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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 would like to continue with our
discussion of solution methods that we use to solve the
finite element equations in nonlinear static analysis.
We considered already in the previous lecture a number of
solution techniques to solve this set of equations.
t plus delta tr is equal to t plus delta tf, where t plus
delta tr is the load vector of externally applied loads at
time t plus delta t, and t plus tf is a nodal point force
vector corresponding to the internal element stresses at
time t plus delta t.
We talked about the full Newton-Raphson method, the
modified Newton-Raphson method, the BFGS method, the
initial stress method, and we discussed
also convergence criteria.
I've summarized here the equations corresponding to the
modified Newton iteration.
Now, a distinguishing feature of these solution techniques
is that the analyst has to prescribe the externally
applied load vector
corresponding to all load steps.
This means schematically that if you have this kind of
displacement load curve, or load displacement curve, shown
here in red, the analyst has to prescribe prior to the
analysis the load levels at which the response is sought.
I've here indicated 1r as a load level corresponding to
load step one, 2r as a load level corresponding to load
step two, et cetera.
Of course, the corresponding solutions that the analyst is
looking for are the displacements.
Now, what might very well happen in practical analysis
is that the analyst chooses certain load levels, and at
the particular load level convergence difficulties are
encountered.
Too many iterations are required to converge, for
example, in a reasonable cost.
And typically, say, if this happens at this load step
here, the analyst decides to restock.
And this means that a solution corresponding to r4 here was
not obtained.
I'm scratching that out.
And the analyst restarts now with a smaller load step,
namely indicated by these green lines here.
The solution is obtained corresponding to that
configuration, corresponding to that configuration, that
configuration, which corresponds in this particular
case now, to the level of r4, because we have three load
steps to reach now r4.
And say, at this point now, again, for example, we may
encounter solution difficulties.
The analyst is not able to obtain this solution here.
Once again, the load step has to be smaller.
To continue the analysis, a restart is necessary at this
particular configuration.
And like this the analyst tries to get closer and closer
to the collapse load.
The conclusion is that we have difficulties in calculating
the collapse loads when we use the techniques that I
described in the last lecture.
Of course, if you restart enough, if you use small
enough to load steps, you will ultimately get very close to
this collapse load, but this is tedious in a practical
analysis, and we would like to have really an automatic
scheme that directly can trace out the collapse load and you
can also go into the post-collapse response, which
is the response beyond this ultimate limit load here.
Well, I've prepared some view graphs to show you, discuss
with you, a solution technique that actually can be employed
to trace automatically load displacement response out, and
go beyond the collapse load, as well.
The idea is that we want to obtain more rapid convergence
in each load step.
We want to have the program automatically select load
increments.
And we want to also be able to solve for the
post-buckling response.
Now, this here means, of course, that a priori, prior
to the analysis, you may not know, in fact you do not know,
for which loads you will obtain the solution.
We get the total load displacement response curve,
but the discrete load levels at which the response was
calculated is being decided by the program.
An effective solution would proceed as follows,
schematically, of course.
Here we have the load axis.
Here we have the displacement axis.
The solution scheme would start with large load
increments in the region where the response is almost linear,
then cut the load increments to a smaller size, and
capture, of course, the ultimate limit load here, and
then decrease the load, as shown here, to go into the
post-buckling, post-collapse response.
And the solution scheme, of course, could continue up this
branch as well.
And the solution scheme should automatically select the load
step sizes, depending on the convergence that has been
encountered in the previous load steps.
We compute now, t plus delta tr using this equation here.
Notice that in this equation, t plus delta t lambda is an
unknown scaler that is going to be determined by the
program, and r is a vector that gives a particular load
distribution.
r is constant.
It may contain the contributions of soft surface
pressures, of concentrated loads, et cetera.
The point is that r is constant and the program will
automatically adjust lambda.
Therefore, we assume in our collapse analysis that the
loads are increasing and decreasing all in the same way
as decided by one scale.
Of course, one could extend such an algorithm by
introducing another vector, calling this r1, introducing a
vector r2, with another scaler.
And then have two scalers that have to be adjusted
automatically by the program.
And perhaps one can even think of three scalers, and so on,
but then the algorithm would become quite complicated.
So we look at a scheme that only has one scaler here, and
one load vector that remains constant
throughout the solution.
As an example here, you see in blue the load
intensity, t lambda r.
Notice a concentrated load here, a pressure load here,
and at the time t plus delta t we have the red intensity, t
plus delta t lambda r, and of course this concentrated load
has increased in the same way as the pressure has increased,
as shown right here.
The basic approach of the solution scheme is shown on
this view graph.
Here we have the low displacement curve that we are
looking for, shown in red again.
Of course, loads plotted vertically, displacements
plotted horizontally.
Now, notice that if this is an equilibrium point, the red
dot, and if you as the analyst were to chose an increment in
load shown by this black line--
I'm putting my pointer now onto it--
then you can see that this black line defined by this
load increment would mean that we have a large number of
iterations necessary.
If you need a large number of iterations, to converge to the
new equilibrium configuration.
In fact, you can see that this red curve is almost parallel
to the black line, so we can anticipate large convergence
difficulties trying to get to a solution with the load
increment being equal from here to that black line.
The important feature of the algorithm that I want to
present to you now, is that this load level is the first
load level which we start in the iteration, but then the
algorithm automatically cuts down this load level, as shown
by this arc here, by the blue arc, until this equilibrium
point is solved for.
And this means that the convergence is much increased.
In other words, the algorithm does not have great
difficulties calculating this particular equilibrium point.
And from here, of course, the same is repeated.
And like this the algorithm goes along this red
low-displacement curve, and traces this whole curve out.
Let us look a bit at the notation that I will be using,
because it's important that we get familiar
with it at this stage.
Notice t lambda r, of course, is the load at time t.
The corresponding displacement is tu.
We used that notation earlier already.
Notice that the increment in displacement from time t to
time t plus delta t is u, shown here in green.
Notice that the increment in the load from time t to time t
plus delta t is, of course, given by the change in this
load factor here.
And that change in the load factor is lambda.
Now, this lambda value is also given in this equation here.
It's buried in there.
Notice this lambda of i here.
We are writing it as a sum all of delta lambda k's We will
solve for all these delta lambda k's And this lambda i,
when i becomes very large, converges to the lambda that I
have up here.
Similarly, this ui which is written as the sum of the
delta uk's this ui converges to the u
that I'm showing here.
So, u and lambda, shown in green here, are the exact
values, the values that you want to converge to with
lambda i and ui.
Notice once again, delta lambda k, summing all these
delta lambda k's up we get lambda i, and summing all the
delta uk's up, we get ui.
We should keep that in mind for the discussion that you
want to go through just now.
The governing equations are now as follows.
On the left hand side we have a tension stiffness matrix.
In the modified Newton iteration we would have tau
equal to t.
Delta ui is our displacement increment vector.
This is here, this scaler, t plus
delta t lambda i, unknown.
This one we know.
And that one we want to calculate in the iteration i.
In other words, this is an increment that is unknown at
the beginning of the iteration.
Here we have the nodal point force vector corresponding to
the elements stresses at time t plus delta t, and at the end
of iteration i minus 1.
Notice that this is a set of linear equations in the
unknowns that are used.
There are n such unknowns, and there's one more unknown here.
So, we have n equations in n, plus 1 unknowns.
We need, therefore, one more equation.
And that equation is given by this constraint equation.
The f here on delta lambda and delta u, some constraint
equation here, gives us the additional equation that we
need for the solution these n plus 1 coupled equations.
The unknowns, of course, are these values here--
n plus 1 unknowns.
To solve these equilibrium equations, we can rewrite them
as shown here.
In other words, the equations on the finite element
displacements can be split up, if you look at the previous
view graph, directly into two sets of
equations as shown here.
The interesting point is that, this equation here does not
involve a delta lambda.
This equation here does not involve a delta lambda either.
Now, having calculated from this equation this vector, and
from that equation that vector, we can directly write
delta u, as shown here, of course, now involving the
delta lambda.
And this is a form of equation for delta u that we will be
using a little later.
And the first constraint equation I'd like to introduce
you to, the one that I briefly mentioned earlier already,
where we talked about an arc which is used to swing the
load level around, so as to get to the low displacement
curve very quickly, and that constraint equation is this
spherical constant arc-length criterion,
which is written here.
Notice here lambda i, here ui.
Of course both of these quantities create, in essence,
beta factor and delta l squared.
Delta l is set at the beginning of the load step.
Delta l, in other words, is constant
throughout the load step.
The value of delta l is chosen based on what has happened in
the previous load steps.
Having chosen delta l, you can calculate the right hand side,
and on the left hand side, we have a constraint between the
increment in lambda, and the increment in delta u.
Remember, lambda I is the sum of all of the delta lambda k's
and ui is the sum of all of the delta uk's I just showed
that on the previous view graph.
Therefore, if we substitute here, we get a constraint
equation between the delta uk, and the delta lambda k.
Well, here we have the definitions once more.
Notice beta is a normalizing factor which is applied in
order to make these terms here dimensionless.
The equation may be solved as follows.
Using that lambda is equal to this equation here.
ui is given as that.
Of course, this here can be expanded using the information
that we discussed earlier.
And substituting for this value and that value into the
constraint equation, we directly obtain a quadratic
equation in delta lambda i Remember that these two
vectors here are, of course, known.
These two vectors are known, and that one is known also,
because that was established in the previous iteration.
If we geometrically interpret this solution scheme, we find
the following.
Here we have our equilibrium point.
Here is delta lambda, which is constant throughout the
solution step.
And let's assume the use of modified
Newton-Raphson iteration.
We start off with this tangent here at this solution point,
the equilibrium point, and we iterate around this arc using
the constant stiffness matrix at each of these points.
In the first iteration, we get to this point.
Second iteration, that point.
Third iteration, that point.
And that signified also looking here to the
displacements.
Notice t plus delta t u1 is the displacement value
calculated in the first iteration.
This is the one calculated in the second iteration.
That is the one calculated in the third iteration.
We use a constant stiffness matrix here.
Of course, we could also change the stiffness matrix,
go in other words, to a stiffness matrix that has been
updated in the iteration corresponding to the current
stress and displacement conditions, and then
convergence would be more rapid.
But of course there would also be a higher cost involved,
because whenever we calculate with a new stiffness matrix,
and we triangularize that stiffness matrix, there's a
considerable cost involved in doing so.
This is one scheme, and we find this scheme quite
effective, except that when we want to calculate limit loads,
we find that near the limit load the scheme can require
quite a number of iterations.
And so, we have been looking for other schemes that might
do better in the range of the limit load.
Well, here we have one other scheme, namely the "constant"
increment of external work criterion.
We have here quotes on the constant, because we are
setting w on the right hand side in the first iteration,
as shown here, and in the next iteration then,
we have a zero here.
Notice w is a value that is calculated in each load step,
and as soon as we find that the first scheme-- and
obviously, constant arc lengths criterion scheme--
has difficulties converging, or marches too slow, then we
switch to this "constant" increment of external work
criterion, with a w from the previous step, the increment
of external work from the previous step.
We use that for the current step, and then this equation
here gives us a constraint on delta lambda 1.
Notice that this equation here means really that we are
looking at is this area here, the shaded area, being w, and
we want to have the delta lambda 1 in such a way that
this shaded area w here is equal to what we have had as
the increment of external work in the previous load step.
Well, this gives us delta lambda (1), and having
obtained delta lambda (1), we can go into the next
iterations, 2 to 3, and so on for i, and as I mentioned
already earlier, we then have a zero on the right hand side.
Otherwise the same left hand side.
And this left hand side has now two solutions.
This solution here is disregarded, because this
solution would reverse the direction of the load, and we
don't want to admit that.
We want to basically have the load increase more and more
until the total collapse of the structure is reached, and
then of course it may decrease to go into the post-buckling
response, but we don't want to have the load totally
reversed, and so this is a solution that we now use for
delta lambda (i).
Notice that for a single degree of freedom system, we
would immediately find that delta u (2) must be zero.
Why?
Because we notice that the load vector is orthogonal to
the displacement vector.
Now, for a single degree of freedom system, of course,
each of these are just numbers, and since the r is
not zero, because that is of course a prescribed value
prior to the analysis, this is not zero, delta
u (2) must be zero.
And that shows already the effectiveness, just show it a
bit, of this algorithm.
All the algorithm altogether is now as follows.
We specify r.
In other words, the analyst has to specify the load
distribution, concentrated loads, distributed loads, that
shall be dealt with in this collapse analysis, and also
the displacement at one degree of freedom corresponding to
delta t lambda.
This is done to just start the algorithm.
Once the analyst has specified the displacement at one degree
of freedom, the program can solve for delta tu--
in other words, all the other displacements--
and this delta tu corresponds to delta t lambda.
We feel that to start the algorithm it is easier for the
analyst to prescribe the displacement at one node
corresponding to delta t lambda--
Rather, to prescribe this displacement than to prescribe
the first load level, because if you don't have much of an
idea how the structure really behaves, it can be very
difficult to give a load level that is reasonable, and is not
too close to the ultimate limit load, or is already
perhaps even beyond the ultimate limit load.
So we feel that this is a good way to proceed.
For example, if you analyze the collapse of a shell, you
may want to pick a node that you know will have non-zero
displacement and assign at that node a displacement, say,
one third of the thickness of the shell.
That displacement then would give you delta t lambda, and
surely would be less than the ultimate
limit load of the shell.
This means, once you have solved for delta t lambda and
delta tu, we can set delta l.
The arc lengths for the next load steps.
And now the program does everything automatically.
It selects the increment in displacements and loads using
this arc-lengths criterion that we just discussed.
We use 1, which is the arc-lengths criterion, for the
next load steps.
We calculate w for each load step.
When w does not change appreciably, or when there are
difficulties with the arc-lengths criterion, with 1,
in other words, then we switch to our constant increment and
external work criterion.
We call that the 2 criterion.
Once again, notice that delta l is calculated in each load
step, and it is adjusted based on the number of iterations
that we have been using in the previous load steps.
In other words, if you go, say, up to load step three,
and you have, say, obtained a solution up to load step
three, the program has used a certain delta in load step
three, and now the program looks at the number of
iterations that were performed to get to the equilibrium
position at the end of load steps three.
Based on these number of iterations, the program then
cuts delta l down, or makes delta l larger, depending on
whether the iterations were very many that you used, or
were very few to get to the equilibrium configuration at
load step three.
So, the a program automatically adjusts delta l.
Also notice that the stiffness matrix is recalculated when
convergence is slow.
In fact, a full Newton-Raphson iteration is performed
automatically, the program switched automatically from
the modified Newton-Raphson iteration to the full
Newton-Raphson iteration when it is
deemed to be more effective.
This is the automatic load incrementation scheme that I
wanted to discuss with you.
And I'd now like to go over to one other topic, a very
important topic as well in the analysis of the nonlinear
response of structures, and this is the linearized
buckling analysis.
In the linearized buckling analysis, we want to predict a
collapse load, the buckling load of the structure via this
criterion here.
The determinant we know at the collapse point of the
stiffness matrix is zero.
The matrix is, another words, singular.
The criterion that we use here determined of k is equal to
zero, of course, means once again
the k matrix is singular.
And that means that we really have a solution to this set of
equations that is nontrivial.
The trivial solution, of course, would always be u star
is equal to zero, but that is not a real solution that we're
interested in.
A solution, a nontrivial solution exists to this set of
equations only in the case when tau k is singular.
And, of course, that is the same as saying that the
determined of k, tau k is equal to zero.
If tau k is singular the structure is unstable, and the
collapse load situation has been reached.
Let's look at what this means physically.
If we have here a beam.
If we look at a beam, pinned at this end, pinned at that
end, subjected to a certain load, then if the load takes
on a certain value, the smallest imbalance of the
load, a very small load this way, would immediately result
in very large displacements.
That's what the buckling criterion tells us.
In other words, at tau r, the buckling load, the structure
is unstable to any smallest load imbalance.
Of course, the material data for the structure are given.
And in this particular case, we use the data
of an elastic beam.
In the linearized buckling analysis we want to predict
the load level, and the buckling load shape
corresponding to the buckling situation, by linearizing
about a particular configuration that is not
necessarily very close to the buckling load level.
In fact, we assume that tau k is given via this relationship
here, and tau r is given via this relationship.
Notice here t minus delta tk and tk are known values, known
stiffness matrices, and similarly, this load vector,
and that load vector, and that one, are known values.
Lambda is a scaler.
So basically, what we are trying to do in the linearized
buckling analysis is to linearize about the
configuration, not necessarily very close to the collapse
load configuration, and establish with this linearized
relationship an equation from which we can calculate an
increment in load that gives us an approximation to the
actual collapse load.
Pictorially, we are doing this here.
We have here the load displacement curve in red.
We have at displacement t minus delta
tu, this load level.
At displacement tu we have that load level.
And if you linearize about these configurations here,
using the stiffness matrices corresponding to these two
load levels, we can predict a collapse load, which because
of this linearization is, of course, not exactly the
collapse load, but we hope that we are close to that
actual collapse load.
Pictorially here, we do the following.
Notice here we have the k plotted and the
lambda value plotted.
Along this axis.
At a particular value, lambda equal to 1, we
would have this point.
Now let's look first along this axis here.
Here we have tk.
Here we have t minus delta tk.
Notice tk is smaller than t minus delta tk.
We are looking here, of course, at a single degree of
freedom case again.
This means that the structure becomes softer, we are getting
closer to the collapse load.
In our linearized buckling analysis we put really through
this point and that point a straight line.
That projects us to a point, lambda 1, at which tau k is
equal to zero.
Corresponding to what we're doing here with the stiffness
matrix, we have here also picture for the loads.
Notice t minus delta tr here, tr there, and a straight line
through it brings us up to tau r.
Tau r is the collapse load.
Tau r is the collapse load, linearized collapse load,
corresponding to the value that we have here
obtained for lambda 1.
The problem of solving for lambda such that the
determinant of tau k is zero is really nothing else than an
eigenproblem.
And the eigenproblem that we have to consider there is
written down here.
Notice phi is an eigenvector, lambda is the eigenvalue--
unknown at this point--
and here we have a difference in matrices that we know, and
once again the eigenvector.
Notice one interesting point.
That to establish this difference in matrix, all you
need to do in the computer program is to store the
previous difference matrix, and subtract from the previous
difference matrix corresponding to load step t
minus delta t, the current difference matrix.
This is interesting to note that because we don't need to
separate out element difference matrices.
We don't need to talk about nonlinear strain stiffness
matrices coming from the elements, and linear strain
stiffness matrices coming from the elements.
We talked about, of course, these particular element
stiffness matrices in previous lectures.
We don't separate any out here.
In fact, what we're doing, we simply take all of the
contributions from the elements into account here,
and all of the contribution towards element into account
here, of course corresponding to time t minus delta t here,
and corresponding to time t here.
This therefore, is a very simple operation in the
computer program.
And having established this eigenvalue problem, we can
schematically look at the eigenvalues that would become
calculated.
Here we have some positive eigenvalues indicated, and
here we have some negative eigenvalues indicated.
Notice that there's a possibility of negative
eigenvalues as well.
A physical problem that shows how you can get a negative
eigenvalue is shown in this view graph.
Here you have a structure, a frame structure, so to say,
that is subjected to a compressive load
and a tensile load.
Now, notice that this compressive load would of
course initiate buckling here, and there would be a
corresponding to a positive eigenvalue.
However, this tensile load has to reverse its sign in order
for this member to buckle, and that would correspond then to
a negative eigenvalue.
Now when we try to solve eigenvalue problems that have
negative and positive eigenvalues, there can be
difficulties with the eigenvalue solution method
that you're using.
It is easier for a solution method to simply deal with
eigenvalues that are only positive.
And for that reason we are reformulating the basic
eigenvalue equation into this form.
Here you have now the eigenvalue problem simply
rewritten, really, with gamma being the eigenvalue, and phi
still being the eigenvector, gamma of course, being a
function of lambda.
With this eigenvalue problem then, we have only positive
eigenvalues on gamma, of course, and we are interested
in finding the smallest gamma value.
Because the smallest gamma value corresponds to the
smallest lambda 1 positive value.
And of course, sometimes you also want to calculate gamma 2
corresponding to the second smallest positive lambda
value, and so on.
The negative lambda values lie over here.
The value of the linearized buckling analysis can be
summarized as follows.
The buckling analysis is not very expensive.
It gives insight into possible modes of failure.
For applicability, however, the pre-buckling displacements
should be small.
And very important it is that the buckling analysis yields
modes, buckling mode shapes, that can be very effectively
used to impose imperfections onto a structure and to study
the sensitivity of the structure to imperfections.
Many structures are very sensitive, particularly shell
structures, are very sensitive to initial imperfections in
the geometry, and this linearized buckling analysis
gives us the mechanisms and means to calculate mode shapes
that we can impose onto the perfect structure as a
geometric imperfection, onto the geometry of the perfect
structure, to study then the behavior of the structure
subject to these imperfections.
But it is very important that the procedure be only employed
with great care, because the results can be quite
misleading.
The buckling loads that you might calculate in the
linearized buckling analysis, these buckling loads may be
way higher than the actual buckling loads that you should
be using in your design.
And we will actually show some examples in the next lecture
pertaining to this particular problem here.
And one should always keep in mind that the procedure really
predicts only physically realistic buckling loads when
we have structures that behave close to the
Euler column type.
In other words, that buckle with very small initial
displacements, or pre-buckling displacements, I should say.
Let's look at an example.
Here we have the example of an arch
subjected to uniform pressure.
The geometric data of the arch are given here.
The material data are given as well.
Notice it's an elastic arch.
And it's an arch with cross section b and h here, being
both equal to 1.
We also want to consider only the two-dimensional
motion of the arch.
In other words, two-dimensional action.
We do not allow out-of-plane bucking for this arch.
Now the finite element model that we select for the arch is
a model of 10 2-node isoparametric beam elements.
We will talk about these isoparametric beam elements--
we call them also isobeam elements--
in a later lecture.
And for the two-dimensional motion of the arch, we model
the complete arch.
The purpose of the analysis is to determine the collapse
mechanism, the collapse load level of the structure.
And not only this particular part, but we also want to
calculate the post-collapse response of the structure.
Let us now go through the solution that was performed
for this arch.
As a first step we calculated the linearized buckling loads
and corresponding mode shapes.
And we calculated two.
Interesting to note that the first mode corresponds to this
pressure, and it's an anti-symmetric mode.
The second buckling mode shape looks like this.
It's a symmetric buckling mode shape.
Notice that, in other words, the anti-symmetric buckling
mode corresponds to a lower load than the symmetry
buckling mode shape.
Having performed the linearized buckling analysis,
we next use our automatic load stepping algorithm to
calculate the displacement response of the arch as the
load increases.
Here on this view graph, we have plotted the pressure
vertically up here, and the displacement of the center of
the arch horizontally.
Notice this black curve here shows a computed response
using about 60 steps in the analysis.
Notice it is a collapse load, of course, here, and this is a
post-collapse response.
It's interesting to note that the collapse load predicted
using the buckling analysis and linearized buckling
analysis given by this blue line lies below this actual
ultimate load predicted for using the automatic load
stepping algorithm.
And we have to ask ourselves why is that the case?
Well, the computed response of the the perfect symmetry arch
does not allow the anti-symmetric displacements
to take place, and it is the anti-symmetric displacements
that, of course, initialize, so to say, anti-symmetric
buckling response.
We have computed the response of a perfect symmetric arch
that is subjected to a perfectly symmetric load, and
this does not allow the anti-symmetric buckling mode
of the arch to come into effect.
However, a real structure will contain imperfections, and
hence it will go into the anti-symmetric behavior, and
the actual collapse load will be below the one that we have
predicted in our first analysis using the automatic
load stepping algorithm, and the results of which
I've just shown you.
Therefore, to really obtain a realistic collapse load, we
have to now impose onto the perfect symmetry arch a
geometric imperfection which allows the anti-symmetric
behavior to take place.
And we do so by adding to the geometry of the arch to the
nodal point coordinates of the arch a multiple of the
anti-symmetric mode shape.
This collapse mode is scaled so that the magnitude of
imperfection is less than 100.
This resulting imperfect arch is, of course, no longer
symmetric, and we now solve for the response of that arch
using our automatic load stepping algorithm again.
This is the response that you'll see.
Pressure and displacement of the arch.
Notice this is well below the linearized buckling analysis
solution, the blue solution, that I showed you on the
earlier response graph.
And this is here a much more realistic estimate for the
collapse load of the arch.
This is the realistic estimate that you would be using, for
example, in the design of the actual structure.
It's also interesting to look at the deflective shape of the
structure, and here we have the deflective shape of the
perfect arch, of being symmetric at this
displacement value.
And here we have a deflective shape of the imperfect arch at
this particular displacement value.
Of course, this one is not a totally
symmetric deflective shape.
Well, we have discussed now quite a number of solution
schemes that we use to solve the finite element equations
in static non-linear analysis.
We have looked in the previous lecture at some schemes-- the
full Newton-Raphson method, the modified Newton-Raphson
method, the BFGS method, the initial stress method--
in which we have to prescribe the load levels.
For each load step we have to prescribe the load level prior
to the analysis, by the input of the computer program, and
in this lecture I have shared with you some experiences
regarding an automatic load stepping scheme regarding
linearized buckling analysis.
What we have not done yet, really, in my opinion is to
look at enough examples, and that's what I would like to do
in the next lecture.
Thank you very much for your attention.