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PROFESSOR: Ladies and gentlemen,
welcome to this

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00:00:23,830 --> 00:00:26,840
lecture on nonlinear finite
element analysis of solids and

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00:00:26,840 --> 00:00:28,150
structures.

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00:00:28,150 --> 00:00:31,050
In the previous lectures, we
talked about the general

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00:00:31,050 --> 00:00:33,740
incremental continuum mechanics
equations that we're

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00:00:33,740 --> 00:00:36,850
using in nonlinear finite
element analysis.

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00:00:36,850 --> 00:00:40,220
In this lecture, I would like,
now, to talk about the finite

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00:00:40,220 --> 00:00:43,570
element matrices that we're
using, actually, in static and

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00:00:43,570 --> 00:00:45,160
dynamic analysis.

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00:00:45,160 --> 00:00:49,000
You want to talk about these
in quite general terms.

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00:00:49,000 --> 00:00:51,510
In the next lecture then, we
will talk more about the

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00:00:51,510 --> 00:00:53,520
details of these matrices.

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00:00:53,520 --> 00:00:56,960
You want to formulate these
finite element matrices.

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00:00:56,960 --> 00:01:00,690
And we want to talk about the
numerical integration that we

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00:01:00,690 --> 00:01:04,170
use to actually evaluate
the matrices.

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00:01:04,170 --> 00:01:07,460
I've prepared some view graphs
regarding this material.

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00:01:07,460 --> 00:01:09,700
And I'd like to share the
information on these view

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00:01:09,700 --> 00:01:10,990
graphs now with you.

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00:01:13,780 --> 00:01:17,000
The derivation of the finite
element matrices, of course,

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00:01:17,000 --> 00:01:20,670
is based on the continuum
mechanics equations that we

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00:01:20,670 --> 00:01:22,340
have developed earlier.

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00:01:22,340 --> 00:01:25,150
And you have seen this equation
earlier when we

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00:01:25,150 --> 00:01:27,790
talked about the total
Lagrangian formulation, the

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00:01:27,790 --> 00:01:29,530
T.L. formulation.

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00:01:29,530 --> 00:01:32,970
We talked about the terms
in this equation.

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00:01:32,970 --> 00:01:36,810
We talked about their meaning,
what they stand for.

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00:01:36,810 --> 00:01:40,400
And we have derived certain
terms here because they have

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00:01:40,400 --> 00:01:43,850
been obtained by linearization
process.

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00:01:43,850 --> 00:01:45,910
This was the governing equation
for the total

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00:01:45,910 --> 00:01:48,730
Lagrangian formulation.

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00:01:48,730 --> 00:01:51,690
The governing continuum
mechanics equation for the

39
00:01:51,690 --> 00:01:54,790
updated Lagrangian formulation
is shown here.

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00:01:54,790 --> 00:01:58,490
Once again, we talked about
each of these terms.

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00:01:58,490 --> 00:02:00,980
We talked about what
their meaning is.

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00:02:00,980 --> 00:02:03,560
And, of course, we arrived
at this equation by

43
00:02:03,560 --> 00:02:04,810
linearization process.

44
00:02:07,310 --> 00:02:09,889
For the total Lagrangian and
the updated Lagrangian

45
00:02:09,889 --> 00:02:13,660
formulation, we recognized
because of the linearization

46
00:02:13,660 --> 00:02:17,310
process that was involved, we
need to iterate to obtain an

47
00:02:17,310 --> 00:02:20,510
accurate solution to the actual
problem of interest.

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00:02:20,510 --> 00:02:23,890
And here on this view graph,
I've summarized the iterative

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00:02:23,890 --> 00:02:29,300
equations that are used in the
total Lagrangian formulation.

50
00:02:29,300 --> 00:02:33,030
We, once again, have seen
this equation before.

51
00:02:33,030 --> 00:02:35,460
We should recognize,
now, these terms.

52
00:02:35,460 --> 00:02:38,890
The iteration count, of course,
being k, as shown here

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00:02:38,890 --> 00:02:40,800
on the left-hand side
of the equation.

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00:02:40,800 --> 00:02:41,960
And k minus one on the

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00:02:41,960 --> 00:02:44,320
right-hand side of the equation.

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00:02:44,320 --> 00:02:48,000
We also recognize this
equation here, now.

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00:02:48,000 --> 00:02:52,370
Namely, the equation where we
update the displacements from

58
00:02:52,370 --> 00:02:56,440
the k minus first iteration
to the k iteration.

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00:02:56,440 --> 00:02:59,640
For this iteration, of course,
we need initial conditions.

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00:02:59,640 --> 00:03:05,240
And those initial conditions we
also talked about earlier.

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00:03:05,240 --> 00:03:07,790
A similar equation is used
in the updated Lagrangian

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00:03:07,790 --> 00:03:09,040
formulation.

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00:03:11,070 --> 00:03:14,990
These are the terms that we
talked about earlier.

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00:03:14,990 --> 00:03:18,500
And, of course, once again, we
are iterating with iteration

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00:03:18,500 --> 00:03:22,100
count of k on the left-hand
side, iteration count of k

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00:03:22,100 --> 00:03:25,190
minus one on the right-hand
side of the equation.

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00:03:25,190 --> 00:03:27,600
The displacements are
updated, as shown

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00:03:27,600 --> 00:03:29,740
here in this equation.

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00:03:29,740 --> 00:03:32,810
And for this iteration, of
course, we need initial

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00:03:32,810 --> 00:03:35,240
conditions, which
are given here.

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00:03:35,240 --> 00:03:38,710
And all of these terms, really,
we discussed in some

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00:03:38,710 --> 00:03:43,490
detail in the earlier
lectures.

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00:03:43,490 --> 00:03:48,430
Well, assuming that loading is
deformation-independent, we

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00:03:48,430 --> 00:03:53,580
also recognize that this here
is the expression for the

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00:03:53,580 --> 00:03:55,430
external virtual work.

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00:03:55,430 --> 00:03:58,650
t plus theta tR, the script
R that I talked

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00:03:58,650 --> 00:03:59,660
about earlier already.

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00:03:59,660 --> 00:04:05,860
And these are the terms that
we already defined earlier.

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00:04:05,860 --> 00:04:11,350
Particularly the body force
term, the surface force term.

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00:04:11,350 --> 00:04:15,910
Notice that, so far, we really
talked about static analysis.

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00:04:15,910 --> 00:04:19,180
And I'd like to now, of course,
also introduce the

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00:04:19,180 --> 00:04:21,339
term that we use in
dynamic analysis.

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00:04:21,339 --> 00:04:23,810
This, of course, is
the inertia term.

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00:04:23,810 --> 00:04:27,530
And in dynamic analysis, this
is the term that is really

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00:04:27,530 --> 00:04:32,330
contained in this term here
as an effect of fV.

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00:04:32,330 --> 00:04:37,850
Of the volume forces, the
forces per unit volume.

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00:04:37,850 --> 00:04:41,580
Notice that this is really
here an integral over the

88
00:04:41,580 --> 00:04:46,600
volume at time t plus theta t of
the mass density at time t

89
00:04:46,600 --> 00:04:50,490
plus theta t times the
accelerations at time t plus

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00:04:50,490 --> 00:04:55,690
theta t, the virtual
displacements, and, of course,

91
00:04:55,690 --> 00:04:58,180
the integrating over the
volume at time t plus

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00:04:58,180 --> 00:04:58,720
[? theta, ?]

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00:04:58,720 --> 00:05:01,610
t plus theta t, as
I said already.

94
00:05:01,610 --> 00:05:05,870
This is, of course, an
"inconvenient" interval.

95
00:05:05,870 --> 00:05:10,970
Inconvenient in quotes because
we don't know this volume.

96
00:05:10,970 --> 00:05:14,610
So we cannot really directly
evaluate this integral.

97
00:05:14,610 --> 00:05:17,820
Instead we'd rather like to work
with the original volume

98
00:05:17,820 --> 00:05:20,150
like we do in the total
Lagrangian formulation.

99
00:05:20,150 --> 00:05:25,140
And transformation of this
integral to an integral over

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00:05:25,140 --> 00:05:28,320
the original volume is achieved
by this equation

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00:05:28,320 --> 00:05:29,720
here, by this term here.

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00:05:29,720 --> 00:05:37,230
Notice that, of course, 0 rho
times 0 dV for a particular

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00:05:37,230 --> 00:05:41,920
set of mass particles must be
equal to t plus theta t rho

104
00:05:41,920 --> 00:05:44,260
times t plus theta tdV.

105
00:05:44,260 --> 00:05:47,770
And this is the reason why we
can directly write down this

106
00:05:47,770 --> 00:05:49,150
equation here.

107
00:05:49,150 --> 00:05:52,770
However, considering this
transformation for finite

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00:05:52,770 --> 00:05:57,060
element analysis, it is
important to realize that we

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00:05:57,060 --> 00:06:01,990
assume here that the
possibilities of the motion,

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00:06:01,990 --> 00:06:08,630
of the material particles as
contained in the finite

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00:06:08,630 --> 00:06:13,030
element interpolations
are the same in this

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00:06:13,030 --> 00:06:16,990
volume as in that volume.

113
00:06:16,990 --> 00:06:19,970
This fortunately, is true
in isoparametric

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00:06:19,970 --> 00:06:21,600
finite element analysis.

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00:06:21,600 --> 00:06:25,450
And therefore, this
transformation from the volume

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00:06:25,450 --> 00:06:31,250
t plus theta t to the volume
V 0 is a very convenient

117
00:06:31,250 --> 00:06:37,520
transformation to perform for
the evaluation of this term.

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00:06:37,520 --> 00:06:40,400
We use that abundantly
in isoparametric

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00:06:40,400 --> 00:06:43,500
finite element analysis.

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00:06:43,500 --> 00:06:48,590
If the external loads are
deformation-dependent, then we

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00:06:48,590 --> 00:06:51,300
have to recognize that
these forces here are

122
00:06:51,300 --> 00:06:56,780
deformation-dependent, and
therefore, we have to evaluate

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00:06:56,780 --> 00:06:58,950
them in the iteration.

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00:06:58,950 --> 00:07:02,420
And that is being shown
here in this equation.

125
00:07:02,420 --> 00:07:06,560
That we are always evaluating
this term new, depending on

126
00:07:06,560 --> 00:07:12,150
the iteration k minus 1, we
integrate this product here

127
00:07:12,150 --> 00:07:14,790
over the volume at time
t plus theta t in

128
00:07:14,790 --> 00:07:16,770
iteration k minus 1.

129
00:07:16,770 --> 00:07:19,920
Similarly, we update, also,
the surface forces.

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00:07:19,920 --> 00:07:24,190
Since the surface area changes
of the body during the last

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00:07:24,190 --> 00:07:28,450
information process, we evaluate
this part here as

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00:07:28,450 --> 00:07:29,210
shown here.

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00:07:29,210 --> 00:07:31,910
Of course, there is an
approximation involved.

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00:07:31,910 --> 00:07:37,790
But if we keep on it iterating
t plus theta tVk minus one,

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00:07:37,790 --> 00:07:40,560
we'll take on the volume
t plus theta tV.

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00:07:40,560 --> 00:07:44,820
And of course then, we really
include here the term that we

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00:07:44,820 --> 00:07:47,820
want to include, namely that
one in the analysis.

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00:07:47,820 --> 00:07:48,840
Similarly here.

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00:07:48,840 --> 00:07:53,610
The surface area, in iteration
k minus 1, as k goes larger

140
00:07:53,610 --> 00:07:54,460
and larger.

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00:07:54,460 --> 00:07:58,560
If we converge, we'll actually
be equal to that surface area,

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00:07:58,560 --> 00:08:02,000
meaning that this expression is
equal to that expression,

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00:08:02,000 --> 00:08:04,190
which of course we
want to include.

144
00:08:04,190 --> 00:08:06,110
It's this expression that
we want to include

145
00:08:06,110 --> 00:08:07,360
in the force vector.

146
00:08:09,880 --> 00:08:13,890
We, in one of the very early
lectures, talked about the

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00:08:13,890 --> 00:08:16,980
materially-nonlinear-only
analysis.

148
00:08:16,980 --> 00:08:20,870
And the equation that is used,
the continuum mechanics

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00:08:20,870 --> 00:08:22,590
equation that is used in the
materially-nonlinear-only

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00:08:22,590 --> 00:08:25,530
analysis is given here.

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00:08:25,530 --> 00:08:33,000
Notice, here we have a stress
strain law tensor, an

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00:08:33,000 --> 00:08:37,880
incremental strain tensor,
the virtual strain.

153
00:08:37,880 --> 00:08:39,799
The incremental strain
tensor is the real

154
00:08:39,799 --> 00:08:41,049
strain tensor here.

155
00:08:41,049 --> 00:08:47,040
All the components of that
tensor have a superscript k.

156
00:08:47,040 --> 00:08:48,370
The external virtual work.

157
00:08:48,370 --> 00:08:51,840
And here, we have the stress
tensor, components of the

158
00:08:51,840 --> 00:08:52,990
stress tensor.

159
00:08:52,990 --> 00:08:56,370
At time t plus theta t in
iteration k minus 1.

160
00:08:56,370 --> 00:08:59,190
And the virtual strains again.

161
00:08:59,190 --> 00:09:03,120
Notice here now, we do not have
anymore subscripts, 0 or

162
00:09:03,120 --> 00:09:10,090
t, on these components because
in materially-nonlinear-only

163
00:09:10,090 --> 00:09:13,750
analysis, we assume that the
deformations are very small.

164
00:09:13,750 --> 00:09:16,650
The displacements are
infinitesimal.

165
00:09:16,650 --> 00:09:19,610
And the stress, the second
Piola-Kirchhoff stress, which

166
00:09:19,610 --> 00:09:22,670
we defined in an earlier lecture
is actually equal to

167
00:09:22,670 --> 00:09:25,930
the Cauchy stress under those
conditions and is equal to the

168
00:09:25,930 --> 00:09:29,540
physical stress that we are
talking about here in the

169
00:09:29,540 --> 00:09:31,640
materially-nonlinear-only
analysis.

170
00:09:31,640 --> 00:09:35,290
In other words, both those
stress measures are equal to

171
00:09:35,290 --> 00:09:38,870
the physical stress that appears
here, which is, of

172
00:09:38,870 --> 00:09:41,520
course, the force
per unit area.

173
00:09:41,520 --> 00:09:43,860
The one that we are so familiar
with in infinitesimal

174
00:09:43,860 --> 00:09:45,110
displacement analysis.

175
00:09:48,640 --> 00:09:51,350
Let us look further at
dynamic analysis.

176
00:09:51,350 --> 00:09:54,430
Dynamic analysis is generally
performed in nonlinear

177
00:09:54,430 --> 00:09:58,790
analysis, using an implicit time
integration scheme or an

178
00:09:58,790 --> 00:10:00,230
explicit time integration
scheme.

179
00:10:00,230 --> 00:10:03,400
And in a later lecture, we
will discuss such time

180
00:10:03,400 --> 00:10:05,070
integration schemes.

181
00:10:05,070 --> 00:10:09,010
In implicit time integration,
we look at the equilibrium

182
00:10:09,010 --> 00:10:13,080
equation at time t plus theta
t to obtain the solution at

183
00:10:13,080 --> 00:10:14,980
time t plus theta t.

184
00:10:14,980 --> 00:10:19,230
And this means that we will
have to evaluate this

185
00:10:19,230 --> 00:10:23,040
left-hand side written here as
given on the right-hand side.

186
00:10:23,040 --> 00:10:26,550
This part here, this external
virtual work, comes from the

187
00:10:26,550 --> 00:10:31,590
external loads that
are not the mass

188
00:10:31,590 --> 00:10:33,180
of the inertia forces.

189
00:10:33,180 --> 00:10:36,220
The inertia forces are taken
care of via this part, via

190
00:10:36,220 --> 00:10:39,520
this integral here,
which I discussed

191
00:10:39,520 --> 00:10:41,090
already a bit earlier.

192
00:10:41,090 --> 00:10:46,180
In explicit time integration,
we are evaluating or we're

193
00:10:46,180 --> 00:10:49,940
looking at the equilibrium
equation at time t to obtain

194
00:10:49,940 --> 00:10:52,020
the solution at time
t plus theta t.

195
00:10:52,020 --> 00:10:54,190
Quite different from implicit
time integration.

196
00:10:54,190 --> 00:10:57,170
And the governing equations are
in the total Lagrangian

197
00:10:57,170 --> 00:11:02,580
formulation right here, in the
U.L., the updated Lagrangian

198
00:11:02,580 --> 00:11:07,680
formulation given here, and in
the materially-nonlinear-only

199
00:11:07,680 --> 00:11:10,960
analysis as given here.

200
00:11:10,960 --> 00:11:15,030
Notice, of course, that R will
involve the inertia forces

201
00:11:15,030 --> 00:11:17,630
evaluated at time t now.

202
00:11:17,630 --> 00:11:19,880
And that will, of course, enable
us to march forward

203
00:11:19,880 --> 00:11:22,940
with a solution as we
will discuss later

204
00:11:22,940 --> 00:11:25,860
on in another lecture.

205
00:11:25,860 --> 00:11:28,930
The finite element equations
corresponding to these

206
00:11:28,930 --> 00:11:32,660
continuum mechanics equations
look as follows.

207
00:11:32,660 --> 00:11:36,390
In materially-nonlinear-only
analysis, let's look first at

208
00:11:36,390 --> 00:11:37,540
static analysis.

209
00:11:37,540 --> 00:11:40,190
We have this basic equation.

210
00:11:40,190 --> 00:11:43,960
Notice a tension stiffness
matrix that does not carry any

211
00:11:43,960 --> 00:11:49,660
subscript 0 or t because we are
talking about the original

212
00:11:49,660 --> 00:11:53,930
volume, the displacements being
infinitesimally small,

213
00:11:53,930 --> 00:11:58,390
the physical stress, no Cauchy
stress really, no second

214
00:11:58,390 --> 00:12:00,330
Piola-Kirchhoff stress needs
to be introduced.

215
00:12:00,330 --> 00:12:02,670
We just talk about the physical
stress that we are so

216
00:12:02,670 --> 00:12:06,190
familiar with in infinitesimal
displacement analysis.

217
00:12:06,190 --> 00:12:11,490
And that, of course, goes into
the evaluation of the K matrix

218
00:12:11,490 --> 00:12:14,260
because the material law will
appear in here, the

219
00:12:14,260 --> 00:12:16,760
incremental displacement vector,
and on the right-hand

220
00:12:16,760 --> 00:12:21,320
side, the load vector, and the
nodal point forces that are

221
00:12:21,320 --> 00:12:25,985
equivalent in the sense of the
principle of virtual work to

222
00:12:25,985 --> 00:12:28,230
the current element stresses.

223
00:12:28,230 --> 00:12:31,690
By current, I mean F time t
plus theta t at the end of

224
00:12:31,690 --> 00:12:34,080
iteration i minus 1.

225
00:12:34,080 --> 00:12:36,400
This equation, of course, looks
very much alike of what

226
00:12:36,400 --> 00:12:38,772
we have seen in the updated
Lagrangian formulation and in

227
00:12:38,772 --> 00:12:41,340
the total Lagrangian
formulation.

228
00:12:41,340 --> 00:12:45,280
In dynamic analysis using
implicit time integration,

229
00:12:45,280 --> 00:12:47,380
this will be the governing
equation.

230
00:12:47,380 --> 00:12:51,960
Now, the mass matrix, the
acceleration vector with an

231
00:12:51,960 --> 00:12:54,530
iteration [? count of ?] i
because we are marching

232
00:12:54,530 --> 00:12:58,310
forward to a situation, to a
configuration which is still

233
00:12:58,310 --> 00:13:03,550
unknown, a tangent stiffness
matrix, the same tangent

234
00:13:03,550 --> 00:13:06,800
stiffness matrix that we see
here, by the way, the

235
00:13:06,800 --> 00:13:09,800
incremental displacement vector,
and on the right-hand

236
00:13:09,800 --> 00:13:15,020
side, the same quantities
that we have here.

237
00:13:15,020 --> 00:13:18,750
In dynamic analysis using
explicit time integration, we

238
00:13:18,750 --> 00:13:22,290
have the mass matrix, the same
mass matrix, generally, that

239
00:13:22,290 --> 00:13:23,600
we have here.

240
00:13:23,600 --> 00:13:26,800
Although in implicit time
integration, we will sometimes

241
00:13:26,800 --> 00:13:29,020
use a lump mass matrix,
sometimes a banded mass

242
00:13:29,020 --> 00:13:31,380
matrix, a consistent
mass matrix.

243
00:13:31,380 --> 00:13:33,780
With an explicit time
integration, we generally use

244
00:13:33,780 --> 00:13:36,160
only the lump mass matrix.

245
00:13:36,160 --> 00:13:39,210
But otherwise, the same
kind of matrix here.

246
00:13:39,210 --> 00:13:41,170
The acceleration vector.

247
00:13:41,170 --> 00:13:45,960
The force vector or the nodal
point forces, corresponding to

248
00:13:45,960 --> 00:13:49,320
time t that are externally
applied to the structure.

249
00:13:49,320 --> 00:13:53,360
And the tF vector, which is the
force vector corresponding

250
00:13:53,360 --> 00:13:57,920
to the stresses in the
elements at time t.

251
00:13:57,920 --> 00:14:01,360
Notice, we are looking here at
equilibrium at time t to

252
00:14:01,360 --> 00:14:03,760
obtain the solution at
time t plus theta t.

253
00:14:03,760 --> 00:14:06,770
We are looking here at
equilibrium, or I should say,

254
00:14:06,770 --> 00:14:09,150
we're iterating for equilibrium
at time

255
00:14:09,150 --> 00:14:09,930
t plus theta t.

256
00:14:09,930 --> 00:14:13,370
And thus, we will obtain
the solution for time

257
00:14:13,370 --> 00:14:14,620
t plus theta t.

258
00:14:17,650 --> 00:14:21,140
In the total Lagrangian
formulation, we have very

259
00:14:21,140 --> 00:14:22,280
similar equations.

260
00:14:22,280 --> 00:14:27,270
In static analysis, a tangent
stiffness matrix, and

261
00:14:27,270 --> 00:14:30,540
otherwise, the same kind of
vectors that I talked about

262
00:14:30,540 --> 00:14:32,000
earlier already.

263
00:14:32,000 --> 00:14:37,160
Notice now, of course, we
have the subscript 0.

264
00:14:37,160 --> 00:14:40,080
Notice also that the total
tangent stiffness matrix is

265
00:14:40,080 --> 00:14:44,310
made up of a part that we might
call a linear strain

266
00:14:44,310 --> 00:14:48,320
stiffness matrix and a part
that we may call nonlinear

267
00:14:48,320 --> 00:14:50,330
strain stiffness matrix.

268
00:14:50,330 --> 00:14:54,430
This is also called the
geometric stiffness matrix.

269
00:14:54,430 --> 00:14:56,840
We will talk about how
we construct these

270
00:14:56,840 --> 00:14:58,610
matrices just now.

271
00:14:58,610 --> 00:15:02,810
In dynamic analysis, we
proceed as in the

272
00:15:02,810 --> 00:15:05,330
material-nonlinear-only
formulation, of course.

273
00:15:05,330 --> 00:15:07,730
And in all the other
formulations, if we use

274
00:15:07,730 --> 00:15:09,630
implicit time integration,
we apply

275
00:15:09,630 --> 00:15:11,165
the equilibrium equation.

276
00:15:11,165 --> 00:15:15,170
We look for the equilibrium
at time t plus theta t, as

277
00:15:15,170 --> 00:15:16,800
expressed here.

278
00:15:16,800 --> 00:15:20,850
And we have vectors, matrices
very similar to what we have

279
00:15:20,850 --> 00:15:24,320
in the material-nonlinear-only
formulation except that once

280
00:15:24,320 --> 00:15:28,000
again, of course, we have now
introduced here the geometric

281
00:15:28,000 --> 00:15:29,370
stiffening affect,
the nonlinear

282
00:15:29,370 --> 00:15:33,000
strain stiffness matrix.

283
00:15:33,000 --> 00:15:36,320
In dynamic analysis using
explicit time integration, we

284
00:15:36,320 --> 00:15:37,980
don't use any K matrix.

285
00:15:37,980 --> 00:15:41,080
We will discuss that also
much more later on.

286
00:15:41,080 --> 00:15:43,060
And this is the governing
equation.

287
00:15:43,060 --> 00:15:45,270
Very much the same as in the
material-nonlinear-only

288
00:15:45,270 --> 00:15:46,520
formulation.

289
00:15:49,030 --> 00:15:53,190
Finally, in the updated
Lagrangian formulation, we

290
00:15:53,190 --> 00:15:56,990
also obtain similar equations
and static analysis.

291
00:15:56,990 --> 00:16:00,290
These are the equations
that we want to solve.

292
00:16:00,290 --> 00:16:03,490
Notice also, the nonlinear
strain to them there or the

293
00:16:03,490 --> 00:16:06,870
nonlinear strain stiffness
matrix.

294
00:16:06,870 --> 00:16:11,940
Otherwise, the matrices and
vectors are very much alike

295
00:16:11,940 --> 00:16:13,620
what we have seen before.

296
00:16:13,620 --> 00:16:19,050
Dynamic analysis implicit time
integration and in dynamic

297
00:16:19,050 --> 00:16:22,180
analysis using explicit
time integration.

298
00:16:22,180 --> 00:16:25,500
Notice that in each of these, we
are always cutting out the

299
00:16:25,500 --> 00:16:32,070
subscript t or t plus theta t
the way I have been talking

300
00:16:32,070 --> 00:16:35,350
about earlier already.

301
00:16:35,350 --> 00:16:38,710
We have seen this equation,
of course, before

302
00:16:38,710 --> 00:16:40,510
in our earlier lectures.

303
00:16:40,510 --> 00:16:43,940
What we now do is we introduce
the mass terms.

304
00:16:43,940 --> 00:16:46,760
And we are talking about
implicit time integration and

305
00:16:46,760 --> 00:16:48,620
explicit time integration.

306
00:16:48,620 --> 00:16:52,410
We should note that these
equations are valid for single

307
00:16:52,410 --> 00:16:53,900
finite element as
well as for an

308
00:16:53,900 --> 00:16:55,410
assemblage of finite elements.

309
00:16:55,410 --> 00:16:59,240
If we have a large number of
elements, then, of course, we

310
00:16:59,240 --> 00:17:03,480
would assemble these as we do it
in linear elastic analysis

311
00:17:03,480 --> 00:17:06,180
using the direct stiffness
method.

312
00:17:06,180 --> 00:17:09,260
Considering an assemblage of
elements, we will see that

313
00:17:09,260 --> 00:17:12,085
different formulations may be
used in different regions of

314
00:17:12,085 --> 00:17:13,220
the structure.

315
00:17:13,220 --> 00:17:17,030
In other words, schematically
here we may have some elements

316
00:17:17,030 --> 00:17:19,410
that are governed by the T.L.,
the total Lagrangian

317
00:17:19,410 --> 00:17:21,940
formulation, some others by
the updated Lagrangian

318
00:17:21,940 --> 00:17:24,839
formulation, and some others by
the material-nonlinear-only

319
00:17:24,839 --> 00:17:25,970
formulation.

320
00:17:25,970 --> 00:17:29,040
Notice that compatibility
between these elements is, of

321
00:17:29,040 --> 00:17:33,330
course, perfectly preserved if
these are compatible elements

322
00:17:33,330 --> 00:17:34,670
as shown here.

323
00:17:34,670 --> 00:17:37,810
Then, there is nothing wrong
with using the U.L.

324
00:17:37,810 --> 00:17:41,990
formulation for certain elements
that are bordering

325
00:17:41,990 --> 00:17:45,540
elements with another
formulation such as this.

326
00:17:45,540 --> 00:17:48,340
It is not true, for example,
that due to the fact that

327
00:17:48,340 --> 00:17:51,860
you're using here different
kinds of formulations, you are

328
00:17:51,860 --> 00:17:54,010
getting an incompatibility
introduced here.

329
00:17:54,010 --> 00:17:55,530
I've heard that sometimes.

330
00:17:55,530 --> 00:17:57,610
That is certainly not
my understanding

331
00:17:57,610 --> 00:17:58,720
of the subject matter.

332
00:17:58,720 --> 00:18:02,260
It does not matter whether
you have the same kind of

333
00:18:02,260 --> 00:18:04,610
formulations or two different
kind of formulations.

334
00:18:04,610 --> 00:18:05,620
That will not effect the

335
00:18:05,620 --> 00:18:08,850
compatibility between two elements.

336
00:18:08,850 --> 00:18:12,480
Let us now concentrate on
the derivation or on the

337
00:18:12,480 --> 00:18:16,260
formulation of a single
element matrix.

338
00:18:16,260 --> 00:18:19,000
To obtain a single element
matrices, we have to

339
00:18:19,000 --> 00:18:21,480
introduce, of course, an
interpolation matrix.

340
00:18:21,480 --> 00:18:24,460
And this matrix interpolates
the internal element

341
00:18:24,460 --> 00:18:28,440
displacements via the nodal
point displacements.

342
00:18:28,440 --> 00:18:32,390
Here, I'm showing a full node
element with these nodal point

343
00:18:32,390 --> 00:18:34,730
displacements.

344
00:18:34,730 --> 00:18:37,350
Notice these are measured in
the Cartesian coordinate

345
00:18:37,350 --> 00:18:39,170
directions.

346
00:18:39,170 --> 00:18:42,240
Notice however, that these nodal
point displacements are

347
00:18:42,240 --> 00:18:44,470
measured into a skewed
direction, an a

348
00:18:44,470 --> 00:18:46,660
b coordinate system.

349
00:18:46,660 --> 00:18:50,620
There's nothing wrong with
introducing different systems

350
00:18:50,620 --> 00:18:56,370
because this is Q system at the
different nodal points.

351
00:18:56,370 --> 00:19:01,630
The nodal point vector, the
vector of nodal point degrees

352
00:19:01,630 --> 00:19:03,720
of freedom is listed here.

353
00:19:03,720 --> 00:19:08,800
Notice that vector carries a hat
to denote the fact that it

354
00:19:08,800 --> 00:19:13,800
contains this great nodal
point displacements.

355
00:19:13,800 --> 00:19:17,950
Notice also, that the subscripts
1, 2 refer to the

356
00:19:17,950 --> 00:19:21,060
Cartesian coordinate
directions for the

357
00:19:21,060 --> 00:19:25,030
superscripts referred
to the nodal point.

358
00:19:25,030 --> 00:19:26,730
Here 1, 2 again.

359
00:19:26,730 --> 00:19:29,670
3 denoting the nodal point.

360
00:19:29,670 --> 00:19:34,890
Notice up here, a b, of course
refer to the skewed

361
00:19:34,890 --> 00:19:36,460
directions.

362
00:19:36,460 --> 00:19:41,390
And the 1 refers to
the nodal point 1.

363
00:19:41,390 --> 00:19:44,230
We want to interpolate the
internal displacements in

364
00:19:44,230 --> 00:19:46,280
terms of the nodal point
displacements.

365
00:19:46,280 --> 00:19:50,480
And that is being achieved by
this relationship here.

366
00:19:50,480 --> 00:19:56,230
U, the internal particle
displacements are given via H,

367
00:19:56,230 --> 00:20:00,370
the displacement interpolation
matrix times U hat.

368
00:20:00,370 --> 00:20:04,240
U hat being this vector, the
one we just discussed.

369
00:20:04,240 --> 00:20:08,260
U being a vector of these
two displacements.

370
00:20:08,260 --> 00:20:11,460
Now, notice that these two
displacements, of course,

371
00:20:11,460 --> 00:20:14,560
depend on which particle
you are looking at.

372
00:20:14,560 --> 00:20:16,890
Here, a particular particle.

373
00:20:16,890 --> 00:20:18,620
This would be the displacement
you want.

374
00:20:18,620 --> 00:20:20,420
That's the displacement U2.

375
00:20:20,420 --> 00:20:24,200
And these displacements vary
over the element, which would

376
00:20:24,200 --> 00:20:27,070
be expressed by this H matrix.

377
00:20:27,070 --> 00:20:29,000
These are the nodal point
displacements.

378
00:20:29,000 --> 00:20:31,970
These are the varying,
continuously varying

379
00:20:31,970 --> 00:20:35,640
displacements of the particles
within the element.

380
00:20:35,640 --> 00:20:40,180
We will, of course, use this
kind of relationship now quite

381
00:20:40,180 --> 00:20:41,270
extensively.

382
00:20:41,270 --> 00:20:44,270
We have, of course, also already
encountered this

383
00:20:44,270 --> 00:20:48,130
relationship in linear
elastic analysis.

384
00:20:48,130 --> 00:20:50,940
Let us now see how we
are formulating

385
00:20:50,940 --> 00:20:52,490
the different matrices.

386
00:20:52,490 --> 00:20:56,070
For all analysis types, in
which we want to include

387
00:20:56,070 --> 00:21:02,880
inertia forces, we evaluate this
integral as shown here.

388
00:21:02,880 --> 00:21:08,660
Notice the displacements and
accelerations are interpolated

389
00:21:08,660 --> 00:21:11,460
via this relationship here.

390
00:21:11,460 --> 00:21:13,550
For the accelerations, of
course, we would have dots

391
00:21:13,550 --> 00:21:16,810
here, dots there denoting
second time derivatives.

392
00:21:16,810 --> 00:21:20,260
And we would have t plus theta
t as a superscript on each of

393
00:21:20,260 --> 00:21:22,210
these variables.

394
00:21:22,210 --> 00:21:25,070
Notice that we also use this
interpolation here for the

395
00:21:25,070 --> 00:21:26,640
virtual displacements.

396
00:21:26,640 --> 00:21:32,050
And the result is given right
here just like in linear

397
00:21:32,050 --> 00:21:33,300
elastic analysis.

398
00:21:35,620 --> 00:21:41,000
The right inside load vector
as evaluated is shown here.

399
00:21:41,000 --> 00:21:45,630
Once again, we introduce
interpolation for U or for the

400
00:21:45,630 --> 00:21:50,120
virtual displacements and the
virtual surface displacements.

401
00:21:50,120 --> 00:21:52,910
This interpolation here
gives us this H

402
00:21:52,910 --> 00:21:55,920
matrix, H transpose matrix.

403
00:21:55,920 --> 00:22:01,830
This here interpolated gives
us the HST matrix.

404
00:22:01,830 --> 00:22:06,300
Notice that HS is the
interpolation matrix for the

405
00:22:06,300 --> 00:22:11,460
surface displacements
as a function of--

406
00:22:11,460 --> 00:22:16,080
or rather it gives the surface
displacements I should say as

407
00:22:16,080 --> 00:22:18,630
a function of the nodal
point displacements.

408
00:22:18,630 --> 00:22:23,840
So HS is really evaluated by
using H, the H matrix I just

409
00:22:23,840 --> 00:22:27,760
talked about, and evaluating
that H matrix on the surface

410
00:22:27,760 --> 00:22:29,460
of the element.

411
00:22:29,460 --> 00:22:31,530
That is how you get HS.

412
00:22:31,530 --> 00:22:33,970
And all of this expression
together

413
00:22:33,970 --> 00:22:35,510
gives us the load vector.

414
00:22:35,510 --> 00:22:39,320
As a matter of fact it is
really the same process

415
00:22:39,320 --> 00:22:42,020
followed that we are using
in linear infinitesimally

416
00:22:42,020 --> 00:22:45,100
displacement analysis.

417
00:22:45,100 --> 00:22:49,270
In material-nonlinear-only
analysis, considering an

418
00:22:49,270 --> 00:22:53,670
incremental displacement UI, we
evaluate this integral here

419
00:22:53,670 --> 00:22:56,340
as shown here.

420
00:22:56,340 --> 00:23:00,480
Notice here the virtual
displacements that are coming

421
00:23:00,480 --> 00:23:03,330
in because we have the virtual
strains there.

422
00:23:03,330 --> 00:23:06,800
Notice here the real
displacements, which are

423
00:23:06,800 --> 00:23:10,420
coming in from these
real strains.

424
00:23:10,420 --> 00:23:12,750
Of course, these are strain
increments and,

425
00:23:12,750 --> 00:23:15,830
correspondingly, displacement
increments.

426
00:23:15,830 --> 00:23:18,190
Nodal point displacement
increments always.

427
00:23:18,190 --> 00:23:25,620
These B matrices, BL matrices,
are obtained by evaluating the

428
00:23:25,620 --> 00:23:29,320
strains from the nodal
point displacements.

429
00:23:29,320 --> 00:23:32,040
And, of course, the
interpolation that is used for

430
00:23:32,040 --> 00:23:33,980
the element goes in here.

431
00:23:33,980 --> 00:23:36,750
The B matrix, of course,
contains derivatives of the

432
00:23:36,750 --> 00:23:40,650
elements of the H matrix.

433
00:23:40,650 --> 00:23:45,410
A vector containing components
of eIj is this one here.

434
00:23:45,410 --> 00:23:47,250
For example, in two dimensional
plane stress

435
00:23:47,250 --> 00:23:51,050
analysis, the entries in this
vector are listed right here.

436
00:23:51,050 --> 00:23:57,420
Notice that there is a 2 here
because e12 is equal to e21.

437
00:23:57,420 --> 00:23:59,720
And the sum of these two,
of course, gives us

438
00:23:59,720 --> 00:24:01,450
a total sheer strain.

439
00:24:01,450 --> 00:24:07,110
And therefore, we simply
put a 2 times e12 here.

440
00:24:07,110 --> 00:24:10,650
This evaluation is performed
much in the same way as in

441
00:24:10,650 --> 00:24:13,360
linear infinitesimal
displacement analysis.

442
00:24:13,360 --> 00:24:17,110
Except that we have to remember
this stress tensor,

443
00:24:17,110 --> 00:24:22,220
this stress strain tensor,
this considerative tensor

444
00:24:22,220 --> 00:24:27,080
varies in the incremental
solution because we have

445
00:24:27,080 --> 00:24:30,470
materially nonlinear
conditions.

446
00:24:30,470 --> 00:24:33,160
So the K matrix here
will change.

447
00:24:33,160 --> 00:24:37,360
And that is indicated, of
course, by the t up there.

448
00:24:37,360 --> 00:24:41,150
Notice the B matrix, in this
incremental analysis, using

449
00:24:41,150 --> 00:24:43,700
the material-nonlinear-only
formulations is constant.

450
00:24:46,610 --> 00:24:47,550
For the material-nonlinear-only

451
00:24:47,550 --> 00:24:50,870
formulation, we also want to
evaluate the F vector.

452
00:24:50,870 --> 00:24:56,290
And that F vector is a result
of this integral here.

453
00:24:56,290 --> 00:24:58,780
We take this integral,

454
00:24:58,780 --> 00:25:02,480
interpolate the virtual strains.

455
00:25:02,480 --> 00:25:06,980
And that is this part here, in
terms of the virtual nodal

456
00:25:06,980 --> 00:25:09,130
point displacements.

457
00:25:09,130 --> 00:25:14,870
And we assemble in this vector
here capital sigma at time t,

458
00:25:14,870 --> 00:25:17,960
the stresses t sigma ij.

459
00:25:17,960 --> 00:25:21,060
We assemble those
in this vector.

460
00:25:21,060 --> 00:25:24,570
Notice, in two dimensions
analysis, this vector is given

461
00:25:24,570 --> 00:25:28,590
down here as these components.

462
00:25:28,590 --> 00:25:31,405
By the way, no two here.

463
00:25:31,405 --> 00:25:32,820
You should think about that.

464
00:25:32,820 --> 00:25:34,070
There should be no two here.

465
00:25:36,700 --> 00:25:39,340
Total Lagrangian formulation.

466
00:25:39,340 --> 00:25:43,570
We have similarly an
integral as in the

467
00:25:43,570 --> 00:25:46,560
material-nonlinear-only
formulation.

468
00:25:46,560 --> 00:25:50,120
We interpolate, once again,
these real incremental

469
00:25:50,120 --> 00:25:52,610
strains, the virtual strains.

470
00:25:52,610 --> 00:25:55,700
And the result is directly
given here.

471
00:25:55,700 --> 00:25:59,710
With the B matrix, now,
defined via this

472
00:25:59,710 --> 00:26:00,960
equation down here.

473
00:26:05,120 --> 00:26:12,420
And, of course, this vector here
contains the components

474
00:26:12,420 --> 00:26:14,077
of the incremental
strain tensor.

475
00:26:17,490 --> 00:26:19,900
However, in the total Lagrangian
formulation, we

476
00:26:19,900 --> 00:26:22,020
know also have an additional
integral.

477
00:26:22,020 --> 00:26:27,580
And that integral is coming in
because of the geometric

478
00:26:27,580 --> 00:26:31,460
stiffening effect of the
nonlinear strain term effect.

479
00:26:31,460 --> 00:26:33,030
Here we have the integral.

480
00:26:33,030 --> 00:26:35,160
And the discretization
is given on the

481
00:26:35,160 --> 00:26:37,070
right-hand side here.

482
00:26:37,070 --> 00:26:42,880
Notice this is a BNL matrix,
nonlinear strain

483
00:26:42,880 --> 00:26:44,320
matrix, we call it.

484
00:26:44,320 --> 00:26:48,410
This here is a matrix of the
second Piola-Kirchhoff

485
00:26:48,410 --> 00:26:51,430
stresses at time t.

486
00:26:51,430 --> 00:26:53,390
And here we have, again,
the BNL matrix.

487
00:26:53,390 --> 00:26:58,350
This product together gives
us the KNL matrix.

488
00:26:58,350 --> 00:27:01,650
One might ask how do you
get these quantities?

489
00:27:01,650 --> 00:27:06,950
Well, actually we construct
this S matrix and the BNL

490
00:27:06,950 --> 00:27:11,610
matrix such that when you take
the product of this whole

491
00:27:11,610 --> 00:27:15,200
right-hand side, you get that.

492
00:27:15,200 --> 00:27:20,110
So these matrices are really
constructed such as to obtain

493
00:27:20,110 --> 00:27:21,550
what we need to get.

494
00:27:21,550 --> 00:27:24,680
And that is this part here.

495
00:27:24,680 --> 00:27:26,970
I will show you later
on specific

496
00:27:26,970 --> 00:27:30,060
examples in another lecture.

497
00:27:30,060 --> 00:27:32,460
Here we have the S
matrix containing

498
00:27:32,460 --> 00:27:34,000
components as I've mentioned.

499
00:27:34,000 --> 00:27:37,820
And this matrix here times
this vector contains the

500
00:27:37,820 --> 00:27:41,770
components of this displacement
derivative.

501
00:27:44,810 --> 00:27:47,300
The right-hand side, of
course, for the total

502
00:27:47,300 --> 00:27:51,136
Lagrangian formulation has the
evaluation of the F vector.

503
00:27:51,136 --> 00:27:54,440
And that one is obtained
from this integral.

504
00:27:54,440 --> 00:27:56,980
Notice we go over.

505
00:27:56,980 --> 00:28:01,580
We evaluate this integral by
this relationship here.

506
00:28:01,580 --> 00:28:05,070
The linear strain displacement
matrix goes in here.

507
00:28:05,070 --> 00:28:07,970
And a vector of the second
Piola-Kirchhoff

508
00:28:07,970 --> 00:28:11,000
stresses goes in here.

509
00:28:11,000 --> 00:28:15,570
Once again, this vector is
constructed in such a way that

510
00:28:15,570 --> 00:28:19,090
this right-hand side here is
equal to that integral.

511
00:28:21,940 --> 00:28:25,240
In the updated Lagrangian
formulation, we proceed much

512
00:28:25,240 --> 00:28:26,750
in the same way.

513
00:28:26,750 --> 00:28:31,270
Considering incremental
displacement UI, we have this

514
00:28:31,270 --> 00:28:33,070
integral to evaluate.

515
00:28:33,070 --> 00:28:36,540
We interpolate the strains via
the strain-displacement

516
00:28:36,540 --> 00:28:41,100
[? interpolation ?] matrix, and
the result is this here.

517
00:28:41,100 --> 00:28:44,270
This is here the linear strain
stiffness matrix.

518
00:28:44,270 --> 00:28:47,470
Here we have a relation very
much alike of what we have in

519
00:28:47,470 --> 00:28:50,850
the total Lagrangian
formulation for the

520
00:28:50,850 --> 00:28:52,100
incremental strains.

521
00:28:54,500 --> 00:28:57,090
The nonlinear strain stiffness
matrix in the updated

522
00:28:57,090 --> 00:29:01,620
Lagrangian formulation is also
very much evaluated like in

523
00:29:01,620 --> 00:29:03,750
the total Lagrangian
formulation.

524
00:29:03,750 --> 00:29:07,010
It is this integral that
we now have to capture.

525
00:29:07,010 --> 00:29:13,240
And we do so by constructing a
BNL matrix, a tall matrix,

526
00:29:13,240 --> 00:29:17,540
such that this total
product here is

527
00:29:17,540 --> 00:29:19,870
equal to this integral.

528
00:29:19,870 --> 00:29:23,540
And what's underlined here in
blue is the matrix that we're

529
00:29:23,540 --> 00:29:25,620
looking for.

530
00:29:25,620 --> 00:29:28,270
We should, of course, also
evaluate in the updated

531
00:29:28,270 --> 00:29:31,260
Lagrangian formulation
the F vector.

532
00:29:31,260 --> 00:29:34,050
And that F vector, which appears
on the right-hand side

533
00:29:34,050 --> 00:29:38,150
of the equation in the updated
Lagrangian formulation, is

534
00:29:38,150 --> 00:29:41,030
evaluated by calculating
this integral.

535
00:29:41,030 --> 00:29:49,240
Notice that this is obtained
by this right-hand side.

536
00:29:49,240 --> 00:29:51,830
The BL, of course, is
the linear strain

537
00:29:51,830 --> 00:29:53,340
displacement matrix.

538
00:29:53,340 --> 00:29:59,850
And here we have a vector, tall
hat, which contains the

539
00:29:59,850 --> 00:30:02,500
stresses, Cauchy the
stresses at time t.

540
00:30:02,500 --> 00:30:08,240
It's constructed such that this
integral here, with this

541
00:30:08,240 --> 00:30:12,710
part in front of it, gives
us exactly that integral.

542
00:30:12,710 --> 00:30:15,355
And what is underlined here in
blue is the actual F vector

543
00:30:15,355 --> 00:30:17,240
that we're looking for.

544
00:30:17,240 --> 00:30:19,780
So what we have seen then, is
that the finite element

545
00:30:19,780 --> 00:30:22,710
matrices for the
material-nonlinear-only, the

546
00:30:22,710 --> 00:30:25,220
total Lagrangian, and the
updated Lagrangian formulation

547
00:30:25,220 --> 00:30:28,920
are formulated by looking
at the individual volume

548
00:30:28,920 --> 00:30:32,460
integrals in these continuum
mechanics formulations.

549
00:30:32,460 --> 00:30:36,660
And by interpolating the
displacements and strains much

550
00:30:36,660 --> 00:30:40,570
in the same way as we are used
to in linear analysis.

551
00:30:40,570 --> 00:30:43,480
Once we have formulated these
matrices, we, of course, have

552
00:30:43,480 --> 00:30:44,700
to evaluate them.

553
00:30:44,700 --> 00:30:49,110
And that is done using numerical
integration, once

554
00:30:49,110 --> 00:30:52,170
again, just very similar
to what we're

555
00:30:52,170 --> 00:30:55,230
doing in linear analysis.

556
00:30:55,230 --> 00:30:58,280
We're using, primarily, Gauss
integration or Newton-Cotes

557
00:30:58,280 --> 00:30:59,640
integration.

558
00:30:59,640 --> 00:31:02,380
Schematically, in
two-dimensional analysis, the

559
00:31:02,380 --> 00:31:06,520
K matrix would be evaluated
as shown here.

560
00:31:06,520 --> 00:31:09,200
Notice that in isoparametric
finite element analysis, we

561
00:31:09,200 --> 00:31:13,320
are integrating from minus 1
to plus 1 over the domain.

562
00:31:13,320 --> 00:31:15,260
Two-dimensional analysis,
of course, two

563
00:31:15,260 --> 00:31:16,790
integrations involved.

564
00:31:16,790 --> 00:31:20,920
That we have a B matrix
transposed, C matrix, B here,

565
00:31:20,920 --> 00:31:24,860
a determinant of a Jacobian
matrix, which comes in because

566
00:31:24,860 --> 00:31:30,330
we are transforming from the x1,
x2 space to the RS space.

567
00:31:30,330 --> 00:31:34,510
And we call that the G matrix's
total product.

568
00:31:34,510 --> 00:31:38,120
And the numerical integrations
then really involves nothing

569
00:31:38,120 --> 00:31:44,870
else but summing a product
of alpha ij, Gij over all

570
00:31:44,870 --> 00:31:46,710
numerical integration points.

571
00:31:46,710 --> 00:31:52,170
Notice the ij here now refers to
the ij's integration point.

572
00:31:52,170 --> 00:31:56,190
This is what we are doing also
in linear analysis and in

573
00:31:56,190 --> 00:31:58,720
nonlinear analysis as well.

574
00:31:58,720 --> 00:32:03,640
Similarly, we would evaluate
the F vector, which, of

575
00:32:03,640 --> 00:32:06,040
course, we have in the
material-nonlinear-only, total

576
00:32:06,040 --> 00:32:09,240
Lagrangian, or updated
Lagrangian formulation as

577
00:32:09,240 --> 00:32:10,660
shown here.

578
00:32:10,660 --> 00:32:12,730
Notice, once again,
integration from

579
00:32:12,730 --> 00:32:15,010
minus 1 to plus 1.

580
00:32:15,010 --> 00:32:19,750
And this part here is what
we might call G again.

581
00:32:19,750 --> 00:32:22,840
F then, is obtained
as shown here.

582
00:32:22,840 --> 00:32:25,830
Of course these are the
integration point weights that

583
00:32:25,830 --> 00:32:29,180
are given to us, which we
simply use in the finite

584
00:32:29,180 --> 00:32:30,900
element solution.

585
00:32:30,900 --> 00:32:34,570
The mass matrix is evaluated
as shown here.

586
00:32:34,570 --> 00:32:36,700
Mass density goes in there.

587
00:32:36,700 --> 00:32:39,900
H transpose H. H, of course,
being the displacement

588
00:32:39,900 --> 00:32:41,490
interpolation matrix.

589
00:32:41,490 --> 00:32:45,010
And this is our G here.

590
00:32:45,010 --> 00:32:49,360
With that G, we should put a bar
under that G here because

591
00:32:49,360 --> 00:32:51,150
it's a matrix.

592
00:32:51,150 --> 00:32:52,220
You put it in.

593
00:32:52,220 --> 00:32:56,660
And if we use that G here
in this formula,

594
00:32:56,660 --> 00:32:59,250
we get the M matrix.

595
00:32:59,250 --> 00:33:03,200
So the numerical integration
is really performed much in

596
00:33:03,200 --> 00:33:07,000
the way as we're doing it
in linear analysis.

597
00:33:07,000 --> 00:33:09,050
Frequently, we use Gauss
integration,

598
00:33:09,050 --> 00:33:10,490
as I mentioned earlier.

599
00:33:10,490 --> 00:33:14,980
And, as a typical example, 3x3
into Gauss integration is

600
00:33:14,980 --> 00:33:16,740
schematically shown here.

601
00:33:16,740 --> 00:33:18,310
Here is our element.

602
00:33:18,310 --> 00:33:20,470
Here is the R coordinate axes.

603
00:33:20,470 --> 00:33:22,500
Here is the s coordinate axes.

604
00:33:22,500 --> 00:33:27,060
This would be the integration
point stations that we are

605
00:33:27,060 --> 00:33:29,270
using for 3x3 integration.

606
00:33:29,270 --> 00:33:32,560
The r and s values are
given as shown here.

607
00:33:32,560 --> 00:33:36,370
Same r and s values as
in linear analysis.

608
00:33:36,370 --> 00:33:40,130
And we notice that these
integration point stations are

609
00:33:40,130 --> 00:33:41,550
all within the element.

610
00:33:41,550 --> 00:33:43,990
That is, of course, one
feature off the Gauss

611
00:33:43,990 --> 00:33:46,620
integration.

612
00:33:46,620 --> 00:33:48,700
As I mentioned earlier they
use also Newton-Cotes

613
00:33:48,700 --> 00:33:53,490
integration, for example for
the integration through the

614
00:33:53,490 --> 00:33:55,140
shell's thickness.

615
00:33:55,140 --> 00:34:00,310
Here is the r direction, which
is a coordinate axis in the

616
00:34:00,310 --> 00:34:01,730
mid-surface of the shell.

617
00:34:01,730 --> 00:34:04,260
And s goes through
the thickness.

618
00:34:04,260 --> 00:34:07,580
Notice here we have five point
Newton-Cotes integration.

619
00:34:07,580 --> 00:34:12,190
And that some integration
points, as a matter of fact

620
00:34:12,190 --> 00:34:16,540
two here, are actually on the
surface of the element.

621
00:34:16,540 --> 00:34:22,460
Because we are including the
surface of the element, we use

622
00:34:22,460 --> 00:34:24,940
Newton-Cotes integration
quite frequently

623
00:34:24,940 --> 00:34:27,230
in nonlinear analysis.

624
00:34:27,230 --> 00:34:32,260
The reason being that if we do
an elasto-plastic analysis, we

625
00:34:32,260 --> 00:34:36,250
find that the larger stresses,
of course, are generally

626
00:34:36,250 --> 00:34:39,219
generated on the surfaces
of the element.

627
00:34:39,219 --> 00:34:44,290
And these are then also the
areas where plasticity sets in

628
00:34:44,290 --> 00:34:49,520
earliest, which means that
we want to pick up this

629
00:34:49,520 --> 00:34:52,880
elasto-plastic response as
quickly as possible.

630
00:34:52,880 --> 00:34:56,469
And integration point stations
on the surface of the element

631
00:34:56,469 --> 00:34:58,890
can be of benefit.

632
00:34:58,890 --> 00:35:02,990
If you compare Gauss with
Newton-Cotes integration, we

633
00:35:02,990 --> 00:35:06,500
recognize that with n Gauss
points, we integrate a

634
00:35:06,500 --> 00:35:10,860
polynomial of order 2n minus 1
exactly, meaning, for example,

635
00:35:10,860 --> 00:35:14,480
with two Gauss points, you
integrate a cubic exactly and

636
00:35:14,480 --> 00:35:16,180
everything below
it, of course.

637
00:35:16,180 --> 00:35:19,370
Whereas with n Newton-Cotes
points, we integrate only a

638
00:35:19,370 --> 00:35:22,040
polynomial of n minus
1 exactly.

639
00:35:22,040 --> 00:35:24,870
So we need really many more
Newton-Cotes integration

640
00:35:24,870 --> 00:35:28,410
points to pick up the same
accuracy in the integration as

641
00:35:28,410 --> 00:35:32,300
you do with the Gauss
point integration.

642
00:35:32,300 --> 00:35:35,620
For this reason, we use,
primarily really, the Gauss

643
00:35:35,620 --> 00:35:39,150
integration, particularly in
the analysis of solids.

644
00:35:39,150 --> 00:35:43,910
Maybe a big, chunky bodies
where there is no need,

645
00:35:43,910 --> 00:35:48,600
really, to pick up the plastic
response say on the surface

646
00:35:48,600 --> 00:35:51,560
directly of the solid.

647
00:35:51,560 --> 00:35:53,790
Newton-Cotes integration
involves points on the

648
00:35:53,790 --> 00:35:54,290
boundaries.

649
00:35:54,290 --> 00:35:55,570
I mentioned that already.

650
00:35:55,570 --> 00:35:58,680
And therefore, this integration
scheme is

651
00:35:58,680 --> 00:36:01,190
effective for structural
elements for the reasons that

652
00:36:01,190 --> 00:36:02,970
I just gave.

653
00:36:02,970 --> 00:36:07,310
In principle, the integration
schemes I employed as in

654
00:36:07,310 --> 00:36:09,046
linear analysis.

655
00:36:09,046 --> 00:36:12,370
The integration order must be
high enough not to have any

656
00:36:12,370 --> 00:36:14,510
spurious energy modes
in the elements.

657
00:36:14,510 --> 00:36:17,170
We will get back to that in
later lectures, particularly

658
00:36:17,170 --> 00:36:19,590
when we talk about structural
elements, beam elements, and

659
00:36:19,590 --> 00:36:21,170
shell elements.

660
00:36:21,170 --> 00:36:24,190
This is a very important
point.

661
00:36:24,190 --> 00:36:28,100
The appropriate integration
order in nonlinear analysis

662
00:36:28,100 --> 00:36:31,470
can sometimes be higher than
in linear analysis, for

663
00:36:31,470 --> 00:36:34,780
example, to model the plasticity
accurately, once

664
00:36:34,780 --> 00:36:40,730
again, in a shell solution
or such type of analysis.

665
00:36:40,730 --> 00:36:44,590
On the other hand, a too high
integration order is also not

666
00:36:44,590 --> 00:36:50,450
of value because remember, that
the maximum displacement

667
00:36:50,450 --> 00:36:53,240
variation, therefore, the
maximum strain variation you

668
00:36:53,240 --> 00:36:56,280
can pick up, is of course, given
by the interpolation

669
00:36:56,280 --> 00:36:57,470
you're using.

670
00:36:57,470 --> 00:37:01,940
So it doesn't make much sense
to go up in very high

671
00:37:01,940 --> 00:37:07,400
integration order in order
to try to pick up a high

672
00:37:07,400 --> 00:37:13,150
variation in strains, plastic
strains, and the corresponding

673
00:37:13,150 --> 00:37:14,240
stresses, of course.

674
00:37:14,240 --> 00:37:17,810
It doesn't make sense to do that
when you are limited by

675
00:37:17,810 --> 00:37:20,660
the actual strain variation
anyway due to the

676
00:37:20,660 --> 00:37:21,710
interpolations on the

677
00:37:21,710 --> 00:37:24,330
displacements that you're using.

678
00:37:24,330 --> 00:37:27,780
Let me show you an example here
that demonstrates some of

679
00:37:27,780 --> 00:37:30,540
the points that I'm
trying to make.

680
00:37:30,540 --> 00:37:34,080
Here we have an eight node
element that models the

681
00:37:34,080 --> 00:37:38,220
response of a cantilever
and the bending moment.

682
00:37:38,220 --> 00:37:41,660
We measure the rotation
phi here.

683
00:37:41,660 --> 00:37:44,380
We have here the
material data.

684
00:37:44,380 --> 00:37:47,440
Notice we are talking about an
elasto-plastic material with

685
00:37:47,440 --> 00:37:48,940
yield stress.

686
00:37:48,940 --> 00:37:50,810
And we apply a bending
moment as shown here.

687
00:37:54,250 --> 00:37:58,280
In linear elastic analysis,
you would get the exact

688
00:37:58,280 --> 00:38:02,820
response to this problem using
one eight node element.

689
00:38:02,820 --> 00:38:04,100
You might have tried
it already.

690
00:38:04,100 --> 00:38:05,570
You may know that.

691
00:38:05,570 --> 00:38:08,300
The reason, of course, being
that this element contains a

692
00:38:08,300 --> 00:38:12,240
parabolic displacement
interpolation, which is the

693
00:38:12,240 --> 00:38:14,120
analytical solution
to this problem.

694
00:38:14,120 --> 00:38:17,260
And therefore, you get
the exact solution.

695
00:38:17,260 --> 00:38:20,800
In elasto-plastic analysis,
however, the solution depends

696
00:38:20,800 --> 00:38:23,910
on the integration order
we are using.

697
00:38:23,910 --> 00:38:27,190
And this is demonstrated
on this view graph.

698
00:38:27,190 --> 00:38:34,585
Here we show on the vertical
axis, the moment normalized to

699
00:38:34,585 --> 00:38:37,210
the moment at first yield.

700
00:38:37,210 --> 00:38:41,810
And on the horizontal axis,
the rotation of the beam

701
00:38:41,810 --> 00:38:47,480
normalized to the rotation at
first yield, respectively.

702
00:38:47,480 --> 00:38:51,850
Now, notice that the linear
elastic response, of course,

703
00:38:51,850 --> 00:38:56,060
would be simply this line here
going up vertically.

704
00:38:56,060 --> 00:38:58,970
In elasto-plastic analysis,
however, the

705
00:38:58,970 --> 00:39:00,940
element starts yielding.

706
00:39:00,940 --> 00:39:05,000
And the yield is picked up,
depending on the integration

707
00:39:05,000 --> 00:39:06,590
order you're using.

708
00:39:06,590 --> 00:39:10,980
With 2 by 2 integration, we
get this solution here.

709
00:39:10,980 --> 00:39:13,520
And this would be
the limit load.

710
00:39:13,520 --> 00:39:16,870
With 3 by 3 integration, we
get this solution for the

711
00:39:16,870 --> 00:39:17,880
limit load.

712
00:39:17,880 --> 00:39:21,730
And with 4 by 4 integration,
we get this solution as a

713
00:39:21,730 --> 00:39:22,520
limit load.

714
00:39:22,520 --> 00:39:25,800
So this solution very much
depends on the integration

715
00:39:25,800 --> 00:39:26,980
order that you're using.

716
00:39:26,980 --> 00:39:30,220
And it shows here that we need
enough integration point

717
00:39:30,220 --> 00:39:34,400
stations through the thickness
of the beam in order to

718
00:39:34,400 --> 00:39:37,700
approximate, appropriately,
the limit load

719
00:39:37,700 --> 00:39:39,970
that we want to calculate.

720
00:39:39,970 --> 00:39:42,390
Let me show you another
problem.

721
00:39:42,390 --> 00:39:44,950
And this is an interesting
problem in which we want to

722
00:39:44,950 --> 00:39:50,200
design a numerical experiment to
test whether an element can

723
00:39:50,200 --> 00:39:54,060
undergo properly large,
rigid body motions.

724
00:39:54,060 --> 00:39:55,890
Here we consider a single

725
00:39:55,890 --> 00:39:59,300
two-dimensional four node element.

726
00:39:59,300 --> 00:40:02,140
It could be plain stress
or plain strain.

727
00:40:02,140 --> 00:40:05,760
And we want to test by a
numerical experiment that you

728
00:40:05,760 --> 00:40:09,260
can perform on a computer
program, whether this element

729
00:40:09,260 --> 00:40:12,680
can actually perform
large, rigid body

730
00:40:12,680 --> 00:40:14,830
motions, of course, properly.

731
00:40:14,830 --> 00:40:17,030
And by properly, I mean there
should be no stresses

732
00:40:17,030 --> 00:40:21,630
generated in the element when it
is subjected to these rigid

733
00:40:21,630 --> 00:40:23,950
body motions.

734
00:40:23,950 --> 00:40:28,050
Well, there are for this element
three rigid body

735
00:40:28,050 --> 00:40:29,570
motions of interest.

736
00:40:29,570 --> 00:40:31,780
Two translations and
one rotation.

737
00:40:31,780 --> 00:40:33,630
It's a two-dimensional
element, so these two

738
00:40:33,630 --> 00:40:38,300
translations, of course, refer
to the horizontal translation

739
00:40:38,300 --> 00:40:40,050
and the vertical translation.

740
00:40:40,050 --> 00:40:43,660
The rotation, of course, refers
to the rotation in the

741
00:40:43,660 --> 00:40:46,130
two-dimensional plane.

742
00:40:46,130 --> 00:40:49,220
To test whether the element
can undergo properly the

743
00:40:49,220 --> 00:40:53,240
horizontal rigid body motion,
we designed this numerical

744
00:40:53,240 --> 00:40:54,380
experiment.

745
00:40:54,380 --> 00:40:57,670
We put here to truss elements.

746
00:40:57,670 --> 00:41:00,640
We call them M.N.O. trusses
because they do not include

747
00:41:00,640 --> 00:41:03,070
then any geometric
nonlinearalities.

748
00:41:03,070 --> 00:41:05,700
They are really just springs.

749
00:41:05,700 --> 00:41:09,430
And we put this element
on a roller here.

750
00:41:09,430 --> 00:41:11,280
Keep it otherwise free.

751
00:41:11,280 --> 00:41:16,680
And we put onto this degree of
freedom a load, R. And on that

752
00:41:16,680 --> 00:41:19,130
degree of freedom a
load, R, as well.

753
00:41:19,130 --> 00:41:20,930
The load is very large.

754
00:41:20,930 --> 00:41:24,790
So the element should move over
stress free by a very

755
00:41:24,790 --> 00:41:26,420
large amount.

756
00:41:26,420 --> 00:41:29,700
This is one rigid body movement
that the fall out

757
00:41:29,700 --> 00:41:32,560
element must, of course,
be able to undergo.

758
00:41:32,560 --> 00:41:39,370
And this test is passed for the
T.L., U.L., and the M.N.O,

759
00:41:39,370 --> 00:41:41,170
or linear analysis, of course.

760
00:41:41,170 --> 00:41:43,590
Similarly, we could perform
this test for

761
00:41:43,590 --> 00:41:45,230
the vertical direction.

762
00:41:45,230 --> 00:41:48,350
And we would find that, once
again, the T.L., U.L.

763
00:41:48,350 --> 00:41:51,510
formulation, and M.N.O.
formulation will pass a test

764
00:41:51,510 --> 00:41:54,940
for the vertical direction
as well.

765
00:41:54,940 --> 00:41:58,760
The interesting test is the
one for the rotation.

766
00:41:58,760 --> 00:42:01,020
Here we have our fall out
element supported at the

767
00:42:01,020 --> 00:42:05,060
left-hand side by a pin, and on
the right-hand side by an

768
00:42:05,060 --> 00:42:05,960
M.N.O. truss.

769
00:42:05,960 --> 00:42:09,080
Once again, a truss element
that does not model any

770
00:42:09,080 --> 00:42:11,010
geometric nonlinearalities.

771
00:42:11,010 --> 00:42:15,030
We are applying to this node
here a force, R, that would,

772
00:42:15,030 --> 00:42:19,760
of course, make this element
here go through the rotation.

773
00:42:19,760 --> 00:42:26,850
Since the force should be
taken in the spring, the

774
00:42:26,850 --> 00:42:29,730
element should rotate
stress free.

775
00:42:29,730 --> 00:42:34,060
Note that because this spring
is and M.N.O. spring, the

776
00:42:34,060 --> 00:42:39,240
force acting onto this node
here must always work

777
00:42:39,240 --> 00:42:41,680
vertically only.

778
00:42:41,680 --> 00:42:47,995
After the load is applied, the
element, originally here, must

779
00:42:47,995 --> 00:42:51,700
have rotated by a very
large amount.

780
00:42:51,700 --> 00:42:54,420
The area must have
not changed.

781
00:42:54,420 --> 00:42:58,600
In other words, the element
size must remain constant.

782
00:42:58,600 --> 00:43:01,320
And the element must
be stress free.

783
00:43:01,320 --> 00:43:06,130
This test is passed by the by
the U.L., and the T.L., the

784
00:43:06,130 --> 00:43:10,100
total Lagrangian formulations,
by the updated Lagrangian, and

785
00:43:10,100 --> 00:43:13,770
the total Lagrangian formations
properly.

786
00:43:13,770 --> 00:43:16,700
But if you were to use the
material-nonlinear-only

787
00:43:16,700 --> 00:43:21,020
formulation, you would see that
this element does not

788
00:43:21,020 --> 00:43:24,520
remain or keep its
original size.

789
00:43:24,520 --> 00:43:28,950
The reason being that this node
here will move up on this

790
00:43:28,950 --> 00:43:32,360
side here because of the
M.N.O. truss there.

791
00:43:32,360 --> 00:43:33,540
We move up here.

792
00:43:33,540 --> 00:43:36,750
And all of the element remain
stress free, the element

793
00:43:36,750 --> 00:43:38,880
actually grows in size.

794
00:43:38,880 --> 00:43:42,440
It's an interesting test that
you can actually perform on a

795
00:43:42,440 --> 00:43:44,570
computer program.

796
00:43:44,570 --> 00:43:46,840
Once again, the total Lagrangian
and the updated

797
00:43:46,840 --> 00:43:50,270
Lagrangian formulations, which,
of course, are designed

798
00:43:50,270 --> 00:43:55,820
to more large quotations and
large strains pass this test

799
00:43:55,820 --> 00:43:58,820
properly, whereas the M.N.O.
Formulation, which is not

800
00:43:58,820 --> 00:44:02,400
designed to model large
quotations, would not pass

801
00:44:02,400 --> 00:44:04,700
this test property.

802
00:44:04,700 --> 00:44:07,730
Well, in this lecture, I've
been trying to give you an

803
00:44:07,730 --> 00:44:11,830
overview of the general element
matrices that we need

804
00:44:11,830 --> 00:44:15,310
in the U.L., T.L., and
M.N.O. formulations.

805
00:44:15,310 --> 00:44:21,060
In the next lectures, we will
derive these element matrices

806
00:44:21,060 --> 00:44:23,980
in detail for different
types of elements.

807
00:44:23,980 --> 00:44:25,380
Thank you very much for
your attention.