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PROFESSOR: Ladies and gentlemen,
welcome to this

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lecture on nonlinear finite
element analysis of solids and

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structures.

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00:00:27,530 --> 00:00:30,360
In this lecture I would like
to discuss with you the

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updated Lagrangian formulation
for general incremental

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nonlinear analysis.

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However, before doing so, I
thought it would be good to

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00:00:37,820 --> 00:00:41,100
take a bit of time and review
some of the material that we

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00:00:41,100 --> 00:00:43,110
have been discussing already.

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00:00:43,110 --> 00:00:46,500
We have been encountering
a large number of quite

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complicated equations, some very
important basic concepts,

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00:00:50,860 --> 00:00:52,480
and some subtleties.

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And to really understand all
of this material, it is

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00:00:56,480 --> 00:00:59,680
necessary to study it, of
course, a lot more than the

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00:00:59,680 --> 00:01:01,670
time that we have allocated
so far.

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00:01:01,670 --> 00:01:04,629
But it might help you to just
review now some of that

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00:01:04,629 --> 00:01:06,460
material once more.

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00:01:06,460 --> 00:01:10,420
I've prepared here on the marker
board the material that

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00:01:10,420 --> 00:01:13,370
relates to the total Lagrangian
formulation, in

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00:01:13,370 --> 00:01:17,350
which really you have been
seeing some of the very

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00:01:17,350 --> 00:01:19,200
important equations.

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00:01:19,200 --> 00:01:23,010
And here we have the basic
equation that is used, of

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00:01:23,010 --> 00:01:25,430
course, in finite element
analysis.

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00:01:25,430 --> 00:01:30,030
This basic equation states, of
course, it is the equation of

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00:01:30,030 --> 00:01:34,610
the principle of virtual work,
states that an internal

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00:01:34,610 --> 00:01:38,800
virtual work, or the internal
virtual work, must be equal to

36
00:01:38,800 --> 00:01:40,230
external virtual work.

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00:01:40,230 --> 00:01:44,210
You remember that we talked
about the Cauchy stress, the

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00:01:44,210 --> 00:01:47,030
stress which is actually
force per unit area.

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00:01:47,030 --> 00:01:49,490
And that's the engineering
stress, that's the stress that

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00:01:49,490 --> 00:01:51,440
we want to solve for.

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00:01:51,440 --> 00:01:54,460
That we talked about
a virtual strain.

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00:01:54,460 --> 00:01:57,960
A virtual strain that
is referred to time

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00:01:57,960 --> 00:01:59,840
t plus delta t.

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00:01:59,840 --> 00:02:02,530
And that is why we have
this subscript here,

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00:02:02,530 --> 00:02:04,220
t plus delta t.

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00:02:04,220 --> 00:02:09,220
And we integrate this product
here over the volume at time t

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00:02:09,220 --> 00:02:10,810
plus delta t.

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00:02:10,810 --> 00:02:14,600
Notice that I'm using here a
green color for the real

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00:02:14,600 --> 00:02:19,710
stress that we want to solve
for, and a red color for the

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virtual quantity that we're
using basically as a mechanism

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00:02:24,390 --> 00:02:26,720
to solve for this stress.

52
00:02:26,720 --> 00:02:29,440
Of course, on the right-hand
side, we identified we have

53
00:02:29,440 --> 00:02:31,260
the external virtual work.

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00:02:31,260 --> 00:02:36,110
And in here, this script R here,
denotes contributions

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00:02:36,110 --> 00:02:40,790
from the surface forces, body
forces that are being applied

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00:02:40,790 --> 00:02:43,710
externally to the body.

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00:02:43,710 --> 00:02:46,500
This is really the basic
equation that we want to

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00:02:46,500 --> 00:02:51,210
operate on, and that we want to
work with to solve for the

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00:02:51,210 --> 00:02:54,660
unknown Cauchy stresses at
time t plus delta t.

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00:02:54,660 --> 00:02:59,450
Now we recall, of course, that
we are here integrating over a

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00:02:59,450 --> 00:03:04,050
volume at time t plus delta t,
and that volume is unknown.

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00:03:04,050 --> 00:03:07,450
Because it is unknown, we cannot
very easily deal with

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00:03:07,450 --> 00:03:10,300
this stress measure, and we
have, therefore, introduced a

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00:03:10,300 --> 00:03:11,610
new stress measure.

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00:03:11,610 --> 00:03:15,740
A stress measure that we call
the second Piola-Kirchhoff

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00:03:15,740 --> 00:03:17,710
stress tensor.

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00:03:17,710 --> 00:03:19,670
We introduce this one here.

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00:03:19,670 --> 00:03:24,320
Notice this is a stress at time
t plus delta t, referred

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00:03:24,320 --> 00:03:28,660
to the original configuration,
0, of the body.

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The capital S, and we have ij
as the components of that

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00:03:33,150 --> 00:03:34,520
stress tensor.

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00:03:34,520 --> 00:03:37,670
Now, this is a new stress tensor
that we know we can

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deal very well with.

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00:03:39,210 --> 00:03:42,600
And we have to also introduce
a new strain tensor.

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00:03:42,600 --> 00:03:44,390
Why do we have to do so?

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00:03:44,390 --> 00:03:49,120
Well because using this stress
tensor, we have to use an

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energy conjugate
strain measure.

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00:03:51,470 --> 00:03:55,650
And this Green-Lagrange strain
tensor, the Green-Lagrange

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00:03:55,650 --> 00:03:59,190
strain tensor is energy
conjugate to the second

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Piola-Kirchhoff stress tensor.

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00:04:01,650 --> 00:04:05,030
Having introduced these two
quantities, and we looked at

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00:04:05,030 --> 00:04:07,470
the components, how they're
constructed, how they're

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00:04:07,470 --> 00:04:11,740
related, for example, in this
case, how this one is related

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to the Cauchy stress.

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00:04:13,410 --> 00:04:17,000
Having now introduced these
two quantities, we can

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00:04:17,000 --> 00:04:22,450
directly work basically with
this equation here.

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00:04:22,450 --> 00:04:28,280
Well to do so, we have
decomposed the stress tensor

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00:04:28,280 --> 00:04:32,540
into a part that we know, and
into an unknown part.

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00:04:32,540 --> 00:04:33,950
And that's an important point.

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00:04:33,950 --> 00:04:38,280
We can decompose this stress
tensor into a stress that we

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00:04:38,280 --> 00:04:41,580
know, and that we don't know,
by simple addition

92
00:04:41,580 --> 00:04:43,250
here on this side.

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00:04:43,250 --> 00:04:46,880
Notice we simply add here
because we had a quantity that

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00:04:46,880 --> 00:04:53,040
is referred to the volume at
time 0, to another quantity

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00:04:53,040 --> 00:04:56,520
that is referred to the
volume at time 0.

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00:04:56,520 --> 00:04:59,930
In other words, the reference
configurations for both of

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00:04:59,930 --> 00:05:03,470
these are the same, and that's
why we can have a simple

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00:05:03,470 --> 00:05:06,120
addition sign here.

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00:05:06,120 --> 00:05:09,930
Please keep in mind that this
quantity is known in our

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00:05:09,930 --> 00:05:11,550
incremental solution.

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00:05:11,550 --> 00:05:15,280
Because we, of course, assume
that we have solved for the

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00:05:15,280 --> 00:05:18,630
configuration of the body
up to a time t.

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00:05:18,630 --> 00:05:22,810
This one is unknown, and this is
the reason why I used here

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00:05:22,810 --> 00:05:26,550
black color to show that this
is an unknown quantity and

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00:05:26,550 --> 00:05:29,210
distinguish it from the
one that is known.

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00:05:29,210 --> 00:05:30,950
We proceed in the same
way with the

107
00:05:30,950 --> 00:05:32,700
Green-Lagrange strain tensor.

108
00:05:32,700 --> 00:05:34,010
Here we have the equations.

109
00:05:34,010 --> 00:05:37,320
The Green-Lagrange strain tensor
at time t plus delta t

110
00:05:37,320 --> 00:05:42,170
is decomposed into a quantity
that we know,

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00:05:42,170 --> 00:05:44,680
plus an unknown quantity.

112
00:05:44,680 --> 00:05:46,760
Once again, in black.

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00:05:46,760 --> 00:05:49,110
So these are the two quantities
that we really want

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00:05:49,110 --> 00:05:51,960
to solve for, the increment in
the second Piola-Kirchhoff

115
00:05:51,960 --> 00:05:55,750
stress, and the increment in
the Green-Lagrange strain.

116
00:05:55,750 --> 00:05:58,260
Now we also discussed that
the increment in the

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00:05:58,260 --> 00:06:01,050
Green-Lagrange strain can
actually be decomposed into

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00:06:01,050 --> 00:06:03,000
two quantities.

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00:06:03,000 --> 00:06:07,300
One that is linear in the
particle incremental

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00:06:07,300 --> 00:06:12,620
displacements ui, and one that
is nonlinear in the particle

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00:06:12,620 --> 00:06:14,770
incremental displacements ui.

122
00:06:14,770 --> 00:06:16,230
Please read it as follows.

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00:06:16,230 --> 00:06:20,600
Linear in ui, nonlinear in ui.

124
00:06:20,600 --> 00:06:24,180
Of course, remember that we have
so far only spoken about

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00:06:24,180 --> 00:06:25,970
the continuum mechanics.

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00:06:25,970 --> 00:06:28,570
In other words, we have not
really, at this point, said

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00:06:28,570 --> 00:06:31,380
anything about the finite
element discretization.

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00:06:31,380 --> 00:06:34,100
We also referred, however, in an
earlier lecture to the fact

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00:06:34,100 --> 00:06:38,800
that if we do interpolate in
finite element analysis, then

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00:06:38,800 --> 00:06:42,230
using continuum elements for
the analysis of solids and

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00:06:42,230 --> 00:06:47,650
structures, the incremental
nodal point variables are

132
00:06:47,650 --> 00:06:50,710
linearly related to these
incremental particle

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00:06:50,710 --> 00:06:51,760
displacements.

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00:06:51,760 --> 00:06:56,440
And for that reason, this is a
true linear strain increment,

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00:06:56,440 --> 00:06:59,910
and that is a true nonlinear
strain increment.

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00:06:59,910 --> 00:07:03,110
On the other hand, if we use
structural elements that also

137
00:07:03,110 --> 00:07:08,920
include or use nodal point
rotations, then the particle

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00:07:08,920 --> 00:07:14,360
displacements, these ui's, are
linearly related to the

139
00:07:14,360 --> 00:07:17,520
incremental nodal point
displacements, but nonlinearly

140
00:07:17,520 --> 00:07:20,670
related to the incremental
nodal point rotations.

141
00:07:20,670 --> 00:07:24,820
And in that case, this is here
still the true linear

142
00:07:24,820 --> 00:07:27,260
increment of the Green-Lagrange

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00:07:27,260 --> 00:07:29,790
strain for the particle.

144
00:07:29,790 --> 00:07:33,750
But this one here is not
containing all the nonlinear

145
00:07:33,750 --> 00:07:37,480
strain incremental parts
for the particle.

146
00:07:37,480 --> 00:07:38,910
There has to be a little bit--

147
00:07:38,910 --> 00:07:42,520
there has to be some amendment
here, some addition here.

148
00:07:42,520 --> 00:07:46,260
And we briefly talked about the
effect of neglecting this

149
00:07:46,260 --> 00:07:47,870
addition as well.

150
00:07:47,870 --> 00:07:51,940
Well the point now is that we
want to use, of course, these

151
00:07:51,940 --> 00:07:57,110
two quantities to substitute
these into the equation of the

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00:07:57,110 --> 00:08:00,900
principle of virtual work, and
we then arrive directly at

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00:08:00,900 --> 00:08:03,250
this relationship here.

154
00:08:03,250 --> 00:08:04,480
Now what do we see here?

155
00:08:04,480 --> 00:08:08,290
We see an integral over the
original volume of the second

156
00:08:08,290 --> 00:08:11,290
Piola-Kirchhoff stress referred
to the original

157
00:08:11,290 --> 00:08:15,740
configuration, and it's a real
stress, the one that we want

158
00:08:15,740 --> 00:08:20,420
to solve for, and that's why I
use the color green again.

159
00:08:20,420 --> 00:08:25,440
This here is a virtual strain,
Green-Lagrange strain.

160
00:08:25,440 --> 00:08:29,230
It's a variation on this
Green-Lagrange strain here.

161
00:08:29,230 --> 00:08:33,770
And this product is integrated
over the original volume,

162
00:08:33,770 --> 00:08:36,049
which is, of course, the
volume that we know.

163
00:08:36,049 --> 00:08:38,190
And this is the reason
why we can so well

164
00:08:38,190 --> 00:08:41,330
deal with this integral.

165
00:08:41,330 --> 00:08:44,280
On the right-hand side, we have
made no modification.

166
00:08:44,280 --> 00:08:47,840
We still have simply the
external virtual work.

167
00:08:47,840 --> 00:08:53,950
Now, please realize that this
equation here is totally

168
00:08:53,950 --> 00:08:57,720
equivalent to the equation that
we discussed earlier.

169
00:08:57,720 --> 00:08:59,180
This equation here.

170
00:08:59,180 --> 00:09:02,150
Here we are talking about
Cauchy stresses, an

171
00:09:02,150 --> 00:09:06,190
infinitesimal virtual strain
integrated over a volume of

172
00:09:06,190 --> 00:09:09,520
time t plus delta t
that is unknown.

173
00:09:09,520 --> 00:09:14,550
This equation has simply been
re-cast into a new form using

174
00:09:14,550 --> 00:09:16,190
continuum mechanics.

175
00:09:16,190 --> 00:09:20,350
No finite elements really, just
continuum mechanics, into

176
00:09:20,350 --> 00:09:23,000
this form here.

177
00:09:23,000 --> 00:09:27,640
And now we have a form that we
can work very well with.

178
00:09:27,640 --> 00:09:31,070
We substitute into this equation
and we linearize, and

179
00:09:31,070 --> 00:09:33,510
we did that in the
earlier lecture.

180
00:09:33,510 --> 00:09:36,950
The result of that substitution
and linearization

181
00:09:36,950 --> 00:09:39,120
is given right here.

182
00:09:39,120 --> 00:09:42,800
Now notice that in each of these
integrals, of course, we

183
00:09:42,800 --> 00:09:45,160
are integrating over a
volume that is known,

184
00:09:45,160 --> 00:09:47,160
the original volume.

185
00:09:47,160 --> 00:09:51,700
Notice that in the first
integral here, we involve,

186
00:09:51,700 --> 00:09:53,550
well a material tensor.

187
00:09:53,550 --> 00:09:55,750
That has to be defined, it
depends on the material that

188
00:09:55,750 --> 00:09:59,460
you're looking at, that
you want to analyze.

189
00:09:59,460 --> 00:10:02,890
This is here the increment in
the Green-Lagrange strain, the

190
00:10:02,890 --> 00:10:04,550
linear part only.

191
00:10:04,550 --> 00:10:10,760
Here is a variation on this
linear increment in green--

192
00:10:10,760 --> 00:10:12,400
sorry, it's in red, of course.

193
00:10:12,400 --> 00:10:13,420
This one was green.

194
00:10:13,420 --> 00:10:14,680
This one is in red.

195
00:10:14,680 --> 00:10:16,690
And this is, of course,
a virtual quantity.

196
00:10:16,690 --> 00:10:18,540
This is the virtual quantity.

197
00:10:18,540 --> 00:10:20,870
Notice that here we are
talking about a

198
00:10:20,870 --> 00:10:22,380
stress that we know.

199
00:10:22,380 --> 00:10:24,640
We're incrementally
decomposing.

200
00:10:24,640 --> 00:10:26,130
Remember?

201
00:10:26,130 --> 00:10:30,260
We're adding to the quantities
at time t, the increment.

202
00:10:30,260 --> 00:10:32,160
The quantities at time
t are known.

203
00:10:32,160 --> 00:10:34,630
So this one is known.

204
00:10:34,630 --> 00:10:39,880
This one here is a variation on
the nonlinear increment of

205
00:10:39,880 --> 00:10:43,050
the Green-Lagrange strain,
total increment.

206
00:10:43,050 --> 00:10:45,260
Well, in red again.

207
00:10:45,260 --> 00:10:47,890
And on the right-hand side we
have the external virtual

208
00:10:47,890 --> 00:10:51,410
work, which of course, is
assumed to be given.

209
00:10:51,410 --> 00:10:54,690
Do you still remember that we
are talking about deformation

210
00:10:54,690 --> 00:10:56,020
independent loading.

211
00:10:56,020 --> 00:10:59,850
So we can calculate this
piece here directly.

212
00:10:59,850 --> 00:11:05,330
And here we have also the stress
that is known, the

213
00:11:05,330 --> 00:11:11,100
stress at time t, and the
virtual strain, linear strain.

214
00:11:11,100 --> 00:11:16,270
This is what we arrived at
in our earlier lecture.

215
00:11:16,270 --> 00:11:19,210
Of course, there was a
linearization involved, and we

216
00:11:19,210 --> 00:11:21,200
know there would be errors
if we simply

217
00:11:21,200 --> 00:11:22,880
operate on this equation.

218
00:11:22,880 --> 00:11:25,270
We don't want to have these
errors in the actual finite

219
00:11:25,270 --> 00:11:26,510
element solution.

220
00:11:26,510 --> 00:11:29,520
And we realize that it
is good to iterate.

221
00:11:29,520 --> 00:11:33,720
So we also looked in the earlier
lecture how we would

222
00:11:33,720 --> 00:11:35,290
iterate typically.

223
00:11:35,290 --> 00:11:38,400
And we developed an equation
where there was something on

224
00:11:38,400 --> 00:11:42,690
the left-hand side, which I
simply denote here by dots.

225
00:11:42,690 --> 00:11:46,120
There were quite some
complicated expressions.

226
00:11:46,120 --> 00:11:48,150
You may want to look
them up again.

227
00:11:48,150 --> 00:11:51,150
But I just denote them by dots
because what is really

228
00:11:51,150 --> 00:11:54,340
important, and that I pointed
out also earlier, is what is

229
00:11:54,340 --> 00:11:55,630
on the right-hand side.

230
00:11:55,630 --> 00:11:57,490
And what do we see on
the right-hand side?

231
00:11:57,490 --> 00:12:02,200
On the right-hand side we see
the external virtual work, and

232
00:12:02,200 --> 00:12:04,860
we are subtracting from
it this integral.

233
00:12:04,860 --> 00:12:06,920
Let's look at this integral
once more.

234
00:12:06,920 --> 00:12:10,250
We have here the second
Piola-Kirchhoff stress at time

235
00:12:10,250 --> 00:12:12,650
t plus delta t.

236
00:12:12,650 --> 00:12:16,060
Of course, always refer to the
original configuration.

237
00:12:16,060 --> 00:12:20,530
Corresponding to iteration
k minus 1.

238
00:12:20,530 --> 00:12:23,810
The iteration here
written in black.

239
00:12:23,810 --> 00:12:27,840
We are operating on the
Green-Lagrange strain at time

240
00:12:27,840 --> 00:12:29,710
t plus delta t.

241
00:12:29,710 --> 00:12:32,230
Refer to the original
configuration in

242
00:12:32,230 --> 00:12:34,190
iteration k minus 1.

243
00:12:34,190 --> 00:12:38,910
Actually, calculated at the end
of iteration k minus 1 is

244
00:12:38,910 --> 00:12:40,160
better to say.

245
00:12:40,160 --> 00:12:43,850
And this product here is
integrated over the original

246
00:12:43,850 --> 00:12:44,920
volume once more.

247
00:12:44,920 --> 00:12:48,870
k runs from 1, 2, 3, onwards.

248
00:12:48,870 --> 00:12:52,510
Well notice that when k is equal
to 1 we have a 0 here,

249
00:12:52,510 --> 00:12:57,050
and we have to ask ourselves
what is this quantity then?

250
00:12:57,050 --> 00:13:00,170
Well if there's a 0 right here,
then we're looking here

251
00:13:00,170 --> 00:13:02,400
at t 0 Sij.

252
00:13:02,400 --> 00:13:04,640
The stress, the second
Piola-Kirchhoff stress

253
00:13:04,640 --> 00:13:08,290
corresponding to the
configuration at time t.

254
00:13:08,290 --> 00:13:12,160
And if we put a 0 here, we
would have also here, the

255
00:13:12,160 --> 00:13:15,870
initial condition being the
Green-Lagrange strain at time

256
00:13:15,870 --> 00:13:19,370
t, of course, referred
to configuration 0.

257
00:13:19,370 --> 00:13:22,120
So we have the write initial
conditions here, and therefore

258
00:13:22,120 --> 00:13:24,500
we can proceed with
the iteration.

259
00:13:24,500 --> 00:13:26,520
We discussed that in the
earlier lecture.

260
00:13:26,520 --> 00:13:30,680
The important point now is
that once this iteration

261
00:13:30,680 --> 00:13:35,170
converges, we have to use, of
course, schemes, solution

262
00:13:35,170 --> 00:13:37,920
procedures that makes the
iteration converge.

263
00:13:37,920 --> 00:13:39,920
We will talk about that
in later lecture.

264
00:13:39,920 --> 00:13:42,490
But once we have obtained
convergence, and let's assume

265
00:13:42,490 --> 00:13:48,020
we have converged, then this R,
the script R, the external

266
00:13:48,020 --> 00:13:51,670
virtual work, is equal
to this integral.

267
00:13:51,670 --> 00:13:53,450
That's how we define
convergence.

268
00:13:53,450 --> 00:13:57,720
And this means then that the
external virtual work is now

269
00:13:57,720 --> 00:14:00,420
equal to the internal
virtual work.

270
00:14:00,420 --> 00:14:01,560
What does this mean?

271
00:14:01,560 --> 00:14:07,460
It means in essence, that this
right-hand side is 0, which

272
00:14:07,460 --> 00:14:11,310
means-- and now I like to go
back to our earlier equation--

273
00:14:11,310 --> 00:14:15,650
that this equation here is
satisfied because notice that

274
00:14:15,650 --> 00:14:18,980
if you look at this equation
and you take this left-hand

275
00:14:18,980 --> 00:14:21,560
side and bring it to the
right-hand side with a minus

276
00:14:21,560 --> 00:14:24,190
in front, then you have
exactly what we

277
00:14:24,190 --> 00:14:26,450
just looked at earlier.

278
00:14:26,450 --> 00:14:30,640
In other words, our iteration
is performed really on an

279
00:14:30,640 --> 00:14:35,060
expression that is nothing else
than this R, this script

280
00:14:35,060 --> 00:14:40,230
R, minus this one on the
right-hand side.

281
00:14:40,230 --> 00:14:42,840
There's a minus sign here and
you take this and bring it

282
00:14:42,840 --> 00:14:44,060
right there.

283
00:14:44,060 --> 00:14:46,040
That's what we just looked at.

284
00:14:46,040 --> 00:14:49,430
Of course, we also remember now
that this expression now

285
00:14:49,430 --> 00:14:53,570
is really totally equivalent
to what we started with.

286
00:14:53,570 --> 00:14:57,170
Namely, this expression
right here.

287
00:14:57,170 --> 00:15:00,860
And this is, of course, what
we really want a solve.

288
00:15:00,860 --> 00:15:03,170
We mentioned earlier that
we introduced the second

289
00:15:03,170 --> 00:15:05,830
Piola-Kirchhoff stress, and that
we would like to get a

290
00:15:05,830 --> 00:15:08,210
physical understanding possibly
of that stress.

291
00:15:08,210 --> 00:15:11,930
And I mentioned that that is
very difficult to obtain.

292
00:15:11,930 --> 00:15:14,640
In the computer program, we
just work with that stress

293
00:15:14,640 --> 00:15:19,260
measure in order to be able to
solve the equation, to operate

294
00:15:19,260 --> 00:15:21,350
well on the principle
of virtual work.

295
00:15:21,350 --> 00:15:24,730
What we really want to do, of
course, is to solve this

296
00:15:24,730 --> 00:15:27,050
equation right here.

297
00:15:27,050 --> 00:15:28,240
This is the equation.

298
00:15:28,240 --> 00:15:32,050
That is the physical stress
that we want to solve for.

299
00:15:32,050 --> 00:15:35,860
Well since this equation is
totally equivalent to that

300
00:15:35,860 --> 00:15:41,810
equation, and looking at this
equation, if we take this

301
00:15:41,810 --> 00:15:45,010
left-hand side, bring it to
the right-hand side--

302
00:15:45,010 --> 00:15:46,950
of course, there's a minus
appearing there--

303
00:15:46,950 --> 00:15:53,870
we do obtain what we see right
here at convergence of course.

304
00:15:53,870 --> 00:15:58,630
We operate properly on the
principle of virtual work.

305
00:15:58,630 --> 00:16:02,380
So this really closes, so to
say, the loop, and I hope it

306
00:16:02,380 --> 00:16:05,730
helps you to have this overview
in understanding of

307
00:16:05,730 --> 00:16:07,560
what I discussed earlier
with you.

308
00:16:07,560 --> 00:16:09,850
But let's go one step further.

309
00:16:09,850 --> 00:16:15,610
In the finite element
discretization, we obtain from

310
00:16:15,610 --> 00:16:17,880
this equation here--

311
00:16:17,880 --> 00:16:20,910
I haven't listed what is on the
left-hand side, you can

312
00:16:20,910 --> 00:16:22,770
look it up--

313
00:16:22,770 --> 00:16:27,330
we obtain directly this
equation here.

314
00:16:27,330 --> 00:16:30,860
Of course, there's quite a
number of steps that we still

315
00:16:30,860 --> 00:16:31,850
have to discuss.

316
00:16:31,850 --> 00:16:35,570
How we obtain this matrix,
we call this the tangent

317
00:16:35,570 --> 00:16:37,430
stiffness matrix.

318
00:16:37,430 --> 00:16:41,130
We have to discuss how we
calculate this vector here,

319
00:16:41,130 --> 00:16:42,480
this vector here.

320
00:16:42,480 --> 00:16:46,200
In fact, the way we actually
introduced it, since it's a

321
00:16:46,200 --> 00:16:49,800
total Lagrangian formulation, I
should really put here a 0.

322
00:16:49,800 --> 00:16:55,200

323
00:16:55,200 --> 00:16:58,330
We have to discuss how we can
calculate this vector for

324
00:16:58,330 --> 00:17:01,165
different elements, et cetera,
and there is a

325
00:17:01,165 --> 00:17:02,490
lot still to be learned.

326
00:17:02,490 --> 00:17:07,400
But remember that up to here, we
only talked about continuum

327
00:17:07,400 --> 00:17:11,020
mechanics, and this is where
the finite element

328
00:17:11,020 --> 00:17:12,400
discretization starts.

329
00:17:12,400 --> 00:17:15,400
And these are then the basic
equations that are being

330
00:17:15,400 --> 00:17:17,170
arrived at.

331
00:17:17,170 --> 00:17:21,819
At convergence, once again, here
we iterate, of course.

332
00:17:21,819 --> 00:17:24,520
Notice that I'm using now
the superscript i.

333
00:17:24,520 --> 00:17:27,250
Of course, this superscript is
a dummy, it could be k, it

334
00:17:27,250 --> 00:17:29,340
could be j, it doesn't matter.

335
00:17:29,340 --> 00:17:32,220
We frequently use i.

336
00:17:32,220 --> 00:17:36,140
I didn't want to use up here i
because we have an i already

337
00:17:36,140 --> 00:17:39,990
as a subscript, that's
why I'm using k here.

338
00:17:39,990 --> 00:17:45,980
But at convergence in this
iteration where we calculate

339
00:17:45,980 --> 00:17:51,120
always new displacements, add
them to the previously

340
00:17:51,120 --> 00:17:54,850
calculated displacements, at
convergence the right-hand

341
00:17:54,850 --> 00:17:59,320
side will be 0, or close
to 0, of course.

342
00:17:59,320 --> 00:18:04,830
Well this means that we arrive
at this equation right here.

343
00:18:04,830 --> 00:18:08,690
Because when the right-hand side
is 0, we satisfy that t

344
00:18:08,690 --> 00:18:12,550
plus delta t R is equal to t
plus delta t F And if you

345
00:18:12,550 --> 00:18:18,180
like, here also, we may want
to introduce this little 0,

346
00:18:18,180 --> 00:18:20,890
signifying that we have used
the total Lagrangian

347
00:18:20,890 --> 00:18:25,530
formulation to calculate
that F.

348
00:18:25,530 --> 00:18:30,140
Now it is important to realize
what we have achieved in the

349
00:18:30,140 --> 00:18:32,590
finite element solution.

350
00:18:32,590 --> 00:18:37,700
We satisfy compatibility in the
finite element solution if

351
00:18:37,700 --> 00:18:40,850
we use compatible
finite elements.

352
00:18:40,850 --> 00:18:43,690
That is buried in here.

353
00:18:43,690 --> 00:18:47,330
We use compatible finite
elements to calculate this F,

354
00:18:47,330 --> 00:18:50,260
therefore we satisfy
compatibility.

355
00:18:50,260 --> 00:18:54,610
We satisfy the stress strain law
properly if we calculate

356
00:18:54,610 --> 00:19:01,810
this F here accurately and
properly through the stress

357
00:19:01,810 --> 00:19:02,850
strain law.

358
00:19:02,850 --> 00:19:05,290
Of course, there is much
to be discussed still.

359
00:19:05,290 --> 00:19:09,390
But let's assume we do that
properly and then we satisfy

360
00:19:09,390 --> 00:19:11,040
the stress strain law.

361
00:19:11,040 --> 00:19:16,430
Equilibrium we also satisfy,
but at the nodal points.

362
00:19:16,430 --> 00:19:20,180
We talked about the fact, in an
earlier lecture, that in a

363
00:19:20,180 --> 00:19:22,910
finite element analysis,
displacement based finite

364
00:19:22,910 --> 00:19:26,040
element analysis, we always
satisfy equilibrium at the

365
00:19:26,040 --> 00:19:26,940
nodal point.

366
00:19:26,940 --> 00:19:29,140
Of course, assuming that the
elements have been properly

367
00:19:29,140 --> 00:19:31,400
formulated.

368
00:19:31,400 --> 00:19:35,650
We would only satisfy local
equilibrium if the mesh is

369
00:19:35,650 --> 00:19:36,440
fine enough.

370
00:19:36,440 --> 00:19:39,940
And we talked earlier, in an
earlier lecture, about how we

371
00:19:39,940 --> 00:19:42,460
can measure whether local
equilibrium is well enough

372
00:19:42,460 --> 00:19:46,980
satisfied using stress jumps,
using pressure band plots.

373
00:19:46,980 --> 00:19:50,660
We talked about that and I
gave you some examples.

374
00:19:50,660 --> 00:19:53,920
The important point though is to
recognize that if we use a

375
00:19:53,920 --> 00:20:02,190
fine enough mesh, then
equilibrium is satisfied, and

376
00:20:02,190 --> 00:20:04,660
the stress strain law would be
satisfied if we calculated

377
00:20:04,660 --> 00:20:07,090
stresses properly from
the given strains.

378
00:20:07,090 --> 00:20:11,070
And compatibility is satisfied
if we use a compatible finite

379
00:20:11,070 --> 00:20:11,830
element mesh.

380
00:20:11,830 --> 00:20:14,870
In other words, the three basic
important requirements

381
00:20:14,870 --> 00:20:19,640
of mechanics are fulfilled when
we are satisfying this

382
00:20:19,640 --> 00:20:22,410
relationship right here.

383
00:20:22,410 --> 00:20:25,750
This basically then rounds
up what we have been

384
00:20:25,750 --> 00:20:27,770
discussing so far.

385
00:20:27,770 --> 00:20:32,870
And we use the total Lagrangian
formulation in this

386
00:20:32,870 --> 00:20:34,800
discussion.

387
00:20:34,800 --> 00:20:38,050
I now like to go over to discuss
this, use the updated

388
00:20:38,050 --> 00:20:40,100
Lagrangian formulation.

389
00:20:40,100 --> 00:20:45,480
The interesting point here is
that we use a stress measure

390
00:20:45,480 --> 00:20:48,885
that is referred to the original
configuration, and so

391
00:20:48,885 --> 00:20:51,930
a strain measure that is
referred to the original

392
00:20:51,930 --> 00:20:53,180
configuration.

393
00:20:53,180 --> 00:20:54,860

394
00:20:54,860 --> 00:20:58,730
I mentioned earlier already that
we want to deal with a

395
00:20:58,730 --> 00:21:00,750
certain configuration
that is known.

396
00:21:00,750 --> 00:21:04,030
The reason, of course, being
that we want to incrementally

397
00:21:04,030 --> 00:21:10,980
decompose this stress into a
quantity that we know, plus a

398
00:21:10,980 --> 00:21:13,930
quantity that we don't know.

399
00:21:13,930 --> 00:21:17,070
And that incremental
decomposition should involve

400
00:21:17,070 --> 00:21:21,010
stress measures that are
referred to a configuration

401
00:21:21,010 --> 00:21:22,250
that is known.

402
00:21:22,250 --> 00:21:24,210
The same for the strains.

403
00:21:24,210 --> 00:21:28,320
Well since we have calculated
all the configurations from

404
00:21:28,320 --> 00:21:33,830
time 0 to time t already, a
natural question to ask is why

405
00:21:33,830 --> 00:21:38,010
don't you use a configuration
other than 0, say, the

406
00:21:38,010 --> 00:21:42,740
configuration at time delta
t to delta t, or t.

407
00:21:42,740 --> 00:21:47,240
well in the updated Lagrangian
formulation, we, indeed, use

408
00:21:47,240 --> 00:21:51,320
the configuration at time t as
a reference configuration.

409
00:21:51,320 --> 00:21:55,670
And so we deal then with the
second Piola-Kirchhoff stress

410
00:21:55,670 --> 00:21:58,140
referred to the configuration
at time t.

411
00:21:58,140 --> 00:22:06,570

412
00:22:06,570 --> 00:22:09,170
This is the stress that
we're dealing with.

413
00:22:09,170 --> 00:22:13,430
Notice t plus delta t here,
still the stress at time t

414
00:22:13,430 --> 00:22:17,030
plus delta t, but now the stress
is referred to the

415
00:22:17,030 --> 00:22:19,260
configuration at time t.

416
00:22:19,260 --> 00:22:22,910
Similarly, we also deal
with a strain.

417
00:22:22,910 --> 00:22:29,240

418
00:22:29,240 --> 00:22:32,940
The Green-Lagrange strain at
time t plus delta t, but

419
00:22:32,940 --> 00:22:37,760
referred to configuration
at time t.

420
00:22:37,760 --> 00:22:39,900
We could, of course,
ask, well why use a

421
00:22:39,900 --> 00:22:41,310
configuration at time t?

422
00:22:41,310 --> 00:22:44,825
Why not use some configuration
in between?

423
00:22:44,825 --> 00:22:47,340
The answer is, and I gave it to
you already in an earlier

424
00:22:47,340 --> 00:22:51,710
lecture, the answer is that
you are losing all of the

425
00:22:51,710 --> 00:22:56,750
advantages of the total
Lagrangian formulation, and

426
00:22:56,750 --> 00:22:59,490
you're losing the advantages
of the updated Lagrangian

427
00:22:59,490 --> 00:23:00,680
formulation.

428
00:23:00,680 --> 00:23:02,500
You are left only with
the disadvantages.

429
00:23:02,500 --> 00:23:06,930
So there's very little point
really in trying to solve the

430
00:23:06,930 --> 00:23:10,752
principle of virtual work using
another configuration

431
00:23:10,752 --> 00:23:14,950
other than 0 or t.

432
00:23:14,950 --> 00:23:17,680
Let us now look at the details
of the updated Lagrangian

433
00:23:17,680 --> 00:23:18,430
formulation.

434
00:23:18,430 --> 00:23:23,470
And here I have now prepared
some view graphs that show all

435
00:23:23,470 --> 00:23:26,450
the equations that we're
operating on.

436
00:23:26,450 --> 00:23:30,660
The principle of virtual work,
once again, is given right on

437
00:23:30,660 --> 00:23:31,980
this first view graph.

438
00:23:31,980 --> 00:23:36,480
We always start with the basic
principle of virtual work, and

439
00:23:36,480 --> 00:23:39,050
that is written here.

440
00:23:39,050 --> 00:23:41,873
This is, once again, the Cauchy
stress, the stress that

441
00:23:41,873 --> 00:23:45,110
we are actually interested
in solving for.

442
00:23:45,110 --> 00:23:49,010
This is the infinitesimal
virtual strain.

443
00:23:49,010 --> 00:23:52,790
Internal virtual work is equal
to external virtual work.

444
00:23:52,790 --> 00:23:57,460
Now, with the stress measure
that I just mentioned to you,

445
00:23:57,460 --> 00:24:02,370
and the corresponding strain
measure, substituted for these

446
00:24:02,370 --> 00:24:06,490
stress and strain measures, we
obtain directly this equation.

447
00:24:06,490 --> 00:24:09,260
Notice that here on the
left-hand side, we have an

448
00:24:09,260 --> 00:24:14,240
integral that is exactly equal
to this integral here.

449
00:24:14,240 --> 00:24:16,010
And that we are not
talking here about

450
00:24:16,010 --> 00:24:17,780
finite elements yet.

451
00:24:17,780 --> 00:24:21,660
It's all continuum mechanics
at this point.

452
00:24:21,660 --> 00:24:25,680
The second Piola-Kirchhoff
stress here refer to time t

453
00:24:25,680 --> 00:24:28,775
is, of course, defined as the
same way as the second

454
00:24:28,775 --> 00:24:32,560
Piola-Kirchhoff stress
as before when it was

455
00:24:32,560 --> 00:24:33,900
referred to time 0.

456
00:24:33,900 --> 00:24:38,610
But what we have to do now is
introduce this t whenever we

457
00:24:38,610 --> 00:24:44,360
use the 0 before in the equation
linking up the Cauchy

458
00:24:44,360 --> 00:24:46,880
stress with the second
Piola-Kirchhoff stress.

459
00:24:46,880 --> 00:24:49,470

460
00:24:49,470 --> 00:24:52,610
We already then know the
solution at time t.

461
00:24:52,610 --> 00:24:56,380
So we know basically these
quantities here.

462
00:24:56,380 --> 00:24:59,910
And we can, therefore, now
decompose directly the

463
00:24:59,910 --> 00:25:06,130
stresses into a stress that
we know, and an increment.

464
00:25:06,130 --> 00:25:09,350
Similarly, we do for
the strains.

465
00:25:09,350 --> 00:25:14,580
It's now interesting to observe
that this stress here

466
00:25:14,580 --> 00:25:17,150
is nothing else than
the Cauchy stress.

467
00:25:17,150 --> 00:25:19,730
If you were to plug in to the
formula that I gave you

468
00:25:19,730 --> 00:25:24,280
earlier, which once again,
expresses the second

469
00:25:24,280 --> 00:25:27,080
Piola-Kirchhoff stress in terms
of the Cauchy stress,

470
00:25:27,080 --> 00:25:30,555
you would simply see that this
quantity is nothing else than

471
00:25:30,555 --> 00:25:31,730
the Cauchy stress.

472
00:25:31,730 --> 00:25:34,270
And therefore, we have
that now here.

473
00:25:34,270 --> 00:25:39,130
And of course, we carry this
part in that equation here.

474
00:25:39,130 --> 00:25:45,840
The Green-Lagrange strain, if
written out, one directly

475
00:25:45,840 --> 00:25:50,120
identifies that this part here
is 0, because we are using

476
00:25:50,120 --> 00:25:53,690
only the increments in
displacements from time t to

477
00:25:53,690 --> 00:25:56,310
time t plus delta t.

478
00:25:56,310 --> 00:26:02,020
And we are left with this part
here, which is unknown.

479
00:26:02,020 --> 00:26:04,310
Let's look at the Green-Lagrange
strain here.

480
00:26:04,310 --> 00:26:07,830
Notice t plus delta
t, t epsilon ij is

481
00:26:07,830 --> 00:26:09,910
written as shown here.

482
00:26:09,910 --> 00:26:16,600
Where I should point out that
we are looking here at the

483
00:26:16,600 --> 00:26:21,250
displacements from time t
to time t plus delta t.

484
00:26:21,250 --> 00:26:26,180
In other words, if I want to
write this out here, I'm doing

485
00:26:26,180 --> 00:26:27,750
it as follows.

486
00:26:27,750 --> 00:26:32,140
I'm saying that it's a partial
of the displacements, t plus

487
00:26:32,140 --> 00:26:42,720
delta t, ui minus tui
with respect to txj.

488
00:26:42,720 --> 00:26:46,680
Notice that here, we are
subtracting from the

489
00:26:46,680 --> 00:26:49,090
displacement at time
t plus delta t, the

490
00:26:49,090 --> 00:26:51,440
displacements tui.

491
00:26:51,440 --> 00:26:53,490
So it's the incremental
displacements that we're

492
00:26:53,490 --> 00:26:55,460
dealing with.

493
00:26:55,460 --> 00:26:58,420
And we're differentiating with
respect to the coordinate at

494
00:26:58,420 --> 00:27:02,760
the time t.

495
00:27:02,760 --> 00:27:07,700
Similarly, of course, we define
these expressions here,

496
00:27:07,700 --> 00:27:11,930
and those expressions, and
those expressions.

497
00:27:11,930 --> 00:27:16,470
Notice, just to emphasize this
point, that this here is

498
00:27:16,470 --> 00:27:20,600
nothing else than ui.

499
00:27:20,600 --> 00:27:25,010
And if we now substitute from
here into there, we directly

500
00:27:25,010 --> 00:27:29,650
recognize that this term is, in
fact, the same as that one,

501
00:27:29,650 --> 00:27:33,090
which I expressed already on
the earlier view graph.

502
00:27:33,090 --> 00:27:37,570
Notice, linear part here,
nonlinear part there.

503
00:27:37,570 --> 00:27:40,730

504
00:27:40,730 --> 00:27:44,140
This part here, of course,
I like to just write out

505
00:27:44,140 --> 00:27:44,950
for you once more.

506
00:27:44,950 --> 00:27:49,590
It's nothing else than partial
ui with respect to txj.

507
00:27:49,590 --> 00:27:53,260

508
00:27:53,260 --> 00:27:57,540
Notice, here we have a product
incremental displacement, and

509
00:27:57,540 --> 00:28:00,260
that's why it is, of
course, nonlinear.

510
00:28:00,260 --> 00:28:02,980
It is interesting to compare
this expression for the

511
00:28:02,980 --> 00:28:05,400
Green-Lagrange strain, for the
incremental Green-Lagrange

512
00:28:05,400 --> 00:28:08,060
strain, to the incremental
Green-Lagrange strain of the

513
00:28:08,060 --> 00:28:10,250
total Lagrangian formulation.

514
00:28:10,250 --> 00:28:14,600
And if you do so, you'll find
that the major difference is

515
00:28:14,600 --> 00:28:17,100
that in the total Lagrangian
formulation you have an

516
00:28:17,100 --> 00:28:19,050
initial displacement effect.

517
00:28:19,050 --> 00:28:22,720
Here, we do not have an initial
displacement effect.

518
00:28:22,720 --> 00:28:26,640
And in fact, that is one of the
advantages of the updated

519
00:28:26,640 --> 00:28:27,890
Lagrangian formulation.

520
00:28:27,890 --> 00:28:30,570

521
00:28:30,570 --> 00:28:35,230
We can now define, just as
in the total Lagrangian

522
00:28:35,230 --> 00:28:38,660
formulation, a linear strain
increment, and a nonlinear

523
00:28:38,660 --> 00:28:39,410
strain increment.

524
00:28:39,410 --> 00:28:43,380
Of course, linear in the
incremental displacement of

525
00:28:43,380 --> 00:28:44,420
the particles.

526
00:28:44,420 --> 00:28:48,280
Nonlinear in the incremental
displacement of the particles.

527
00:28:48,280 --> 00:28:50,580
And this means that
directly we obtain

528
00:28:50,580 --> 00:28:52,180
this expression here.

529
00:28:52,180 --> 00:28:56,260
And the variation on here,
of course, on this total

530
00:28:56,260 --> 00:28:58,660
Green-Lagrange strain increment
is simply given by

531
00:28:58,660 --> 00:28:59,910
these two additions.

532
00:28:59,910 --> 00:29:03,060

533
00:29:03,060 --> 00:29:07,270
The equation of the principle
of virtual work now becomes

534
00:29:07,270 --> 00:29:13,180
substituting from what I just
discussed with you, directly

535
00:29:13,180 --> 00:29:14,540
this equation.

536
00:29:14,540 --> 00:29:18,220
Notice we have here now, the
external virtual work, just as

537
00:29:18,220 --> 00:29:23,490
before, and we have here the
internal virtual work

538
00:29:23,490 --> 00:29:27,640
corresponding to the
stresses at time t.

539
00:29:27,640 --> 00:29:28,950
And these are actually
the Cauchy

540
00:29:28,950 --> 00:29:31,520
stresses, the real stresses.

541
00:29:31,520 --> 00:29:33,540
This is here again our
out of balance

542
00:29:33,540 --> 00:29:36,380
virtual work, so to say.

543
00:29:36,380 --> 00:29:41,980
On the left-hand side, we have
one part that includes an

544
00:29:41,980 --> 00:29:43,490
increment in the stress.

545
00:29:43,490 --> 00:29:47,870
Refer to time t now, because
the t configuration is our

546
00:29:47,870 --> 00:29:50,890
reference configuration.

547
00:29:50,890 --> 00:29:55,600
And we are multiplying it by
the variation on the total

548
00:29:55,600 --> 00:29:58,200
Green-Lagrange strain
increment--

549
00:29:58,200 --> 00:29:59,340
that's one integral--

550
00:29:59,340 --> 00:30:01,280
over the volume at time t.

551
00:30:01,280 --> 00:30:06,290
And here we have the Cauchy
stress times the variation on

552
00:30:06,290 --> 00:30:08,140
the nonlinear part of the

553
00:30:08,140 --> 00:30:09,390
Green-Lagrange strain increment.

554
00:30:09,390 --> 00:30:11,940

555
00:30:11,940 --> 00:30:15,490
Given a variation in
displacement data ui, of

556
00:30:15,490 --> 00:30:19,750
course, we can calculate
the right-hand side.

557
00:30:19,750 --> 00:30:23,560
But we recognize that on the
left-hand side, we have

558
00:30:23,560 --> 00:30:26,420
unknown displacement increments
and there would be

559
00:30:26,420 --> 00:30:30,390
nonlinearities occurring
on the left-hand side.

560
00:30:30,390 --> 00:30:34,430
Remember, that so far no
approximations have been made.

561
00:30:34,430 --> 00:30:37,290
This is a statement that is
valid for any amount of

562
00:30:37,290 --> 00:30:40,160
deformation, any amount
of displacements,

563
00:30:40,160 --> 00:30:41,410
rotations, and strains.

564
00:30:41,410 --> 00:30:44,040

565
00:30:44,040 --> 00:30:49,110
Just as in the total Lagrangian
formulation, the

566
00:30:49,110 --> 00:30:51,790
equation of the principle of
virtual work in the updated

567
00:30:51,790 --> 00:30:54,240
Lagrangian formulation is, of
course, a very complicated

568
00:30:54,240 --> 00:30:57,280
equation and highly nonlinear
in the incremental

569
00:30:57,280 --> 00:30:58,340
displacement.

570
00:30:58,340 --> 00:31:01,770
And we somehow have
to linearize.

571
00:31:01,770 --> 00:31:05,790
Because we want to obtain a
finite element solution, we

572
00:31:05,790 --> 00:31:10,090
linearize first the principle of
virtual work equation, and

573
00:31:10,090 --> 00:31:14,580
then we are ready to obtain an
equation as shown here, which

574
00:31:14,580 --> 00:31:18,530
is very much like the equation
which we are operating on

575
00:31:18,530 --> 00:31:20,550
using the total Lagrangian
formulation.

576
00:31:20,550 --> 00:31:23,960
Notice, because we're dealing
with the updated Lagrangian

577
00:31:23,960 --> 00:31:30,410
formulation, we now have a t
here, and we have a t there,

578
00:31:30,410 --> 00:31:33,400
meaning the reference
configuration is the time t

579
00:31:33,400 --> 00:31:38,010
configuration, and
here as well.

580
00:31:38,010 --> 00:31:42,290
We begin to linearize the terms
just as we have done it

581
00:31:42,290 --> 00:31:44,940
earlier in the total Lagrangian
formulation.

582
00:31:44,940 --> 00:31:48,060
And we recognize that
this term here is

583
00:31:48,060 --> 00:31:50,970
already linear in ui.

584
00:31:50,970 --> 00:31:54,140
That is quite simply seen
because the stress does not

585
00:31:54,140 --> 00:31:57,886
contain ui, it's a
known quantity.

586
00:31:57,886 --> 00:32:04,480
The variation on this part here
gives us this expression.

587
00:32:04,480 --> 00:32:09,200
And we recognize that this is
linear in ui because here we

588
00:32:09,200 --> 00:32:11,290
have the unknown ui appearing.

589
00:32:11,290 --> 00:32:14,790
This is going to be a constant
for a given variation in

590
00:32:14,790 --> 00:32:15,860
displacement.

591
00:32:15,860 --> 00:32:18,260
And this is here also
linear and ui.

592
00:32:18,260 --> 00:32:20,810
This is going to be a constant
for a given variation in

593
00:32:20,810 --> 00:32:21,770
displacement.

594
00:32:21,770 --> 00:32:25,460
And so we have a linear
part here in ui.

595
00:32:25,460 --> 00:32:27,900
No linearization
required here.

596
00:32:27,900 --> 00:32:30,440
Let us now look at
the next term.

597
00:32:30,440 --> 00:32:32,910
Here we have the increment in
the second Piola-Kirchhoff

598
00:32:32,910 --> 00:32:35,680
stress times the variation
on the increment in the

599
00:32:35,680 --> 00:32:38,160
Green-Lagrange strain
integrated over the

600
00:32:38,160 --> 00:32:40,000
volume at time t.

601
00:32:40,000 --> 00:32:43,930
We recognize that tSij is a
nonlinear function in general

602
00:32:43,930 --> 00:32:47,110
of the increment in the
Green-Lagrange strain.

603
00:32:47,110 --> 00:32:49,000
If we take the variation
on that increment and

604
00:32:49,000 --> 00:32:52,440
Green-Lagrange strain, we obtain
these two terms, and we

605
00:32:52,440 --> 00:32:57,140
recognize that this is also
a linear function of ui.

606
00:32:57,140 --> 00:33:00,350
Therefore, in this product, we
have to neglect now all higher

607
00:33:00,350 --> 00:33:02,830
order terms in ui.

608
00:33:02,830 --> 00:33:06,690
First we recognize now that
tSij can be written as a

609
00:33:06,690 --> 00:33:10,660
Taylor series expansion,
as shown here.

610
00:33:10,660 --> 00:33:14,610
Notice that tSij is equal to
one first term plus higher

611
00:33:14,610 --> 00:33:15,610
order terms.

612
00:33:15,610 --> 00:33:17,980
These higher order
terms we neglect.

613
00:33:17,980 --> 00:33:21,340
If you look at this term here,
we recognize that we can

614
00:33:21,340 --> 00:33:26,440
substitute for this term,
as shown here.

615
00:33:26,440 --> 00:33:29,830
And this one here is, of course,
the constitutive

616
00:33:29,830 --> 00:33:31,320
relation, the tangent
constitutive

617
00:33:31,320 --> 00:33:33,580
relation for the material.

618
00:33:33,580 --> 00:33:37,550
Recognizing that this one here
is quadratic in ui, we drop

619
00:33:37,550 --> 00:33:40,760
that term and we get this
linearized result.

620
00:33:40,760 --> 00:33:44,260
Notice that here we have a
constitutive tensor referred

621
00:33:44,260 --> 00:33:46,240
to the configuration
at time t.

622
00:33:46,240 --> 00:33:49,490
In the total Lagrangian
formulation we had here a 0

623
00:33:49,490 --> 00:33:51,240
instead of a t.

624
00:33:51,240 --> 00:33:55,820
Well if we now look at the
product that we wanted to

625
00:33:55,820 --> 00:34:00,380
linearize again, we have this
product on the left-hand side,

626
00:34:00,380 --> 00:34:03,780
which is now approximately
equal to this term here.

627
00:34:03,780 --> 00:34:08,310
This already we derived as a
linearization to that term.

628
00:34:08,310 --> 00:34:12,500
Here we are substituting
for that term here.

629
00:34:12,500 --> 00:34:17,199
Notice that if we multiply out,
we directly obtain this

630
00:34:17,199 --> 00:34:18,960
relationship here.

631
00:34:18,960 --> 00:34:22,210
This one here does not
contain ui of course.

632
00:34:22,210 --> 00:34:25,199
And this one here
is linear in ui.

633
00:34:25,199 --> 00:34:29,020
This means that since this one
also is linear in ui, that we

634
00:34:29,020 --> 00:34:31,600
have here a quadratic
term in ui.

635
00:34:31,600 --> 00:34:35,280
Notice this term now, totally
linear, this one quadratic, we

636
00:34:35,280 --> 00:34:39,870
drop that one, and we are left
with the linearized result,

637
00:34:39,870 --> 00:34:43,050
the result that we
wanted to obtain.

638
00:34:43,050 --> 00:34:46,260
The final linearized equation
is then given

639
00:34:46,260 --> 00:34:47,420
on this view graph.

640
00:34:47,420 --> 00:34:50,210
Notice we have here the
constitutive relation times

641
00:34:50,210 --> 00:34:53,830
the linear strain increment in
the total Green-Lagrange

642
00:34:53,830 --> 00:34:57,360
strain increment, the
variation on it.

643
00:34:57,360 --> 00:34:59,370
And of course, this one again
integrated over the

644
00:34:59,370 --> 00:35:01,520
volume at time t.

645
00:35:01,520 --> 00:35:04,530
Here we have a term
where there was no

646
00:35:04,530 --> 00:35:07,450
linearization necessary.

647
00:35:07,450 --> 00:35:09,830
And here we have a term
also there was no

648
00:35:09,830 --> 00:35:11,210
linearization necessary.

649
00:35:11,210 --> 00:35:15,870
These two terms were obtained
already earlier.

650
00:35:15,870 --> 00:35:20,430
These here, this term be
obtained by linearization.

651
00:35:20,430 --> 00:35:24,330
Notice that when we now use
finite element interpolations,

652
00:35:24,330 --> 00:35:30,020
we will express this total term
here via this term here

653
00:35:30,020 --> 00:35:37,330
where we now have a tangent
finite element matrix, which

654
00:35:37,330 --> 00:35:42,870
is denoted by the t up there
at time t referred to the

655
00:35:42,870 --> 00:35:45,180
configuration at time t.

656
00:35:45,180 --> 00:35:48,820
This t here refers to the
configuration at time t.

657
00:35:48,820 --> 00:35:51,460
There's an increment in the
displacement that we want to

658
00:35:51,460 --> 00:35:54,000
solve for, the real displacement
increment.

659
00:35:54,000 --> 00:35:58,140
And this is here, the virtual
displacement vector.

660
00:35:58,140 --> 00:36:02,580
The right-hand side gives us a
virtual displacement vector,

661
00:36:02,580 --> 00:36:06,830
of course, transposed times this
vector, and here we have

662
00:36:06,830 --> 00:36:09,670
the externally applied loads
coming from the external

663
00:36:09,670 --> 00:36:14,940
virtual work minus the nodal
point force vector

664
00:36:14,940 --> 00:36:19,060
corresponding to the internal
element stresses.

665
00:36:19,060 --> 00:36:23,180
That is this term here results
into this nodal point force

666
00:36:23,180 --> 00:36:26,820
vector corresponding to the
internal element stresses.

667
00:36:26,820 --> 00:36:31,890
So here we have the continuum
mechanics relation in black,

668
00:36:31,890 --> 00:36:36,400
linearized from the general
principle of virtual work, and

669
00:36:36,400 --> 00:36:40,130
here we have the finite element
expression of that.

670
00:36:40,130 --> 00:36:42,590
Of course, the actual finite
element solution is then

671
00:36:42,590 --> 00:36:46,990
obtained by actually setting
this vector here equal to the

672
00:36:46,990 --> 00:36:51,830
identity matrix so that we are
left with K times delta U is

673
00:36:51,830 --> 00:36:54,470
equal to t plus delta
tR minus tF.

674
00:36:54,470 --> 00:36:57,930

675
00:36:57,930 --> 00:37:01,540
An important point in this
development is to recognize

676
00:37:01,540 --> 00:37:05,400
that this term here is, of
course, the virtual work due

677
00:37:05,400 --> 00:37:08,980
to the element internal
stresses at time t.

678
00:37:08,980 --> 00:37:13,460
And we interpret this part here
as the out of balance

679
00:37:13,460 --> 00:37:16,560
virtual work just in the same
way as we have been doing it

680
00:37:16,560 --> 00:37:17,825
in the total Lagrangian
formulation.

681
00:37:17,825 --> 00:37:21,680

682
00:37:21,680 --> 00:37:24,440
Let's look now at the total
solution using the updated

683
00:37:24,440 --> 00:37:26,800
Lagrangian formulation.

684
00:37:26,800 --> 00:37:30,000
The displacement iteration is
much the same way as in the

685
00:37:30,000 --> 00:37:31,970
total Lagrangian formulation.

686
00:37:31,970 --> 00:37:37,530
We take the old value
corresponding to k minus 1

687
00:37:37,530 --> 00:37:40,490
iteration, and add an increment
that we are solving

688
00:37:40,490 --> 00:37:44,650
for to obtain a new estimate
on the displacements.

689
00:37:44,650 --> 00:37:46,930
k being the iteration counter.

690
00:37:46,930 --> 00:37:49,640
Notice, of course, in this
iteration, we have to have

691
00:37:49,640 --> 00:37:52,330
initial conditions, and these
initial conditions are given

692
00:37:52,330 --> 00:37:53,060
right here.

693
00:37:53,060 --> 00:37:57,800
In other words, when k is equal
to 1, we would have here

694
00:37:57,800 --> 00:38:03,690
t plus delta t ui 0, and
therefore we need this value,

695
00:38:03,690 --> 00:38:08,110
and that value is given right
here as t plus delta tui, 0

696
00:38:08,110 --> 00:38:10,360
equal to tui.

697
00:38:10,360 --> 00:38:11,970
In other words, the displacement
at the

698
00:38:11,970 --> 00:38:15,460
configuration t.

699
00:38:15,460 --> 00:38:19,020
If you look now at the total
continuum mechanics equations

700
00:38:19,020 --> 00:38:24,120
that we are operating on, I
should say one continuum

701
00:38:24,120 --> 00:38:26,770
mechanics equation really,
which, of course, gives us

702
00:38:26,770 --> 00:38:29,870
then in the finite element
discretization, the set of

703
00:38:29,870 --> 00:38:34,770
simultaneous equations that
we will be working with.

704
00:38:34,770 --> 00:38:38,120
This continuum mechanics
equation looks as follows.

705
00:38:38,120 --> 00:38:41,550
On the left-hand side we have
the tangent constitutive

706
00:38:41,550 --> 00:38:47,770
relationship times an increment
in real strains that

707
00:38:47,770 --> 00:38:50,990
we want to solve for
in iteration k.

708
00:38:50,990 --> 00:38:54,030
These are the virtual strains.

709
00:38:54,030 --> 00:38:57,650
We integrate that product over
the volume at time t.

710
00:38:57,650 --> 00:39:00,590
That, of course, holds for any
one of these integrals.

711
00:39:00,590 --> 00:39:03,500
And here we have now the actual
stresses, the real

712
00:39:03,500 --> 00:39:06,880
stresses that we know already,
which correspond to the

713
00:39:06,880 --> 00:39:10,020
configuration at time t.

714
00:39:10,020 --> 00:39:15,150
Then a variation on the
incremental Green-Lagrange

715
00:39:15,150 --> 00:39:19,610
strain, but I should add its
only the nonlinear increment

716
00:39:19,610 --> 00:39:24,750
in the Green-Lagrange strain
corresponding to iteration k.

717
00:39:24,750 --> 00:39:29,290
Once again, integrated over
the volume at time t.

718
00:39:29,290 --> 00:39:33,520
And on the right-hand side, we
have the external virtual work

719
00:39:33,520 --> 00:39:35,290
and this expression.

720
00:39:35,290 --> 00:39:37,770
Let's look at this expression
carefully.

721
00:39:37,770 --> 00:39:41,845
Here we have an integral over
the volume at time t plus

722
00:39:41,845 --> 00:39:44,930
delta t in the iteration
k minus 1.

723
00:39:44,930 --> 00:39:48,490
Therefore, we are updating that
volume as we go along in

724
00:39:48,490 --> 00:39:50,120
the iteration.

725
00:39:50,120 --> 00:39:53,730
Notice that here we have the
stresses corresponding to

726
00:39:53,730 --> 00:39:55,500
iteration k minus 1.

727
00:39:55,500 --> 00:39:58,040
We are updating these stresses,
of course, also in

728
00:39:58,040 --> 00:39:59,280
the iteration.

729
00:39:59,280 --> 00:40:02,580
Here we have the virtual strains
corresponding to

730
00:40:02,580 --> 00:40:07,870
iteration at the end of
iteration k minus 1.

731
00:40:07,870 --> 00:40:09,600
We are updating this as well.

732
00:40:09,600 --> 00:40:13,350
And this product here, once
again, is integrated over the

733
00:40:13,350 --> 00:40:17,510
volume at time t plus delta t
corresponding to the end of

734
00:40:17,510 --> 00:40:22,060
iteration k minus 1 or the
beginning of iteration k.

735
00:40:22,060 --> 00:40:27,290
Well it's important to realize
that when this right-hand side

736
00:40:27,290 --> 00:40:34,140
is 0, we have fulfilled, or we
have basically solved, the

737
00:40:34,140 --> 00:40:37,070
equation of the principle of
virtual work, because the

738
00:40:37,070 --> 00:40:42,530
external virtual work is equal
to the internal virtual work.

739
00:40:42,530 --> 00:40:44,790
And then, of course, we have
obtained the solution.

740
00:40:44,790 --> 00:40:49,290
So we iterate with k equals 1,
2, 3, and so on, until the

741
00:40:49,290 --> 00:40:54,370
right-hand side is 0, and then
we are satisfying that the

742
00:40:54,370 --> 00:40:59,880
external virtual work is equal
to the internal virtual work.

743
00:40:59,880 --> 00:41:05,180
When we discretize this
continuum mechanics equation

744
00:41:05,180 --> 00:41:10,410
via finite elements, we obtain
this equation here.

745
00:41:10,410 --> 00:41:14,590
ttK, once again, the tangent
stiffness matrix.

746
00:41:14,590 --> 00:41:17,590
Here, the increment in the
nodal point displacements

747
00:41:17,590 --> 00:41:19,420
corresponding to iteration k.

748
00:41:19,420 --> 00:41:21,930
On the right-hand side
the R vector of

749
00:41:21,930 --> 00:41:23,730
externally applied loads.

750
00:41:23,730 --> 00:41:27,370
And here, the nodal point force
vector corresponding to

751
00:41:27,370 --> 00:41:31,510
the internal element stresses
at time t plus delta t in

752
00:41:31,510 --> 00:41:34,050
iteration k minus 1.

753
00:41:34,050 --> 00:41:36,080
At the end of iteration
k minus 1 is

754
00:41:36,080 --> 00:41:37,670
more precisely said.

755
00:41:37,670 --> 00:41:40,810
Notice that we are having a t
plus delta t down here because

756
00:41:40,810 --> 00:41:42,440
we are always updating
the actual

757
00:41:42,440 --> 00:41:44,920
configuration of reference.

758
00:41:44,920 --> 00:41:47,160
Here we are updating the
configuration of reference

759
00:41:47,160 --> 00:41:51,470
because we are working with the
Cauchy stresses at time t

760
00:41:51,470 --> 00:41:55,390
plus delta t, and in iteration
at the end of

761
00:41:55,390 --> 00:41:57,540
iteration k minus 1.

762
00:41:57,540 --> 00:42:01,010
Of course, this vector is
calculated from the

763
00:42:01,010 --> 00:42:06,080
displacement at time
t plus delta t, and

764
00:42:06,080 --> 00:42:07,540
iteration k minus 1.

765
00:42:07,540 --> 00:42:10,160
These, of course, are known.

766
00:42:10,160 --> 00:42:14,710
This is a known quantity,
whereas the unknown quantity

767
00:42:14,710 --> 00:42:16,840
are the increments in
the displacements.

768
00:42:16,840 --> 00:42:19,020
These we want to calculate.

769
00:42:19,020 --> 00:42:23,560
We iterate until, with k equals
1, 2, 3, and so on,

770
00:42:23,560 --> 00:42:26,130
until the right-hand
side is 0.

771
00:42:26,130 --> 00:42:30,380
Of course, this is nothing else
than an expression of, in

772
00:42:30,380 --> 00:42:33,960
finite element analysis, of the
principle of virtual work

773
00:42:33,960 --> 00:42:36,540
where we had also the right-hand
side, this was the

774
00:42:36,540 --> 00:42:39,620
external virtual work, and this
was the internal virtual

775
00:42:39,620 --> 00:42:44,850
work corresponding to the
current element stresses.

776
00:42:44,850 --> 00:42:50,738
Notice that the displacements
that we are talking about here

777
00:42:50,738 --> 00:42:56,500
are obtained by adding all the
incremental displacement that

778
00:42:56,500 --> 00:42:58,815
we're dealing with here.

779
00:42:58,815 --> 00:43:02,840
We are taking these and add them
all up as expressed in

780
00:43:02,840 --> 00:43:08,660
this summation sign here. j goes
from 1 to k to get, and

781
00:43:08,660 --> 00:43:11,800
we take that sum, of course,
and add it to tU, the

782
00:43:11,800 --> 00:43:15,820
displacement corresponding to
configuration time t, to get

783
00:43:15,820 --> 00:43:17,310
the displacements

784
00:43:17,310 --> 00:43:21,750
corresponding to the last iteration.

785
00:43:21,750 --> 00:43:24,878
The total iterative process is
summarized on this view graph.

786
00:43:24,878 --> 00:43:27,510

787
00:43:27,510 --> 00:43:31,060
We start with a given tU, the
displacement at time t are

788
00:43:31,060 --> 00:43:35,340
given, and the external loads
are also given corresponding

789
00:43:35,340 --> 00:43:37,260
to time t plus delta t.

790
00:43:37,260 --> 00:43:41,470
We compute the tangent stiffness
matrix corresponding

791
00:43:41,470 --> 00:43:45,770
to time t, and the nodal point
force vector corresponding to

792
00:43:45,770 --> 00:43:50,640
the internal element
stresses at time t.

793
00:43:50,640 --> 00:43:54,440
This nodal point force vector
provides initial conditions

794
00:43:54,440 --> 00:43:57,510
for this vector, because that is
the vector which will enter

795
00:43:57,510 --> 00:43:59,550
actually into the iteration.

796
00:43:59,550 --> 00:44:01,600
This displacement vector
provides the initial

797
00:44:01,600 --> 00:44:03,890
conditions for that vector.

798
00:44:03,890 --> 00:44:07,900
We set the iteration counter
k equal to 1.

799
00:44:07,900 --> 00:44:13,690
We go in here, and notice now we
are solving this equation,

800
00:44:13,690 --> 00:44:16,680
which we just discussed.

801
00:44:16,680 --> 00:44:20,410
When k is equal to 1, of course,
we have here F0, and

802
00:44:20,410 --> 00:44:24,520
that F0 is nothing else
than that vector here.

803
00:44:24,520 --> 00:44:28,340
Because that's how we
initialized it.

804
00:44:28,340 --> 00:44:30,030
So we can calculate
this incremental

805
00:44:30,030 --> 00:44:31,880
displacement vector.

806
00:44:31,880 --> 00:44:34,580
We add that increment and
displacement vector to the

807
00:44:34,580 --> 00:44:36,270
displacement which
we had already.

808
00:44:36,270 --> 00:44:40,160
When k is equal to 1, of course,
this vector here is

809
00:44:40,160 --> 00:44:45,525
nothing else than that vector,
which we are given already.

810
00:44:45,525 --> 00:44:48,130
We add on and get a new
estimate for the

811
00:44:48,130 --> 00:44:49,650
displacements.

812
00:44:49,650 --> 00:44:51,670
We then check for convergence.

813
00:44:51,670 --> 00:44:54,330
This means we are checking
whether equilibrium is

814
00:44:54,330 --> 00:44:57,280
satisfied sufficiently, meaning
whether the right-hand

815
00:44:57,280 --> 00:45:04,080
side is 0 with this new
displacement estimate.

816
00:45:04,080 --> 00:45:10,010
If we have not converged, we go
into the next calculation

817
00:45:10,010 --> 00:45:14,150
of the force vector, the nodal
point force vector

818
00:45:14,150 --> 00:45:17,440
corresponding to the current
internal element stresses.

819
00:45:17,440 --> 00:45:25,060
And notice that, of course, now
this force vector, these

820
00:45:25,060 --> 00:45:29,520
elements stresses that go in
here, are computed based on

821
00:45:29,520 --> 00:45:33,560
the displacement at time t plus
delta t at the end of

822
00:45:33,560 --> 00:45:35,400
iteration k.

823
00:45:35,400 --> 00:45:38,700
In other words, these are the
displacements that we have

824
00:45:38,700 --> 00:45:42,380
just been calculating
right there.

825
00:45:42,380 --> 00:45:45,355
We increase the iteration
counter, and keep

826
00:45:45,355 --> 00:45:47,420
on looping as before.

827
00:45:47,420 --> 00:45:52,750
So this is the loop that we go
through in the iteration until

828
00:45:52,750 --> 00:45:56,760
we converge, until
we converge.

829
00:45:56,760 --> 00:46:02,370
If you compare this view graph
with the view graph that we

830
00:46:02,370 --> 00:46:05,220
discussed in the total
Lagrangian formulation, you

831
00:46:05,220 --> 00:46:07,350
will find a great similarity.

832
00:46:07,350 --> 00:46:13,260
You will find that basically all
of these quantities have

833
00:46:13,260 --> 00:46:15,760
already been discussed in the
total Lagrangian formulation,

834
00:46:15,760 --> 00:46:18,440
except that in the total
Lagrangian formulation we

835
00:46:18,440 --> 00:46:25,350
always carry it down here and
down there as 0 instead of t,

836
00:46:25,350 --> 00:46:28,280
t plus delta t here.

837
00:46:28,280 --> 00:46:29,830
We carry it as 0.

838
00:46:29,830 --> 00:46:33,510
Well because the reference
configuration was the 0

839
00:46:33,510 --> 00:46:34,690
configuration.

840
00:46:34,690 --> 00:46:36,370
Now the reference configuration
for the

841
00:46:36,370 --> 00:46:39,770
stiffness matrix is the
t configuration.

842
00:46:39,770 --> 00:46:43,010
The reference configuration for
the F vector is the t plus

843
00:46:43,010 --> 00:46:48,700
delta t configuration after
iteration k minus 1.

844
00:46:48,700 --> 00:46:52,970
So there's a great similarity
in the iteration process, in

845
00:46:52,970 --> 00:46:56,260
the finite element matrices
that we are setting up.

846
00:46:56,260 --> 00:47:01,345
And of course, we have to ask
ourselves what comparison is

847
00:47:01,345 --> 00:47:03,650
there between the T.L.
and U.L. formulation?

848
00:47:03,650 --> 00:47:06,960
Why would one want to use the
U.L. formulation if we have

849
00:47:06,960 --> 00:47:09,680
the T.L. formulation,
for example.

850
00:47:09,680 --> 00:47:14,130
Well in the T.L. formulation,
all the derivatives are

851
00:47:14,130 --> 00:47:17,710
defined with respect to the
original configuration.

852
00:47:17,710 --> 00:47:22,000
In the U.L. formulation, they
are defined with respect to

853
00:47:22,000 --> 00:47:25,030
the current configuration.

854
00:47:25,030 --> 00:47:27,330
And that will provide
differences, that does provide

855
00:47:27,330 --> 00:47:28,180
differences.

856
00:47:28,180 --> 00:47:32,510
For example, in the U.L.
formulation, we will have to

857
00:47:32,510 --> 00:47:37,280
calculate the derivatives
newly in each iteration,

858
00:47:37,280 --> 00:47:41,040
because the configuration
is continuously updated.

859
00:47:41,040 --> 00:47:43,940
We don't need to do it, of
course, for the k matrix if we

860
00:47:43,940 --> 00:47:46,860
keep the k matrix constant
on the left-hand side.

861
00:47:46,860 --> 00:47:49,660
But for the F vector on the
right-hand side, we will have

862
00:47:49,660 --> 00:47:53,795
to always calculate new
derivatives because the F

863
00:47:53,795 --> 00:47:57,200
vector involves the Cauchy
stresses times the

864
00:47:57,200 --> 00:48:01,620
infinitesimal virtual strains
referred to the current

865
00:48:01,620 --> 00:48:07,320
configuration, t plus delta t,
iteration k minus 1, which is

866
00:48:07,320 --> 00:48:08,180
being updated.

867
00:48:08,180 --> 00:48:11,240
And so we would always calculate
new derivatives

868
00:48:11,240 --> 00:48:13,370
right there.

869
00:48:13,370 --> 00:48:15,270
There, of course, is an
expense involved.

870
00:48:15,270 --> 00:48:18,790
On the other hand, the U.L.
formulation does not have this

871
00:48:18,790 --> 00:48:21,600
initial displacement effect that
I mentioned earlier, so

872
00:48:21,600 --> 00:48:23,460
there's an advantage.

873
00:48:23,460 --> 00:48:25,630
In the U.L. formulation,
we also directly

874
00:48:25,630 --> 00:48:27,440
deal with Cauchy stresses.

875
00:48:27,440 --> 00:48:29,430
These are, of course, the
physical stresses that we are

876
00:48:29,430 --> 00:48:32,080
actually interested in, and that
we want to get printed

877
00:48:32,080 --> 00:48:33,880
out from the computer program.

878
00:48:33,880 --> 00:48:37,810
Hence, there's no transformation
necessary from

879
00:48:37,810 --> 00:48:39,700
some stress measure, and I'm
thinking of the second

880
00:48:39,700 --> 00:48:41,690
Piola-Kirchhoff stress that we
use in the total Lagrangian

881
00:48:41,690 --> 00:48:44,850
formulation, to the Cauchy
stresses that we actually want

882
00:48:44,850 --> 00:48:47,400
to obtain in the computer
program, and that we as

883
00:48:47,400 --> 00:48:50,750
engineers want to get, of
course, printed out.

884
00:48:50,750 --> 00:48:55,420
Well if we now look at one other
important point, one

885
00:48:55,420 --> 00:48:59,990
important point, regarding the
T.L. and U.L. formulation, we

886
00:48:59,990 --> 00:49:02,490
recognize that if we make
the assumption, the same

887
00:49:02,490 --> 00:49:05,020
assumptions, in both of these
two formulations in the

888
00:49:05,020 --> 00:49:08,520
linearization, as we, of course,
have done, really, the

889
00:49:08,520 --> 00:49:10,160
way I've presented it.

890
00:49:10,160 --> 00:49:15,210
Then the U.L. and T.L.
formulation, provided the

891
00:49:15,210 --> 00:49:18,960
appropriate transformations
are made, certain

892
00:49:18,960 --> 00:49:22,610
transformation rules are
followed for the stresses, the

893
00:49:22,610 --> 00:49:24,040
strains, and so on.

894
00:49:24,040 --> 00:49:26,740
Those rules we will be talking
about further later on in

895
00:49:26,740 --> 00:49:27,830
later lectures.

896
00:49:27,830 --> 00:49:33,130
Then, indeed, exactly the same
stiffness matrix and force

897
00:49:33,130 --> 00:49:37,580
vector, F vector, vectors I
should say, are calculated.

898
00:49:37,580 --> 00:49:42,040
In other words, provided we
follow certain transformation

899
00:49:42,040 --> 00:49:48,000
rules, indeed, the same
stiffness matrices and force

900
00:49:48,000 --> 00:49:53,340
vectors are calculated in both
of these formulations.

901
00:49:53,340 --> 00:49:56,230
Of course, the actual numerical
arithmetic required

902
00:49:56,230 --> 00:49:58,730
to obtain those matrices
and vectors

903
00:49:58,730 --> 00:50:00,020
are, of course, different.

904
00:50:00,020 --> 00:50:03,090
But the final result
is the same.

905
00:50:03,090 --> 00:50:05,900
We can demonstrate that in the
general case, as I will do

906
00:50:05,900 --> 00:50:09,720
later on in a lecture, and we
can also demonstrate that on

907
00:50:09,720 --> 00:50:11,410
simple examples.

908
00:50:11,410 --> 00:50:14,020
In fact, we will look in one
lecture at the truss element

909
00:50:14,020 --> 00:50:17,850
where we can directly show
that the T.L. and U.L.

910
00:50:17,850 --> 00:50:21,020
formulation really give exactly
the same result for

911
00:50:21,020 --> 00:50:23,120
the k matrix, as well
as the F vector.

912
00:50:23,120 --> 00:50:25,890
So there we have an example,
but as I mentioned, we can

913
00:50:25,890 --> 00:50:28,380
also show it in the general
case, and that we do

914
00:50:28,380 --> 00:50:30,480
as well later on.

915
00:50:30,480 --> 00:50:32,180
Thank you very much for
your attention.

916
00:50:32,180 --> 00:50:33,579