other words, what we're
doing really here is we

integrate over all of the angle,
alpha k and beta k, to

get a more accurate
approximation for the nodal

point vector, of the director
vector, I should say, at time

t plus delta t.

Notice that if you do this
integration in one step with

the Euler forward method, you
get back the equation that we

had earlier on the viewgraph.

And which I tried to explain or
discuss with you using this

picture that you saw.

And I mentioned also that after
this integration of

course we want to normalize the
lengths of this vector to

make it always a unit
length vector.

We recognize that with this
approach we have only five

degrees of freedom per node.

Three translations in the
Cartesian coordinate

directions which are stationary,
and two rotations

refer to the local nodal point
vectors v 1 and v 2 at time t.

Now notice in geometric
nonlinear analysis, of course,

this vector and that vector,
t v 1 and t v 2, change

direction so our alpha k and
beta k are rotations that are

measured about changing
directions.

That is an important
point to recognize.

Let's look at one pictorial

representation of what's happening.

Here we have node k.

We have a smooth shell, say,
that is discretized using four

shell elements.

I've taken one shell element
away as shown here so that we

can look into the shell.

And at that node we have, as
shown now, v 1 and v 2 at time

t and the director vector
corresponding to a time t.

We measure at that time
alpha k and beta k

about v 1 and v 2.

And we also measure the
displacements of the node, u

1, u 2, u 3.

So notice that this node
moves as shown.

These are the three
translational degrees of

freedom and this director vector
here moves to a new

position and also changes
direction.

And that new director
vector of course is

given here in red.

Notice that there is no
physical stiffness

corresponding to the rotation
about the director vector.

No physical stiffness.

The five degrees of freedom
that the element very

naturally carries, u 1, u
2, u 3, alpha k, beta k.

Alpha, beta at every node k.

If only shell elements connect
to node k and the node k is

not subjected to
boundary-prescribed rotations

then we only need to assign
these five degrees of freedom

to the node and only work with
these five degrees of freedom

at that node.

However if we deal with a node
to which also beam element is

connected, which of course in
general has three rotational

degrees of freedom, or a node
on which a boundary rotation

other than alpha k or beta k is
imposed then we transform

the two nodal rotations, alpha
k and beta k, to the three

Cartesian axes.

Because this way we can directly
deal then with the

connection here and
the imposition of

the boundary rotation.

We can do so directly using
these three rotations now

measured in the Cartesian
axis directions.

I mentioned already that the
above interpolations for 0 x

i, t x i and u i, in other
words, for the original

geometry, the current geometry,
and the incremental

displacements the way we have
developed them, they can

directly be used to obtain
the strain displacement

transformation matrices.

And we really do so in the same
way as for the 3-D solid

elements which we discussed
in an earlier lecture.

However, there's one method to
recognize, and I briefly

pointed it out also already in
the earlier lecture, that

using this expression here--

and I now must refer to the
earlier material that we

discussed--

using this expression to obtain
the strain displacement

matrix, we realize that we
obtain the exact linear strain

displacement matrix.

However, using this expression
here to develop the nonlinear

strain displacement matrix, t 0
b n l, for the shell element

only in approximation to the
exact second-order strain

displacement rotation expression
is obtained because

the internal element
displacements depend

nonlinearly on the nodal
point rotations.

I pointed that out earlier.

Please refer back to that
discussion to obtain a deeper

understanding of what
I mean here.

The important point, of course,
is that we do obtain

the exact linear strain
displacement matrix, so at

convergence in an iteration, k
delta u equals delta r, when

we have converged we actually
have the exact solution to the

model that we're using,
of course.

So this is important that we
obtain the appropriate and

exact t 0 b l matrix.

The effect of what we are
neglecting here was earlier

mentioned and please refer
back to that lecture.

We finally need to still impose
the condition that the

stress in the direction
normal to the shell

mid-surface is zero.

Remember this was one further
assumption that we discussed

at the beginning of
this lecture.

We use a direction
of the director

vector as a normal direction.

This means that at each Gauss
integration point within the

element, and which we want to
evaluate the stress-strain

law, we first of all set up a
system of vectors, e bar r, e

bar s, e t, that are mutually
perpendicular.

Now let's look here into the
picture above and we see here

at a particular Gauss point,
schematically shown, the

vector e r and e s which are
vectors corresponding to the r

and s axes.

e t is a vector corresponding
to the direction of the

director vector at that point.

e t we accept as the normal
direction, that's for the

shell at that point, and we
construct e bar s and e bar r

to be vectors perpendicular
to e t and to themselves.

And that is achieved by this
relationship here.

Having now established e t, e
bar r and e bar s, we use this

stress-strain law.

And by this I mean let's
look first what's

in these round brackets.

We use this stress-strain law
corresponding to these

directions.

In other words, the directions e
bar r, e bar s and e t where

this is the normal direction,
this corresponds to the normal

direction e t.

Notice by putting zeroes here
we impose the fact that into

the direction e t we
have zero stress.

Of course this matrix
is symmetric.

Notice we have k here which is
a shear correction factor

applied to the transfer shear
stresses in the shell.

New of course is Poisson's
ratio and

e is Young's modulus.

Now this is the material law
corresponding to e t, e

bar r, e bar s.

And what we now have to do is
transform this material law at

every integration point to
the global directions.

Global directions because for
the global x1, x2, x3

directions we have established
the b, the strain

displacement matrices.

And that gives us this material,
or c, shell.

This q s h t, q s h is, I
think, quite well known.

Let me show you the form
of it briefly.

It's a transformation matrix
where we show some of the

terms here, as you can see.

And these terms, l 1, m
1, n 1, et cetera, are

defined down here.

And l 1, for example, is nothing
else but the cosine of

the angle between
e 1 and e bar r.

In other words, it's a
transformation matrix that you

are probably quite familiar
with in linear analysis.

It's the matrix that transforms
the stress and

strain components from one
coordinate system to the other

coordinate system.

And of course the coordinate
systems that we are

transforming, that we're using
here, are the bar, e bar, r, e

bar s, e t coordinate system on
the one side and e 1, e 2,

e 3 on the other side.

Well, using this matrix we
assure that the columns and

rows 1 to 3 in c s h reflect
that the stress normal to the

shell mid-surface is zero.

This holds true because,
remember if we go back once

more to the viewgraph, we
have set this column and

corresponding row to
zero elements.

And of course this means that
the stress normal to the shell

surface is zero.

I should also briefly point
out that we have a plane

stress condition corresponding
to the other direction of

stresses, in other words,
the e bar r and e

bar s direction stresses.

We have a plane stress condition
as you can see here.

And of course this term would
go with it as well.

This term and these terms here
reflect the plane stress

situation in the plane
of the shell, zero

stress through the thickness.

And all that we are transforming
to the global

system now here--

that fact, that physical
fact--

of course must still be
reflected in c s h.

And that's what's being said
here at that point.

If you want to do plastic
analysis, creep analysis, you

proceed in the same way.

But you calculate then once
again the stress-strain

metrics as in the analysis
of 3-D solids.

And having got that
stress-strain matrix, six by

six matrix, you impose the
condition that the stress

normal to the mid-surface is
zero in much the same way as

what we have done here for the
elastic material condition.

Finally, regarding the kinematic
descriptions that we

talked about for the shell
elements, it is interesting to

note that also transition
elements can be developed.

These can be quite useful in
practical analysis because

they are elements with some
mid-surface nodes that carry,

in other words, associate
director vectors and five

degrees of freedom per node
and some top and bottom

surface nodes with three
translational degrees of

freedom per node.

These elements would be used,
for example, to model

shell-to-solid transitions or to
model shell intersections.

And here you see one such
typical element just

schematically shown.

Here we have a mid-surface
node, three translational

degrees of freedom corresponding
to the

stationary coordinate frame,
two rotational degrees of

freedom the way we talked about
it just now, mid-surface

node here and top and bottom
nodes here with three

translational degrees
of freedom for

each of these nodes.

Notice we can directly couple
3-D solids to this phase here

and shell elements to
this phase here.

We haven't shown mid-surface
nodes here, but you would,

say, have all the mid-surface
nodes here and then surely

shell element can directly be
coupled into here in such

situation as schematically shown
right down here on this

viewgraph where we have solid
elements on this side and

transition element right
there and shell

elements on that side.

Because here you would continue
with shell elements,

you would continue with
solid elements here.

And once again three
translational degrees of

freedom for these top and bottom
surface nodes and five

degrees of freedom at shell
mid-surface nodes as shown for

that node right here.

Notice we can also use these
transition elements to model

shell intersections very nicely
as exemplified up here.

Well, this brings me to the
end of what I wanted to

discuss with you in
this lecture.

Of course what we have not done
yet is to look at example

solutions and what I'd like to
do in the next lecture is

first of all talk with you
about beam elements, the

isoparametric beam element,
which is formulated much in

the same way as the
shell elements

that we just discussed.

And then I'd like to show you
applications of the beam

elements as well as the
shell elements.

So please, if you are interested
in this subject

matter, continue looking at the
second tape, part two of

this set of lectures.

Thank you very much for
your attention.