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 --> 00:00:19,510 hundreds of MIT courses visit MIT OpenCourseWare at 7 00:00:19,510 --> 00:00:21,862 ocw.mit.edu. 8 00:00:21,862 --> 00:00:23,990 PROFESSOR: Ladies and gentlemen, welcome to this 9 00:00:23,990 --> 00:00:26,720 lecture on nonlinear finite element analysis of solids and 10 00:00:26,720 --> 00:00:28,130 structures. 11 00:00:28,130 --> 00:00:29,590 In this lecture I'd like to discuss with 12 00:00:29,590 --> 00:00:31,350 you structural elements. 13 00:00:31,350 --> 00:00:34,340 Structural elements are, of course, employed to model 14 00:00:34,340 --> 00:00:36,940 beam, plate and shell structures and are most 15 00:00:36,940 --> 00:00:38,910 important elements. 16 00:00:38,910 --> 00:00:41,430 Because of their importance in engineering practice much 17 00:00:41,430 --> 00:00:44,400 research and development effort has been focused on the 18 00:00:44,400 --> 00:00:47,220 development of efficient structural elements. 19 00:00:47,220 --> 00:00:50,170 In this and the next lecture I'd like to discuss with you 20 00:00:50,170 --> 00:00:53,190 some modern effective elements. 21 00:00:53,190 --> 00:00:57,200 We will first discuss shell elements and then beam 22 00:00:57,200 --> 00:01:00,930 elements and then we go back to use the shell elements, a 23 00:01:00,930 --> 00:01:05,400 concept that we discussed in some actual applications. 24 00:01:05,400 --> 00:01:10,490 When we do structural analysis we should keep one method in 25 00:01:10,490 --> 00:01:13,460 mind, namely that in a geometrically nonlinear 26 00:01:13,460 --> 00:01:21,140 analysis, a flat shell, referred to as a plate, goes 27 00:01:21,140 --> 00:01:24,280 very rapidly over into the behavior of a shell because of 28 00:01:24,280 --> 00:01:27,660 the curvature that develops as the plate deforms. 29 00:01:27,660 --> 00:01:32,710 Therefore, to analyze geometrically nonlinear plates 30 00:01:32,710 --> 00:01:37,110 we really are quite well off using general shell elements. 31 00:01:37,110 --> 00:01:40,480 And I'd like to now focus our attention on the development 32 00:01:40,480 --> 00:01:43,840 of general shell elements that we actually then employ to 33 00:01:43,840 --> 00:01:49,010 analyze plates as well as general shells. 34 00:01:49,010 --> 00:01:51,810 There are various solution approaches that one can follow 35 00:01:51,810 --> 00:01:54,610 for the development of efficient elements. 36 00:01:54,610 --> 00:01:59,120 And one such approach is the use of general 37 00:01:59,120 --> 00:02:01,180 beam and shell theories. 38 00:02:01,180 --> 00:02:04,020 I'm thinking of beams as well as shells although I want to 39 00:02:04,020 --> 00:02:08,220 focus our attention on shells, really, in this first lecture. 40 00:02:08,220 --> 00:02:15,070 One starts with beam or shell theory as general as possible 41 00:02:15,070 --> 00:02:18,790 and develops the governing differential equations. 42 00:02:18,790 --> 00:02:21,150 From the governing differential equation one 43 00:02:21,150 --> 00:02:24,120 develops variational formulations and one 44 00:02:24,120 --> 00:02:26,780 discretizes those variation formulations using finite 45 00:02:26,780 --> 00:02:29,080 element interpolations. 46 00:02:29,080 --> 00:02:31,130 This is the approach that is taken. 47 00:02:31,130 --> 00:02:35,640 The disadvantage of that approach is, in general, lack 48 00:02:35,640 --> 00:02:38,200 of generality of the approach. 49 00:02:38,200 --> 00:02:40,670 You're starting off with a particular shell 50 00:02:40,670 --> 00:02:43,160 theory, beam theory. 51 00:02:43,160 --> 00:02:46,880 You of course would like to have that shell theory as 52 00:02:46,880 --> 00:02:53,020 general as possible but you will find that really, the 53 00:02:53,020 --> 00:02:55,760 shell theory that you're starting of will mostly only 54 00:02:55,760 --> 00:02:58,380 be applicable to a certain class of shells. 55 00:02:58,380 --> 00:03:01,000 Therefore your finite elements that you're developing would 56 00:03:01,000 --> 00:03:04,680 also be only applicable to that same class of shells. 57 00:03:04,680 --> 00:03:10,010 And that means in engineering practice, in actual usage, the 58 00:03:10,010 --> 00:03:14,270 user has to be very familiar for which the particular 59 00:03:14,270 --> 00:03:17,480 element developed, as shown here. 60 00:03:17,480 --> 00:03:21,400 I'll discuss now for what kinds of shells this 61 00:03:21,400 --> 00:03:24,190 particular shell element is really applicable. 62 00:03:24,190 --> 00:03:26,500 We would rather like to have shell elements that are 63 00:03:26,500 --> 00:03:29,130 applicable to any shell. 64 00:03:29,130 --> 00:03:31,460 Of course this is a very big aim, very 65 00:03:31,460 --> 00:03:32,500 difficult to achieve. 66 00:03:32,500 --> 00:03:34,580 But this is what we would like to have ideally 67 00:03:34,580 --> 00:03:36,570 in engineering practice. 68 00:03:36,570 --> 00:03:44,320 The other difficulty with this approach is that frequently a 69 00:03:44,320 --> 00:03:48,120 large number of nodal degrees of freedom have to be carried 70 00:03:48,120 --> 00:03:51,250 along in the development of the shell elements. 71 00:03:51,250 --> 00:03:55,860 And what I'm thinking of there is that you don't just have 72 00:03:55,860 --> 00:03:58,790 translations and rotations at the nodal points, which has 73 00:03:58,790 --> 00:04:01,560 engineering degrees of freedom that we would like to have and 74 00:04:01,560 --> 00:04:05,360 see in for the element, but you also have to have 75 00:04:05,360 --> 00:04:09,010 additional degrees of freedoms relating to the curvatures in 76 00:04:09,010 --> 00:04:09,790 the elements. 77 00:04:09,790 --> 00:04:10,370 And so on. 78 00:04:10,370 --> 00:04:17,160 So this of course would mean, or does mean, difficult use of 79 00:04:17,160 --> 00:04:21,560 such elements and these reasons, these two reasons 80 00:04:21,560 --> 00:04:25,740 here have really driven the research and development 81 00:04:25,740 --> 00:04:28,780 efforts in different directions. 82 00:04:28,780 --> 00:04:33,600 Another approach is to use simple elements. 83 00:04:33,600 --> 00:04:37,660 Simple elements, but a large number of elements, then to 84 00:04:37,660 --> 00:04:41,770 model very complex beam and shell structures. 85 00:04:41,770 --> 00:04:45,360 As an example I'd like to just refer you to a 3-node 86 00:04:45,360 --> 00:04:50,580 triangle, a flat element in which the plate-bending 87 00:04:50,580 --> 00:04:53,620 behavior is modeled in a particular way and the plane 88 00:04:53,620 --> 00:04:57,380 stress behavior is modeled as a constant strain, constant 89 00:04:57,380 --> 00:04:58,560 stress element. 90 00:04:58,560 --> 00:05:01,010 These two are superimposed, these two behaviors are 91 00:05:01,010 --> 00:05:05,850 superimposed and you have a very simple shell element that 92 00:05:05,850 --> 00:05:09,760 in certain analyses can be quite effective. 93 00:05:09,760 --> 00:05:12,720 Of course we have to recognize in this approach that the 94 00:05:12,720 --> 00:05:16,560 coupling between the membrane and bending action is only 95 00:05:16,560 --> 00:05:18,750 introduced at the element nodes. 96 00:05:18,750 --> 00:05:21,020 That is a major disadvantage. 97 00:05:21,020 --> 00:05:25,080 And that is the reason why you need so many elements to model 98 00:05:25,080 --> 00:05:31,610 a shell and that if we use a triangular 3-noded element 99 00:05:31,610 --> 00:05:37,080 with just a constant strain, constant stress element to 100 00:05:37,080 --> 00:05:41,500 model the membrane behavior, the membrane action, of course 101 00:05:41,500 --> 00:05:43,980 the membrane action is quite poorly approximated. 102 00:05:43,980 --> 00:05:48,590 And that is another reason why we need so many elements to 103 00:05:48,590 --> 00:05:51,940 model a complex shell behavior. 104 00:05:51,940 --> 00:05:57,490 Here on this viewgraph I'm showing a picture of the 105 00:05:57,490 --> 00:05:59,530 element that I just referred to. 106 00:05:59,530 --> 00:06:02,790 Here you have a triangular element in the three 107 00:06:02,790 --> 00:06:05,000 dimensional space. 108 00:06:05,000 --> 00:06:11,565 Notice we introduce a local coordinate system, x bar 2, x 109 00:06:11,565 --> 00:06:16,070 bar 3, x bar 1. 110 00:06:16,070 --> 00:06:19,130 Local coordinate system, and in this local coordinate 111 00:06:19,130 --> 00:06:25,260 system we measure the displacements and rotations. 112 00:06:25,260 --> 00:06:30,290 And we superimpose for this element the bending behavior, 113 00:06:30,290 --> 00:06:34,060 the bending behavior corresponding to this degree 114 00:06:34,060 --> 00:06:37,480 of freedom, that degree of freedom and this degree of 115 00:06:37,480 --> 00:06:40,940 freedom at every node, of course. 116 00:06:40,940 --> 00:06:45,150 We add, or superimpose onto this bending behavior the 117 00:06:45,150 --> 00:06:48,500 membrane behavior which corresponds to these two 118 00:06:48,500 --> 00:06:51,140 degrees of freedom. 119 00:06:51,140 --> 00:06:54,080 And we immediately notice that corresponding to this degree 120 00:06:54,080 --> 00:06:58,480 of freedom we don't have a real physical stiffness and we 121 00:06:58,480 --> 00:07:04,360 introduce a little artificial stiffness as shown down here. 122 00:07:04,360 --> 00:07:11,930 Now this artificial stiffness has to be selected and you 123 00:07:11,930 --> 00:07:15,560 want to select it such as to take out the singularity out 124 00:07:15,560 --> 00:07:21,650 of the system but yet make the stiffness not too big so as to 125 00:07:21,650 --> 00:07:24,020 destroy the behavior of the element. 126 00:07:24,020 --> 00:07:27,190 Because it's an artificial stiffness you really want to 127 00:07:27,190 --> 00:07:30,140 make it as small as possible just to take out the 128 00:07:30,140 --> 00:07:33,020 singularity out of this element when you apply this 129 00:07:33,020 --> 00:07:36,940 element in the modeling of general shell structures. 130 00:07:36,940 --> 00:07:41,480 This artificial stiffness actually is quite bothersome 131 00:07:41,480 --> 00:07:43,520 in nonlinear analysis. 132 00:07:43,520 --> 00:07:47,610 It can provide problems and we really don't like it. 133 00:07:47,610 --> 00:07:52,750 But if we use this approach we have to introduce it and well, 134 00:07:52,750 --> 00:07:57,250 we have to, so to say, live with it, with the difficulties 135 00:07:57,250 --> 00:07:58,100 that we encounter. 136 00:07:58,100 --> 00:08:02,140 But the other approach that I want to discuss with you 137 00:08:02,140 --> 00:08:04,520 really quite extensively in these two lectures, namely 138 00:08:04,520 --> 00:08:08,510 using the isoparametric elements, having curved 139 00:08:08,510 --> 00:08:12,340 elements there as well, we don't introduce this 140 00:08:12,340 --> 00:08:15,880 artificial stiffness anymore because our experiences of 141 00:08:15,880 --> 00:08:21,295 introducing it in nonlinear analysis show us that there 142 00:08:21,295 --> 00:08:24,850 are many difficulties that come into the analysis 143 00:08:24,850 --> 00:08:27,820 procedure if you have this artificial stiffness. 144 00:08:27,820 --> 00:08:29,590 So we got rid of it. 145 00:08:29,590 --> 00:08:33,179 We don't need to use it any longer for the more modern 146 00:08:33,179 --> 00:08:37,240 elements that I will be talking about just now. 147 00:08:37,240 --> 00:08:42,549 The approach for these modern elements is to use 148 00:08:42,549 --> 00:08:47,660 isoparametric interpolations and we talk then about the 149 00:08:47,660 --> 00:08:50,330 isoparametric (degenerate) beams and shell elements. 150 00:08:50,330 --> 00:08:54,750 "Degenerate" because we degenerate these elements, or 151 00:08:54,750 --> 00:08:58,140 we obtain these elements I should say, by degeneration 152 00:08:58,140 --> 00:08:59,980 from three dimensional behavior. 153 00:08:59,980 --> 00:09:03,020 We will talk much more about it just now. 154 00:09:03,020 --> 00:09:07,970 But in essence we are saying we take the 3-D continuum 155 00:09:07,970 --> 00:09:12,490 equations and we degenerate those equations to the 156 00:09:12,490 --> 00:09:16,150 particular shell behavior and beam behavior for beams that 157 00:09:16,150 --> 00:09:18,950 we would like to capture. 158 00:09:18,950 --> 00:09:23,000 The resulting elements can be used to model quite general 159 00:09:23,000 --> 00:09:24,470 beam and shell structures. 160 00:09:24,470 --> 00:09:28,570 And that is, of course, a very large advantage in engineering 161 00:09:28,570 --> 00:09:32,290 practice if you can use the same elements to model a 162 00:09:32,290 --> 00:09:36,140 variety of structures. 163 00:09:36,140 --> 00:09:43,160 The basic approach of this isoparametric interpolation is 164 00:09:43,160 --> 00:09:46,450 to use the total and updated Lagrangian formulations that 165 00:09:46,450 --> 00:09:48,480 we developed earlier. 166 00:09:48,480 --> 00:09:51,740 We talked in the earlier lectures quite extensively 167 00:09:51,740 --> 00:09:54,810 about the total updated Lagrangian formulation, the 168 00:09:54,810 --> 00:09:58,840 continuum mechanics equations, as well as the finite element 169 00:09:58,840 --> 00:10:01,850 discretization of the continuum mechanics equations. 170 00:10:01,850 --> 00:10:06,000 But we applied the final discretization only to 2-D and 171 00:10:06,000 --> 00:10:07,330 3-D solid elements. 172 00:10:07,330 --> 00:10:11,150 Now we want to do the same for shell elements. 173 00:10:11,150 --> 00:10:16,670 We recall that in the T.L. formulation the governing 174 00:10:16,670 --> 00:10:21,640 equation is this one here, which is nothing else than the 175 00:10:21,640 --> 00:10:24,020 principal of virtual work operating on the second 176 00:10:24,020 --> 00:10:26,520 Piola-Kirchhoff stress and the variation on the 177 00:10:26,520 --> 00:10:28,600 Green-Lagrange strain. 178 00:10:28,600 --> 00:10:32,510 This integral has taken over the original volume of the 179 00:10:32,510 --> 00:10:34,720 structure, of the element, when we 180 00:10:34,720 --> 00:10:37,420 develop a finite element. 181 00:10:37,420 --> 00:10:40,050 And this of course, the internal virtual work, and 182 00:10:40,050 --> 00:10:42,770 here we have the external virtual work. 183 00:10:42,770 --> 00:10:47,200 Notice the linearization of this left integral here 184 00:10:47,200 --> 00:10:50,360 resulted into these three integrals. 185 00:10:50,360 --> 00:10:52,540 We went through that linearization in quite some 186 00:10:52,540 --> 00:10:56,150 detail and we talked about the individual terms. 187 00:10:56,150 --> 00:10:58,670 I don't think it's now necessity to review that 188 00:10:58,670 --> 00:11:00,640 material anymore. 189 00:11:00,640 --> 00:11:03,070 Please refer to the earlier lectures. 190 00:11:03,070 --> 00:11:05,930 The same approach of course we used also for the U.L. 191 00:11:05,930 --> 00:11:07,350 formulation. 192 00:11:07,350 --> 00:11:10,800 Here is the general starting point, the principle of 193 00:11:10,800 --> 00:11:14,130 virtual work now, using second Piola-Kirchhoff stresses refer 194 00:11:14,130 --> 00:11:16,600 to the configuration at time t. 195 00:11:16,600 --> 00:11:18,620 Variations in the Green-Lagrange strains refer 196 00:11:18,620 --> 00:11:21,250 to the time configuration at time t. 197 00:11:21,250 --> 00:11:24,930 This gives the internal virtual work corresponding to 198 00:11:24,930 --> 00:11:30,200 time t plus delta t, t plus delta t. 199 00:11:30,200 --> 00:11:33,020 And this is the external virtual work, same external 200 00:11:33,020 --> 00:11:35,730 virtual work of course that we are having in the total 201 00:11:35,730 --> 00:11:37,080 Lagrangian formulation. 202 00:11:37,080 --> 00:11:41,100 The linearization of the left hand side integral here 203 00:11:41,100 --> 00:11:44,200 results into these three integrals. 204 00:11:44,200 --> 00:11:46,700 Once again we talked about the linearization quiet 205 00:11:46,700 --> 00:11:50,940 extensively and if you refer to the earlier lectures surely 206 00:11:50,940 --> 00:11:54,070 you recognize individual terms that you're seeing here now. 207 00:11:54,070 --> 00:11:58,500 We use these governing equations for the total 208 00:11:58,500 --> 00:12:00,580 Lagrangian and updated Lagrangian formulation to 209 00:12:00,580 --> 00:12:03,472 develop the general shell elements. 210 00:12:03,472 --> 00:12:06,580 And what we have to do now is to impose on these equations 211 00:12:06,580 --> 00:12:10,290 the basic assumptions of beam and shell action. 212 00:12:10,290 --> 00:12:13,630 And let us go now through these basic assumptions one by 213 00:12:13,630 --> 00:12:15,180 one carefully. 214 00:12:15,180 --> 00:12:17,890 The first assumption is that the material particles 215 00:12:17,890 --> 00:12:21,225 originally on a straight line normal to the mid-surface of 216 00:12:21,225 --> 00:12:24,810 the beam or the shell remain on that straight line 217 00:12:24,810 --> 00:12:27,470 throughout the response history. 218 00:12:27,470 --> 00:12:30,050 This is one most important assumption. 219 00:12:30,050 --> 00:12:33,160 You must have encountered this assumption or at least some 220 00:12:33,160 --> 00:12:38,000 form of it already earlier in your discussion of beam 221 00:12:38,000 --> 00:12:39,940 theories and possibly shell theories. 222 00:12:39,940 --> 00:12:43,360 Let's look at this assumption more closely, so. 223 00:12:43,360 --> 00:12:47,890 For beams we would say plane sections initially normal to 224 00:12:47,890 --> 00:12:50,600 the mid-surface remain plane sections during 225 00:12:50,600 --> 00:12:53,460 the response history. 226 00:12:53,460 --> 00:12:55,690 This is basically saying the same thing 227 00:12:55,690 --> 00:12:58,200 what I just said earlier. 228 00:12:58,200 --> 00:13:01,440 And if you look at this closely you recognize that we 229 00:13:01,440 --> 00:13:06,430 do not say that the plane sections initially normal to 230 00:13:06,430 --> 00:13:10,810 the mid-surface remain plane sections during the response 231 00:13:10,810 --> 00:13:14,600 history and remain normal to the mid-surface. 232 00:13:14,600 --> 00:13:16,680 We don't say that. 233 00:13:16,680 --> 00:13:20,410 That's of course being said when you use the 234 00:13:20,410 --> 00:13:22,430 Euler-Bernoulli beam theory. 235 00:13:22,430 --> 00:13:27,840 We don't say this, that the plane sections remain normal 236 00:13:27,840 --> 00:13:31,190 to the mid-surface throughout the response history. 237 00:13:31,190 --> 00:13:34,710 We don't say that and because we don't say that we in effect 238 00:13:34,710 --> 00:13:38,600 include in an approximate way shear deformations. 239 00:13:38,600 --> 00:13:42,530 In other words, we look here at the effect of transverse 240 00:13:42,530 --> 00:13:46,570 shear deformations is included, and hence the lines 241 00:13:46,570 --> 00:13:49,160 initially normal to the mid-surface do not remain 242 00:13:49,160 --> 00:13:52,810 normal to the mid-surface during the deformations. 243 00:13:52,810 --> 00:13:56,440 Let's look at what this means pictorially. 244 00:13:56,440 --> 00:14:04,750 Here we have a section of a beam at time 0 and we draw a 245 00:14:04,750 --> 00:14:09,900 line normal, that is, at 90 degrees to the mid-surface, 246 00:14:09,900 --> 00:14:11,950 shown as a dashed line. 247 00:14:11,950 --> 00:14:15,570 And we identify particles on that line. 248 00:14:15,570 --> 00:14:17,730 Here we have four such [? rad ?] 249 00:14:17,730 --> 00:14:19,620 material particles. 250 00:14:19,620 --> 00:14:23,770 Now the beam will move, deform, go through large 251 00:14:23,770 --> 00:14:27,260 displacements, large rotations. 252 00:14:27,260 --> 00:14:31,350 But actually we assume small strains. 253 00:14:31,350 --> 00:14:34,770 And we see that these material particles which were 254 00:14:34,770 --> 00:14:38,230 originally up here are now down here. 255 00:14:38,230 --> 00:14:42,630 We identify that these material particles are still 256 00:14:42,630 --> 00:14:44,790 on a straight line. 257 00:14:44,790 --> 00:14:49,360 But this straight line is not anymore normal, that it is at 258 00:14:49,360 --> 00:14:51,980 90 degrees to the mid-surface. 259 00:14:51,980 --> 00:14:56,600 And because it is not any more normal to the mid-surface we 260 00:14:56,600 --> 00:15:00,360 do include shear deformations approximately because we 261 00:15:00,360 --> 00:15:03,350 assume that the shear deformations are constant 262 00:15:03,350 --> 00:15:06,450 throughout the thickness of the beam. 263 00:15:06,450 --> 00:15:09,390 This is a most important assumption. 264 00:15:09,390 --> 00:15:14,210 We're looking here what looks like a beam but actually, if 265 00:15:14,210 --> 00:15:18,010 you think of another dimension here, you can directly see 266 00:15:18,010 --> 00:15:21,330 that the same picture is also applicable to 267 00:15:21,330 --> 00:15:23,770 the motion of a shell. 268 00:15:23,770 --> 00:15:27,910 The second important assumption is that the stress 269 00:15:27,910 --> 00:15:31,560 in the direction normal to the beam or shell mid-surface is 270 00:15:31,560 --> 00:15:35,750 zero throughout the response history. 271 00:15:35,750 --> 00:15:40,050 In other words, there is no stress developed normal to the 272 00:15:40,050 --> 00:15:40,920 mid-surface. 273 00:15:40,920 --> 00:15:44,230 But notice that here is a stress along the material 274 00:15:44,230 --> 00:15:46,680 fiber that is initially normal to the mid-surface is 275 00:15:46,680 --> 00:15:47,780 considered. 276 00:15:47,780 --> 00:15:52,190 Now this material fiber which is initially normal to the 277 00:15:52,190 --> 00:15:55,235 mid-surface will not remain normal to the 278 00:15:55,235 --> 00:15:57,700 mid-surface as I just said. 279 00:15:57,700 --> 00:16:02,760 And in the motion we will consider always this stress in 280 00:16:02,760 --> 00:16:07,080 the direction of that material fiber which was originally 281 00:16:07,080 --> 00:16:08,830 normal to the mid-surface. 282 00:16:08,830 --> 00:16:12,320 So after motion has taken place we are not really 283 00:16:12,320 --> 00:16:14,770 talking anymore exactly-- 284 00:16:14,770 --> 00:16:17,710 you want to look in great detail at what's happening-- 285 00:16:17,710 --> 00:16:21,010 we don't talk anymore exactly about the stress, that is 286 00:16:21,010 --> 00:16:24,530 normal to the current mid-surface. 287 00:16:24,530 --> 00:16:28,420 But we always talk about the stress in the direction of the 288 00:16:28,420 --> 00:16:32,630 fiber that was initially normal to the mid-surface. 289 00:16:32,630 --> 00:16:33,880 That's being said here. 290 00:16:36,490 --> 00:16:41,230 And the third assumption, also most important assumption, is 291 00:16:41,230 --> 00:16:44,420 that the thickness of the beam or shell remains constant. 292 00:16:44,420 --> 00:16:48,540 Here then we clearly identify that we are using really a 293 00:16:48,540 --> 00:16:51,690 small-- we are assuming small strain conditions, but we 294 00:16:51,690 --> 00:16:54,285 allow for very large displacements and rotations. 295 00:16:56,960 --> 00:17:01,280 Well, with these three kinematic and static 296 00:17:01,280 --> 00:17:04,829 assumptions clearly identified we are now ready to actually 297 00:17:04,829 --> 00:17:08,010 develop the shell element interpolations. 298 00:17:08,010 --> 00:17:10,960 And let's go at that. 299 00:17:10,960 --> 00:17:14,500 The first point is that we incorporate the geometric 300 00:17:14,500 --> 00:17:15,810 assumptions. 301 00:17:15,810 --> 00:17:20,660 Straight lines normal to the mid-surface remain straight, 302 00:17:20,660 --> 00:17:24,230 put here in quotes, that geometric assumption and the 303 00:17:24,230 --> 00:17:26,750 geometric assumption that the shell thickness remains 304 00:17:26,750 --> 00:17:29,730 constant throughout the whole motion. 305 00:17:29,730 --> 00:17:32,320 These were two assumptions that we just discussed. 306 00:17:32,320 --> 00:17:36,640 We incorporate that into our shell element formulation by 307 00:17:36,640 --> 00:17:39,580 using the appropriate geometric and displacement 308 00:17:39,580 --> 00:17:42,320 interpolations. 309 00:17:42,320 --> 00:17:46,800 We incorporate the condition of zero stress normal to the 310 00:17:46,800 --> 00:17:50,030 mid-surface, I put it in quotes here, because remember, 311 00:17:50,030 --> 00:17:52,690 we are talking about the stress in the direction of the 312 00:17:52,690 --> 00:17:56,370 fiber that was originally normal to the mid-surface. 313 00:17:56,370 --> 00:18:00,240 We incorporate this condition by using the appropriate 314 00:18:00,240 --> 00:18:02,230 stress-strain law. 315 00:18:02,230 --> 00:18:06,980 Let's talk first about this assumption and then about this 316 00:18:06,980 --> 00:18:10,250 assumption, how we're using these to actually develop our 317 00:18:10,250 --> 00:18:12,810 shell elements. 318 00:18:12,810 --> 00:18:17,740 To focus our attention I'd like to talk 319 00:18:17,740 --> 00:18:20,570 about a 9-node element. 320 00:18:20,570 --> 00:18:21,600 However. 321 00:18:21,600 --> 00:18:26,030 we will later on see that in practice actually we don't use 322 00:18:26,030 --> 00:18:27,680 the 9-node element very much. 323 00:18:27,680 --> 00:18:31,760 We actually recommend the use of a 16-node element and a 324 00:18:31,760 --> 00:18:36,190 4-node element but this 9-node element in some 325 00:18:36,190 --> 00:18:38,220 analysis is also used. 326 00:18:38,220 --> 00:18:43,560 And it certainly is an element with which I can discuss with 327 00:18:43,560 --> 00:18:46,810 you, share with you all the experiences regarding the 328 00:18:46,810 --> 00:18:49,860 formulation of the elements because what we're talking 329 00:18:49,860 --> 00:18:53,640 about now really is applicable to any of the elements. 330 00:18:53,640 --> 00:18:55,970 In fact, we're talking about variable number nodes 331 00:18:55,970 --> 00:19:01,020 elements, where the number of nodes can be selected by the 332 00:19:01,020 --> 00:19:05,110 analyst and the geometric assumptions that we're now 333 00:19:05,110 --> 00:19:08,660 talking about are the same for any one of these elements. 334 00:19:08,660 --> 00:19:10,610 How do we go about the formulation? 335 00:19:10,610 --> 00:19:16,380 Well, one important point is that we introduce at each node 336 00:19:16,380 --> 00:19:21,260 lying on the mid-surface, and here we see such 9-nodes, a 337 00:19:21,260 --> 00:19:29,180 director vector, a director vector t v n k, t referring 338 00:19:29,180 --> 00:19:32,740 already to the geometry at time t. 339 00:19:32,740 --> 00:19:38,130 Of course this director vector is actually input by the 340 00:19:38,130 --> 00:19:42,810 analyst for time 0 and then it evolves with the 341 00:19:42,810 --> 00:19:45,130 motion of the shell. 342 00:19:45,130 --> 00:19:48,670 Also we're introducing for the analysis, and that's being 343 00:19:48,670 --> 00:19:51,420 done automatically in the computer program, these two 344 00:19:51,420 --> 00:19:57,990 vectors here, t v 1 k and t v 2 k, which are normal to the 345 00:19:57,990 --> 00:19:59,240 director vector. 346 00:20:01,270 --> 00:20:03,510 These vectors are calculated automatically in 347 00:20:03,510 --> 00:20:04,430 the computer program. 348 00:20:04,430 --> 00:20:06,450 We talk more about it a little later. 349 00:20:06,450 --> 00:20:11,780 Notice that the thickness here at this node is a k, and 350 00:20:11,780 --> 00:20:16,720 notice that v n k at node k-- k of course 351 00:20:16,720 --> 00:20:17,950 stands for that node-- 352 00:20:17,950 --> 00:20:21,050 acts into the direction off the thickness here. 353 00:20:21,050 --> 00:20:23,990 Notice that such [? triad ?] 354 00:20:23,990 --> 00:20:27,940 of vectors is of course being worked with at 355 00:20:27,940 --> 00:20:29,590 each of these nodes. 356 00:20:29,590 --> 00:20:34,940 And notice that the thickness at the nodes can change. 357 00:20:34,940 --> 00:20:39,390 The element is defined as follows. 358 00:20:39,390 --> 00:20:44,060 As far as the analyst is concerned the initial nodal 359 00:20:44,060 --> 00:20:48,390 point coordinates of all the nodal points on the 360 00:20:48,390 --> 00:20:51,600 mid-surface must be input. 361 00:20:51,600 --> 00:20:56,080 Also the initial director vector must be input, here now 362 00:20:56,080 --> 00:21:00,750 you see the zero, and the thickness at every 363 00:21:00,750 --> 00:21:02,800 node must be input. 364 00:21:02,800 --> 00:21:05,950 Notice if these director vectors at all of the nodes 365 00:21:05,950 --> 00:21:08,790 are known, with the thicknesses at the nodes of 366 00:21:08,790 --> 00:21:13,630 course, then we can interpolate the thickness at 367 00:21:13,630 --> 00:21:18,120 any point of the mid-surface of the shell and we can 368 00:21:18,120 --> 00:21:23,240 interpolate the director vector corresponding to any 369 00:21:23,240 --> 00:21:24,900 point on the mid-surface. 370 00:21:24,900 --> 00:21:27,640 Here such point on the mid-surface we get the 371 00:21:27,640 --> 00:21:30,910 thickness at that point, from the thicknesses that we have 372 00:21:30,910 --> 00:21:34,170 here, and from the director vectors. 373 00:21:34,170 --> 00:21:36,470 And of course we are also getting the director vector at 374 00:21:36,470 --> 00:21:38,615 this point from these director vectors. 375 00:21:42,040 --> 00:21:46,680 So the analyst must put in the nodal point coordinates of the 376 00:21:46,680 --> 00:21:50,200 mid-surface nodes and the direction cosines of these 377 00:21:50,200 --> 00:21:51,340 director vectors. 378 00:21:51,340 --> 00:21:53,330 Much of it of course can be generated 379 00:21:53,330 --> 00:21:56,190 in a practical analysis. 380 00:21:56,190 --> 00:21:59,090 We use an isoparametric coordinate system with 381 00:21:59,090 --> 00:22:01,490 coordinates r, s, and t. 382 00:22:01,490 --> 00:22:06,360 The coordinates r and s correspond to a measure in the 383 00:22:06,360 --> 00:22:07,610 mid-surface. 384 00:22:09,220 --> 00:22:14,700 The coordinate t is measured in the direction of the 385 00:22:14,700 --> 00:22:15,950 director vectors. 386 00:22:18,800 --> 00:22:25,570 The geometry at time 0 is interpolated as shown here in 387 00:22:25,570 --> 00:22:27,130 this equation. 388 00:22:27,130 --> 00:22:32,750 0 x i gives us the coordinates, three, i goes 389 00:22:32,750 --> 00:22:38,310 from one to three, of any material particle in the 390 00:22:38,310 --> 00:22:39,680 stationary coordinate frame. 391 00:22:39,680 --> 00:22:43,500 I should point out once more very strongly that we use a 392 00:22:43,500 --> 00:22:48,210 stationary Cartesian coordinate frame, x1, x2, x3, 393 00:22:48,210 --> 00:22:51,820 to describe the geometry of the element and to work with 394 00:22:51,820 --> 00:22:53,490 our element. 395 00:22:53,490 --> 00:22:57,410 This coordinate frame, x1, x2, x3, is stationary. 396 00:22:57,410 --> 00:23:00,230 In that stationary coordinate frame of course we are 397 00:23:00,230 --> 00:23:03,780 measuring the coordinates of any material particle 398 00:23:03,780 --> 00:23:06,070 corresponding to time 0, corresponding 399 00:23:06,070 --> 00:23:07,790 to time t, et cetera. 400 00:23:07,790 --> 00:23:10,790 The same way as we discussed it in earlier lectures when we 401 00:23:10,790 --> 00:23:14,870 talked about the analysis of solids, 2-D and 3-D solids, 402 00:23:14,870 --> 00:23:17,960 and when we talked about the continuum mechanics equations. 403 00:23:17,960 --> 00:23:25,030 So here these coordinates of the material particles, as the 404 00:23:25,030 --> 00:23:27,860 material particles are moving through the stationary 405 00:23:27,860 --> 00:23:31,460 coordinate frame, are given by the right hand side. 406 00:23:31,460 --> 00:23:33,020 And what do we see here? 407 00:23:33,020 --> 00:23:36,150 k is 1 to n, n is the number of nodes for the element that 408 00:23:36,150 --> 00:23:39,910 I've shown you and would be nine. 409 00:23:39,910 --> 00:23:44,910 h k are the interpolation functions corresponding to the 410 00:23:44,910 --> 00:23:50,160 two-dimensional surface of the element. 411 00:23:50,160 --> 00:23:53,220 In other words, these h k's are really the 2-D 412 00:23:53,220 --> 00:23:58,060 interpolation functions as we are used to see them for plane 413 00:23:58,060 --> 00:24:00,970 stress, plane strain, and axisymmetric analysis. 414 00:24:00,970 --> 00:24:03,390 Same interpolation functions. 415 00:24:03,390 --> 00:24:09,790 These are the nodal point coordinates at time 0. 416 00:24:09,790 --> 00:24:13,150 Here we have t, that is the third isoparametric 417 00:24:13,150 --> 00:24:14,320 coordinate. 418 00:24:14,320 --> 00:24:16,390 We talked about it just now. 419 00:24:16,390 --> 00:24:18,380 k, going from 1 to n again. 420 00:24:18,380 --> 00:24:20,980 a k are the thicknesses at the nodal points. 421 00:24:20,980 --> 00:24:25,870 h k here is exactly the same h k that you see here. 422 00:24:25,870 --> 00:24:28,460 And these are the direction cosines of 423 00:24:28,460 --> 00:24:30,460 the director vectors. 424 00:24:30,460 --> 00:24:37,090 Director vector n means director vector k, or normal. 425 00:24:37,090 --> 00:24:39,470 n really stands for normal, but it's really the director 426 00:24:39,470 --> 00:24:42,030 vector referring to the director vector k, 427 00:24:42,030 --> 00:24:43,300 of course, a node. 428 00:24:43,300 --> 00:24:47,580 0 means time 0 and i means the components, the three 429 00:24:47,580 --> 00:24:49,790 components of the director vector. 430 00:24:49,790 --> 00:24:51,360 That's what we're looking at here. 431 00:24:51,360 --> 00:24:58,130 Now if you leave this term out then you would have simply the 432 00:24:58,130 --> 00:25:02,570 interpolation of the mid-surface as for a membrane 433 00:25:02,570 --> 00:25:07,310 element, of course curved mid-surface. 434 00:25:07,310 --> 00:25:12,500 This term here is added in to take into account the effect 435 00:25:12,500 --> 00:25:13,750 of the shell thickness. 436 00:25:16,780 --> 00:25:22,100 Similarly at time t we have, applying the same 437 00:25:22,100 --> 00:25:27,040 interpolation t, x, i here now. 438 00:25:27,040 --> 00:25:31,590 Here we have t, x, i, k, same h k that we talked about 439 00:25:31,590 --> 00:25:35,610 earlier, same summing that we talked about just now. 440 00:25:35,610 --> 00:25:40,790 Here now t, v, n, i, k, the direction cosines, also 441 00:25:40,790 --> 00:25:43,350 director vectors. 442 00:25:43,350 --> 00:25:48,610 Director vector k at time t now. 443 00:25:48,610 --> 00:25:54,410 Notice that all we have done in this term and in that term 444 00:25:54,410 --> 00:25:58,030 is to substitute for the 0 that we had here and that we 445 00:25:58,030 --> 00:26:00,550 had there, the t now. 446 00:26:00,550 --> 00:26:04,520 Of course this t means time t, this t here 447 00:26:04,520 --> 00:26:06,710 means coordinate t. 448 00:26:06,710 --> 00:26:10,080 That's why we put a circle around it and wrote it out 449 00:26:10,080 --> 00:26:10,760 there once more. 450 00:26:10,760 --> 00:26:14,070 This is the coordinate, the isoparametric coordinate t. 451 00:26:14,070 --> 00:26:19,260 r s coordinates go in here, t coordinate goes in there. 452 00:26:19,260 --> 00:26:23,800 What has happened here is that originally our director vector 453 00:26:23,800 --> 00:26:27,520 might look like shown in this picture here. 454 00:26:27,520 --> 00:26:35,880 The node coordinates are given here, 0 x i k, and in the time 455 00:26:35,880 --> 00:26:41,800 from time 0 to time t this director vector moves to look 456 00:26:41,800 --> 00:26:43,200 as shown now here. 457 00:26:43,200 --> 00:26:48,850 And of course the nodal point itself has moved as well. 458 00:26:48,850 --> 00:26:53,770 So it's these quantities here, these [? rad ?] quantities 459 00:26:53,770 --> 00:26:56,380 that we're using here, which carries a curve. 460 00:26:59,260 --> 00:27:02,870 Well, to obtain the displacement of any material 461 00:27:02,870 --> 00:27:06,220 particle within the shell we proceed now exactly the same 462 00:27:06,220 --> 00:27:08,610 way as we have proceeded in the development 463 00:27:08,610 --> 00:27:10,150 of 3-D solid elements. 464 00:27:10,150 --> 00:27:14,600 We take the interpolation for the geometry at time t, 465 00:27:14,600 --> 00:27:17,810 subtract the interpolation for the geometry at time 0 and we 466 00:27:17,810 --> 00:27:20,240 get the displacements corresponding to time t. 467 00:27:20,240 --> 00:27:23,310 The result is this equation here. 468 00:27:23,310 --> 00:27:25,470 Here we have the displacements of any material 469 00:27:25,470 --> 00:27:26,970 particle in the shell. 470 00:27:26,970 --> 00:27:30,230 Here we have the nodal point displacements from 471 00:27:30,230 --> 00:27:32,810 time 0 to time t. 472 00:27:32,810 --> 00:27:38,490 Here we have the director vectors at time t, so to say, 473 00:27:38,490 --> 00:27:40,980 minus the director vectors at time 0. 474 00:27:40,980 --> 00:27:45,520 Really these are the direction cosines corresponding to the 475 00:27:45,520 --> 00:27:49,082 director vector at time t, minus the direction cosines of 476 00:27:49,082 --> 00:27:54,370 the director vector corresponding at time 0. 477 00:27:54,370 --> 00:27:58,340 Of course this quantity here is obtained by this equation 478 00:27:58,340 --> 00:28:02,950 and that quantity here is simply obtained by subtracting 479 00:28:02,950 --> 00:28:07,340 the right hand sides corresponding to t x i and 0 x 480 00:28:07,340 --> 00:28:11,950 i the way you just have seen them on the previous 481 00:28:11,950 --> 00:28:14,870 viewgraphs. 482 00:28:14,870 --> 00:28:18,150 The incremental displacement from time t to time t plus 483 00:28:18,150 --> 00:28:23,960 delta t is similarly obtained from this relationship here. 484 00:28:23,960 --> 00:28:28,290 And the result is shown here where now we have here the 485 00:28:28,290 --> 00:28:32,350 increments in the nodal point displacements. 486 00:28:32,350 --> 00:28:35,760 And here we have the increments in the direction 487 00:28:35,760 --> 00:28:41,250 cosines of the director vectors from time t to time t 488 00:28:41,250 --> 00:28:43,900 plus delta t. 489 00:28:43,900 --> 00:28:45,900 The equation is given right there. 490 00:28:48,980 --> 00:28:51,550 Well, with the equations that we have developed so far we 491 00:28:51,550 --> 00:28:55,090 are almost ready to establish the strain displacement 492 00:28:55,090 --> 00:28:59,630 matrices for the T.L. and U.L. formulations of the element. 493 00:28:59,630 --> 00:29:02,150 We have the coordinate interpolations for the 494 00:29:02,150 --> 00:29:03,320 material particles. 495 00:29:03,320 --> 00:29:05,460 We discussed those. 496 00:29:05,460 --> 00:29:10,540 And we have at the moment also the interpolation of the 497 00:29:10,540 --> 00:29:13,640 incremental displacements of the material particles in the 498 00:29:13,640 --> 00:29:16,750 shell element in terms of nodal point incremental 499 00:29:16,750 --> 00:29:21,050 displacements and the increments of the direction 500 00:29:21,050 --> 00:29:24,210 cosines of the director vectors at the nodal points. 501 00:29:24,210 --> 00:29:28,520 What we, however, want is to have the incremental 502 00:29:28,520 --> 00:29:33,370 displacements in terms of the nodal point displacements and 503 00:29:33,370 --> 00:29:36,170 nodal point rotations. 504 00:29:36,170 --> 00:29:39,210 The nodal point rotations because the incremental nodal 505 00:29:39,210 --> 00:29:42,040 point displacements and nodal point rotations are the 506 00:29:42,040 --> 00:29:47,090 engineering type quantities that we can nicely deal with 507 00:29:47,090 --> 00:29:51,280 in a computer program when we analyze shell structures. 508 00:29:51,280 --> 00:29:56,420 So what we want is to express v n i k, the increments in the 509 00:29:56,420 --> 00:30:01,170 direction cosines of the director vectors, from time t 510 00:30:01,170 --> 00:30:03,370 to time t plus delta t in terms of 511 00:30:03,370 --> 00:30:04,930 the nodal point rotations. 512 00:30:04,930 --> 00:30:08,710 And that is achieved as follows. 513 00:30:08,710 --> 00:30:16,302 Here we have the stationary coordinate frame x1, x2, x3. 514 00:30:16,302 --> 00:30:21,510 e 1 is a vector into the x1 direction, e 2 the vector into 515 00:30:21,510 --> 00:30:25,500 the x2 direction, e 3 the vector into the x3 direction. 516 00:30:25,500 --> 00:30:30,090 Here we show v n k for nodal point k. 517 00:30:30,090 --> 00:30:32,910 In other words, a director vector, at time 0. 518 00:30:32,910 --> 00:30:37,730 This one is input by the analyst. 519 00:30:37,730 --> 00:30:42,430 These two are calculated in the program. 520 00:30:42,430 --> 00:30:47,890 And of course there are such two for every nodal point k. 521 00:30:47,890 --> 00:30:51,950 For every nodal point k we also put in a director vector. 522 00:30:51,950 --> 00:30:56,080 Now the convention that is used, that can be used for the 523 00:30:56,080 --> 00:31:01,670 v 1 and v 2 calculations, that convention is given down here. 524 00:31:01,670 --> 00:31:09,690 Notice that if v n points into the e 3 direction then v 1 525 00:31:09,690 --> 00:31:12,910 points into the 1 direction and v 2 526 00:31:12,910 --> 00:31:14,760 points into the 2 direction. 527 00:31:14,760 --> 00:31:19,620 By that I mean into the e 1 and into the e 2 directions. 528 00:31:19,620 --> 00:31:25,900 Notice, this is a detail, that when v n points into the e 2 529 00:31:25,900 --> 00:31:29,740 direction this formula breaks down and you need to use some 530 00:31:29,740 --> 00:31:30,670 other convention. 531 00:31:30,670 --> 00:31:31,940 But that is a detail. 532 00:31:31,940 --> 00:31:34,800 We don't really need to discuss that 533 00:31:34,800 --> 00:31:37,470 very much here now. 534 00:31:37,470 --> 00:31:42,610 Anyways, let us say then at every nodal point k, v n has 535 00:31:42,610 --> 00:31:46,240 been input by the user, v 1, v 2, calculated by 536 00:31:46,240 --> 00:31:47,610 the computer program. 537 00:31:47,610 --> 00:31:49,630 We notice that these two are once again 538 00:31:49,630 --> 00:31:51,950 perpendicular to v n. 539 00:31:51,950 --> 00:31:58,860 Then we can directly say that the increment in the direction 540 00:31:58,860 --> 00:32:07,740 cosines of the vector, the director vector, is given via 541 00:32:07,740 --> 00:32:09,550 this relationship here. 542 00:32:09,550 --> 00:32:13,000 Now I've written this down already for time t. 543 00:32:13,000 --> 00:32:16,480 Of course it also holds for time 0, all you do is 544 00:32:16,480 --> 00:32:19,690 substitute for t, 0. 545 00:32:19,690 --> 00:32:23,390 It holds in fact for any time of the motion that we are 546 00:32:23,390 --> 00:32:25,040 considering. 547 00:32:25,040 --> 00:32:28,100 Let's see why this holds. 548 00:32:28,100 --> 00:32:31,790 Well, this picture here shows what's happening. 549 00:32:31,790 --> 00:32:37,360 Here we have v n, here we have v 2, here we have v 1. 550 00:32:37,360 --> 00:32:42,600 Notice alpha is the rotation about v 1, beta is the 551 00:32:42,600 --> 00:32:45,860 rotation about v 2. 552 00:32:45,860 --> 00:32:52,870 Notice that with the rotation beta about v 2 here, we of 553 00:32:52,870 --> 00:32:59,130 course have an increment in this vector shown by this 554 00:32:59,130 --> 00:33:06,210 component because this here is now v n at time t plus delta t 555 00:33:06,210 --> 00:33:08,650 when alpha k is 0. 556 00:33:08,650 --> 00:33:09,860 When alpha k is 0. 557 00:33:09,860 --> 00:33:12,710 Now of course you would also have the alpha k component 558 00:33:12,710 --> 00:33:16,930 coming in and that means you have to add also this term. 559 00:33:16,930 --> 00:33:21,860 Notice that once you have obtained this vector you want 560 00:33:21,860 --> 00:33:24,250 to normalize its lengths again. 561 00:33:24,250 --> 00:33:29,240 But this picture here shows why this term is a correct 562 00:33:29,240 --> 00:33:32,910 term to use in this formula and you can extend this 563 00:33:32,910 --> 00:33:37,140 picture to also include this term. 564 00:33:37,140 --> 00:33:42,440 With this relationship we can substitute for v n and that's 565 00:33:42,440 --> 00:33:45,580 now done in this equation. 566 00:33:45,580 --> 00:33:51,430 We have substituted for v n and have now our increments in 567 00:33:51,430 --> 00:33:55,400 the displacements for the material particle within the 568 00:33:55,400 --> 00:34:02,940 shell in terms of nodal point incremental displacements and 569 00:34:02,940 --> 00:34:09,139 rotations about these v 1 and v 2 axes that we defined. 570 00:34:12,310 --> 00:34:18,290 Well, having established this interpolation for the 571 00:34:18,290 --> 00:34:22,340 incremental displacements and, of course, the interpolations 572 00:34:22,340 --> 00:34:26,760 for the geometries of the element at time 0, at time t, 573 00:34:26,760 --> 00:34:30,210 we can directly establish the strain displacement matrices. 574 00:34:30,210 --> 00:34:32,940 And we will see we can then set up the k matrix, the f 575 00:34:32,940 --> 00:34:37,699 vector, the elements that go into the equilibrium equations 576 00:34:37,699 --> 00:34:41,230 the way that we discussed it in the earlier lectures. 577 00:34:41,230 --> 00:34:47,100 And from the solution of k u equals r or k delta u equals 578 00:34:47,100 --> 00:34:50,719 delta r in nonlinear analysis, we of course get our nodal 579 00:34:50,719 --> 00:34:54,500 point rotations, alpha k and beta k. 580 00:34:54,500 --> 00:34:58,230 And once we have calculated these nodal point rotations we 581 00:34:58,230 --> 00:35:03,110 obtain by this relationship here the v n at time 582 00:35:03,110 --> 00:35:04,220 t plus delta t. 583 00:35:04,220 --> 00:35:08,330 In other words, what we're doing really here is we 584 00:35:08,330 --> 00:35:13,620 integrate over all of the angle, alpha k and beta k, to 585 00:35:13,620 --> 00:35:17,600 get a more accurate approximation for the nodal 586 00:35:17,600 --> 00:35:21,780 point vector, of the director vector, I should say, at time 587 00:35:21,780 --> 00:35:23,050 t plus delta t. 588 00:35:23,050 --> 00:35:27,810 Notice that if you do this integration in one step with 589 00:35:27,810 --> 00:35:31,270 the Euler forward method, you get back the equation that we 590 00:35:31,270 --> 00:35:33,010 had earlier on the viewgraph. 591 00:35:33,010 --> 00:35:39,250 And which I tried to explain or discuss with you using this 592 00:35:39,250 --> 00:35:41,900 picture that you saw. 593 00:35:41,900 --> 00:35:44,250 And I mentioned also that after this integration of 594 00:35:44,250 --> 00:35:47,210 course we want to normalize the lengths of this vector to 595 00:35:47,210 --> 00:35:51,430 make it always a unit length vector. 596 00:35:51,430 --> 00:35:55,630 We recognize that with this approach we have only five 597 00:35:55,630 --> 00:35:57,260 degrees of freedom per node. 598 00:35:57,260 --> 00:35:59,780 Three translations in the Cartesian coordinate 599 00:35:59,780 --> 00:36:04,050 directions which are stationary, and two rotations 600 00:36:04,050 --> 00:36:09,350 refer to the local nodal point vectors v 1 and v 2 at time t. 601 00:36:09,350 --> 00:36:12,850 Now notice in geometric nonlinear analysis, of course, 602 00:36:12,850 --> 00:36:18,620 this vector and that vector, t v 1 and t v 2, change 603 00:36:18,620 --> 00:36:24,370 direction so our alpha k and beta k are rotations that are 604 00:36:24,370 --> 00:36:26,285 measured about changing directions. 605 00:36:29,190 --> 00:36:31,420 That is an important point to recognize. 606 00:36:31,420 --> 00:36:35,860 Let's look at one pictorial 607 00:36:35,860 --> 00:36:38,060 representation of what's happening. 608 00:36:38,060 --> 00:36:39,930 Here we have node k. 609 00:36:39,930 --> 00:36:44,650 We have a smooth shell, say, that is discretized using four 610 00:36:44,650 --> 00:36:45,370 shell elements. 611 00:36:45,370 --> 00:36:48,320 I've taken one shell element away as shown here so that we 612 00:36:48,320 --> 00:36:51,190 can look into the shell. 613 00:36:51,190 --> 00:37:02,870 And at that node we have, as shown now, v 1 and v 2 at time 614 00:37:02,870 --> 00:37:08,160 t and the director vector corresponding to a time t. 615 00:37:08,160 --> 00:37:13,730 We measure at that time alpha k and beta k 616 00:37:13,730 --> 00:37:17,980 about v 1 and v 2. 617 00:37:17,980 --> 00:37:25,170 And we also measure the displacements of the node, u 618 00:37:25,170 --> 00:37:28,760 1, u 2, u 3. 619 00:37:28,760 --> 00:37:34,380 So notice that this node moves as shown. 620 00:37:34,380 --> 00:37:36,200 These are the three translational degrees of 621 00:37:36,200 --> 00:37:41,930 freedom and this director vector here moves to a new 622 00:37:41,930 --> 00:37:46,580 position and also changes direction. 623 00:37:46,580 --> 00:37:49,420 And that new director vector of course is 624 00:37:49,420 --> 00:37:52,780 given here in red. 625 00:37:52,780 --> 00:37:55,270 Notice that there is no physical stiffness 626 00:37:55,270 --> 00:37:59,740 corresponding to the rotation about the director vector. 627 00:37:59,740 --> 00:38:00,980 No physical stiffness. 628 00:38:00,980 --> 00:38:04,590 The five degrees of freedom that the element very 629 00:38:04,590 --> 00:38:14,150 naturally carries, u 1, u 2, u 3, alpha k, beta k. 630 00:38:14,150 --> 00:38:16,900 Alpha, beta at every node k. 631 00:38:20,110 --> 00:38:25,840 If only shell elements connect to node k and the node k is 632 00:38:25,840 --> 00:38:29,510 not subjected to boundary-prescribed rotations 633 00:38:29,510 --> 00:38:33,830 then we only need to assign these five degrees of freedom 634 00:38:33,830 --> 00:38:37,050 to the node and only work with these five degrees of freedom 635 00:38:37,050 --> 00:38:38,230 at that node. 636 00:38:38,230 --> 00:38:42,020 However if we deal with a node to which also beam element is 637 00:38:42,020 --> 00:38:45,720 connected, which of course in general has three rotational 638 00:38:45,720 --> 00:38:50,140 degrees of freedom, or a node on which a boundary rotation 639 00:38:50,140 --> 00:38:55,760 other than alpha k or beta k is imposed then we transform 640 00:38:55,760 --> 00:39:00,580 the two nodal rotations, alpha k and beta k, to the three 641 00:39:00,580 --> 00:39:02,310 Cartesian axes. 642 00:39:02,310 --> 00:39:07,610 Because this way we can directly deal then with the 643 00:39:07,610 --> 00:39:09,880 connection here and the imposition of 644 00:39:09,880 --> 00:39:13,210 the boundary rotation. 645 00:39:13,210 --> 00:39:17,280 We can do so directly using these three rotations now 646 00:39:17,280 --> 00:39:21,700 measured in the Cartesian axis directions. 647 00:39:21,700 --> 00:39:28,740 I mentioned already that the above interpolations for 0 x 648 00:39:28,740 --> 00:39:32,580 i, t x i and u i, in other words, for the original 649 00:39:32,580 --> 00:39:35,040 geometry, the current geometry, and the incremental 650 00:39:35,040 --> 00:39:39,460 displacements the way we have developed them, they can 651 00:39:39,460 --> 00:39:42,420 directly be used to obtain the strain displacement 652 00:39:42,420 --> 00:39:43,960 transformation matrices. 653 00:39:43,960 --> 00:39:48,630 And we really do so in the same way as for the 3-D solid 654 00:39:48,630 --> 00:39:53,610 elements which we discussed in an earlier lecture. 655 00:39:53,610 --> 00:39:57,450 However, there's one method to recognize, and I briefly 656 00:39:57,450 --> 00:40:01,440 pointed it out also already in the earlier lecture, that 657 00:40:01,440 --> 00:40:04,830 using this expression here-- 658 00:40:04,830 --> 00:40:08,840 and I now must refer to the earlier material that we 659 00:40:08,840 --> 00:40:10,850 discussed-- 660 00:40:10,850 --> 00:40:15,900 using this expression to obtain the strain displacement 661 00:40:15,900 --> 00:40:19,700 matrix, we realize that we obtain the exact linear strain 662 00:40:19,700 --> 00:40:22,300 displacement matrix. 663 00:40:22,300 --> 00:40:27,330 However, using this expression here to develop the nonlinear 664 00:40:27,330 --> 00:40:32,510 strain displacement matrix, t 0 b n l, for the shell element 665 00:40:32,510 --> 00:40:35,590 only in approximation to the exact second-order strain 666 00:40:35,590 --> 00:40:39,610 displacement rotation expression is obtained because 667 00:40:39,610 --> 00:40:42,900 the internal element displacements depend 668 00:40:42,900 --> 00:40:45,120 nonlinearly on the nodal point rotations. 669 00:40:45,120 --> 00:40:47,600 I pointed that out earlier. 670 00:40:47,600 --> 00:40:52,810 Please refer back to that discussion to obtain a deeper 671 00:40:52,810 --> 00:40:55,790 understanding of what I mean here. 672 00:40:55,790 --> 00:41:00,010 The important point, of course, is that we do obtain 673 00:41:00,010 --> 00:41:04,010 the exact linear strain displacement matrix, so at 674 00:41:04,010 --> 00:41:12,220 convergence in an iteration, k delta u equals delta r, when 675 00:41:12,220 --> 00:41:17,480 we have converged we actually have the exact solution to the 676 00:41:17,480 --> 00:41:20,340 model that we're using, of course. 677 00:41:20,340 --> 00:41:24,240 So this is important that we obtain the appropriate and 678 00:41:24,240 --> 00:41:27,640 exact t 0 b l matrix. 679 00:41:27,640 --> 00:41:33,120 The effect of what we are neglecting here was earlier 680 00:41:33,120 --> 00:41:37,800 mentioned and please refer back to that lecture. 681 00:41:37,800 --> 00:41:41,160 We finally need to still impose the condition that the 682 00:41:41,160 --> 00:41:43,710 stress in the direction normal to the shell 683 00:41:43,710 --> 00:41:44,450 mid-surface is zero. 684 00:41:44,450 --> 00:41:47,770 Remember this was one further assumption that we discussed 685 00:41:47,770 --> 00:41:49,670 at the beginning of this lecture. 686 00:41:49,670 --> 00:41:51,800 We use a direction of the director 687 00:41:51,800 --> 00:41:53,155 vector as a normal direction. 688 00:41:55,670 --> 00:42:02,530 This means that at each Gauss integration point within the 689 00:42:02,530 --> 00:42:05,180 element, and which we want to evaluate the stress-strain 690 00:42:05,180 --> 00:42:11,355 law, we first of all set up a system of vectors, e bar r, e 691 00:42:11,355 --> 00:42:16,450 bar s, e t, that are mutually perpendicular. 692 00:42:16,450 --> 00:42:20,260 Now let's look here into the picture above and we see here 693 00:42:20,260 --> 00:42:23,950 at a particular Gauss point, schematically shown, the 694 00:42:23,950 --> 00:42:30,000 vector e r and e s which are vectors corresponding to the r 695 00:42:30,000 --> 00:42:31,260 and s axes. 696 00:42:33,770 --> 00:42:37,890 e t is a vector corresponding to the direction of the 697 00:42:37,890 --> 00:42:41,730 director vector at that point. 698 00:42:41,730 --> 00:42:46,590 e t we accept as the normal direction, that's for the 699 00:42:46,590 --> 00:42:53,830 shell at that point, and we construct e bar s and e bar r 700 00:42:53,830 --> 00:43:00,640 to be vectors perpendicular to e t and to themselves. 701 00:43:00,640 --> 00:43:05,070 And that is achieved by this relationship here. 702 00:43:05,070 --> 00:43:13,440 Having now established e t, e bar r and e bar s, we use this 703 00:43:13,440 --> 00:43:16,420 stress-strain law. 704 00:43:16,420 --> 00:43:19,280 And by this I mean let's look first what's 705 00:43:19,280 --> 00:43:21,870 in these round brackets. 706 00:43:21,870 --> 00:43:25,080 We use this stress-strain law corresponding to these 707 00:43:25,080 --> 00:43:27,200 directions. 708 00:43:27,200 --> 00:43:32,090 In other words, the directions e bar r, e bar s and e t where 709 00:43:32,090 --> 00:43:35,490 this is the normal direction, this corresponds to the normal 710 00:43:35,490 --> 00:43:36,780 direction e t. 711 00:43:36,780 --> 00:43:41,790 Notice by putting zeroes here we impose the fact that into 712 00:43:41,790 --> 00:43:45,220 the direction e t we have zero stress. 713 00:43:45,220 --> 00:43:47,280 Of course this matrix is symmetric. 714 00:43:47,280 --> 00:43:51,620 Notice we have k here which is a shear correction factor 715 00:43:51,620 --> 00:43:56,610 applied to the transfer shear stresses in the shell. 716 00:43:56,610 --> 00:43:59,130 New of course is Poisson's ratio and 717 00:43:59,130 --> 00:44:00,390 e is Young's modulus. 718 00:44:00,390 --> 00:44:04,497 Now this is the material law corresponding to e t, e 719 00:44:04,497 --> 00:44:07,050 bar r, e bar s. 720 00:44:07,050 --> 00:44:10,950 And what we now have to do is transform this material law at 721 00:44:10,950 --> 00:44:15,870 every integration point to the global directions. 722 00:44:15,870 --> 00:44:19,370 Global directions because for the global x1, x2, x3 723 00:44:19,370 --> 00:44:22,680 directions we have established the b, the strain 724 00:44:22,680 --> 00:44:25,160 displacement matrices. 725 00:44:25,160 --> 00:44:28,830 And that gives us this material, or c, shell. 726 00:44:28,830 --> 00:44:35,230 This q s h t, q s h is, I think, quite well known. 727 00:44:35,230 --> 00:44:38,590 Let me show you the form of it briefly. 728 00:44:38,590 --> 00:44:42,100 It's a transformation matrix where we show some of the 729 00:44:42,100 --> 00:44:44,520 terms here, as you can see. 730 00:44:47,040 --> 00:44:53,820 And these terms, l 1, m 1, n 1, et cetera, are 731 00:44:53,820 --> 00:44:56,240 defined down here. 732 00:44:56,240 --> 00:45:02,580 And l 1, for example, is nothing else but the cosine of 733 00:45:02,580 --> 00:45:07,210 the angle between e 1 and e bar r. 734 00:45:07,210 --> 00:45:09,600 In other words, it's a transformation matrix that you 735 00:45:09,600 --> 00:45:13,080 are probably quite familiar with in linear analysis. 736 00:45:13,080 --> 00:45:17,100 It's the matrix that transforms the stress and 737 00:45:17,100 --> 00:45:20,900 strain components from one coordinate system to the other 738 00:45:20,900 --> 00:45:22,050 coordinate system. 739 00:45:22,050 --> 00:45:24,170 And of course the coordinate systems that we are 740 00:45:24,170 --> 00:45:30,290 transforming, that we're using here, are the bar, e bar, r, e 741 00:45:30,290 --> 00:45:35,840 bar s, e t coordinate system on the one side and e 1, e 2, 742 00:45:35,840 --> 00:45:38,860 e 3 on the other side. 743 00:45:38,860 --> 00:45:44,530 Well, using this matrix we assure that the columns and 744 00:45:44,530 --> 00:45:50,970 rows 1 to 3 in c s h reflect that the stress normal to the 745 00:45:50,970 --> 00:45:53,610 shell mid-surface is zero. 746 00:45:53,610 --> 00:45:59,270 This holds true because, remember if we go back once 747 00:45:59,270 --> 00:46:04,230 more to the viewgraph, we have set this column and 748 00:46:04,230 --> 00:46:07,860 corresponding row to zero elements. 749 00:46:07,860 --> 00:46:11,390 And of course this means that the stress normal to the shell 750 00:46:11,390 --> 00:46:12,940 surface is zero. 751 00:46:12,940 --> 00:46:15,790 I should also briefly point out that we have a plane 752 00:46:15,790 --> 00:46:20,410 stress condition corresponding to the other direction of 753 00:46:20,410 --> 00:46:24,360 stresses, in other words, the e bar r and e 754 00:46:24,360 --> 00:46:26,240 bar s direction stresses. 755 00:46:26,240 --> 00:46:30,390 We have a plane stress condition as you can see here. 756 00:46:30,390 --> 00:46:33,390 And of course this term would go with it as well. 757 00:46:33,390 --> 00:46:36,820 This term and these terms here reflect the plane stress 758 00:46:36,820 --> 00:46:40,700 situation in the plane of the shell, zero 759 00:46:40,700 --> 00:46:42,230 stress through the thickness. 760 00:46:42,230 --> 00:46:45,480 And all that we are transforming to the global 761 00:46:45,480 --> 00:46:46,820 system now here-- 762 00:46:46,820 --> 00:46:48,980 that fact, that physical fact-- 763 00:46:48,980 --> 00:46:52,900 of course must still be reflected in c s h. 764 00:46:52,900 --> 00:46:58,670 And that's what's being said here at that point. 765 00:46:58,670 --> 00:47:03,970 If you want to do plastic analysis, creep analysis, you 766 00:47:03,970 --> 00:47:06,580 proceed in the same way. 767 00:47:06,580 --> 00:47:09,370 But you calculate then once again the stress-strain 768 00:47:09,370 --> 00:47:12,850 metrics as in the analysis of 3-D solids. 769 00:47:12,850 --> 00:47:17,340 And having got that stress-strain matrix, six by 770 00:47:17,340 --> 00:47:21,950 six matrix, you impose the condition that the stress 771 00:47:21,950 --> 00:47:25,040 normal to the mid-surface is zero in much the same way as 772 00:47:25,040 --> 00:47:31,240 what we have done here for the elastic material condition. 773 00:47:31,240 --> 00:47:34,770 Finally, regarding the kinematic descriptions that we 774 00:47:34,770 --> 00:47:38,260 talked about for the shell elements, it is interesting to 775 00:47:38,260 --> 00:47:41,940 note that also transition elements can be developed. 776 00:47:41,940 --> 00:47:47,560 These can be quite useful in practical analysis because 777 00:47:47,560 --> 00:47:52,060 they are elements with some mid-surface nodes that carry, 778 00:47:52,060 --> 00:47:55,110 in other words, associate director vectors and five 779 00:47:55,110 --> 00:47:58,760 degrees of freedom per node and some top and bottom 780 00:47:58,760 --> 00:48:01,900 surface nodes with three translational degrees of 781 00:48:01,900 --> 00:48:03,780 freedom per node. 782 00:48:03,780 --> 00:48:06,570 These elements would be used, for example, to model 783 00:48:06,570 --> 00:48:10,640 shell-to-solid transitions or to model shell intersections. 784 00:48:10,640 --> 00:48:14,400 And here you see one such typical element just 785 00:48:14,400 --> 00:48:16,220 schematically shown. 786 00:48:16,220 --> 00:48:19,210 Here we have a mid-surface node, three translational 787 00:48:19,210 --> 00:48:21,920 degrees of freedom corresponding to the 788 00:48:21,920 --> 00:48:25,350 stationary coordinate frame, two rotational degrees of 789 00:48:25,350 --> 00:48:30,900 freedom the way we talked about it just now, mid-surface 790 00:48:30,900 --> 00:48:34,420 node here and top and bottom nodes here with three 791 00:48:34,420 --> 00:48:36,560 translational degrees of freedom for 792 00:48:36,560 --> 00:48:38,200 each of these nodes. 793 00:48:38,200 --> 00:48:45,180 Notice we can directly couple 3-D solids to this phase here 794 00:48:45,180 --> 00:48:49,110 and shell elements to this phase here. 795 00:48:49,110 --> 00:48:51,560 We haven't shown mid-surface nodes here, but you would, 796 00:48:51,560 --> 00:48:54,860 say, have all the mid-surface nodes here and then surely 797 00:48:54,860 --> 00:48:58,540 shell element can directly be coupled into here in such 798 00:48:58,540 --> 00:49:02,840 situation as schematically shown right down here on this 799 00:49:02,840 --> 00:49:08,190 viewgraph where we have solid elements on this side and 800 00:49:08,190 --> 00:49:11,180 transition element right there and shell 801 00:49:11,180 --> 00:49:13,590 elements on that side. 802 00:49:13,590 --> 00:49:16,200 Because here you would continue with shell elements, 803 00:49:16,200 --> 00:49:17,920 you would continue with solid elements here. 804 00:49:17,920 --> 00:49:20,600 And once again three translational degrees of 805 00:49:20,600 --> 00:49:24,900 freedom for these top and bottom surface nodes and five 806 00:49:24,900 --> 00:49:30,350 degrees of freedom at shell mid-surface nodes as shown for 807 00:49:30,350 --> 00:49:31,760 that node right here. 808 00:49:31,760 --> 00:49:34,950 Notice we can also use these transition elements to model 809 00:49:34,950 --> 00:49:40,665 shell intersections very nicely as exemplified up here. 810 00:49:40,665 --> 00:49:42,860 Well, this brings me to the end of what I wanted to 811 00:49:42,860 --> 00:49:44,890 discuss with you in this lecture. 812 00:49:44,890 --> 00:49:47,360 Of course what we have not done yet is to look at example 813 00:49:47,360 --> 00:49:50,640 solutions and what I'd like to do in the next lecture is 814 00:49:50,640 --> 00:49:56,020 first of all talk with you about beam elements, the 815 00:49:56,020 --> 00:49:58,050 isoparametric beam element, which is formulated much in 816 00:49:58,050 --> 00:49:59,950 the same way as the shell elements 817 00:49:59,950 --> 00:50:01,670 that we just discussed. 818 00:50:01,670 --> 00:50:04,870 And then I'd like to show you applications of the beam 819 00:50:04,870 --> 00:50:07,800 elements as well as the shell elements. 820 00:50:07,800 --> 00:50:10,720 So please, if you are interested in this subject 821 00:50:10,720 --> 00:50:16,580 matter, continue looking at the second tape, part two of 822 00:50:16,580 --> 00:50:17,620 this set of lectures. 823 00:50:17,620 --> 00:50:19,390 Thank you very much for your attention.