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PROFESSOR: Ladies and gentlemen,
welcome to this
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lecture on nonlinear finite
element analysis of solids and
00:00:26.840 --> 00:00:28.150
structures.
00:00:28.150 --> 00:00:31.050
In the previous lectures, we
talked about the general
00:00:31.050 --> 00:00:33.740
incremental continuum mechanics
equations that we're
00:00:33.740 --> 00:00:36.850
using in nonlinear finite
element analysis.
00:00:36.850 --> 00:00:40.220
In this lecture, I would like,
now, to talk about the finite
00:00:40.220 --> 00:00:43.570
element matrices that we're
using, actually, in static and
00:00:43.570 --> 00:00:45.160
dynamic analysis.
00:00:45.160 --> 00:00:49.000
You want to talk about these
in quite general terms.
00:00:49.000 --> 00:00:51.510
In the next lecture then, we
will talk more about the
00:00:51.510 --> 00:00:53.520
details of these matrices.
00:00:53.520 --> 00:00:56.960
You want to formulate these
finite element matrices.
00:00:56.960 --> 00:01:00.690
And we want to talk about the
numerical integration that we
00:01:00.690 --> 00:01:04.170
use to actually evaluate
the matrices.
00:01:04.170 --> 00:01:07.460
I've prepared some view graphs
regarding this material.
00:01:07.460 --> 00:01:09.700
And I'd like to share the
information on these view
00:01:09.700 --> 00:01:10.990
graphs now with you.
00:01:13.780 --> 00:01:17.000
The derivation of the finite
element matrices, of course,
00:01:17.000 --> 00:01:20.670
is based on the continuum
mechanics equations that we
00:01:20.670 --> 00:01:22.340
have developed earlier.
00:01:22.340 --> 00:01:25.150
And you have seen this equation
earlier when we
00:01:25.150 --> 00:01:27.790
talked about the total
Lagrangian formulation, the
00:01:27.790 --> 00:01:29.530
T.L. formulation.
00:01:29.530 --> 00:01:32.970
We talked about the terms
in this equation.
00:01:32.970 --> 00:01:36.810
We talked about their meaning,
what they stand for.
00:01:36.810 --> 00:01:40.400
And we have derived certain
terms here because they have
00:01:40.400 --> 00:01:43.850
been obtained by linearization
process.
00:01:43.850 --> 00:01:45.910
This was the governing equation
for the total
00:01:45.910 --> 00:01:48.730
Lagrangian formulation.
00:01:48.730 --> 00:01:51.690
The governing continuum
mechanics equation for the
00:01:51.690 --> 00:01:54.790
updated Lagrangian formulation
is shown here.
00:01:54.790 --> 00:01:58.490
Once again, we talked about
each of these terms.
00:01:58.490 --> 00:02:00.980
We talked about what
their meaning is.
00:02:00.980 --> 00:02:03.560
And, of course, we arrived
at this equation by
00:02:03.560 --> 00:02:04.810
linearization process.
00:02:07.310 --> 00:02:09.889
For the total Lagrangian and
the updated Lagrangian
00:02:09.889 --> 00:02:13.660
formulation, we recognized
because of the linearization
00:02:13.660 --> 00:02:17.310
process that was involved, we
need to iterate to obtain an
00:02:17.310 --> 00:02:20.510
accurate solution to the actual
problem of interest.
00:02:20.510 --> 00:02:23.890
And here on this view graph,
I've summarized the iterative
00:02:23.890 --> 00:02:29.300
equations that are used in the
total Lagrangian formulation.
00:02:29.300 --> 00:02:33.030
We, once again, have seen
this equation before.
00:02:33.030 --> 00:02:35.460
We should recognize,
now, these terms.
00:02:35.460 --> 00:02:38.890
The iteration count, of course,
being k, as shown here
00:02:38.890 --> 00:02:40.800
on the left-hand side
of the equation.
00:02:40.800 --> 00:02:41.960
And k minus one on the
00:02:41.960 --> 00:02:44.320
right-hand side of the equation.
00:02:44.320 --> 00:02:48.000
We also recognize this
equation here, now.
00:02:48.000 --> 00:02:52.370
Namely, the equation where we
update the displacements from
00:02:52.370 --> 00:02:56.440
the k minus first iteration
to the k iteration.
00:02:56.440 --> 00:02:59.640
For this iteration, of course,
we need initial conditions.
00:02:59.640 --> 00:03:05.240
And those initial conditions we
also talked about earlier.
00:03:05.240 --> 00:03:07.790
A similar equation is used
in the updated Lagrangian
00:03:07.790 --> 00:03:09.040
formulation.
00:03:11.070 --> 00:03:14.990
These are the terms that we
talked about earlier.
00:03:14.990 --> 00:03:18.500
And, of course, once again, we
are iterating with iteration
00:03:18.500 --> 00:03:22.100
count of k on the left-hand
side, iteration count of k
00:03:22.100 --> 00:03:25.190
minus one on the right-hand
side of the equation.
00:03:25.190 --> 00:03:27.600
The displacements are
updated, as shown
00:03:27.600 --> 00:03:29.740
here in this equation.
00:03:29.740 --> 00:03:32.810
And for this iteration, of
course, we need initial
00:03:32.810 --> 00:03:35.240
conditions, which
are given here.
00:03:35.240 --> 00:03:38.710
And all of these terms, really,
we discussed in some
00:03:38.710 --> 00:03:43.490
detail in the earlier
lectures.
00:03:43.490 --> 00:03:48.430
Well, assuming that loading is
deformation-independent, we
00:03:48.430 --> 00:03:53.580
also recognize that this here
is the expression for the
00:03:53.580 --> 00:03:55.430
external virtual work.
00:03:55.430 --> 00:03:58.650
t plus theta tR, the script
R that I talked
00:03:58.650 --> 00:03:59.660
about earlier already.
00:03:59.660 --> 00:04:05.860
And these are the terms that
we already defined earlier.
00:04:05.860 --> 00:04:11.350
Particularly the body force
term, the surface force term.
00:04:11.350 --> 00:04:15.910
Notice that, so far, we really
talked about static analysis.
00:04:15.910 --> 00:04:19.180
And I'd like to now, of course,
also introduce the
00:04:19.180 --> 00:04:21.339
term that we use in
dynamic analysis.
00:04:21.339 --> 00:04:23.810
This, of course, is
the inertia term.
00:04:23.810 --> 00:04:27.530
And in dynamic analysis, this
is the term that is really
00:04:27.530 --> 00:04:32.330
contained in this term here
as an effect of fV.
00:04:32.330 --> 00:04:37.850
Of the volume forces, the
forces per unit volume.
00:04:37.850 --> 00:04:41.580
Notice that this is really
here an integral over the
00:04:41.580 --> 00:04:46.600
volume at time t plus theta t of
the mass density at time t
00:04:46.600 --> 00:04:50.490
plus theta t times the
accelerations at time t plus
00:04:50.490 --> 00:04:55.690
theta t, the virtual
displacements, and, of course,
00:04:55.690 --> 00:04:58.180
the integrating over the
volume at time t plus
00:04:58.180 --> 00:04:58.720
[? theta, ?]
00:04:58.720 --> 00:05:01.610
t plus theta t, as
I said already.
00:05:01.610 --> 00:05:05.870
This is, of course, an
"inconvenient" interval.
00:05:05.870 --> 00:05:10.970
Inconvenient in quotes because
we don't know this volume.
00:05:10.970 --> 00:05:14.610
So we cannot really directly
evaluate this integral.
00:05:14.610 --> 00:05:17.820
Instead we'd rather like to work
with the original volume
00:05:17.820 --> 00:05:20.150
like we do in the total
Lagrangian formulation.
00:05:20.150 --> 00:05:25.140
And transformation of this
integral to an integral over
00:05:25.140 --> 00:05:28.320
the original volume is achieved
by this equation
00:05:28.320 --> 00:05:29.720
here, by this term here.
00:05:29.720 --> 00:05:37.230
Notice that, of course, 0 rho
times 0 dV for a particular
00:05:37.230 --> 00:05:41.920
set of mass particles must be
equal to t plus theta t rho
00:05:41.920 --> 00:05:44.260
times t plus theta tdV.
00:05:44.260 --> 00:05:47.770
And this is the reason why we
can directly write down this
00:05:47.770 --> 00:05:49.150
equation here.
00:05:49.150 --> 00:05:52.770
However, considering this
transformation for finite
00:05:52.770 --> 00:05:57.060
element analysis, it is
important to realize that we
00:05:57.060 --> 00:06:01.990
assume here that the
possibilities of the motion,
00:06:01.990 --> 00:06:08.630
of the material particles as
contained in the finite
00:06:08.630 --> 00:06:13.030
element interpolations
are the same in this
00:06:13.030 --> 00:06:16.990
volume as in that volume.
00:06:16.990 --> 00:06:19.970
This fortunately, is true
in isoparametric
00:06:19.970 --> 00:06:21.600
finite element analysis.
00:06:21.600 --> 00:06:25.450
And therefore, this
transformation from the volume
00:06:25.450 --> 00:06:31.250
t plus theta t to the volume
V 0 is a very convenient
00:06:31.250 --> 00:06:37.520
transformation to perform for
the evaluation of this term.
00:06:37.520 --> 00:06:40.400
We use that abundantly
in isoparametric
00:06:40.400 --> 00:06:43.500
finite element analysis.
00:06:43.500 --> 00:06:48.590
If the external loads are
deformation-dependent, then we
00:06:48.590 --> 00:06:51.300
have to recognize that
these forces here are
00:06:51.300 --> 00:06:56.780
deformation-dependent, and
therefore, we have to evaluate
00:06:56.780 --> 00:06:58.950
them in the iteration.
00:06:58.950 --> 00:07:02.420
And that is being shown
here in this equation.
00:07:02.420 --> 00:07:06.560
That we are always evaluating
this term new, depending on
00:07:06.560 --> 00:07:12.150
the iteration k minus 1, we
integrate this product here
00:07:12.150 --> 00:07:14.790
over the volume at time
t plus theta t in
00:07:14.790 --> 00:07:16.770
iteration k minus 1.
00:07:16.770 --> 00:07:19.920
Similarly, we update, also,
the surface forces.
00:07:19.920 --> 00:07:24.190
Since the surface area changes
of the body during the last
00:07:24.190 --> 00:07:28.450
information process, we evaluate
this part here as
00:07:28.450 --> 00:07:29.210
shown here.
00:07:29.210 --> 00:07:31.910
Of course, there is an
approximation involved.
00:07:31.910 --> 00:07:37.790
But if we keep on it iterating
t plus theta tVk minus one,
00:07:37.790 --> 00:07:40.560
we'll take on the volume
t plus theta tV.
00:07:40.560 --> 00:07:44.820
And of course then, we really
include here the term that we
00:07:44.820 --> 00:07:47.820
want to include, namely that
one in the analysis.
00:07:47.820 --> 00:07:48.840
Similarly here.
00:07:48.840 --> 00:07:53.610
The surface area, in iteration
k minus 1, as k goes larger
00:07:53.610 --> 00:07:54.460
and larger.
00:07:54.460 --> 00:07:58.560
If we converge, we'll actually
be equal to that surface area,
00:07:58.560 --> 00:08:02.000
meaning that this expression is
equal to that expression,
00:08:02.000 --> 00:08:04.190
which of course we
want to include.
00:08:04.190 --> 00:08:06.110
It's this expression that
we want to include
00:08:06.110 --> 00:08:07.360
in the force vector.
00:08:09.880 --> 00:08:13.890
We, in one of the very early
lectures, talked about the
00:08:13.890 --> 00:08:16.980
materially-nonlinear-only
analysis.
00:08:16.980 --> 00:08:20.870
And the equation that is used,
the continuum mechanics
00:08:20.870 --> 00:08:22.590
equation that is used in the
materially-nonlinear-only
00:08:22.590 --> 00:08:25.530
analysis is given here.
00:08:25.530 --> 00:08:33.000
Notice, here we have a stress
strain law tensor, an
00:08:33.000 --> 00:08:37.880
incremental strain tensor,
the virtual strain.
00:08:37.880 --> 00:08:39.799
The incremental strain
tensor is the real
00:08:39.799 --> 00:08:41.049
strain tensor here.
00:08:41.049 --> 00:08:47.040
All the components of that
tensor have a superscript k.
00:08:47.040 --> 00:08:48.370
The external virtual work.
00:08:48.370 --> 00:08:51.840
And here, we have the stress
tensor, components of the
00:08:51.840 --> 00:08:52.990
stress tensor.
00:08:52.990 --> 00:08:56.370
At time t plus theta t in
iteration k minus 1.
00:08:56.370 --> 00:08:59.190
And the virtual strains again.
00:08:59.190 --> 00:09:03.120
Notice here now, we do not have
anymore subscripts, 0 or
00:09:03.120 --> 00:09:10.090
t, on these components because
in materially-nonlinear-only
00:09:10.090 --> 00:09:13.750
analysis, we assume that the
deformations are very small.
00:09:13.750 --> 00:09:16.650
The displacements are
infinitesimal.
00:09:16.650 --> 00:09:19.610
And the stress, the second
Piola-Kirchhoff stress, which
00:09:19.610 --> 00:09:22.670
we defined in an earlier lecture
is actually equal to
00:09:22.670 --> 00:09:25.930
the Cauchy stress under those
conditions and is equal to the
00:09:25.930 --> 00:09:29.540
physical stress that we are
talking about here in the
00:09:29.540 --> 00:09:31.640
materially-nonlinear-only
analysis.
00:09:31.640 --> 00:09:35.290
In other words, both those
stress measures are equal to
00:09:35.290 --> 00:09:38.870
the physical stress that appears
here, which is, of
00:09:38.870 --> 00:09:41.520
course, the force
per unit area.
00:09:41.520 --> 00:09:43.860
The one that we are so familiar
with in infinitesimal
00:09:43.860 --> 00:09:45.110
displacement analysis.
00:09:48.640 --> 00:09:51.350
Let us look further at
dynamic analysis.
00:09:51.350 --> 00:09:54.430
Dynamic analysis is generally
performed in nonlinear
00:09:54.430 --> 00:09:58.790
analysis, using an implicit time
integration scheme or an
00:09:58.790 --> 00:10:00.230
explicit time integration
scheme.
00:10:00.230 --> 00:10:03.400
And in a later lecture, we
will discuss such time
00:10:03.400 --> 00:10:05.070
integration schemes.
00:10:05.070 --> 00:10:09.010
In implicit time integration,
we look at the equilibrium
00:10:09.010 --> 00:10:13.080
equation at time t plus theta
t to obtain the solution at
00:10:13.080 --> 00:10:14.980
time t plus theta t.
00:10:14.980 --> 00:10:19.230
And this means that we will
have to evaluate this
00:10:19.230 --> 00:10:23.040
left-hand side written here as
given on the right-hand side.
00:10:23.040 --> 00:10:26.550
This part here, this external
virtual work, comes from the
00:10:26.550 --> 00:10:31.590
external loads that
are not the mass
00:10:31.590 --> 00:10:33.180
of the inertia forces.
00:10:33.180 --> 00:10:36.220
The inertia forces are taken
care of via this part, via
00:10:36.220 --> 00:10:39.520
this integral here,
which I discussed
00:10:39.520 --> 00:10:41.090
already a bit earlier.
00:10:41.090 --> 00:10:46.180
In explicit time integration,
we are evaluating or we're
00:10:46.180 --> 00:10:49.940
looking at the equilibrium
equation at time t to obtain
00:10:49.940 --> 00:10:52.020
the solution at time
t plus theta t.
00:10:52.020 --> 00:10:54.190
Quite different from implicit
time integration.
00:10:54.190 --> 00:10:57.170
And the governing equations are
in the total Lagrangian
00:10:57.170 --> 00:11:02.580
formulation right here, in the
U.L., the updated Lagrangian
00:11:02.580 --> 00:11:07.680
formulation given here, and in
the materially-nonlinear-only
00:11:07.680 --> 00:11:10.960
analysis as given here.
00:11:10.960 --> 00:11:15.030
Notice, of course, that R will
involve the inertia forces
00:11:15.030 --> 00:11:17.630
evaluated at time t now.
00:11:17.630 --> 00:11:19.880
And that will, of course, enable
us to march forward
00:11:19.880 --> 00:11:22.940
with a solution as we
will discuss later
00:11:22.940 --> 00:11:25.860
on in another lecture.
00:11:25.860 --> 00:11:28.930
The finite element equations
corresponding to these
00:11:28.930 --> 00:11:32.660
continuum mechanics equations
look as follows.
00:11:32.660 --> 00:11:36.390
In materially-nonlinear-only
analysis, let's look first at
00:11:36.390 --> 00:11:37.540
static analysis.
00:11:37.540 --> 00:11:40.190
We have this basic equation.
00:11:40.190 --> 00:11:43.960
Notice a tension stiffness
matrix that does not carry any
00:11:43.960 --> 00:11:49.660
subscript 0 or t because we are
talking about the original
00:11:49.660 --> 00:11:53.930
volume, the displacements being
infinitesimally small,
00:11:53.930 --> 00:11:58.390
the physical stress, no Cauchy
stress really, no second
00:11:58.390 --> 00:12:00.330
Piola-Kirchhoff stress needs
to be introduced.
00:12:00.330 --> 00:12:02.670
We just talk about the physical
stress that we are so
00:12:02.670 --> 00:12:06.190
familiar with in infinitesimal
displacement analysis.
00:12:06.190 --> 00:12:11.490
And that, of course, goes into
the evaluation of the K matrix
00:12:11.490 --> 00:12:14.260
because the material law will
appear in here, the
00:12:14.260 --> 00:12:16.760
incremental displacement vector,
and on the right-hand
00:12:16.760 --> 00:12:21.320
side, the load vector, and the
nodal point forces that are
00:12:21.320 --> 00:12:25.985
equivalent in the sense of the
principle of virtual work to
00:12:25.985 --> 00:12:28.230
the current element stresses.
00:12:28.230 --> 00:12:31.690
By current, I mean F time t
plus theta t at the end of
00:12:31.690 --> 00:12:34.080
iteration i minus 1.
00:12:34.080 --> 00:12:36.400
This equation, of course, looks
very much alike of what
00:12:36.400 --> 00:12:38.772
we have seen in the updated
Lagrangian formulation and in
00:12:38.772 --> 00:12:41.340
the total Lagrangian
formulation.
00:12:41.340 --> 00:12:45.280
In dynamic analysis using
implicit time integration,
00:12:45.280 --> 00:12:47.380
this will be the governing
equation.
00:12:47.380 --> 00:12:51.960
Now, the mass matrix, the
acceleration vector with an
00:12:51.960 --> 00:12:54.530
iteration [? count of ?] i
because we are marching
00:12:54.530 --> 00:12:58.310
forward to a situation, to a
configuration which is still
00:12:58.310 --> 00:13:03.550
unknown, a tangent stiffness
matrix, the same tangent
00:13:03.550 --> 00:13:06.800
stiffness matrix that we see
here, by the way, the
00:13:06.800 --> 00:13:09.800
incremental displacement vector,
and on the right-hand
00:13:09.800 --> 00:13:15.020
side, the same quantities
that we have here.
00:13:15.020 --> 00:13:18.750
In dynamic analysis using
explicit time integration, we
00:13:18.750 --> 00:13:22.290
have the mass matrix, the same
mass matrix, generally, that
00:13:22.290 --> 00:13:23.600
we have here.
00:13:23.600 --> 00:13:26.800
Although in implicit time
integration, we will sometimes
00:13:26.800 --> 00:13:29.020
use a lump mass matrix,
sometimes a banded mass
00:13:29.020 --> 00:13:31.380
matrix, a consistent
mass matrix.
00:13:31.380 --> 00:13:33.780
With an explicit time
integration, we generally use
00:13:33.780 --> 00:13:36.160
only the lump mass matrix.
00:13:36.160 --> 00:13:39.210
But otherwise, the same
kind of matrix here.
00:13:39.210 --> 00:13:41.170
The acceleration vector.
00:13:41.170 --> 00:13:45.960
The force vector or the nodal
point forces, corresponding to
00:13:45.960 --> 00:13:49.320
time t that are externally
applied to the structure.
00:13:49.320 --> 00:13:53.360
And the tF vector, which is the
force vector corresponding
00:13:53.360 --> 00:13:57.920
to the stresses in the
elements at time t.
00:13:57.920 --> 00:14:01.360
Notice, we are looking here at
equilibrium at time t to
00:14:01.360 --> 00:14:03.760
obtain the solution at
time t plus theta t.
00:14:03.760 --> 00:14:06.770
We are looking here at
equilibrium, or I should say,
00:14:06.770 --> 00:14:09.150
we're iterating for equilibrium
at time
00:14:09.150 --> 00:14:09.930
t plus theta t.
00:14:09.930 --> 00:14:13.370
And thus, we will obtain
the solution for time
00:14:13.370 --> 00:14:14.620
t plus theta t.
00:14:17.650 --> 00:14:21.140
In the total Lagrangian
formulation, we have very
00:14:21.140 --> 00:14:22.280
similar equations.
00:14:22.280 --> 00:14:27.270
In static analysis, a tangent
stiffness matrix, and
00:14:27.270 --> 00:14:30.540
otherwise, the same kind of
vectors that I talked about
00:14:30.540 --> 00:14:32.000
earlier already.
00:14:32.000 --> 00:14:37.160
Notice now, of course, we
have the subscript 0.
00:14:37.160 --> 00:14:40.080
Notice also that the total
tangent stiffness matrix is
00:14:40.080 --> 00:14:44.310
made up of a part that we might
call a linear strain
00:14:44.310 --> 00:14:48.320
stiffness matrix and a part
that we may call nonlinear
00:14:48.320 --> 00:14:50.330
strain stiffness matrix.
00:14:50.330 --> 00:14:54.430
This is also called the
geometric stiffness matrix.
00:14:54.430 --> 00:14:56.840
We will talk about how
we construct these
00:14:56.840 --> 00:14:58.610
matrices just now.
00:14:58.610 --> 00:15:02.810
In dynamic analysis, we
proceed as in the
00:15:02.810 --> 00:15:05.330
material-nonlinear-only
formulation, of course.
00:15:05.330 --> 00:15:07.730
And in all the other
formulations, if we use
00:15:07.730 --> 00:15:09.630
implicit time integration,
we apply
00:15:09.630 --> 00:15:11.165
the equilibrium equation.
00:15:11.165 --> 00:15:15.170
We look for the equilibrium
at time t plus theta t, as
00:15:15.170 --> 00:15:16.800
expressed here.
00:15:16.800 --> 00:15:20.850
And we have vectors, matrices
very similar to what we have
00:15:20.850 --> 00:15:24.320
in the material-nonlinear-only
formulation except that once
00:15:24.320 --> 00:15:28.000
again, of course, we have now
introduced here the geometric
00:15:28.000 --> 00:15:29.370
stiffening affect,
the nonlinear
00:15:29.370 --> 00:15:33.000
strain stiffness matrix.
00:15:33.000 --> 00:15:36.320
In dynamic analysis using
explicit time integration, we
00:15:36.320 --> 00:15:37.980
don't use any K matrix.
00:15:37.980 --> 00:15:41.080
We will discuss that also
much more later on.
00:15:41.080 --> 00:15:43.060
And this is the governing
equation.
00:15:43.060 --> 00:15:45.270
Very much the same as in the
material-nonlinear-only
00:15:45.270 --> 00:15:46.520
formulation.
00:15:49.030 --> 00:15:53.190
Finally, in the updated
Lagrangian formulation, we
00:15:53.190 --> 00:15:56.990
also obtain similar equations
and static analysis.
00:15:56.990 --> 00:16:00.290
These are the equations
that we want to solve.
00:16:00.290 --> 00:16:03.490
Notice also, the nonlinear
strain to them there or the
00:16:03.490 --> 00:16:06.870
nonlinear strain stiffness
matrix.
00:16:06.870 --> 00:16:11.940
Otherwise, the matrices and
vectors are very much alike
00:16:11.940 --> 00:16:13.620
what we have seen before.
00:16:13.620 --> 00:16:19.050
Dynamic analysis implicit time
integration and in dynamic
00:16:19.050 --> 00:16:22.180
analysis using explicit
time integration.
00:16:22.180 --> 00:16:25.500
Notice that in each of these, we
are always cutting out the
00:16:25.500 --> 00:16:32.070
subscript t or t plus theta t
the way I have been talking
00:16:32.070 --> 00:16:35.350
about earlier already.
00:16:35.350 --> 00:16:38.710
We have seen this equation,
of course, before
00:16:38.710 --> 00:16:40.510
in our earlier lectures.
00:16:40.510 --> 00:16:43.940
What we now do is we introduce
the mass terms.
00:16:43.940 --> 00:16:46.760
And we are talking about
implicit time integration and
00:16:46.760 --> 00:16:48.620
explicit time integration.
00:16:48.620 --> 00:16:52.410
We should note that these
equations are valid for single
00:16:52.410 --> 00:16:53.900
finite element as
well as for an
00:16:53.900 --> 00:16:55.410
assemblage of finite elements.
00:16:55.410 --> 00:16:59.240
If we have a large number of
elements, then, of course, we
00:16:59.240 --> 00:17:03.480
would assemble these as we do it
in linear elastic analysis
00:17:03.480 --> 00:17:06.180
using the direct stiffness
method.
00:17:06.180 --> 00:17:09.260
Considering an assemblage of
elements, we will see that
00:17:09.260 --> 00:17:12.085
different formulations may be
used in different regions of
00:17:12.085 --> 00:17:13.220
the structure.
00:17:13.220 --> 00:17:17.030
In other words, schematically
here we may have some elements
00:17:17.030 --> 00:17:19.410
that are governed by the T.L.,
the total Lagrangian
00:17:19.410 --> 00:17:21.940
formulation, some others by
the updated Lagrangian
00:17:21.940 --> 00:17:24.839
formulation, and some others by
the material-nonlinear-only
00:17:24.839 --> 00:17:25.970
formulation.
00:17:25.970 --> 00:17:29.040
Notice that compatibility
between these elements is, of
00:17:29.040 --> 00:17:33.330
course, perfectly preserved if
these are compatible elements
00:17:33.330 --> 00:17:34.670
as shown here.
00:17:34.670 --> 00:17:37.810
Then, there is nothing wrong
with using the U.L.
00:17:37.810 --> 00:17:41.990
formulation for certain elements
that are bordering
00:17:41.990 --> 00:17:45.540
elements with another
formulation such as this.
00:17:45.540 --> 00:17:48.340
It is not true, for example,
that due to the fact that
00:17:48.340 --> 00:17:51.860
you're using here different
kinds of formulations, you are
00:17:51.860 --> 00:17:54.010
getting an incompatibility
introduced here.
00:17:54.010 --> 00:17:55.530
I've heard that sometimes.
00:17:55.530 --> 00:17:57.610
That is certainly not
my understanding
00:17:57.610 --> 00:17:58.720
of the subject matter.
00:17:58.720 --> 00:18:02.260
It does not matter whether
you have the same kind of
00:18:02.260 --> 00:18:04.610
formulations or two different
kind of formulations.
00:18:04.610 --> 00:18:05.620
That will not effect the
00:18:05.620 --> 00:18:08.850
compatibility between two elements.
00:18:08.850 --> 00:18:12.480
Let us now concentrate on
the derivation or on the
00:18:12.480 --> 00:18:16.260
formulation of a single
element matrix.
00:18:16.260 --> 00:18:19.000
To obtain a single element
matrices, we have to
00:18:19.000 --> 00:18:21.480
introduce, of course, an
interpolation matrix.
00:18:21.480 --> 00:18:24.460
And this matrix interpolates
the internal element
00:18:24.460 --> 00:18:28.440
displacements via the nodal
point displacements.
00:18:28.440 --> 00:18:32.390
Here, I'm showing a full node
element with these nodal point
00:18:32.390 --> 00:18:34.730
displacements.
00:18:34.730 --> 00:18:37.350
Notice these are measured in
the Cartesian coordinate
00:18:37.350 --> 00:18:39.170
directions.
00:18:39.170 --> 00:18:42.240
Notice however, that these nodal
point displacements are
00:18:42.240 --> 00:18:44.470
measured into a skewed
direction, an a
00:18:44.470 --> 00:18:46.660
b coordinate system.
00:18:46.660 --> 00:18:50.620
There's nothing wrong with
introducing different systems
00:18:50.620 --> 00:18:56.370
because this is Q system at the
different nodal points.
00:18:56.370 --> 00:19:01.630
The nodal point vector, the
vector of nodal point degrees
00:19:01.630 --> 00:19:03.720
of freedom is listed here.
00:19:03.720 --> 00:19:08.800
Notice that vector carries a hat
to denote the fact that it
00:19:08.800 --> 00:19:13.800
contains this great nodal
point displacements.
00:19:13.800 --> 00:19:17.950
Notice also, that the subscripts
1, 2 refer to the
00:19:17.950 --> 00:19:21.060
Cartesian coordinate
directions for the
00:19:21.060 --> 00:19:25.030
superscripts referred
to the nodal point.
00:19:25.030 --> 00:19:26.730
Here 1, 2 again.
00:19:26.730 --> 00:19:29.670
3 denoting the nodal point.
00:19:29.670 --> 00:19:34.890
Notice up here, a b, of course
refer to the skewed
00:19:34.890 --> 00:19:36.460
directions.
00:19:36.460 --> 00:19:41.390
And the 1 refers to
the nodal point 1.
00:19:41.390 --> 00:19:44.230
We want to interpolate the
internal displacements in
00:19:44.230 --> 00:19:46.280
terms of the nodal point
displacements.
00:19:46.280 --> 00:19:50.480
And that is being achieved by
this relationship here.
00:19:50.480 --> 00:19:56.230
U, the internal particle
displacements are given via H,
00:19:56.230 --> 00:20:00.370
the displacement interpolation
matrix times U hat.
00:20:00.370 --> 00:20:04.240
U hat being this vector, the
one we just discussed.
00:20:04.240 --> 00:20:08.260
U being a vector of these
two displacements.
00:20:08.260 --> 00:20:11.460
Now, notice that these two
displacements, of course,
00:20:11.460 --> 00:20:14.560
depend on which particle
you are looking at.
00:20:14.560 --> 00:20:16.890
Here, a particular particle.
00:20:16.890 --> 00:20:18.620
This would be the displacement
you want.
00:20:18.620 --> 00:20:20.420
That's the displacement U2.
00:20:20.420 --> 00:20:24.200
And these displacements vary
over the element, which would
00:20:24.200 --> 00:20:27.070
be expressed by this H matrix.
00:20:27.070 --> 00:20:29.000
These are the nodal point
displacements.
00:20:29.000 --> 00:20:31.970
These are the varying,
continuously varying
00:20:31.970 --> 00:20:35.640
displacements of the particles
within the element.
00:20:35.640 --> 00:20:40.180
We will, of course, use this
kind of relationship now quite
00:20:40.180 --> 00:20:41.270
extensively.
00:20:41.270 --> 00:20:44.270
We have, of course, also already
encountered this
00:20:44.270 --> 00:20:48.130
relationship in linear
elastic analysis.
00:20:48.130 --> 00:20:50.940
Let us now see how we
are formulating
00:20:50.940 --> 00:20:52.490
the different matrices.
00:20:52.490 --> 00:20:56.070
For all analysis types, in
which we want to include
00:20:56.070 --> 00:21:02.880
inertia forces, we evaluate this
integral as shown here.
00:21:02.880 --> 00:21:08.660
Notice the displacements and
accelerations are interpolated
00:21:08.660 --> 00:21:11.460
via this relationship here.
00:21:11.460 --> 00:21:13.550
For the accelerations, of
course, we would have dots
00:21:13.550 --> 00:21:16.810
here, dots there denoting
second time derivatives.
00:21:16.810 --> 00:21:20.260
And we would have t plus theta
t as a superscript on each of
00:21:20.260 --> 00:21:22.210
these variables.
00:21:22.210 --> 00:21:25.070
Notice that we also use this
interpolation here for the
00:21:25.070 --> 00:21:26.640
virtual displacements.
00:21:26.640 --> 00:21:32.050
And the result is given right
here just like in linear
00:21:32.050 --> 00:21:33.300
elastic analysis.
00:21:35.620 --> 00:21:41.000
The right inside load vector
as evaluated is shown here.
00:21:41.000 --> 00:21:45.630
Once again, we introduce
interpolation for U or for the
00:21:45.630 --> 00:21:50.120
virtual displacements and the
virtual surface displacements.
00:21:50.120 --> 00:21:52.910
This interpolation here
gives us this H
00:21:52.910 --> 00:21:55.920
matrix, H transpose matrix.
00:21:55.920 --> 00:22:01.830
This here interpolated gives
us the HST matrix.
00:22:01.830 --> 00:22:06.300
Notice that HS is the
interpolation matrix for the
00:22:06.300 --> 00:22:11.460
surface displacements
as a function of--
00:22:11.460 --> 00:22:16.080
or rather it gives the surface
displacements I should say as
00:22:16.080 --> 00:22:18.630
a function of the nodal
point displacements.
00:22:18.630 --> 00:22:23.840
So HS is really evaluated by
using H, the H matrix I just
00:22:23.840 --> 00:22:27.760
talked about, and evaluating
that H matrix on the surface
00:22:27.760 --> 00:22:29.460
of the element.
00:22:29.460 --> 00:22:31.530
That is how you get HS.
00:22:31.530 --> 00:22:33.970
And all of this expression
together
00:22:33.970 --> 00:22:35.510
gives us the load vector.
00:22:35.510 --> 00:22:39.320
As a matter of fact it is
really the same process
00:22:39.320 --> 00:22:42.020
followed that we are using
in linear infinitesimally
00:22:42.020 --> 00:22:45.100
displacement analysis.
00:22:45.100 --> 00:22:49.270
In material-nonlinear-only
analysis, considering an
00:22:49.270 --> 00:22:53.670
incremental displacement UI, we
evaluate this integral here
00:22:53.670 --> 00:22:56.340
as shown here.
00:22:56.340 --> 00:23:00.480
Notice here the virtual
displacements that are coming
00:23:00.480 --> 00:23:03.330
in because we have the virtual
strains there.
00:23:03.330 --> 00:23:06.800
Notice here the real
displacements, which are
00:23:06.800 --> 00:23:10.420
coming in from these
real strains.
00:23:10.420 --> 00:23:12.750
Of course, these are strain
increments and,
00:23:12.750 --> 00:23:15.830
correspondingly, displacement
increments.
00:23:15.830 --> 00:23:18.190
Nodal point displacement
increments always.
00:23:18.190 --> 00:23:25.620
These B matrices, BL matrices,
are obtained by evaluating the
00:23:25.620 --> 00:23:29.320
strains from the nodal
point displacements.
00:23:29.320 --> 00:23:32.040
And, of course, the
interpolation that is used for
00:23:32.040 --> 00:23:33.980
the element goes in here.
00:23:33.980 --> 00:23:36.750
The B matrix, of course,
contains derivatives of the
00:23:36.750 --> 00:23:40.650
elements of the H matrix.
00:23:40.650 --> 00:23:45.410
A vector containing components
of eIj is this one here.
00:23:45.410 --> 00:23:47.250
For example, in two dimensional
plane stress
00:23:47.250 --> 00:23:51.050
analysis, the entries in this
vector are listed right here.
00:23:51.050 --> 00:23:57.420
Notice that there is a 2 here
because e12 is equal to e21.
00:23:57.420 --> 00:23:59.720
And the sum of these two,
of course, gives us
00:23:59.720 --> 00:24:01.450
a total sheer strain.
00:24:01.450 --> 00:24:07.110
And therefore, we simply
put a 2 times e12 here.
00:24:07.110 --> 00:24:10.650
This evaluation is performed
much in the same way as in
00:24:10.650 --> 00:24:13.360
linear infinitesimal
displacement analysis.
00:24:13.360 --> 00:24:17.110
Except that we have to remember
this stress tensor,
00:24:17.110 --> 00:24:22.220
this stress strain tensor,
this considerative tensor
00:24:22.220 --> 00:24:27.080
varies in the incremental
solution because we have
00:24:27.080 --> 00:24:30.470
materially nonlinear
conditions.
00:24:30.470 --> 00:24:33.160
So the K matrix here
will change.
00:24:33.160 --> 00:24:37.360
And that is indicated, of
course, by the t up there.
00:24:37.360 --> 00:24:41.150
Notice the B matrix, in this
incremental analysis, using
00:24:41.150 --> 00:24:43.700
the material-nonlinear-only
formulations is constant.
00:24:46.610 --> 00:24:47.550
For the material-nonlinear-only
00:24:47.550 --> 00:24:50.870
formulation, we also want to
evaluate the F vector.
00:24:50.870 --> 00:24:56.290
And that F vector is a result
of this integral here.
00:24:56.290 --> 00:24:58.780
We take this integral,
00:24:58.780 --> 00:25:02.480
interpolate the virtual strains.
00:25:02.480 --> 00:25:06.980
And that is this part here, in
terms of the virtual nodal
00:25:06.980 --> 00:25:09.130
point displacements.
00:25:09.130 --> 00:25:14.870
And we assemble in this vector
here capital sigma at time t,
00:25:14.870 --> 00:25:17.960
the stresses t sigma ij.
00:25:17.960 --> 00:25:21.060
We assemble those
in this vector.
00:25:21.060 --> 00:25:24.570
Notice, in two dimensions
analysis, this vector is given
00:25:24.570 --> 00:25:28.590
down here as these components.
00:25:28.590 --> 00:25:31.405
By the way, no two here.
00:25:31.405 --> 00:25:32.820
You should think about that.
00:25:32.820 --> 00:25:34.070
There should be no two here.
00:25:36.700 --> 00:25:39.340
Total Lagrangian formulation.
00:25:39.340 --> 00:25:43.570
We have similarly an
integral as in the
00:25:43.570 --> 00:25:46.560
material-nonlinear-only
formulation.
00:25:46.560 --> 00:25:50.120
We interpolate, once again,
these real incremental
00:25:50.120 --> 00:25:52.610
strains, the virtual strains.
00:25:52.610 --> 00:25:55.700
And the result is directly
given here.
00:25:55.700 --> 00:25:59.710
With the B matrix, now,
defined via this
00:25:59.710 --> 00:26:00.960
equation down here.
00:26:05.120 --> 00:26:12.420
And, of course, this vector here
contains the components
00:26:12.420 --> 00:26:14.077
of the incremental
strain tensor.
00:26:17.490 --> 00:26:19.900
However, in the total Lagrangian
formulation, we
00:26:19.900 --> 00:26:22.020
know also have an additional
integral.
00:26:22.020 --> 00:26:27.580
And that integral is coming in
because of the geometric
00:26:27.580 --> 00:26:31.460
stiffening effect of the
nonlinear strain term effect.
00:26:31.460 --> 00:26:33.030
Here we have the integral.
00:26:33.030 --> 00:26:35.160
And the discretization
is given on the
00:26:35.160 --> 00:26:37.070
right-hand side here.
00:26:37.070 --> 00:26:42.880
Notice this is a BNL matrix,
nonlinear strain
00:26:42.880 --> 00:26:44.320
matrix, we call it.
00:26:44.320 --> 00:26:48.410
This here is a matrix of the
second Piola-Kirchhoff
00:26:48.410 --> 00:26:51.430
stresses at time t.
00:26:51.430 --> 00:26:53.390
And here we have, again,
the BNL matrix.
00:26:53.390 --> 00:26:58.350
This product together gives
us the KNL matrix.
00:26:58.350 --> 00:27:01.650
One might ask how do you
get these quantities?
00:27:01.650 --> 00:27:06.950
Well, actually we construct
this S matrix and the BNL
00:27:06.950 --> 00:27:11.610
matrix such that when you take
the product of this whole
00:27:11.610 --> 00:27:15.200
right-hand side, you get that.
00:27:15.200 --> 00:27:20.110
So these matrices are really
constructed such as to obtain
00:27:20.110 --> 00:27:21.550
what we need to get.
00:27:21.550 --> 00:27:24.680
And that is this part here.
00:27:24.680 --> 00:27:26.970
I will show you later
on specific
00:27:26.970 --> 00:27:30.060
examples in another lecture.
00:27:30.060 --> 00:27:32.460
Here we have the S
matrix containing
00:27:32.460 --> 00:27:34.000
components as I've mentioned.
00:27:34.000 --> 00:27:37.820
And this matrix here times
this vector contains the
00:27:37.820 --> 00:27:41.770
components of this displacement
derivative.
00:27:44.810 --> 00:27:47.300
The right-hand side, of
course, for the total
00:27:47.300 --> 00:27:51.136
Lagrangian formulation has the
evaluation of the F vector.
00:27:51.136 --> 00:27:54.440
And that one is obtained
from this integral.
00:27:54.440 --> 00:27:56.980
Notice we go over.
00:27:56.980 --> 00:28:01.580
We evaluate this integral by
this relationship here.
00:28:01.580 --> 00:28:05.070
The linear strain displacement
matrix goes in here.
00:28:05.070 --> 00:28:07.970
And a vector of the second
Piola-Kirchhoff
00:28:07.970 --> 00:28:11.000
stresses goes in here.
00:28:11.000 --> 00:28:15.570
Once again, this vector is
constructed in such a way that
00:28:15.570 --> 00:28:19.090
this right-hand side here is
equal to that integral.
00:28:21.940 --> 00:28:25.240
In the updated Lagrangian
formulation, we proceed much
00:28:25.240 --> 00:28:26.750
in the same way.
00:28:26.750 --> 00:28:31.270
Considering incremental
displacement UI, we have this
00:28:31.270 --> 00:28:33.070
integral to evaluate.
00:28:33.070 --> 00:28:36.540
We interpolate the strains via
the strain-displacement
00:28:36.540 --> 00:28:41.100
[? interpolation ?] matrix, and
the result is this here.
00:28:41.100 --> 00:28:44.270
This is here the linear strain
stiffness matrix.
00:28:44.270 --> 00:28:47.470
Here we have a relation very
much alike of what we have in
00:28:47.470 --> 00:28:50.850
the total Lagrangian
formulation for the
00:28:50.850 --> 00:28:52.100
incremental strains.
00:28:54.500 --> 00:28:57.090
The nonlinear strain stiffness
matrix in the updated
00:28:57.090 --> 00:29:01.620
Lagrangian formulation is also
very much evaluated like in
00:29:01.620 --> 00:29:03.750
the total Lagrangian
formulation.
00:29:03.750 --> 00:29:07.010
It is this integral that
we now have to capture.
00:29:07.010 --> 00:29:13.240
And we do so by constructing a
BNL matrix, a tall matrix,
00:29:13.240 --> 00:29:17.540
such that this total
product here is
00:29:17.540 --> 00:29:19.870
equal to this integral.
00:29:19.870 --> 00:29:23.540
And what's underlined here in
blue is the matrix that we're
00:29:23.540 --> 00:29:25.620
looking for.
00:29:25.620 --> 00:29:28.270
We should, of course, also
evaluate in the updated
00:29:28.270 --> 00:29:31.260
Lagrangian formulation
the F vector.
00:29:31.260 --> 00:29:34.050
And that F vector, which appears
on the right-hand side
00:29:34.050 --> 00:29:38.150
of the equation in the updated
Lagrangian formulation, is
00:29:38.150 --> 00:29:41.030
evaluated by calculating
this integral.
00:29:41.030 --> 00:29:49.240
Notice that this is obtained
by this right-hand side.
00:29:49.240 --> 00:29:51.830
The BL, of course, is
the linear strain
00:29:51.830 --> 00:29:53.340
displacement matrix.
00:29:53.340 --> 00:29:59.850
And here we have a vector, tall
hat, which contains the
00:29:59.850 --> 00:30:02.500
stresses, Cauchy the
stresses at time t.
00:30:02.500 --> 00:30:08.240
It's constructed such that this
integral here, with this
00:30:08.240 --> 00:30:12.710
part in front of it, gives
us exactly that integral.
00:30:12.710 --> 00:30:15.355
And what is underlined here in
blue is the actual F vector
00:30:15.355 --> 00:30:17.240
that we're looking for.
00:30:17.240 --> 00:30:19.780
So what we have seen then, is
that the finite element
00:30:19.780 --> 00:30:22.710
matrices for the
material-nonlinear-only, the
00:30:22.710 --> 00:30:25.220
total Lagrangian, and the
updated Lagrangian formulation
00:30:25.220 --> 00:30:28.920
are formulated by looking
at the individual volume
00:30:28.920 --> 00:30:32.460
integrals in these continuum
mechanics formulations.
00:30:32.460 --> 00:30:36.660
And by interpolating the
displacements and strains much
00:30:36.660 --> 00:30:40.570
in the same way as we are used
to in linear analysis.
00:30:40.570 --> 00:30:43.480
Once we have formulated these
matrices, we, of course, have
00:30:43.480 --> 00:30:44.700
to evaluate them.
00:30:44.700 --> 00:30:49.110
And that is done using numerical
integration, once
00:30:49.110 --> 00:30:52.170
again, just very similar
to what we're
00:30:52.170 --> 00:30:55.230
doing in linear analysis.
00:30:55.230 --> 00:30:58.280
We're using, primarily, Gauss
integration or Newton-Cotes
00:30:58.280 --> 00:30:59.640
integration.
00:30:59.640 --> 00:31:02.380
Schematically, in
two-dimensional analysis, the
00:31:02.380 --> 00:31:06.520
K matrix would be evaluated
as shown here.
00:31:06.520 --> 00:31:09.200
Notice that in isoparametric
finite element analysis, we
00:31:09.200 --> 00:31:13.320
are integrating from minus 1
to plus 1 over the domain.
00:31:13.320 --> 00:31:15.260
Two-dimensional analysis,
of course, two
00:31:15.260 --> 00:31:16.790
integrations involved.
00:31:16.790 --> 00:31:20.920
That we have a B matrix
transposed, C matrix, B here,
00:31:20.920 --> 00:31:24.860
a determinant of a Jacobian
matrix, which comes in because
00:31:24.860 --> 00:31:30.330
we are transforming from the x1,
x2 space to the RS space.
00:31:30.330 --> 00:31:34.510
And we call that the G matrix's
total product.
00:31:34.510 --> 00:31:38.120
And the numerical integrations
then really involves nothing
00:31:38.120 --> 00:31:44.870
else but summing a product
of alpha ij, Gij over all
00:31:44.870 --> 00:31:46.710
numerical integration points.
00:31:46.710 --> 00:31:52.170
Notice the ij here now refers to
the ij's integration point.
00:31:52.170 --> 00:31:56.190
This is what we are doing also
in linear analysis and in
00:31:56.190 --> 00:31:58.720
nonlinear analysis as well.
00:31:58.720 --> 00:32:03.640
Similarly, we would evaluate
the F vector, which, of
00:32:03.640 --> 00:32:06.040
course, we have in the
material-nonlinear-only, total
00:32:06.040 --> 00:32:09.240
Lagrangian, or updated
Lagrangian formulation as
00:32:09.240 --> 00:32:10.660
shown here.
00:32:10.660 --> 00:32:12.730
Notice, once again,
integration from
00:32:12.730 --> 00:32:15.010
minus 1 to plus 1.
00:32:15.010 --> 00:32:19.750
And this part here is what
we might call G again.
00:32:19.750 --> 00:32:22.840
F then, is obtained
as shown here.
00:32:22.840 --> 00:32:25.830
Of course these are the
integration point weights that
00:32:25.830 --> 00:32:29.180
are given to us, which we
simply use in the finite
00:32:29.180 --> 00:32:30.900
element solution.
00:32:30.900 --> 00:32:34.570
The mass matrix is evaluated
as shown here.
00:32:34.570 --> 00:32:36.700
Mass density goes in there.
00:32:36.700 --> 00:32:39.900
H transpose H. H, of course,
being the displacement
00:32:39.900 --> 00:32:41.490
interpolation matrix.
00:32:41.490 --> 00:32:45.010
And this is our G here.
00:32:45.010 --> 00:32:49.360
With that G, we should put a bar
under that G here because
00:32:49.360 --> 00:32:51.150
it's a matrix.
00:32:51.150 --> 00:32:52.220
You put it in.
00:32:52.220 --> 00:32:56.660
And if we use that G here
in this formula,
00:32:56.660 --> 00:32:59.250
we get the M matrix.
00:32:59.250 --> 00:33:03.200
So the numerical integration
is really performed much in
00:33:03.200 --> 00:33:07.000
the way as we're doing it
in linear analysis.
00:33:07.000 --> 00:33:09.050
Frequently, we use Gauss
integration,
00:33:09.050 --> 00:33:10.490
as I mentioned earlier.
00:33:10.490 --> 00:33:14.980
And, as a typical example, 3x3
into Gauss integration is
00:33:14.980 --> 00:33:16.740
schematically shown here.
00:33:16.740 --> 00:33:18.310
Here is our element.
00:33:18.310 --> 00:33:20.470
Here is the R coordinate axes.
00:33:20.470 --> 00:33:22.500
Here is the s coordinate axes.
00:33:22.500 --> 00:33:27.060
This would be the integration
point stations that we are
00:33:27.060 --> 00:33:29.270
using for 3x3 integration.
00:33:29.270 --> 00:33:32.560
The r and s values are
given as shown here.
00:33:32.560 --> 00:33:36.370
Same r and s values as
in linear analysis.
00:33:36.370 --> 00:33:40.130
And we notice that these
integration point stations are
00:33:40.130 --> 00:33:41.550
all within the element.
00:33:41.550 --> 00:33:43.990
That is, of course, one
feature off the Gauss
00:33:43.990 --> 00:33:46.620
integration.
00:33:46.620 --> 00:33:48.700
As I mentioned earlier they
use also Newton-Cotes
00:33:48.700 --> 00:33:53.490
integration, for example for
the integration through the
00:33:53.490 --> 00:33:55.140
shell's thickness.
00:33:55.140 --> 00:34:00.310
Here is the r direction, which
is a coordinate axis in the
00:34:00.310 --> 00:34:01.730
mid-surface of the shell.
00:34:01.730 --> 00:34:04.260
And s goes through
the thickness.
00:34:04.260 --> 00:34:07.580
Notice here we have five point
Newton-Cotes integration.
00:34:07.580 --> 00:34:12.190
And that some integration
points, as a matter of fact
00:34:12.190 --> 00:34:16.540
two here, are actually on the
surface of the element.
00:34:16.540 --> 00:34:22.460
Because we are including the
surface of the element, we use
00:34:22.460 --> 00:34:24.940
Newton-Cotes integration
quite frequently
00:34:24.940 --> 00:34:27.230
in nonlinear analysis.
00:34:27.230 --> 00:34:32.260
The reason being that if we do
an elasto-plastic analysis, we
00:34:32.260 --> 00:34:36.250
find that the larger stresses,
of course, are generally
00:34:36.250 --> 00:34:39.219
generated on the surfaces
of the element.
00:34:39.219 --> 00:34:44.290
And these are then also the
areas where plasticity sets in
00:34:44.290 --> 00:34:49.520
earliest, which means that
we want to pick up this
00:34:49.520 --> 00:34:52.880
elasto-plastic response as
quickly as possible.
00:34:52.880 --> 00:34:56.469
And integration point stations
on the surface of the element
00:34:56.469 --> 00:34:58.890
can be of benefit.
00:34:58.890 --> 00:35:02.990
If you compare Gauss with
Newton-Cotes integration, we
00:35:02.990 --> 00:35:06.500
recognize that with n Gauss
points, we integrate a
00:35:06.500 --> 00:35:10.860
polynomial of order 2n minus 1
exactly, meaning, for example,
00:35:10.860 --> 00:35:14.480
with two Gauss points, you
integrate a cubic exactly and
00:35:14.480 --> 00:35:16.180
everything below
it, of course.
00:35:16.180 --> 00:35:19.370
Whereas with n Newton-Cotes
points, we integrate only a
00:35:19.370 --> 00:35:22.040
polynomial of n minus
1 exactly.
00:35:22.040 --> 00:35:24.870
So we need really many more
Newton-Cotes integration
00:35:24.870 --> 00:35:28.410
points to pick up the same
accuracy in the integration as
00:35:28.410 --> 00:35:32.300
you do with the Gauss
point integration.
00:35:32.300 --> 00:35:35.620
For this reason, we use,
primarily really, the Gauss
00:35:35.620 --> 00:35:39.150
integration, particularly in
the analysis of solids.
00:35:39.150 --> 00:35:43.910
Maybe a big, chunky bodies
where there is no need,
00:35:43.910 --> 00:35:48.600
really, to pick up the plastic
response say on the surface
00:35:48.600 --> 00:35:51.560
directly of the solid.
00:35:51.560 --> 00:35:53.790
Newton-Cotes integration
involves points on the
00:35:53.790 --> 00:35:54.290
boundaries.
00:35:54.290 --> 00:35:55.570
I mentioned that already.
00:35:55.570 --> 00:35:58.680
And therefore, this integration
scheme is
00:35:58.680 --> 00:36:01.190
effective for structural
elements for the reasons that
00:36:01.190 --> 00:36:02.970
I just gave.
00:36:02.970 --> 00:36:07.310
In principle, the integration
schemes I employed as in
00:36:07.310 --> 00:36:09.046
linear analysis.
00:36:09.046 --> 00:36:12.370
The integration order must be
high enough not to have any
00:36:12.370 --> 00:36:14.510
spurious energy modes
in the elements.
00:36:14.510 --> 00:36:17.170
We will get back to that in
later lectures, particularly
00:36:17.170 --> 00:36:19.590
when we talk about structural
elements, beam elements, and
00:36:19.590 --> 00:36:21.170
shell elements.
00:36:21.170 --> 00:36:24.190
This is a very important
point.
00:36:24.190 --> 00:36:28.100
The appropriate integration
order in nonlinear analysis
00:36:28.100 --> 00:36:31.470
can sometimes be higher than
in linear analysis, for
00:36:31.470 --> 00:36:34.780
example, to model the plasticity
accurately, once
00:36:34.780 --> 00:36:40.730
again, in a shell solution
or such type of analysis.
00:36:40.730 --> 00:36:44.590
On the other hand, a too high
integration order is also not
00:36:44.590 --> 00:36:50.450
of value because remember, that
the maximum displacement
00:36:50.450 --> 00:36:53.240
variation, therefore, the
maximum strain variation you
00:36:53.240 --> 00:36:56.280
can pick up, is of course, given
by the interpolation
00:36:56.280 --> 00:36:57.470
you're using.
00:36:57.470 --> 00:37:01.940
So it doesn't make much sense
to go up in very high
00:37:01.940 --> 00:37:07.400
integration order in order
to try to pick up a high
00:37:07.400 --> 00:37:13.150
variation in strains, plastic
strains, and the corresponding
00:37:13.150 --> 00:37:14.240
stresses, of course.
00:37:14.240 --> 00:37:17.810
It doesn't make sense to do that
when you are limited by
00:37:17.810 --> 00:37:20.660
the actual strain variation
anyway due to the
00:37:20.660 --> 00:37:21.710
interpolations on the
00:37:21.710 --> 00:37:24.330
displacements that you're using.
00:37:24.330 --> 00:37:27.780
Let me show you an example here
that demonstrates some of
00:37:27.780 --> 00:37:30.540
the points that I'm
trying to make.
00:37:30.540 --> 00:37:34.080
Here we have an eight node
element that models the
00:37:34.080 --> 00:37:38.220
response of a cantilever
and the bending moment.
00:37:38.220 --> 00:37:41.660
We measure the rotation
phi here.
00:37:41.660 --> 00:37:44.380
We have here the
material data.
00:37:44.380 --> 00:37:47.440
Notice we are talking about an
elasto-plastic material with
00:37:47.440 --> 00:37:48.940
yield stress.
00:37:48.940 --> 00:37:50.810
And we apply a bending
moment as shown here.
00:37:54.250 --> 00:37:58.280
In linear elastic analysis,
you would get the exact
00:37:58.280 --> 00:38:02.820
response to this problem using
one eight node element.
00:38:02.820 --> 00:38:04.100
You might have tried
it already.
00:38:04.100 --> 00:38:05.570
You may know that.
00:38:05.570 --> 00:38:08.300
The reason, of course, being
that this element contains a
00:38:08.300 --> 00:38:12.240
parabolic displacement
interpolation, which is the
00:38:12.240 --> 00:38:14.120
analytical solution
to this problem.
00:38:14.120 --> 00:38:17.260
And therefore, you get
the exact solution.
00:38:17.260 --> 00:38:20.800
In elasto-plastic analysis,
however, the solution depends
00:38:20.800 --> 00:38:23.910
on the integration order
we are using.
00:38:23.910 --> 00:38:27.190
And this is demonstrated
on this view graph.
00:38:27.190 --> 00:38:34.585
Here we show on the vertical
axis, the moment normalized to
00:38:34.585 --> 00:38:37.210
the moment at first yield.
00:38:37.210 --> 00:38:41.810
And on the horizontal axis,
the rotation of the beam
00:38:41.810 --> 00:38:47.480
normalized to the rotation at
first yield, respectively.
00:38:47.480 --> 00:38:51.850
Now, notice that the linear
elastic response, of course,
00:38:51.850 --> 00:38:56.060
would be simply this line here
going up vertically.
00:38:56.060 --> 00:38:58.970
In elasto-plastic analysis,
however, the
00:38:58.970 --> 00:39:00.940
element starts yielding.
00:39:00.940 --> 00:39:05.000
And the yield is picked up,
depending on the integration
00:39:05.000 --> 00:39:06.590
order you're using.
00:39:06.590 --> 00:39:10.980
With 2 by 2 integration, we
get this solution here.
00:39:10.980 --> 00:39:13.520
And this would be
the limit load.
00:39:13.520 --> 00:39:16.870
With 3 by 3 integration, we
get this solution for the
00:39:16.870 --> 00:39:17.880
limit load.
00:39:17.880 --> 00:39:21.730
And with 4 by 4 integration,
we get this solution as a
00:39:21.730 --> 00:39:22.520
limit load.
00:39:22.520 --> 00:39:25.800
So this solution very much
depends on the integration
00:39:25.800 --> 00:39:26.980
order that you're using.
00:39:26.980 --> 00:39:30.220
And it shows here that we need
enough integration point
00:39:30.220 --> 00:39:34.400
stations through the thickness
of the beam in order to
00:39:34.400 --> 00:39:37.700
approximate, appropriately,
the limit load
00:39:37.700 --> 00:39:39.970
that we want to calculate.
00:39:39.970 --> 00:39:42.390
Let me show you another
problem.
00:39:42.390 --> 00:39:44.950
And this is an interesting
problem in which we want to
00:39:44.950 --> 00:39:50.200
design a numerical experiment to
test whether an element can
00:39:50.200 --> 00:39:54.060
undergo properly large,
rigid body motions.
00:39:54.060 --> 00:39:55.890
Here we consider a single
00:39:55.890 --> 00:39:59.300
two-dimensional four node element.
00:39:59.300 --> 00:40:02.140
It could be plain stress
or plain strain.
00:40:02.140 --> 00:40:05.760
And we want to test by a
numerical experiment that you
00:40:05.760 --> 00:40:09.260
can perform on a computer
program, whether this element
00:40:09.260 --> 00:40:12.680
can actually perform
large, rigid body
00:40:12.680 --> 00:40:14.830
motions, of course, properly.
00:40:14.830 --> 00:40:17.030
And by properly, I mean there
should be no stresses
00:40:17.030 --> 00:40:21.630
generated in the element when it
is subjected to these rigid
00:40:21.630 --> 00:40:23.950
body motions.
00:40:23.950 --> 00:40:28.050
Well, there are for this element
three rigid body
00:40:28.050 --> 00:40:29.570
motions of interest.
00:40:29.570 --> 00:40:31.780
Two translations and
one rotation.
00:40:31.780 --> 00:40:33.630
It's a two-dimensional
element, so these two
00:40:33.630 --> 00:40:38.300
translations, of course, refer
to the horizontal translation
00:40:38.300 --> 00:40:40.050
and the vertical translation.
00:40:40.050 --> 00:40:43.660
The rotation, of course, refers
to the rotation in the
00:40:43.660 --> 00:40:46.130
two-dimensional plane.
00:40:46.130 --> 00:40:49.220
To test whether the element
can undergo properly the
00:40:49.220 --> 00:40:53.240
horizontal rigid body motion,
we designed this numerical
00:40:53.240 --> 00:40:54.380
experiment.
00:40:54.380 --> 00:40:57.670
We put here to truss elements.
00:40:57.670 --> 00:41:00.640
We call them M.N.O. trusses
because they do not include
00:41:00.640 --> 00:41:03.070
then any geometric
nonlinearalities.
00:41:03.070 --> 00:41:05.700
They are really just springs.
00:41:05.700 --> 00:41:09.430
And we put this element
on a roller here.
00:41:09.430 --> 00:41:11.280
Keep it otherwise free.
00:41:11.280 --> 00:41:16.680
And we put onto this degree of
freedom a load, R. And on that
00:41:16.680 --> 00:41:19.130
degree of freedom a
load, R, as well.
00:41:19.130 --> 00:41:20.930
The load is very large.
00:41:20.930 --> 00:41:24.790
So the element should move over
stress free by a very
00:41:24.790 --> 00:41:26.420
large amount.
00:41:26.420 --> 00:41:29.700
This is one rigid body movement
that the fall out
00:41:29.700 --> 00:41:32.560
element must, of course,
be able to undergo.
00:41:32.560 --> 00:41:39.370
And this test is passed for the
T.L., U.L., and the M.N.O,
00:41:39.370 --> 00:41:41.170
or linear analysis, of course.
00:41:41.170 --> 00:41:43.590
Similarly, we could perform
this test for
00:41:43.590 --> 00:41:45.230
the vertical direction.
00:41:45.230 --> 00:41:48.350
And we would find that, once
again, the T.L., U.L.
00:41:48.350 --> 00:41:51.510
formulation, and M.N.O.
formulation will pass a test
00:41:51.510 --> 00:41:54.940
for the vertical direction
as well.
00:41:54.940 --> 00:41:58.760
The interesting test is the
one for the rotation.
00:41:58.760 --> 00:42:01.020
Here we have our fall out
element supported at the
00:42:01.020 --> 00:42:05.060
left-hand side by a pin, and on
the right-hand side by an
00:42:05.060 --> 00:42:05.960
M.N.O. truss.
00:42:05.960 --> 00:42:09.080
Once again, a truss element
that does not model any
00:42:09.080 --> 00:42:11.010
geometric nonlinearalities.
00:42:11.010 --> 00:42:15.030
We are applying to this node
here a force, R, that would,
00:42:15.030 --> 00:42:19.760
of course, make this element
here go through the rotation.
00:42:19.760 --> 00:42:26.850
Since the force should be
taken in the spring, the
00:42:26.850 --> 00:42:29.730
element should rotate
stress free.
00:42:29.730 --> 00:42:34.060
Note that because this spring
is and M.N.O. spring, the
00:42:34.060 --> 00:42:39.240
force acting onto this node
here must always work
00:42:39.240 --> 00:42:41.680
vertically only.
00:42:41.680 --> 00:42:47.995
After the load is applied, the
element, originally here, must
00:42:47.995 --> 00:42:51.700
have rotated by a very
large amount.
00:42:51.700 --> 00:42:54.420
The area must have
not changed.
00:42:54.420 --> 00:42:58.600
In other words, the element
size must remain constant.
00:42:58.600 --> 00:43:01.320
And the element must
be stress free.
00:43:01.320 --> 00:43:06.130
This test is passed by the by
the U.L., and the T.L., the
00:43:06.130 --> 00:43:10.100
total Lagrangian formulations,
by the updated Lagrangian, and
00:43:10.100 --> 00:43:13.770
the total Lagrangian formations
properly.
00:43:13.770 --> 00:43:16.700
But if you were to use the
material-nonlinear-only
00:43:16.700 --> 00:43:21.020
formulation, you would see that
this element does not
00:43:21.020 --> 00:43:24.520
remain or keep its
original size.
00:43:24.520 --> 00:43:28.950
The reason being that this node
here will move up on this
00:43:28.950 --> 00:43:32.360
side here because of the
M.N.O. truss there.
00:43:32.360 --> 00:43:33.540
We move up here.
00:43:33.540 --> 00:43:36.750
And all of the element remain
stress free, the element
00:43:36.750 --> 00:43:38.880
actually grows in size.
00:43:38.880 --> 00:43:42.440
It's an interesting test that
you can actually perform on a
00:43:42.440 --> 00:43:44.570
computer program.
00:43:44.570 --> 00:43:46.840
Once again, the total Lagrangian
and the updated
00:43:46.840 --> 00:43:50.270
Lagrangian formulations, which,
of course, are designed
00:43:50.270 --> 00:43:55.820
to more large quotations and
large strains pass this test
00:43:55.820 --> 00:43:58.820
properly, whereas the M.N.O.
Formulation, which is not
00:43:58.820 --> 00:44:02.400
designed to model large
quotations, would not pass
00:44:02.400 --> 00:44:04.700
this test property.
00:44:04.700 --> 00:44:07.730
Well, in this lecture, I've
been trying to give you an
00:44:07.730 --> 00:44:11.830
overview of the general element
matrices that we need
00:44:11.830 --> 00:44:15.310
in the U.L., T.L., and
M.N.O. formulations.
00:44:15.310 --> 00:44:21.060
In the next lectures, we will
derive these element matrices
00:44:21.060 --> 00:44:23.980
in detail for different
types of elements.
00:44:23.980 --> 00:44:25.380
Thank you very much for
your attention.