1 00:00:00,040 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:03,880 Commons license. 3 00:00:03,880 --> 00:00:06,920 Your support will help MIT OpenCourseWare continue to 4 00:00:06,920 --> 00:00:10,570 offer high quality educational resources for free. 5 00:00:10,570 --> 00:00:13,470 To make a donation or view additional materials from 6 00:00:13,470 --> 00:00:17,400 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,400 --> 00:00:18,650 ocw.mit.edu. 8 00:00:21,980 --> 00:00:23,920 PROFESSOR: Ladies and gentlemen, welcome to this 9 00:00:23,920 --> 00:00:27,240 lecture on nonlinear finite element analysis of solids and 10 00:00:27,240 --> 00:00:28,430 structures. 11 00:00:28,430 --> 00:00:31,350 In the previous lectures, we considered the general 12 00:00:31,350 --> 00:00:34,100 continuum mechanics formulations that we use for 13 00:00:34,100 --> 00:00:36,230 nonlinear finite element analysis. 14 00:00:36,230 --> 00:00:39,940 And we also introduced briefly the finite element matrices. 15 00:00:39,940 --> 00:00:42,530 In the coming lectures, I would like now to discuss with 16 00:00:42,530 --> 00:00:47,470 you these finite element matrices in more detail. 17 00:00:47,470 --> 00:00:51,640 Finite element matrices can generally be categorized as 18 00:00:51,640 --> 00:00:53,240 continuum elements. 19 00:00:53,240 --> 00:00:57,060 We call them sometimes also solid elements. 20 00:00:57,060 --> 00:00:58,510 The truss element, for example, would 21 00:00:58,510 --> 00:01:00,500 be a continuum element. 22 00:01:00,500 --> 00:01:04,050 The 2D element, the 3D elements would 23 00:01:04,050 --> 00:01:05,560 be continuum elements. 24 00:01:05,560 --> 00:01:10,370 The 2D and 3D elements we may also call solid elements. 25 00:01:10,370 --> 00:01:14,300 And as another category, we have the structural elements. 26 00:01:14,300 --> 00:01:16,920 Structure elements are beam elements, plate elements, 27 00:01:16,920 --> 00:01:18,030 shell elements. 28 00:01:18,030 --> 00:01:20,490 Of course, a distinguishing feature between the structural 29 00:01:20,490 --> 00:01:23,860 elements and the continuum elements is that continuum 30 00:01:23,860 --> 00:01:27,880 elements carry only nodal point displacements as degrees 31 00:01:27,880 --> 00:01:30,880 of freedom, whereas the structural elements have also 32 00:01:30,880 --> 00:01:34,140 rotational degrees of freedom at their nodes. 33 00:01:34,140 --> 00:01:37,920 In this lecture, I like to talk about the 2D continuum 34 00:01:37,920 --> 00:01:41,820 elements, the 2D plane stress, plane strain, and 35 00:01:41,820 --> 00:01:43,700 axisymmetric elements. 36 00:01:43,700 --> 00:01:47,230 These elements are used very, very widely in the engineering 37 00:01:47,230 --> 00:01:50,460 professions for all sorts of analyses-- 38 00:01:50,460 --> 00:01:54,620 plane stress analyses of plates, plane strain analysis 39 00:01:54,620 --> 00:01:58,670 all dams, axisymmetric analysis of shells, and 40 00:01:58,670 --> 00:02:01,120 so on and so on. 41 00:02:01,120 --> 00:02:05,140 The elements are very general, and can be used for geometric 42 00:02:05,140 --> 00:02:07,770 and material nonlinear analyses. 43 00:02:07,770 --> 00:02:11,290 I also like to then, at the end of the lecture, talk 44 00:02:11,290 --> 00:02:14,780 briefly about the 3D elements that are also very widely 45 00:02:14,780 --> 00:02:18,950 used, and that are really formulated in the same way as 46 00:02:18,950 --> 00:02:20,220 the 2D elements. 47 00:02:20,220 --> 00:02:24,010 Therefore, once you understand to 2D elements very well, it 48 00:02:24,010 --> 00:02:27,030 is fairly easy to generalize these concepts or use these 49 00:02:27,030 --> 00:02:31,200 concepts also to construct and formulate 3D elements. 50 00:02:31,200 --> 00:02:35,110 Let me now go over to my view graph and discuss it with you 51 00:02:35,110 --> 00:02:37,850 the information that I have on these view graphs. 52 00:02:37,850 --> 00:02:41,310 Once again, I like to talk about plane stress, plane 53 00:02:41,310 --> 00:02:43,130 strain elements, and axisymmetric 54 00:02:43,130 --> 00:02:45,410 elements in this lecture. 55 00:02:45,410 --> 00:02:49,450 And these derivations that we will be discussing, as I said 56 00:02:49,450 --> 00:02:53,090 already, are directly applicable also, or can 57 00:02:53,090 --> 00:02:56,260 directly be extended to three dimensional elements. 58 00:02:56,260 --> 00:02:58,560 Let's look at a typical 2D element, 59 00:02:58,560 --> 00:03:00,010 two dimensional element. 60 00:03:00,010 --> 00:03:03,720 This is a nine-node element in the stationary coordinate 61 00:03:03,720 --> 00:03:05,960 frame, x1, x2. 62 00:03:05,960 --> 00:03:09,430 x times 0, we would see this element here. 63 00:03:09,430 --> 00:03:12,790 Notice there are nine nodes, 1 to 9. 64 00:03:12,790 --> 00:03:14,970 Notice that we will be talking about the 65 00:03:14,970 --> 00:03:17,550 isoparametric elements. 66 00:03:17,550 --> 00:03:22,790 And these have the R and F auxiliary coordinate system, 67 00:03:22,790 --> 00:03:27,960 the natural coordinate system, just like in linear analysis. 68 00:03:27,960 --> 00:03:30,570 X times 0, to the element is here. 69 00:03:30,570 --> 00:03:33,370 And at time t, the element is here. 70 00:03:33,370 --> 00:03:36,800 Notice that the element has undergone large displacements, 71 00:03:36,800 --> 00:03:38,730 large rotations. 72 00:03:38,730 --> 00:03:41,180 You don't see very large rotations here, but the 73 00:03:41,180 --> 00:03:43,390 rotations could be very large. 74 00:03:43,390 --> 00:03:45,820 And certainly also large strains. 75 00:03:45,820 --> 00:03:48,920 You can see directly that the element here has grown from 76 00:03:48,920 --> 00:03:50,660 its size, so certainly it must have been 77 00:03:50,660 --> 00:03:53,300 subjected to large strains. 78 00:03:53,300 --> 00:03:55,730 So we consider really a very general motion. 79 00:03:55,730 --> 00:03:59,950 But remember, once again, that the coordinate frame, x1, x2, 80 00:03:59,950 --> 00:04:03,120 the Cartesian coordinate frame, remains stationary, as 81 00:04:03,120 --> 00:04:07,070 we have discussed in the previous lectures. 82 00:04:07,070 --> 00:04:10,610 Because the elements are isoparametric elements, we can 83 00:04:10,610 --> 00:04:12,710 directly write these expressions. 84 00:04:12,710 --> 00:04:17,310 That 0x1's, the coordinates of the material points in the 85 00:04:17,310 --> 00:04:21,740 elements at time 0, are given via this interpolation. 86 00:04:21,740 --> 00:04:25,730 The hk are the interpolation functions that we also use in 87 00:04:25,730 --> 00:04:27,190 linear analysis. 88 00:04:27,190 --> 00:04:31,370 The 0x1k are the nodal point coordinates. 89 00:04:31,370 --> 00:04:34,120 k refers to the nodal point. 90 00:04:34,120 --> 00:04:36,995 1 refers to the coordinate direction. 91 00:04:36,995 --> 00:04:40,010 0 refers to the fact that they're looking at the 92 00:04:40,010 --> 00:04:41,900 configuration at time 0. 93 00:04:41,900 --> 00:04:44,750 This 0 is, of course, the same 0 that we see here. 94 00:04:44,750 --> 00:04:47,720 This one is the same one that we see there. 95 00:04:47,720 --> 00:04:49,830 We are summing, of course, of all the nodes. 96 00:04:49,830 --> 00:04:52,020 And for the element that I just had on the previous view 97 00:04:52,020 --> 00:04:56,770 graph, it would be nine nodes, so n is equal to 9. 98 00:04:56,770 --> 00:04:58,830 We have a similar expression for the 99 00:04:58,830 --> 00:05:02,420 x2 coordinate direction. 100 00:05:02,420 --> 00:05:05,630 In other words, 0x2 is given like that. 101 00:05:05,630 --> 00:05:08,410 This is the x2 coordinate of a material particle, and it's 102 00:05:08,410 --> 00:05:11,900 expressed in terms of the nodal point coordinates of the 103 00:05:11,900 --> 00:05:16,540 elements expressed in terms of the nodal point coordinates of 104 00:05:16,540 --> 00:05:18,030 the element. 105 00:05:18,030 --> 00:05:21,820 The same expression is also applicable at time t. 106 00:05:21,820 --> 00:05:27,596 Notice all we have exchanged is the 0 to a t, 0 to a t. 107 00:05:27,596 --> 00:05:31,090 And similarly here for the x2 coordinate. 108 00:05:31,090 --> 00:05:33,900 0 to t, and similar here, 0 to t. 109 00:05:36,740 --> 00:05:38,800 Let us look at an example. 110 00:05:38,800 --> 00:05:41,890 Here we have depicted a four-node element. 111 00:05:41,890 --> 00:05:43,950 The original element lies here. 112 00:05:43,950 --> 00:05:45,400 It's black. 113 00:05:45,400 --> 00:05:47,780 The R and F system, of course, is the 114 00:05:47,780 --> 00:05:49,480 natural coordinate system. 115 00:05:49,480 --> 00:05:52,590 This is a configuration of the element at time 0. 116 00:05:52,590 --> 00:05:57,200 It moves into that configuration to time t, or it 117 00:05:57,200 --> 00:06:00,620 is at, in this configuration, at time t. 118 00:06:00,620 --> 00:06:04,530 Notice that these are the interpolations that I just 119 00:06:04,530 --> 00:06:06,620 introduced you to. 120 00:06:06,620 --> 00:06:08,830 The hk functions are, of course, 121 00:06:08,830 --> 00:06:10,250 interpolation functions. 122 00:06:10,250 --> 00:06:15,800 Once again, these are here, the nodal point coordinates. 123 00:06:15,800 --> 00:06:21,200 k is the nodal point, t is the time that we're looking at, xi 124 00:06:21,200 --> 00:06:23,090 means the i's direction. 125 00:06:23,090 --> 00:06:26,610 i, of course, in this particular case, 1 or 2. 126 00:06:26,610 --> 00:06:30,240 Similarly, for the original geometry of the element. 127 00:06:30,240 --> 00:06:34,640 The hk's are listed out here on the right-hand side. 128 00:06:34,640 --> 00:06:37,490 Notice these are the same interpolation functions that 129 00:06:37,490 --> 00:06:39,540 we are using in linear analysis. 130 00:06:39,540 --> 00:06:40,720 No difference there. 131 00:06:40,720 --> 00:06:44,270 For example, h1 is given right here. 132 00:06:44,270 --> 00:06:46,980 And this is, of course, the interpolation function 133 00:06:46,980 --> 00:06:53,410 corresponding to nodal point 1, as shown right there. 134 00:06:53,410 --> 00:06:55,570 You are probably very familiar with these interpolation 135 00:06:55,570 --> 00:06:58,790 functions, and I don't need to go into details there. 136 00:06:58,790 --> 00:07:03,130 But let us look now at the following, namely what had 137 00:07:03,130 --> 00:07:07,440 happened in the motion to a nodal point. 138 00:07:07,440 --> 00:07:11,670 A typical nodal point would be the second nodal point here. 139 00:07:11,670 --> 00:07:16,630 The original coordinates are as shown. 140 00:07:16,630 --> 00:07:21,500 And these original coordinates have grown, or have become 141 00:07:21,500 --> 00:07:26,370 larger as shown here, because this node here has moved to 142 00:07:26,370 --> 00:07:27,650 that new position. 143 00:07:27,650 --> 00:07:33,030 Notice this is now here, the coordinate of node 2, of 144 00:07:33,030 --> 00:07:34,840 course, x1 coordinate. 145 00:07:34,840 --> 00:07:40,480 This superscript 2 means node 2, this 1 means 1 direction. 146 00:07:40,480 --> 00:07:42,560 t means time t. 147 00:07:42,560 --> 00:07:46,940 Here we have 1 and 2 and time 0. 148 00:07:46,940 --> 00:07:50,690 Notice here, 2, 2 times 0. 149 00:07:50,690 --> 00:07:56,330 This 2 here, the bottom 2, means coordinate direction 2. 150 00:07:56,330 --> 00:07:59,210 This top 2 means nodal point 2. 151 00:07:59,210 --> 00:08:02,600 This is a convention that we want to use. 152 00:08:02,600 --> 00:08:07,730 It is a bit heavy, but we have to somehow use a convention to 153 00:08:07,730 --> 00:08:13,040 label our coordinates, and this is the one that I chose 154 00:08:13,040 --> 00:08:14,740 some time ago. 155 00:08:14,740 --> 00:08:19,810 Similar here for the 2 coordinate, x2 156 00:08:19,810 --> 00:08:23,210 coordinate, at time t. 157 00:08:23,210 --> 00:08:27,120 tx2 at nodal point 2. 158 00:08:27,120 --> 00:08:31,050 That upper 2, once again, be the nodal point 2. 159 00:08:31,050 --> 00:08:35,450 And of course, this would also apply for all 160 00:08:35,450 --> 00:08:36,740 the other nodal points. 161 00:08:39,760 --> 00:08:44,179 If we look at the motion of a material particle that is in 162 00:08:44,179 --> 00:08:50,775 the element, we would obtain that motion from the motion of 163 00:08:50,775 --> 00:08:52,050 the nodal points. 164 00:08:52,050 --> 00:08:55,950 Here we have now, in a nine-node element that is 165 00:08:55,950 --> 00:09:00,500 originality here, and moves into this position. 166 00:09:00,500 --> 00:09:03,310 At time t it is in this position. 167 00:09:03,310 --> 00:09:08,200 Let's look at one particular particle within the element. 168 00:09:08,200 --> 00:09:14,120 Here we have one particle right there. 169 00:09:14,120 --> 00:09:18,050 Notice that this particle here is given via this 170 00:09:18,050 --> 00:09:20,112 relationship here. 171 00:09:20,112 --> 00:09:23,810 r and s are both 0.5-- 172 00:09:23,810 --> 00:09:27,050 you see r positive means this direction, s positive means 173 00:09:27,050 --> 00:09:31,170 that direction, r and s 0.5 is there. 174 00:09:31,170 --> 00:09:38,220 We would use r and s equal to 0.5, substitute into hk. 175 00:09:38,220 --> 00:09:45,670 And then, of course, we have 9 such hk's, substitute r and s 176 00:09:45,670 --> 00:09:48,820 equal to 0.5 into each of these hk's. 177 00:09:48,820 --> 00:09:54,410 And sum out, this right-hand side, to get the coordinate, 178 00:09:54,410 --> 00:09:55,380 the coordinates-- 179 00:09:55,380 --> 00:09:56,980 there are two, of course-- 180 00:09:56,980 --> 00:10:02,390 of this point here, this material particle, at time 0. 181 00:10:02,390 --> 00:10:05,655 That's how we would obtain the coordinates of 182 00:10:05,655 --> 00:10:07,180 that material particle. 183 00:10:07,180 --> 00:10:10,950 Now, at time t, we proceed much in the same way. 184 00:10:10,950 --> 00:10:13,790 Here we have the equation. 185 00:10:13,790 --> 00:10:19,850 We would, again, take hk at r equal to 0.5, s equal to 0.5, 186 00:10:19,850 --> 00:10:27,500 for all case, and multiply these hk values by these 187 00:10:27,500 --> 00:10:32,100 values here, which of course, are given because we must know 188 00:10:32,100 --> 00:10:36,210 where these nodal points have arrived at. 189 00:10:36,210 --> 00:10:39,240 So we can evaluate the right-hand side to directly 190 00:10:39,240 --> 00:10:43,470 get these two values, there are two such values, tx1 and 191 00:10:43,470 --> 00:10:48,600 tx2, which gives then the position of this material 192 00:10:48,600 --> 00:10:51,330 particle at time t. 193 00:10:51,330 --> 00:10:55,110 Notice that the isoparametric coordinates of a material 194 00:10:55,110 --> 00:10:57,210 particle never change. 195 00:10:57,210 --> 00:11:00,510 You put that here in red because that is very important 196 00:11:00,510 --> 00:11:02,160 to keep in mind. 197 00:11:02,160 --> 00:11:05,680 Of course, the actual coordinates of the particle 198 00:11:05,680 --> 00:11:08,300 change, because that particle moves through space in a 199 00:11:08,300 --> 00:11:11,150 stationary coordinate frame, x1 and x2. 200 00:11:11,150 --> 00:11:13,290 But we are in s coordinates. 201 00:11:13,290 --> 00:11:17,490 The natural coordinates do not change. 202 00:11:17,490 --> 00:11:22,020 Well a major advantage of the isoparametric finite element 203 00:11:22,020 --> 00:11:26,040 analysis is that we can directly write, of course, at 204 00:11:26,040 --> 00:11:30,600 the displacements, are given as shown here via the nodal 205 00:11:30,600 --> 00:11:32,460 point displacements. 206 00:11:32,460 --> 00:11:36,180 tU1 is the displacement of the material particle 207 00:11:36,180 --> 00:11:37,810 into the one direction. 208 00:11:37,810 --> 00:11:41,260 The hk's are the isoparametric interpolation functions. 209 00:11:41,260 --> 00:11:44,340 And these are the nodal point displacements. 210 00:11:44,340 --> 00:11:46,970 k being the nodal point displacements 211 00:11:46,970 --> 00:11:49,680 of nodal point k. 212 00:11:49,680 --> 00:11:53,110 Similarly, for the 2 direction. 213 00:11:53,110 --> 00:11:57,390 And this holds at time t, and it also holds for the 214 00:11:57,390 --> 00:12:02,220 incremental displacements from time t to time t plus delta t, 215 00:12:02,220 --> 00:12:03,470 as written right here. 216 00:12:05,880 --> 00:12:09,430 That this is, in fact, 2 can easily be shown from the 217 00:12:09,430 --> 00:12:11,160 coordinate interpolations. 218 00:12:11,160 --> 00:12:15,850 You see we had already these two interpolations, and all we 219 00:12:15,850 --> 00:12:20,150 need to do now is subtract on the left-hand side, and on the 220 00:12:20,150 --> 00:12:25,020 right-hand side, to obtain this equation here. 221 00:12:25,020 --> 00:12:27,910 And what are left with here, of course, must be the 222 00:12:27,910 --> 00:12:30,790 displacement of the material particle that 223 00:12:30,790 --> 00:12:32,730 we're looking at-- 224 00:12:32,730 --> 00:12:34,850 tUi. 225 00:12:34,850 --> 00:12:38,450 And here, we must have the displacement 226 00:12:38,450 --> 00:12:40,960 of the nodal points. 227 00:12:40,960 --> 00:12:44,980 And these are denoted as tUik. 228 00:12:44,980 --> 00:12:48,220 And that is exactly the relationships that we were 229 00:12:48,220 --> 00:12:54,110 just dealing with, tUi is equal to hk tUik. 230 00:12:54,110 --> 00:12:57,490 There is one very important point that I like to 231 00:12:57,490 --> 00:12:58,790 point out to you. 232 00:12:58,790 --> 00:13:07,250 Namely, that these equations show directly that if we use a 233 00:13:07,250 --> 00:13:12,270 finite element mesh that is originally compatible, in 234 00:13:12,270 --> 00:13:16,780 other words, compatible in a linear analysis, than this 235 00:13:16,780 --> 00:13:20,320 finite element mesh will remain compatible throughout 236 00:13:20,320 --> 00:13:24,000 the motion, throughout the large deformation motion. 237 00:13:24,000 --> 00:13:25,980 And that is a very important point. 238 00:13:25,980 --> 00:13:31,090 That we can say that the mesh which originally is compatible 239 00:13:31,090 --> 00:13:33,730 will remain compatible throughout the analysis. 240 00:13:33,730 --> 00:13:38,120 That follows directly from these equations. 241 00:13:38,120 --> 00:13:41,150 The element matrices that we need, of course, require 242 00:13:41,150 --> 00:13:45,930 derivatives, and these are obtained much in the same way 243 00:13:45,930 --> 00:13:48,030 as in linear analysis. 244 00:13:48,030 --> 00:13:53,250 We need this derivative here, partial tUi, with respect to 245 00:13:53,250 --> 00:13:56,410 the original coordinates. 246 00:13:56,410 --> 00:13:58,000 This is the actual derivative. 247 00:13:58,000 --> 00:14:00,170 This is the abbreviation that we used 248 00:14:00,170 --> 00:14:01,840 in the earlier lectures. 249 00:14:01,840 --> 00:14:05,050 And we obtained this derivative by taking the 250 00:14:05,050 --> 00:14:09,260 differentiation of the hk's with respect to the original 251 00:14:09,260 --> 00:14:10,190 coordinates. 252 00:14:10,190 --> 00:14:11,340 Of course, these are numbers. 253 00:14:11,340 --> 00:14:14,470 These are the nodal point displacements. 254 00:14:14,470 --> 00:14:17,800 So it's this one that we really need to evaluate this 255 00:14:17,800 --> 00:14:19,020 derivative. 256 00:14:19,020 --> 00:14:21,720 Similarly, for the incremental displacements. 257 00:14:21,720 --> 00:14:24,020 We want to take the differentiation of the 258 00:14:24,020 --> 00:14:26,700 incremental displacement with respect to the original 259 00:14:26,700 --> 00:14:27,550 coordinates. 260 00:14:27,550 --> 00:14:29,050 It's achieved this way. 261 00:14:29,050 --> 00:14:32,600 Once again, here we have an expression that we need to 262 00:14:32,600 --> 00:14:36,870 evaluate, which also goes in here, of course. 263 00:14:36,870 --> 00:14:40,000 And as we evaluate, as we will just now see, much in the same 264 00:14:40,000 --> 00:14:41,870 way as in linear analysis. 265 00:14:41,870 --> 00:14:45,320 Notice, here we have written down the partial of Ui with 266 00:14:45,320 --> 00:14:48,560 respect to the current coordinates 267 00:14:48,560 --> 00:14:50,890 obtained as given here. 268 00:14:50,890 --> 00:14:54,810 Of course, these are the nodal point displacement increments, 269 00:14:54,810 --> 00:14:58,010 and here we have the differentiation of the hk's 270 00:14:58,010 --> 00:15:00,660 with respect to the current coordinates now. 271 00:15:00,660 --> 00:15:06,330 So once we have these evaluated, these expressions 272 00:15:06,330 --> 00:15:10,160 evaluated, we can obtain all of the derivatives that go 273 00:15:10,160 --> 00:15:15,190 into the strain displacement matrices that we want to have 274 00:15:15,190 --> 00:15:17,620 for the element. 275 00:15:17,620 --> 00:15:22,275 The derivatives are evaluated using the chain rule, just as 276 00:15:22,275 --> 00:15:24,570 in linear analysis. 277 00:15:24,570 --> 00:15:28,760 We are using that partial hk with respect to r, is given as 278 00:15:28,760 --> 00:15:33,395 partial hk with respect to x1 times partial x1 with respect 279 00:15:33,395 --> 00:15:36,060 to r, et cetera. 280 00:15:36,060 --> 00:15:38,300 Of course, these are here the derivatives 281 00:15:38,300 --> 00:15:41,120 that we want to calculate. 282 00:15:41,120 --> 00:15:44,240 Notice, this is what we need to calculate in 283 00:15:44,240 --> 00:15:46,240 order to get these. 284 00:15:46,240 --> 00:15:50,340 And these here, just like in linear analysis, go into the 285 00:15:50,340 --> 00:15:53,660 Jacobian matrix written down here. 286 00:15:53,660 --> 00:15:56,460 Here we have the Jacobian matrix. 287 00:15:56,460 --> 00:15:58,540 And this is what we want. 288 00:15:58,540 --> 00:16:01,970 Therefore, we need to invert the Jacobian matrix, as in 289 00:16:01,970 --> 00:16:03,820 linear analysis. 290 00:16:03,820 --> 00:16:07,160 And we obtain via this relationship here, the 291 00:16:07,160 --> 00:16:09,860 required derivatives. 292 00:16:09,860 --> 00:16:13,560 The entries in this matrix involve 293 00:16:13,560 --> 00:16:16,410 derivatives of this form. 294 00:16:16,410 --> 00:16:20,000 Partial x1, 0x1 with respect to r, with 295 00:16:20,000 --> 00:16:22,310 respect to s and so on. 296 00:16:22,310 --> 00:16:27,910 And those are obtained here as shown on the right-hand side. 297 00:16:27,910 --> 00:16:31,070 Notice here, of course, we only need differentiations 298 00:16:31,070 --> 00:16:33,190 with respect to r, with respect to the natural 299 00:16:33,190 --> 00:16:33,830 coordinates. 300 00:16:33,830 --> 00:16:39,440 And since the functions hk are a function of r and f, we can 301 00:16:39,440 --> 00:16:44,660 directly evaluate these kinds of expressions that go in the 302 00:16:44,660 --> 00:16:48,520 Jacobian matrix, and the inverse, of course. 303 00:16:48,520 --> 00:16:51,420 If we want to take derivatives with respect to the current 304 00:16:51,420 --> 00:16:54,390 coordinates, we proceed much in the same way. 305 00:16:54,390 --> 00:16:57,070 This is the relationship that we arrive at by simply 306 00:16:57,070 --> 00:17:04,900 substituting instead of the 0xi, the txi in to the 307 00:17:04,900 --> 00:17:09,319 Jacobian matrix, and of course, into the expressions 308 00:17:09,319 --> 00:17:11,319 that are in here. 309 00:17:11,319 --> 00:17:16,599 So here we have a Jacobian matrix that is giving the 310 00:17:16,599 --> 00:17:20,260 derivatives of the current coordinates at time t with 311 00:17:20,260 --> 00:17:22,859 respect to the natural coordinates. 312 00:17:22,859 --> 00:17:25,349 Such an element is obtained as shown here. 313 00:17:25,349 --> 00:17:28,240 It involves again only the differentiation of the 314 00:17:28,240 --> 00:17:32,100 interpolation functions with respect to the natural 315 00:17:32,100 --> 00:17:33,040 coordinates. 316 00:17:33,040 --> 00:17:36,440 Here, of course with respect to s, because we want to 317 00:17:36,440 --> 00:17:38,590 differentiate with respect to s. 318 00:17:38,590 --> 00:17:42,200 Of course, these are the nodal point coordinate at time t, 319 00:17:42,200 --> 00:17:43,550 which are known. 320 00:17:43,550 --> 00:17:47,090 We invert this relationship here to obtain the 321 00:17:47,090 --> 00:17:51,350 differentiation that we need to have. 322 00:17:51,350 --> 00:17:54,620 We are now ready to compute the required element matrices 323 00:17:54,620 --> 00:17:57,820 for the total Lagrangian formulation. 324 00:17:57,820 --> 00:18:02,100 And the element matrices that we want to compute, of course, 325 00:18:02,100 --> 00:18:06,195 our those established in the early lectures. 326 00:18:06,195 --> 00:18:11,130 The matrices that go into evaluating these matrices are 327 00:18:11,130 --> 00:18:13,860 listed here. 328 00:18:13,860 --> 00:18:17,020 Here, of course, the tangent constitutive relationship, 329 00:18:17,020 --> 00:18:20,260 which we will talk about much later. 330 00:18:20,260 --> 00:18:21,690 Not in this lecture. 331 00:18:21,690 --> 00:18:24,120 Here we have the linear strain displacement matrix, the 332 00:18:24,120 --> 00:18:27,570 nonlinear strain displacement matrix. 333 00:18:27,570 --> 00:18:30,750 Here again, the linear strain displacement matrix, a stress 334 00:18:30,750 --> 00:18:33,650 vector, and a stress matrix. 335 00:18:33,650 --> 00:18:36,680 Let's see now how these matrices look for the two 336 00:18:36,680 --> 00:18:39,010 dimensional case. 337 00:18:39,010 --> 00:18:43,690 The constitutive relation, just very briefly-- 338 00:18:43,690 --> 00:18:47,960 again, much more detail will be given in a later lecture-- 339 00:18:47,960 --> 00:18:51,480 would be relating the increment in stress to the 340 00:18:51,480 --> 00:18:52,810 increments in strain. 341 00:18:52,810 --> 00:18:58,410 Notice there's a 2 here, and this is the matrix which is 342 00:18:58,410 --> 00:19:01,080 established from this relationship here. 343 00:19:01,080 --> 00:19:02,800 In the early lectures, we used this 344 00:19:02,800 --> 00:19:05,800 relationship, the tensor notation. 345 00:19:05,800 --> 00:19:08,960 Well we have now a matrix notation, and this is the 346 00:19:08,960 --> 00:19:11,680 relationship with that matrix notation. 347 00:19:11,680 --> 00:19:16,030 C for a linear elastic material would look as shown 348 00:19:16,030 --> 00:19:20,210 here, and you are familiar with this relationship from 349 00:19:20,210 --> 00:19:21,200 linear analysis. 350 00:19:21,200 --> 00:19:25,280 It's the same matrix that we encounter in linear analysis. 351 00:19:25,280 --> 00:19:27,300 Of course, e being Young's modulus, nu 352 00:19:27,300 --> 00:19:28,575 being Poisson's ratio. 353 00:19:31,700 --> 00:19:35,700 We also derived in the early lectures the expression for 354 00:19:35,700 --> 00:19:37,710 the incremental strain terms. 355 00:19:37,710 --> 00:19:39,510 And here they are listed out. 356 00:19:39,510 --> 00:19:42,640 We have derived these right-hand side expressions. 357 00:19:42,640 --> 00:19:48,110 We notice that what is here underscored by a blue line was 358 00:19:48,110 --> 00:19:50,400 the initial displacement effect. 359 00:19:50,400 --> 00:19:52,900 And is this initial displacement effect, of 360 00:19:52,900 --> 00:19:56,340 course, is a particular in gradient off the total 361 00:19:56,340 --> 00:19:58,310 Lagrangian formulation, as we discussed 362 00:19:58,310 --> 00:19:59,560 in the early lectures. 363 00:20:02,560 --> 00:20:05,800 The nonlinear strain terms that we also derived in the 364 00:20:05,800 --> 00:20:08,640 earlier lectures, are listed here for the 365 00:20:08,640 --> 00:20:12,160 two dimensional case. 366 00:20:12,160 --> 00:20:15,470 We have seen all these expressions on the earlier 367 00:20:15,470 --> 00:20:19,100 view graph, except that we use then additional notation, in 368 00:20:19,100 --> 00:20:24,970 other words, a notation that involved k and j's and i's, 369 00:20:24,970 --> 00:20:26,945 and we have to sum over k. 370 00:20:26,945 --> 00:20:30,320 Well if you do so, you directly arrive at these 371 00:20:30,320 --> 00:20:31,570 relationships here. 372 00:20:35,250 --> 00:20:39,935 It's an interesting point to derive this expression and the 373 00:20:39,935 --> 00:20:43,540 linear strain part for the hoop strain-- this is called 374 00:20:43,540 --> 00:20:44,910 the hoop strain-- 375 00:20:44,910 --> 00:20:48,230 in the axisymmetric case. 376 00:20:48,230 --> 00:20:50,690 And let us look at that derivation 377 00:20:50,690 --> 00:20:52,360 a bit more in detail. 378 00:20:52,360 --> 00:20:57,500 Here we have an axisymmetric element in its original 379 00:20:57,500 --> 00:20:59,260 configuration. 380 00:20:59,260 --> 00:21:04,830 At time 0 it has moved into this configuration up to time 381 00:21:04,830 --> 00:21:06,290 t plus delta t. 382 00:21:06,290 --> 00:21:10,380 Since we want to get the incremental strain from time t 383 00:21:10,380 --> 00:21:13,420 to time t plus delta t, we're looking at the configuration 384 00:21:13,420 --> 00:21:17,040 of time t plus delta t in this derivation. 385 00:21:17,040 --> 00:21:21,310 Notice that this here is the axis of revolution, which we 386 00:21:21,310 --> 00:21:27,140 denote as x2, x1 is this axis. 387 00:21:27,140 --> 00:21:33,470 And if we look as a plan view onto this element, we would 388 00:21:33,470 --> 00:21:39,720 see this x1, x2 coming out of the view graph, and x3 going 389 00:21:39,720 --> 00:21:42,950 down like this. 390 00:21:42,950 --> 00:21:48,980 Notice this here we label as 0ds, the initial length of a 391 00:21:48,980 --> 00:21:50,980 hoop fiber, so to say. 392 00:21:50,980 --> 00:21:56,150 In fact, this hoop fiber starts right there at 0x1, 393 00:21:56,150 --> 00:21:59,980 which is, and I have to go now upwards to the upper graph 394 00:21:59,980 --> 00:22:03,140 again, which is nothing else than the starting point of 395 00:22:03,140 --> 00:22:05,470 that fiber. 396 00:22:05,470 --> 00:22:10,250 In other words, this green dot, this green dot is nothing 397 00:22:10,250 --> 00:22:15,230 else than the start of this arrow. 398 00:22:15,230 --> 00:22:19,660 I could say let's line them up like that. 399 00:22:19,660 --> 00:22:25,590 So this arrow here goes really into the view graph up there. 400 00:22:25,590 --> 00:22:28,800 If you look further to the right here we see that we have 401 00:22:28,800 --> 00:22:33,300 also a blue arrow, of course, curved. 402 00:22:33,300 --> 00:22:36,310 We call it the hoop, it's a circle really, the radius 403 00:22:36,310 --> 00:22:41,430 around this origin that we're looking at, like that. 404 00:22:41,430 --> 00:22:47,130 And notice that the start of this arrow here is this blue 405 00:22:47,130 --> 00:22:48,380 dot right there. 406 00:22:50,650 --> 00:22:53,505 A material fiber, a material particle, let's put it this 407 00:22:53,505 --> 00:22:58,310 way, a material particle that has moved from here to there 408 00:22:58,310 --> 00:23:05,270 causes, in axisymmetric analysis, this fiber here to 409 00:23:05,270 --> 00:23:07,090 take on this length. 410 00:23:10,650 --> 00:23:14,330 And of course, the change in this length 411 00:23:14,330 --> 00:23:17,960 gives us the hoop strain. 412 00:23:17,960 --> 00:23:22,440 Let us look at this relationship here, because 413 00:23:22,440 --> 00:23:26,070 that gives us a relationship between the change in the 414 00:23:26,070 --> 00:23:31,960 length of the fiber, green fiber, blue fiber, so to say, 415 00:23:31,960 --> 00:23:34,020 in plan view. 416 00:23:34,020 --> 00:23:40,420 And that is nothing else than 0x1 dividing t plus delta x1. 417 00:23:40,420 --> 00:23:41,590 Why is that the case? 418 00:23:41,590 --> 00:23:46,210 Well you see it by geometry from this picture. 419 00:23:46,210 --> 00:23:48,930 And we will use this expression here now to 420 00:23:48,930 --> 00:23:50,700 evaluate the actual strain. 421 00:23:54,520 --> 00:23:57,800 We find that the Green-Lagrangian strain can be 422 00:23:57,800 --> 00:24:01,160 written as in this form. 423 00:24:01,160 --> 00:24:05,820 We substitute the expression that we just obtained for this 424 00:24:05,820 --> 00:24:08,110 relationship. 425 00:24:08,110 --> 00:24:11,430 We substitute the displacements. 426 00:24:11,430 --> 00:24:14,750 Displacements to configuration t, and incremental 427 00:24:14,750 --> 00:24:20,430 displacement from t to t plus delta t divided by the 0x1, of 428 00:24:20,430 --> 00:24:22,300 course, still here. 429 00:24:22,300 --> 00:24:26,690 We then go through a number of steps of arithmetic, I'm sure 430 00:24:26,690 --> 00:24:31,300 you can easily do those, and you arrive at this result. 431 00:24:31,300 --> 00:24:35,820 If we look at this, we find that this here expression only 432 00:24:35,820 --> 00:24:40,330 involves the displacements to time t. 433 00:24:40,330 --> 00:24:42,950 And that must, therefore, be the Green-Lagrangian 434 00:24:42,950 --> 00:24:46,590 strain at time t. 435 00:24:46,590 --> 00:24:51,070 Notice that this expression here involves incremental 436 00:24:51,070 --> 00:24:54,330 displacements, linear incremental displacement, no 437 00:24:54,330 --> 00:24:57,000 products of them, and the initial 438 00:24:57,000 --> 00:24:59,610 displacement at time t. 439 00:24:59,610 --> 00:25:02,990 This is the initial displacement effect, which we 440 00:25:02,990 --> 00:25:06,160 always have in the total Lagrangian formulation. 441 00:25:06,160 --> 00:25:10,570 This is a linear strain term involving only the incremental 442 00:25:10,570 --> 00:25:11,540 displacement. 443 00:25:11,540 --> 00:25:16,370 And this the total linear strain increment. 444 00:25:16,370 --> 00:25:20,140 Notice this is a total nonlinear strain incremental. 445 00:25:20,140 --> 00:25:24,380 It involves the incremental displacement u1 squared, and 446 00:25:24,380 --> 00:25:27,420 that's why it is a nonlinear strain increment. 447 00:25:27,420 --> 00:25:32,650 Of course, this total here is the incremental strain. 448 00:25:32,650 --> 00:25:36,310 This total is the incremental strain. 449 00:25:36,310 --> 00:25:41,590 It's a nice derivation that gives some insight into how 450 00:25:41,590 --> 00:25:44,270 these expressions that I had already on the earlier view 451 00:25:44,270 --> 00:25:46,450 graph are arrived at. 452 00:25:46,450 --> 00:25:50,135 Well we are now ready to construct the B matrix. 453 00:25:52,750 --> 00:25:54,300 On the left-hand side, we would have, 454 00:25:54,300 --> 00:25:56,590 of course, the strains. 455 00:25:56,590 --> 00:26:00,850 If you look at the linear strain displacement matrix, we 456 00:26:00,850 --> 00:26:03,090 have the linear strains here. 457 00:26:03,090 --> 00:26:08,660 Notice there's a 2 here because of the 0e12 being 458 00:26:08,660 --> 00:26:10,540 equal to 0e21. 459 00:26:10,540 --> 00:26:15,000 We simply put 2 times 0e12 in here. 460 00:26:15,000 --> 00:26:18,580 And this is a total linear strain increment. 461 00:26:18,580 --> 00:26:22,670 Here we have a sum of two matrices giving us a total 462 00:26:22,670 --> 00:26:25,380 linear strain displacement matrix. 463 00:26:25,380 --> 00:26:27,760 This is the one that does not include the initial 464 00:26:27,760 --> 00:26:28,910 displacement effect. 465 00:26:28,910 --> 00:26:31,310 That one includes initial displacement effect. 466 00:26:31,310 --> 00:26:35,680 And here we have the nodal point incremental displacement 467 00:26:35,680 --> 00:26:37,860 the way we defined them in an earlier lecture. 468 00:26:40,540 --> 00:26:48,030 Well the entries in t0BL0 are shown here, involving of 469 00:26:48,030 --> 00:26:52,200 course, four-nodal point k derivatives of the 470 00:26:52,200 --> 00:26:54,080 interpolation functions corresponding 471 00:26:54,080 --> 00:26:55,330 to nodal point k. 472 00:26:57,680 --> 00:27:02,290 Notice that this last row is only included if we are 473 00:27:02,290 --> 00:27:06,420 dealing with an axisymmetric analysis. 474 00:27:06,420 --> 00:27:11,990 And notice that these are then exactly the terms that we also 475 00:27:11,990 --> 00:27:15,490 have in linear analysis. 476 00:27:15,490 --> 00:27:18,590 So no surprises here. 477 00:27:18,590 --> 00:27:20,980 No new entries, as a matter of fact, when 478 00:27:20,980 --> 00:27:23,300 compared to linear analysis. 479 00:27:23,300 --> 00:27:27,110 Except that we see a 0 here, meaning, of course, that we're 480 00:27:27,110 --> 00:27:29,440 taking a differentiation with respect to the original 481 00:27:29,440 --> 00:27:31,660 coordinates. 482 00:27:31,660 --> 00:27:34,590 Generally, in linear analysis, if you look at the book, of 483 00:27:34,590 --> 00:27:37,700 course, you would not see this 0 here because it's not 484 00:27:37,700 --> 00:27:39,330 necessary to have that 0 there. 485 00:27:39,330 --> 00:27:41,290 We always take differentiations with respect 486 00:27:41,290 --> 00:27:43,770 to the original geometry. 487 00:27:43,770 --> 00:27:48,140 Maybe a quick word also how we want to read this here. 488 00:27:48,140 --> 00:27:52,630 Notice these two entries here, of course, nothing else in 489 00:27:52,630 --> 00:27:54,360 these two blue entries. 490 00:27:54,360 --> 00:27:58,540 Because these two entries multiply these two columns, 491 00:27:58,540 --> 00:28:02,780 for readability I put this entry there, put this entry 492 00:28:02,780 --> 00:28:05,470 there, so that you directly see this column here 493 00:28:05,470 --> 00:28:10,420 corresponds to u1k, and this column corresponds to u2k. 494 00:28:10,420 --> 00:28:13,430 Of course, both columns correspond to node k. 495 00:28:16,700 --> 00:28:21,170 If we look at the matrix t0BL1, which includes now the 496 00:28:21,170 --> 00:28:24,500 initial displacement effect, in fact, introduces the 497 00:28:24,500 --> 00:28:28,150 initial displacement effect to the total strain displacement 498 00:28:28,150 --> 00:28:32,430 matrix, it looks this way. 499 00:28:32,430 --> 00:28:35,670 Once again, this is a contribution coming 500 00:28:35,670 --> 00:28:39,080 corresponding to u1k This is the contribution 501 00:28:39,080 --> 00:28:41,270 corresponding to u2k. 502 00:28:41,270 --> 00:28:44,050 And notice here you have the initial displacement effect 503 00:28:44,050 --> 00:28:47,820 appearing right here. 504 00:28:47,820 --> 00:28:51,270 All initial displacement effects. 505 00:28:51,270 --> 00:28:54,120 And similarly here, initial displacement effect. 506 00:28:57,070 --> 00:29:01,320 Once again, if you don't have an axisymmetric analysis, in 507 00:29:01,320 --> 00:29:04,000 other words, you have a plane stress, plane strain case, you 508 00:29:04,000 --> 00:29:06,020 would drop this last row. 509 00:29:10,400 --> 00:29:16,000 We construct the t0BNL and T0S matrix. 510 00:29:16,000 --> 00:29:19,770 Next for the geometric stiffness matrix, and we 511 00:29:19,770 --> 00:29:25,470 talked about this expression in the earlier lecture, notice 512 00:29:25,470 --> 00:29:30,790 that this here is giving up, of course, the matrix, the k 513 00:29:30,790 --> 00:29:34,830 matrix, that we're looking for, this part here. 514 00:29:34,830 --> 00:29:37,840 And the F matrix looks as shown here. 515 00:29:40,720 --> 00:29:44,810 I pointed out very strongly in the earlier lecture that we 516 00:29:44,810 --> 00:29:50,740 construct the S and the BNL such that when this product is 517 00:29:50,740 --> 00:29:54,020 taken, we get directly this one. 518 00:29:54,020 --> 00:29:55,650 Because this is basic. 519 00:29:55,650 --> 00:29:57,520 This is obtained from the continuum mechanics 520 00:29:57,520 --> 00:29:59,190 formulation. 521 00:29:59,190 --> 00:30:01,110 And we want to evaluate this. 522 00:30:01,110 --> 00:30:05,650 Therefore, we construct the BNL and S such that this 523 00:30:05,650 --> 00:30:08,970 product, indeed, gives us this contribution. 524 00:30:08,970 --> 00:30:11,870 And as is constructed as shown here. 525 00:30:11,870 --> 00:30:15,420 The BNL is constructed as shown here. 526 00:30:15,420 --> 00:30:19,160 Notice the u2k contributions, the u1k 527 00:30:19,160 --> 00:30:21,670 contribution for node k. 528 00:30:27,840 --> 00:30:33,670 Finally, we also need the t0S vector, hat vector. 529 00:30:33,670 --> 00:30:37,380 I pointed out in the earlier lecture that this vector and 530 00:30:37,380 --> 00:30:43,140 this matrix are constructed in such a way that this product 531 00:30:43,140 --> 00:30:47,260 here gives us exactly this expression here. 532 00:30:47,260 --> 00:30:51,720 This is basic coming from continuum mechanics, and this 533 00:30:51,720 --> 00:30:53,680 is what we have to capture. 534 00:30:53,680 --> 00:30:56,110 The entries in F hat are given here. 535 00:31:00,040 --> 00:31:05,220 And with this then, we would be ready to actually calculate 536 00:31:05,220 --> 00:31:08,630 all the matrices and vectors for the total Lagrangian 537 00:31:08,630 --> 00:31:09,860 formulation. 538 00:31:09,860 --> 00:31:16,220 Let us now look at an example to reinforce our understanding 539 00:31:16,220 --> 00:31:21,070 of how all of these matrices are constructed, 540 00:31:21,070 --> 00:31:23,030 how they are evaluated. 541 00:31:23,030 --> 00:31:25,900 Here we have the following case. 542 00:31:25,900 --> 00:31:30,330 A four-node element originally in the 543 00:31:30,330 --> 00:31:32,230 configuration shown in black. 544 00:31:35,110 --> 00:31:39,310 Four nodes, as you can see, one, two, three, four. 545 00:31:39,310 --> 00:31:44,830 This element moves from time 0 to time t into this 546 00:31:44,830 --> 00:31:46,570 configuration. 547 00:31:46,570 --> 00:31:48,370 The RAD configuration. 548 00:31:48,370 --> 00:31:54,170 Node 0.1 move there, node 0.2 move there. 549 00:31:54,170 --> 00:32:00,520 Notice that the element has stretched and sheared over, 550 00:32:00,520 --> 00:32:04,160 but it has only stretched into the vertical direction, not 551 00:32:04,160 --> 00:32:07,200 into the horizontal direction. 552 00:32:07,200 --> 00:32:13,140 Let us identify the lengths, values that we 553 00:32:13,140 --> 00:32:14,190 have to deal with. 554 00:32:14,190 --> 00:32:16,930 0.2 here, 0.2 there. 555 00:32:16,930 --> 00:32:20,000 Notice that the displacement upward is 0.1. 556 00:32:20,000 --> 00:32:24,640 And the shearing over, so to say, is 0.1. 557 00:32:24,640 --> 00:32:27,550 We want to consider plane strain conditions. 558 00:32:27,550 --> 00:32:33,030 And the problem that we like to pose is calculate these two 559 00:32:33,030 --> 00:32:36,510 matrices, the linear strain, and the nonlinear strain 560 00:32:36,510 --> 00:32:41,440 displacement matrix for this particular case. 561 00:32:41,440 --> 00:32:45,740 Let's first look now a bit at what's happening here to the 562 00:32:45,740 --> 00:32:48,530 material fiber. 563 00:32:48,530 --> 00:32:54,100 If we look at the horizontal material fiber, shown here in 564 00:32:54,100 --> 00:32:59,100 black in the original configuration, we notice that 565 00:32:59,100 --> 00:33:06,050 these material fibers are simply translated rigidly over 566 00:33:06,050 --> 00:33:07,690 horizontally. 567 00:33:07,690 --> 00:33:13,160 As shown here via the red arrows. 568 00:33:13,160 --> 00:33:18,830 So these black material fibers lying horizontally are simply 569 00:33:18,830 --> 00:33:23,870 sheared over rigidly, as shown by the red arrows. 570 00:33:23,870 --> 00:33:25,860 2 time t, of course. 571 00:33:25,860 --> 00:33:31,060 Let's look now at the vertical fibers in this problem. 572 00:33:31,060 --> 00:33:33,140 Let's see what has happened to them. 573 00:33:33,140 --> 00:33:38,990 Here we have the vertical fiber shown in black in the 574 00:33:38,990 --> 00:33:41,020 original configuration. 575 00:33:41,020 --> 00:33:45,050 Let's see how they end up in their final configuration. 576 00:33:45,050 --> 00:33:49,290 We see that these fibers have actually stretched, they've 577 00:33:49,290 --> 00:33:52,470 become longer, and they have also rotated over. 578 00:33:56,570 --> 00:34:00,040 Well this is the kinematics that we are looking at. 579 00:34:00,040 --> 00:34:05,080 And let us now attack the problem of constructing, once 580 00:34:05,080 --> 00:34:10,090 again, the linear strain displacement matrix, and the 581 00:34:10,090 --> 00:34:13,469 nonlinear strain displacement matrix. 582 00:34:13,469 --> 00:34:18,520 We do so by first identifying which is the isoparametric 583 00:34:18,520 --> 00:34:20,790 coordinate system that we use. 584 00:34:20,790 --> 00:34:25,780 That coordinate system is shown here, r and s. 585 00:34:29,360 --> 00:34:33,670 Let us then now start this problem. 586 00:34:33,670 --> 00:34:38,130 And we identify simply by inspection really because it's 587 00:34:38,130 --> 00:34:41,540 a fairly simple geometry that we're dealing with. 588 00:34:41,540 --> 00:34:45,540 That this differentiation is nothing else than 0.1, this 589 00:34:45,540 --> 00:34:50,580 differentiation is 0, that one is 0, and this one is 0.1. 590 00:34:50,580 --> 00:34:53,940 Of course, these are the elements that go into the 591 00:34:53,940 --> 00:34:57,160 Jacobian matrix, as I discussed earlier. 592 00:34:57,160 --> 00:35:02,180 We put these elements into this 2 by 2 matrix, calculate 593 00:35:02,180 --> 00:35:07,660 the determinant, interesting to note what the value is. 594 00:35:07,660 --> 00:35:10,550 That one is needed, of course, later on in the numerical 595 00:35:10,550 --> 00:35:14,960 integration of the matrices. 596 00:35:14,960 --> 00:35:20,330 And we also recognize that to obtain this derivative of an 597 00:35:20,330 --> 00:35:22,770 interpolation function, we need to 598 00:35:22,770 --> 00:35:24,960 invert this matrix here. 599 00:35:24,960 --> 00:35:28,400 This 0.1 inverted gives us 10, and that is 600 00:35:28,400 --> 00:35:30,310 this 10 right there. 601 00:35:30,310 --> 00:35:34,780 Therefore, we have now partial with respect to x1, 0x1 is 10 602 00:35:34,780 --> 00:35:37,810 times partial with respect to R. Of course, in the actual 603 00:35:37,810 --> 00:35:43,110 expression, we would put hk's in here, and hk's in there. 604 00:35:43,110 --> 00:35:45,190 Similar we get the differentiation 605 00:35:45,190 --> 00:35:47,290 with respect to x2. 606 00:35:47,290 --> 00:35:50,360 We will use this expression here, putting the actual hk's 607 00:35:50,360 --> 00:35:53,280 in there and in there. 608 00:35:53,280 --> 00:35:56,700 Well this is some preparatory work to actually complete the 609 00:35:56,700 --> 00:36:01,520 problem of constructing the linear strain and nonlinear 610 00:36:01,520 --> 00:36:03,430 strain displacement matrix. 611 00:36:03,430 --> 00:36:07,270 And I think this is actually a very good point where you 612 00:36:07,270 --> 00:36:11,320 might want to sit back and try to complete the whole problem. 613 00:36:44,360 --> 00:36:46,770 Ladies and gentlemen, welcome again. 614 00:36:46,770 --> 00:36:49,610 I hope you've had a close look at the example, and surely I 615 00:36:49,610 --> 00:36:52,170 would be very interested in knowing how it went. 616 00:36:52,170 --> 00:36:54,870 But let us now look at the solution. 617 00:36:54,870 --> 00:36:58,420 We already discussed that for this example, the Jacobian is 618 00:36:58,420 --> 00:37:01,790 given by this matrix here. 619 00:37:01,790 --> 00:37:04,770 And therefore, these are the differentiations that we can 620 00:37:04,770 --> 00:37:06,710 directly use. 621 00:37:06,710 --> 00:37:09,130 Notice, of course, we need the differentiation with respect 622 00:37:09,130 --> 00:37:13,090 to the x1 coordinate, and with respect to the x2 coordinate 623 00:37:13,090 --> 00:37:17,330 of all the interpolation functions. 624 00:37:17,330 --> 00:37:22,400 One way to proceed now is to make a little table where we 625 00:37:22,400 --> 00:37:27,900 have here in this column the nodes, we have in this column 626 00:37:27,900 --> 00:37:31,400 here the differentiations that we need, partial hk with 627 00:37:31,400 --> 00:37:33,600 respect to 0x1. 628 00:37:33,600 --> 00:37:39,260 Further differentiations here, partial hk with respect to x2. 629 00:37:39,260 --> 00:37:43,710 Notice that the Jacobian, the inverse of the Jacobian matrix 630 00:37:43,710 --> 00:37:48,812 being 10, makes a 1/4 equal to 2.-- 631 00:37:48,812 --> 00:37:52,350 1/4 times 10 being equal to 2.5. 632 00:37:52,350 --> 00:37:56,420 So that's why you see the 2.5 here. 633 00:37:56,420 --> 00:37:59,700 These are the nodal point displacements, which, of 634 00:37:59,700 --> 00:38:02,340 course, are given for this particular case. 635 00:38:02,340 --> 00:38:05,420 And with these nodal point displacements given, and these 636 00:38:05,420 --> 00:38:09,580 differentiations calculated as listed here, we can now 637 00:38:09,580 --> 00:38:12,980 evaluate these products here. 638 00:38:12,980 --> 00:38:17,620 And that is done as shown in these columns. 639 00:38:17,620 --> 00:38:22,870 Notice the sum of these gives us these differentiations 640 00:38:22,870 --> 00:38:27,960 here, which go into the initial displacement effect of 641 00:38:27,960 --> 00:38:29,210 the strain displacement matrix. 642 00:38:31,710 --> 00:38:35,150 Well like that, of course, you can also calculate the 643 00:38:35,150 --> 00:38:39,780 differentiations of the t0u21 and the t0u22. 644 00:38:42,670 --> 00:38:46,400 And both these terms are also required in the initial 645 00:38:46,400 --> 00:38:50,280 displacement effect of the strain displacement matrix. 646 00:38:53,000 --> 00:38:56,050 Another way to proceed, to get this initial displacement 647 00:38:56,050 --> 00:39:00,700 effect, is to evaluate the deformation gradient. 648 00:39:00,700 --> 00:39:03,400 In an earlier lecture, we talked about the deformation 649 00:39:03,400 --> 00:39:06,170 gradient and it's listed right here. 650 00:39:06,170 --> 00:39:08,700 It's a 2 by 2 matrix because we're talking about a two 651 00:39:08,700 --> 00:39:10,660 dimensional motion. 652 00:39:10,660 --> 00:39:14,100 And here you have, for example, partial tx2 with 653 00:39:14,100 --> 00:39:15,450 respect to 0x1. 654 00:39:15,450 --> 00:39:21,100 Here you have partial tx1 with respect to 0x1. 655 00:39:21,100 --> 00:39:25,190 0x1, et cetera. 656 00:39:25,190 --> 00:39:29,960 And if we take this deformation gradient or the 657 00:39:29,960 --> 00:39:32,910 elements of that information gradient, we can directly 658 00:39:32,910 --> 00:39:36,110 obtain these differentiations here. 659 00:39:36,110 --> 00:39:41,190 In other words, t0x11 minus 1 must give us 660 00:39:41,190 --> 00:39:43,270 this expression here. 661 00:39:43,270 --> 00:39:50,900 t0x12, being that one here, gives us tu1 with respect to 662 00:39:50,900 --> 00:39:53,960 0x2, in shorthand notation written as 663 00:39:53,960 --> 00:39:57,400 shown here, et cetera. 664 00:39:57,400 --> 00:40:00,430 So we can obtain, in other words, these elements also 665 00:40:00,430 --> 00:40:01,945 from the deformation gradient. 666 00:40:05,210 --> 00:40:08,090 Let us now look at how the columns of the strain 667 00:40:08,090 --> 00:40:10,240 displacement matrices look. 668 00:40:10,240 --> 00:40:13,540 We simply substitute into the general expressions that I 669 00:40:13,540 --> 00:40:21,780 gave you earlier, and here we have the t0 be a 0 entry for 670 00:40:21,780 --> 00:40:23,910 node three. 671 00:40:23,910 --> 00:40:27,430 Of course there are eight columns altogether because we 672 00:40:27,430 --> 00:40:29,230 have a four-node element. 673 00:40:29,230 --> 00:40:32,400 We just showed two such columns here for-- 674 00:40:32,400 --> 00:40:34,750 namely, those corresponding to node three. 675 00:40:34,750 --> 00:40:39,550 Similarly, for t0BL1, we get these entries here. 676 00:40:43,040 --> 00:40:47,330 Once again, also for node three, of course. 677 00:40:47,330 --> 00:40:50,280 Similarly, we can also construct the corresponding 678 00:40:50,280 --> 00:40:53,600 columns in the t0BNL matrix. 679 00:40:53,600 --> 00:40:56,170 And these columns are given right there. 680 00:40:59,110 --> 00:41:01,060 Once again, for node three. 681 00:41:01,060 --> 00:41:06,050 This is, of course, also a matrix that have altogether 682 00:41:06,050 --> 00:41:09,540 eight columns, because there are four nodes, eight degrees 683 00:41:09,540 --> 00:41:12,100 of freedom. 684 00:41:12,100 --> 00:41:15,870 Let us now, as a next step, consider the updated 685 00:41:15,870 --> 00:41:21,340 Lagrangian formulation in the general plane stress, plane 686 00:41:21,340 --> 00:41:23,650 strain, axisymmetric case. 687 00:41:23,650 --> 00:41:26,870 In the updated Lagrangian formulation, we have 688 00:41:26,870 --> 00:41:30,290 identified earlier, in an earlier lecture, that we need 689 00:41:30,290 --> 00:41:33,290 these matrices and that vector. 690 00:41:33,290 --> 00:41:36,860 This is, of course, a linear strain stiffness matrix, 691 00:41:36,860 --> 00:41:39,110 that's a nonlinear strain stiffness matrix, and that is 692 00:41:39,110 --> 00:41:42,350 the force vector that corresponds to the internet 693 00:41:42,350 --> 00:41:44,710 element stresses. 694 00:41:44,710 --> 00:41:48,930 The matrices that go into the calculation of these matrices 695 00:41:48,930 --> 00:41:50,770 are listed here. 696 00:41:50,770 --> 00:41:54,070 The tangent material relationship, the linear 697 00:41:54,070 --> 00:41:56,770 strain displacement matrix go into the 698 00:41:56,770 --> 00:41:59,620 calculation of this k matrix. 699 00:41:59,620 --> 00:42:04,410 This stress matrix, and that strain displacement matrix, 700 00:42:04,410 --> 00:42:07,060 the nonlinear strain displacement matrix, these two 701 00:42:07,060 --> 00:42:10,720 quantities go into the calculation of this k matrix. 702 00:42:10,720 --> 00:42:14,720 And this stress vector, and that linear strain 703 00:42:14,720 --> 00:42:16,810 displacement matrix goes into the 704 00:42:16,810 --> 00:42:20,050 calculation of that F vector. 705 00:42:20,050 --> 00:42:24,990 We already derived that all earlier in an earlier lecture. 706 00:42:24,990 --> 00:42:30,240 The stress strain matrix is listed here. 707 00:42:30,240 --> 00:42:35,870 Notice that we have the incremental stresses. 708 00:42:35,870 --> 00:42:38,500 Of course, these are increments in the second 709 00:42:38,500 --> 00:42:41,890 Piola-Kirchhoff stresses from time t onwards. 710 00:42:41,890 --> 00:42:44,400 That's why you have the t here. 711 00:42:44,400 --> 00:42:47,880 That's why this t corresponds, of course, to the updated 712 00:42:47,880 --> 00:42:49,680 Lagrangian formulation. 713 00:42:49,680 --> 00:42:52,270 Here we have the material tensor, here we have the 714 00:42:52,270 --> 00:42:53,160 strain terms. 715 00:42:53,160 --> 00:42:56,160 Once again, the 2 that I pointed out early already for 716 00:42:56,160 --> 00:42:58,710 the total Lagrangian case as well. 717 00:42:58,710 --> 00:43:04,500 Of course this matrix here, contains the elements of the 718 00:43:04,500 --> 00:43:09,810 material tensor, tCijrs, which we used earlier when we talked 719 00:43:09,810 --> 00:43:11,495 about the continuum mechanics formations. 720 00:43:14,360 --> 00:43:18,550 Well for the linear elastic case, we would simply use the 721 00:43:18,550 --> 00:43:22,740 very familiar stress strain law that we are using also in 722 00:43:22,740 --> 00:43:23,990 linear analysis. 723 00:43:23,990 --> 00:43:26,550 Young's modulus, Poisson's ratio. 724 00:43:26,550 --> 00:43:31,210 So this is one typical case for this C matrix. 725 00:43:31,210 --> 00:43:34,960 Of course, we will discuss later on in later lectures how 726 00:43:34,960 --> 00:43:37,620 we construct the C matrix for all sorts of material 727 00:43:37,620 --> 00:43:38,870 conditions. 728 00:43:40,900 --> 00:43:45,330 We need for the B matrix, for the linear strain displacement 729 00:43:45,330 --> 00:43:50,410 matrix, these entries here, meaning we need these 730 00:43:50,410 --> 00:43:52,830 differentiations. 731 00:43:52,830 --> 00:43:57,270 Notice partial u1 with respect to tx1 is in shorthand 732 00:43:57,270 --> 00:43:59,590 notation written as shown here. 733 00:43:59,590 --> 00:44:06,270 te22 is equal to that element, and similarly we go on. 734 00:44:06,270 --> 00:44:08,380 Notice this, of course, is again, the hoop strain. 735 00:44:11,300 --> 00:44:14,890 For the nonlinear strain displacement matrix, we need 736 00:44:14,890 --> 00:44:17,540 to look at the nonlinear strain terms. 737 00:44:17,540 --> 00:44:22,580 And the nonlinear strain terms are listed out here now. 738 00:44:22,580 --> 00:44:26,890 We identified these strain terms earlier when we 739 00:44:26,890 --> 00:44:29,790 discussed the updated Lagrangian formulation in an 740 00:44:29,790 --> 00:44:31,080 earlier lecture. 741 00:44:31,080 --> 00:44:34,150 Of course, at that time, the strain terms were represented 742 00:44:34,150 --> 00:44:37,590 in terms of ijk indices. 743 00:44:37,590 --> 00:44:43,110 Now we have to simply put i and j equal to 1, for example, 744 00:44:43,110 --> 00:44:47,530 and k equal to 1, and equal to 2, and you will directly 745 00:44:47,530 --> 00:44:52,410 obtain from the earlier expression that I presented to 746 00:44:52,410 --> 00:44:54,420 you this term here. 747 00:44:54,420 --> 00:44:58,500 And similarly, you obtain all the other terms as well. 748 00:44:58,500 --> 00:45:01,880 By the way, the hoop strain, the linear, and here we see 749 00:45:01,880 --> 00:45:05,760 the nonlinear hoop strain, expression would be derived in 750 00:45:05,760 --> 00:45:09,565 the same way as we have done it in this lecture for the 751 00:45:09,565 --> 00:45:11,720 total Lagrangian formulation. 752 00:45:11,720 --> 00:45:14,710 It would actually be a good exercise for you to do so one 753 00:45:14,710 --> 00:45:18,330 and see how that goes. 754 00:45:18,330 --> 00:45:24,220 We construct the ttBL matrix to capture the total strain, 755 00:45:24,220 --> 00:45:29,670 total linear strain, listed in this vector here as shown by 756 00:45:29,670 --> 00:45:30,540 this equation. 757 00:45:30,540 --> 00:45:34,000 Of course, here we have the nodal point displacements. 758 00:45:34,000 --> 00:45:38,160 The nodal point displacements are in this vector. 759 00:45:38,160 --> 00:45:41,210 Denote always that these are the discrete nodal point 760 00:45:41,210 --> 00:45:43,760 displacements. 761 00:45:43,760 --> 00:45:48,050 This last term, of course, we only introduced in 762 00:45:48,050 --> 00:45:50,740 axisymmetric analysis. 763 00:45:50,740 --> 00:45:56,370 The entries in this ttBL matrix are shown here for a 764 00:45:56,370 --> 00:45:58,530 typical node k. 765 00:45:58,530 --> 00:46:01,290 We use the same kind of picture as for the total 766 00:46:01,290 --> 00:46:02,620 Lagrangian formulation. 767 00:46:02,620 --> 00:46:05,410 Notice here are the actual displacements, the way they 768 00:46:05,410 --> 00:46:07,440 would appear in a vector. 769 00:46:07,440 --> 00:46:12,110 But since this element here hits this column so to say, we 770 00:46:12,110 --> 00:46:14,470 have written it here once more in blue. 771 00:46:14,470 --> 00:46:17,230 This element here hits this column, and we have therefore 772 00:46:17,230 --> 00:46:19,030 written it here once more in blue. 773 00:46:19,030 --> 00:46:25,060 So this is really the column corresponding to the case node 774 00:46:25,060 --> 00:46:29,070 and the u1, the 1 direction. 775 00:46:29,070 --> 00:46:32,470 This is the column corresponding to the case node 776 00:46:32,470 --> 00:46:35,510 and the 2 direction. 777 00:46:35,510 --> 00:46:40,550 If you compare this matrix with the matrix that we use in 778 00:46:40,550 --> 00:46:45,040 linear analysis, you would see that the matrix looks very 779 00:46:45,040 --> 00:46:48,860 similar, except that in linear analysis you would not have 780 00:46:48,860 --> 00:46:53,880 this t, or you don't put that t there in general because the 781 00:46:53,880 --> 00:46:56,980 t would, of course, be actually the 0 configuration 782 00:46:56,980 --> 00:46:59,090 because all the differentiations are referred 783 00:46:59,090 --> 00:47:02,120 to 0 configuration anyway. 784 00:47:02,120 --> 00:47:05,910 And it is common to not have an index down here. 785 00:47:05,910 --> 00:47:09,810 If you look at this textbook, chapter five, you would, for 786 00:47:09,810 --> 00:47:13,290 example, see there expressions such as this 787 00:47:13,290 --> 00:47:17,090 without this t there. 788 00:47:17,090 --> 00:47:20,200 So there's really not much of a surprise right here. 789 00:47:20,200 --> 00:47:24,120 You will surely see that this is the right relationship to 790 00:47:24,120 --> 00:47:28,430 use in the B matrix based on your 791 00:47:28,430 --> 00:47:32,480 knowledge of linear analysis. 792 00:47:32,480 --> 00:47:37,530 The expression of the nonlinear strain displacement 793 00:47:37,530 --> 00:47:43,450 matrix and the stress matrix are these two expressions are 794 00:47:43,450 --> 00:47:47,950 constructed in such a way as to have that this product on 795 00:47:47,950 --> 00:47:52,010 the left-hand side is equal to that expression there. 796 00:47:52,010 --> 00:47:57,100 I pointed that out already earlier, that on the 797 00:47:57,100 --> 00:48:00,380 right-hand side this is the continuum mechanics variable, 798 00:48:00,380 --> 00:48:02,670 and this continuum mechanics variable is, of course, the 799 00:48:02,670 --> 00:48:06,610 basic quantity, the basic quantity that we actually want 800 00:48:06,610 --> 00:48:07,730 to capture. 801 00:48:07,730 --> 00:48:12,240 And we construct this matrix and that matrix 802 00:48:12,240 --> 00:48:14,880 such as to do so. 803 00:48:14,880 --> 00:48:19,480 I should just point out there should be no bar here. 804 00:48:19,480 --> 00:48:22,840 Of course, this is a tensor quantity, it's not a matrix, 805 00:48:22,840 --> 00:48:24,490 and so there should be no bar there. 806 00:48:24,490 --> 00:48:27,130 That was a little error on my part. 807 00:48:27,130 --> 00:48:30,060 The entries in the t tau are given right here. 808 00:48:34,860 --> 00:48:38,430 The last row and column are, of course, only included if 809 00:48:38,430 --> 00:48:41,470 you deal with an axisymmetric analysis. 810 00:48:41,470 --> 00:48:47,650 ttBNL is shown here. 811 00:48:47,650 --> 00:48:50,920 Notice again, differentiations with respect to time t. 812 00:48:53,810 --> 00:48:59,670 And the last row is only included if you have an 813 00:48:59,670 --> 00:49:00,920 axisymmetric analysis. 814 00:49:04,240 --> 00:49:09,740 Finally, we want to also construct our stress vector 815 00:49:09,740 --> 00:49:14,620 such that when it is entered here with the BL matrix that 816 00:49:14,620 --> 00:49:19,820 we already have defined, we capture via this product here 817 00:49:19,820 --> 00:49:21,450 exactly that term. 818 00:49:21,450 --> 00:49:25,120 This is the term that is basic, that we have derived 819 00:49:25,120 --> 00:49:27,100 from the continuum mechanics formulation. 820 00:49:27,100 --> 00:49:30,680 This is what we want to capture, and we do so by this 821 00:49:30,680 --> 00:49:32,110 expression. 822 00:49:32,110 --> 00:49:35,230 And t tau hat is given right here. 823 00:49:35,230 --> 00:49:38,490 It lists all the stresses really, including the hoop 824 00:49:38,490 --> 00:49:42,100 stress if you have an axisymmetric analysis. 825 00:49:42,100 --> 00:49:44,260 This completes what I wanted to say about the two 826 00:49:44,260 --> 00:49:45,480 dimensional elements. 827 00:49:45,480 --> 00:49:48,420 We talked about the total Lagrangian formulation, and 828 00:49:48,420 --> 00:49:51,260 the updated Lagrangian formulation of these elements. 829 00:49:51,260 --> 00:49:53,880 In other words, we really presented the matrices 830 00:49:53,880 --> 00:49:55,680 corresponding to these formulations 831 00:49:55,680 --> 00:49:58,370 in quite some detail. 832 00:49:58,370 --> 00:50:00,550 These details that we discussed for the two 833 00:50:00,550 --> 00:50:04,470 dimensional elements are also almost directly applicable for 834 00:50:04,470 --> 00:50:06,270 the three dimensional elements. 835 00:50:06,270 --> 00:50:11,080 And here we have a typical eight-node element in a 836 00:50:11,080 --> 00:50:16,180 stationary coordinate frame, x1, x2, x3, in its original 837 00:50:16,180 --> 00:50:17,430 configuration. 838 00:50:19,300 --> 00:50:22,070 The nodal point coordinates are listed here. 839 00:50:22,070 --> 00:50:26,130 Notice k, of course, stands again for the node k. 840 00:50:26,130 --> 00:50:29,940 1, 2, 3 stands for the directions 1, 2, 3. 841 00:50:29,940 --> 00:50:34,460 And the left superscript here stands for the time 0, the 842 00:50:34,460 --> 00:50:36,080 configuration 0. 843 00:50:36,080 --> 00:50:41,500 The element moves from time 0 to time t up to this 844 00:50:41,500 --> 00:50:43,240 configuration here. 845 00:50:43,240 --> 00:50:47,190 And the nodal points now are tx1k-- 846 00:50:47,190 --> 00:50:49,480 nodal point coordinate I should have said. 847 00:50:49,480 --> 00:50:52,410 Now tx1k, tx2k, tx3k. 848 00:50:55,290 --> 00:51:00,170 The notation is much the same as we have been using in two 849 00:51:00,170 --> 00:51:01,310 dimensional analysis. 850 00:51:01,310 --> 00:51:05,590 Of course, we now have a third coordinate, the x3 coordinate 851 00:51:05,590 --> 00:51:08,055 also in the formulation. 852 00:51:08,055 --> 00:51:13,500 We now use interpolations much the same way as for the two 853 00:51:13,500 --> 00:51:16,040 dimensional elements. 854 00:51:16,040 --> 00:51:18,740 For the original configuration, we have these 855 00:51:18,740 --> 00:51:19,990 interpolations. 856 00:51:21,350 --> 00:51:24,530 Notice, the third direction enters now. 857 00:51:24,530 --> 00:51:27,480 Once again, these are the interpolation functions. 858 00:51:27,480 --> 00:51:32,050 These are the nodal point coordinates corresponding to 859 00:51:32,050 --> 00:51:34,710 the 1 direction. 860 00:51:34,710 --> 00:51:38,170 These are the nodal point coordinates corresponding to 861 00:51:38,170 --> 00:51:40,570 the 2 direction, and so on. 862 00:51:40,570 --> 00:51:44,490 Of course, we now also have to introduce the nodal point 863 00:51:44,490 --> 00:51:47,160 coordinates corresponding to the 3 direction. 864 00:51:47,160 --> 00:51:49,340 And if the number of nodes, if you have an eight-node 865 00:51:49,340 --> 00:51:52,190 element, and of course is 8. 866 00:51:52,190 --> 00:51:55,395 These are the interpolations used for the original geometry 867 00:51:55,395 --> 00:52:01,290 of the element, and for the configuration at time t, we 868 00:52:01,290 --> 00:52:02,540 use these interpolations. 869 00:52:05,110 --> 00:52:10,240 Notice these are now the nodal points coordinates 870 00:52:10,240 --> 00:52:11,820 corresponding to time t. 871 00:52:14,480 --> 00:52:16,180 We use still the same interpolation 872 00:52:16,180 --> 00:52:17,440 functions, of course. 873 00:52:17,440 --> 00:52:23,570 And N is the same number as for the 0 interpolation, or 874 00:52:23,570 --> 00:52:27,230 rather for the interpolation of the original geometry. 875 00:52:27,230 --> 00:52:29,410 In other words, for the eight-node element that we 876 00:52:29,410 --> 00:52:33,520 briefly considered, N is in each case equal to 8. 877 00:52:33,520 --> 00:52:40,440 We use these expressions, we subtract from tx1 the 0x1 from 878 00:52:40,440 --> 00:52:45,930 tx2, the 0x2 expression from tx3, the 0x3 expression, and 879 00:52:45,930 --> 00:52:49,040 directly obtain the interpolations for the 880 00:52:49,040 --> 00:52:51,220 displacement of the elements. 881 00:52:51,220 --> 00:52:54,390 And once we have the displacement interpolation, 882 00:52:54,390 --> 00:52:58,050 we, of course, can directly find the derivative of these 883 00:52:58,050 --> 00:53:00,910 displacement interpolations to obtain the strain 884 00:53:00,910 --> 00:53:02,220 interpolations. 885 00:53:02,220 --> 00:53:06,260 And these expressions then, the derivatives of the 886 00:53:06,260 --> 00:53:09,920 displacement interpolation, enter into the construction of 887 00:53:09,920 --> 00:53:12,760 the strain displacement matrices. 888 00:53:12,760 --> 00:53:16,250 So we see that basically, the same procedure that we 889 00:53:16,250 --> 00:53:18,840 discussed with two dimensional analysis is directly 890 00:53:18,840 --> 00:53:21,130 applicable to the three dimensional analysis, the only 891 00:53:21,130 --> 00:53:25,090 difference being that the third direction, x3, has to be 892 00:53:25,090 --> 00:53:28,230 interpolated, the displacements have to be 893 00:53:28,230 --> 00:53:29,850 carried along corresponding to the third 894 00:53:29,850 --> 00:53:32,310 direction, et cetera. 895 00:53:32,310 --> 00:53:35,210 This then brings me to the end of what I wanted to say in 896 00:53:35,210 --> 00:53:36,040 this lecture. 897 00:53:36,040 --> 00:53:37,800 Thank you very much for your attention.