1 00:00:00,030 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:04,000 Commons license. 3 00:00:04,000 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high quality educational resources for free. 5 00:00:10,690 --> 00:00:13,310 To make a donation or view additional materials 6 00:00:13,310 --> 00:00:17,025 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,025 --> 00:00:17,650 at ocw.mit.edu. 8 00:00:27,035 --> 00:00:28,410 LORNA GIBSON: So I was just going 9 00:00:28,410 --> 00:00:31,584 to be here to answer questions. 10 00:00:31,584 --> 00:00:33,740 AUDIENCE: Just clarifying, What was the material 11 00:00:33,740 --> 00:00:34,656 that we were covering? 12 00:00:36,327 --> 00:00:37,410 LORNA GIBSON: In the test? 13 00:00:37,410 --> 00:00:38,290 AUDIENCE: Yeah. 14 00:00:38,290 --> 00:00:40,570 LORNA GIBSON: So the test covers everything up 15 00:00:40,570 --> 00:00:43,410 to the end of the part on modeling foams, 16 00:00:43,410 --> 00:00:45,886 but not the bit on the performance indices, 17 00:00:45,886 --> 00:00:49,733 and the material selection charts for foams. 18 00:00:49,733 --> 00:00:52,274 So I think up to the end of the fractured toughness of foams. 19 00:00:52,274 --> 00:00:53,202 AUDIENCE: I see. 20 00:00:53,202 --> 00:00:54,130 OK. 21 00:00:54,130 --> 00:00:58,009 So not covering past [INAUDIBLE]. 22 00:00:58,009 --> 00:01:00,092 LORNA GIBSON: Not covering thermal properties, no. 23 00:01:00,092 --> 00:01:02,020 It doesn't cover thermal properties. 24 00:01:17,144 --> 00:01:19,484 Here you go. 25 00:01:19,484 --> 00:01:22,160 So you know I got this MacVicar award, 26 00:01:22,160 --> 00:01:26,310 and we had a lunch on Friday at the Catalyst restaurant. 27 00:01:26,310 --> 00:01:27,709 And I had to get up and speak. 28 00:01:27,709 --> 00:01:29,250 And just as I got up and spoke, there 29 00:01:29,250 --> 00:01:31,285 was a red-tailed hawk swooped by the window. 30 00:01:31,285 --> 00:01:31,910 It was perfect. 31 00:01:31,910 --> 00:01:35,063 It was perfect. 32 00:01:35,063 --> 00:01:35,562 Yeah. 33 00:01:35,562 --> 00:01:36,960 Hi. 34 00:01:36,960 --> 00:01:38,970 So I'm just here to answer questions. 35 00:01:38,970 --> 00:01:40,300 So come on. 36 00:01:40,300 --> 00:01:41,920 Somebody must have questions. 37 00:01:41,920 --> 00:01:44,280 It's all perfectly clear? 38 00:01:44,280 --> 00:01:45,100 You want me to do-- 39 00:01:45,100 --> 00:01:50,565 AUDIENCE: Talk through test one from last year a little bit. 40 00:01:50,565 --> 00:01:52,190 LORNA GIBSON: You would have to give me 41 00:01:52,190 --> 00:01:53,295 test one from last year. 42 00:01:53,295 --> 00:01:54,040 I didn't bring it with me. 43 00:01:54,040 --> 00:01:55,030 I just brought the problem sets. 44 00:01:55,030 --> 00:01:55,910 AUDIENCE: I have it on my computer, 45 00:01:55,910 --> 00:01:57,760 and I could read you the problems 46 00:01:57,760 --> 00:01:59,652 or just hand you the laptop. 47 00:01:59,652 --> 00:02:00,620 Whichever you prefer. 48 00:02:00,620 --> 00:02:02,536 LORNA GIBSON: Why don't you hand me the laptop 49 00:02:02,536 --> 00:02:03,780 and I'll try to do it. 50 00:02:03,780 --> 00:02:06,110 Is that OK? 51 00:02:06,110 --> 00:02:06,760 OK. 52 00:02:06,760 --> 00:02:11,200 So the question is about the test for 2014. 53 00:02:11,200 --> 00:02:12,250 OK. 54 00:02:12,250 --> 00:02:14,970 So the first question was, describe four processes 55 00:02:14,970 --> 00:02:18,930 for making honeycombs, and comment on the type of material 56 00:02:18,930 --> 00:02:20,880 usually used for each process. 57 00:02:20,880 --> 00:02:22,580 So I did post the solutions, right? 58 00:02:22,580 --> 00:02:23,486 Did you look at that? 59 00:02:23,486 --> 00:02:24,860 AUDIENCE: Yeah, I looked at them. 60 00:02:24,860 --> 00:02:29,500 I guess I just feel like I don't fully understand 61 00:02:29,500 --> 00:02:31,649 why things are there. 62 00:02:31,649 --> 00:02:32,940 But I can look at it some more. 63 00:02:32,940 --> 00:02:35,582 LORNA GIBSON: Well, I can go over it, if you want. 64 00:02:35,582 --> 00:02:37,290 AUDIENCE: I'll just look at it some more. 65 00:02:37,290 --> 00:02:38,440 LORNA GIBSON: No, I can go over it. 66 00:02:38,440 --> 00:02:39,390 So four processes. 67 00:02:39,390 --> 00:02:40,020 Let's see. 68 00:02:40,020 --> 00:02:43,466 So there's the expansion process, where you take sheets 69 00:02:43,466 --> 00:02:44,840 and you glue the sheets together, 70 00:02:44,840 --> 00:02:46,590 and then you pull them apart. 71 00:02:46,590 --> 00:02:48,600 So you can only really use that process 72 00:02:48,600 --> 00:02:50,280 for materials that are going to have 73 00:02:50,280 --> 00:02:52,860 large plastic deformations. 74 00:02:52,860 --> 00:02:56,260 So you could use it for metals, some polymers. 75 00:02:56,260 --> 00:02:58,420 But you couldn't really use it for ceramics. 76 00:02:58,420 --> 00:03:00,700 You couldn't use it for glass because as soon 77 00:03:00,700 --> 00:03:03,571 as you yanked on it, you'd break the sheets, right? 78 00:03:03,571 --> 00:03:04,070 So-- 79 00:03:04,070 --> 00:03:06,860 AUDIENCE: Is it rigid polymers that you can use it for? 80 00:03:06,860 --> 00:03:09,360 LORNA GIBSON: Well, something like nylon you can use it for. 81 00:03:09,360 --> 00:03:10,660 Something that's got a little yield, 82 00:03:10,660 --> 00:03:12,326 that will have some sort of yield point. 83 00:03:12,326 --> 00:03:15,250 Not like if you had epoxy, you couldn't use it for epoxy. 84 00:03:15,250 --> 00:03:16,304 So, OK. 85 00:03:16,304 --> 00:03:17,220 So that's one process. 86 00:03:17,220 --> 00:03:19,410 Another process is a corrugation process 87 00:03:19,410 --> 00:03:23,610 where you have a wheel that has little gear knobs on it. 88 00:03:23,610 --> 00:03:26,440 And you run your flat sheet through that 89 00:03:26,440 --> 00:03:28,860 and it comes out with the half hexagonal profile, 90 00:03:28,860 --> 00:03:30,310 and you glue those together. 91 00:03:30,310 --> 00:03:31,760 So again, you need something that's going to yield. 92 00:03:31,760 --> 00:03:33,420 So that would typically be a metal 93 00:03:33,420 --> 00:03:35,600 that you would use that with. 94 00:03:35,600 --> 00:03:36,150 Let's see. 95 00:03:36,150 --> 00:03:39,250 Another processor making honeycombs is 3D printing. 96 00:03:39,250 --> 00:03:41,550 You can 3D print honeycombs. 97 00:03:41,550 --> 00:03:44,480 And there's different ways to do it. 98 00:03:44,480 --> 00:03:46,420 One way is by having an ink. 99 00:03:46,420 --> 00:03:50,440 So if you want to print in ink, typically that's 100 00:03:50,440 --> 00:03:52,860 some sort of polymer that you're printing. 101 00:03:52,860 --> 00:03:55,902 I suppose physically it's possible to print glass 102 00:03:55,902 --> 00:03:57,360 or to print a metal, but you'd have 103 00:03:57,360 --> 00:04:01,290 to have some very high temperature setup to do that. 104 00:04:01,290 --> 00:04:05,810 So typically a resin of some sort. 105 00:04:05,810 --> 00:04:06,500 Let's see. 106 00:04:06,500 --> 00:04:08,320 Other ways to make honeycombs. 107 00:04:08,320 --> 00:04:10,000 You can extrude honeycombs. 108 00:04:10,000 --> 00:04:12,310 So the ceramic honeycombs we saw were made 109 00:04:12,310 --> 00:04:14,810 by extruding a ceramic slurry. 110 00:04:14,810 --> 00:04:17,180 And typically, you would do that with a ceramic 111 00:04:17,180 --> 00:04:18,570 that's a slurry and a powder. 112 00:04:18,570 --> 00:04:20,510 You wouldn't necessarily do that with a metal. 113 00:04:20,510 --> 00:04:21,979 I don't think I've seen any metal honeycombs that 114 00:04:21,979 --> 00:04:23,440 are extruded like that. 115 00:04:23,440 --> 00:04:23,959 OK? 116 00:04:23,959 --> 00:04:24,500 AUDIENCE: OK. 117 00:04:24,500 --> 00:04:26,250 LORNA GIBSON: Are we good with number one? 118 00:04:26,250 --> 00:04:27,992 AUDIENCE: Yeah, that makes sense. 119 00:04:27,992 --> 00:04:29,658 LORNA GIBSON: We now need your password. 120 00:04:29,658 --> 00:04:30,546 AUDIENCE: Oh, sorry. 121 00:04:35,402 --> 00:04:38,825 So is there an actual difference between 3D printing 122 00:04:38,825 --> 00:04:40,790 and the extrusion process? 123 00:04:40,790 --> 00:04:41,780 LORNA GIBSON: Yeah. 124 00:04:41,780 --> 00:04:45,840 So the extrusion you have a die, and you squeeze the material 125 00:04:45,840 --> 00:04:47,520 through the die, right? 126 00:04:47,520 --> 00:04:50,230 So extrusion's kind of like the toothpastey thing. 127 00:04:50,230 --> 00:04:52,780 And 3D printing, you can have an ink 128 00:04:52,780 --> 00:04:58,850 or you can do 3D printing where you have, let's say, a powder. 129 00:04:58,850 --> 00:05:01,080 And then you print the binder. 130 00:05:01,080 --> 00:05:05,550 And then you heat it up some way to get the binder to cure. 131 00:05:05,550 --> 00:05:07,790 And then you get rid of the powder that's not bound. 132 00:05:07,790 --> 00:05:10,110 That's another way to do the 3D printing. 133 00:05:10,110 --> 00:05:11,130 OK? 134 00:05:11,130 --> 00:05:13,239 AUDIENCE: Can you also pour it into a mold? 135 00:05:13,239 --> 00:05:14,030 LORNA GIBSON: Yeah. 136 00:05:14,030 --> 00:05:16,900 Yeah, those silicon rubber honeycombs that I showed you, 137 00:05:16,900 --> 00:05:19,560 those are all made by pouring a liquid into a mold 138 00:05:19,560 --> 00:05:20,390 and then curing it. 139 00:05:20,390 --> 00:05:20,890 Yeah. 140 00:05:20,890 --> 00:05:23,060 Yeah, there's other ways, it just asked for four, 141 00:05:23,060 --> 00:05:24,800 so I randomly thought of four. 142 00:05:24,800 --> 00:05:26,330 OK. 143 00:05:26,330 --> 00:05:29,640 OK, are we good? 144 00:05:29,640 --> 00:05:32,740 So the next one is a hexagonal titanium alloy 145 00:05:32,740 --> 00:05:39,390 honeycomb has h/l is two, theta is 45, and t/l is 0.05. 146 00:05:39,390 --> 00:05:41,890 It says, the end constraint factor for elastic buckling 147 00:05:41,890 --> 00:05:45,020 is n equals 0.806. 148 00:05:45,020 --> 00:05:47,870 The titanium has a modulus of 110 gigapascals, 149 00:05:47,870 --> 00:05:49,802 and yield strength of 880. 150 00:05:49,802 --> 00:05:51,760 And then you have to calculate some properties. 151 00:05:51,760 --> 00:05:53,896 So would you like me to do that? 152 00:05:53,896 --> 00:05:54,590 AUDIENCE: Yeah. 153 00:05:54,590 --> 00:05:55,490 LORNA GIBSON: Yeah, you would? 154 00:05:55,490 --> 00:05:56,182 OK. 155 00:05:56,182 --> 00:05:58,382 AUDIENCE: Does anybody else want that? 156 00:05:58,382 --> 00:06:00,340 LORNA GIBSON: I don't see a lot of other people 157 00:06:00,340 --> 00:06:04,110 wanting anything else, so I might as well do that. 158 00:06:04,110 --> 00:06:04,680 Let's see. 159 00:06:04,680 --> 00:06:06,880 I would need a piece of chalk. 160 00:06:06,880 --> 00:06:08,850 Here we go. 161 00:06:08,850 --> 00:06:09,540 OK. 162 00:06:09,540 --> 00:06:17,150 So it's a titanium alloy honeycomb, 163 00:06:17,150 --> 00:06:30,810 and we're told h/l is 2, theta's 45, and t/l is 0.05. 164 00:06:30,810 --> 00:06:34,120 And we're told that n-- why don't I put the n over here-- n 165 00:06:34,120 --> 00:06:38,135 is 0.806. 166 00:06:38,135 --> 00:06:46,270 And we're told the modulus of the solid is 110 gigapascals, 167 00:06:46,270 --> 00:06:54,131 and the yield strength of the solid is 880 mega pascals. 168 00:06:54,131 --> 00:06:54,630 OK. 169 00:06:54,630 --> 00:06:56,230 So it says calculate the value of 170 00:06:56,230 --> 00:06:58,750 and describe the mechanism of deformation failure 171 00:06:58,750 --> 00:07:02,980 for-- and the first part is the Young's modulus in the two 172 00:07:02,980 --> 00:07:05,440 direction, e star 2. 173 00:07:05,440 --> 00:07:06,500 OK. 174 00:07:06,500 --> 00:07:09,950 So I don't remember these formulas either, 175 00:07:09,950 --> 00:07:13,210 so I need to look at my notes. 176 00:07:13,210 --> 00:07:16,415 And oh, I don't have the formula for e 2 in my notes. 177 00:07:20,260 --> 00:07:20,760 Let me see. 178 00:07:20,760 --> 00:07:22,102 Is it in any of the problems? 179 00:07:22,102 --> 00:07:23,560 AUDIENCE: I have the formula sheet. 180 00:07:23,560 --> 00:07:25,268 LORNA GIBSON: You have the formula sheet? 181 00:07:25,268 --> 00:07:27,910 I didn't bring the formula sheet with me. 182 00:07:27,910 --> 00:07:28,634 You have it? 183 00:07:32,586 --> 00:07:34,028 Yeah, I think it's there. 184 00:07:34,028 --> 00:07:37,940 I'm pretty sure it's there. 185 00:07:37,940 --> 00:07:40,770 OK, so this is just like substituting, 186 00:07:40,770 --> 00:07:44,450 and there's nothing complicated about this. 187 00:07:44,450 --> 00:07:56,580 So it's equal to Es times t/l cubed times h/l plus sine theta 188 00:07:56,580 --> 00:08:01,230 divided by cos cubed theta. 189 00:08:01,230 --> 00:08:02,730 So then you just plug everything in. 190 00:08:02,730 --> 00:08:05,360 So this is 110 gigapascals. 191 00:08:05,360 --> 00:08:09,580 And I'm going to put 110,000 mega pascals, 192 00:08:09,580 --> 00:08:12,406 because it's probably going to be less than a gigapascal. 193 00:08:12,406 --> 00:08:14,480 Then t/l is 0.05. 194 00:08:14,480 --> 00:08:19,120 So that's 0.05 cubed. 195 00:08:19,120 --> 00:08:28,980 And then h/l is 2 plus sign of 45 is 0.707. 196 00:08:28,980 --> 00:08:35,480 And then we divide by cos theta cubed, 0.707 cubed. 197 00:08:35,480 --> 00:08:39,510 And then I'm not going to work it out, but that's OK. 198 00:08:39,510 --> 00:08:41,580 I assume that's what's in the solution. 199 00:08:41,580 --> 00:08:42,387 Are we good? 200 00:08:42,387 --> 00:08:43,470 So it's just substituting. 201 00:08:43,470 --> 00:08:44,880 That's all it is. 202 00:08:44,880 --> 00:08:48,530 In the second part-- OK. 203 00:08:48,530 --> 00:08:52,080 You need to change the time on this, because it just 204 00:08:52,080 --> 00:08:53,065 keeps timing out. 205 00:08:53,065 --> 00:08:56,540 AUDIENCE: Sorry, I don't know how to change it, but I'll try. 206 00:08:56,540 --> 00:08:58,650 LORNA GIBSON: Oh, is that what this is? 207 00:08:58,650 --> 00:09:00,850 OK, this is the test. 208 00:09:00,850 --> 00:09:03,970 Oh, this is from 2013. 209 00:09:03,970 --> 00:09:06,890 Oh, the next-- if we keep going. 210 00:09:06,890 --> 00:09:07,610 Here we go. 211 00:09:07,610 --> 00:09:09,345 He's got it, so you take your computer. 212 00:09:09,345 --> 00:09:10,780 AUDIENCE: That's probably better. 213 00:09:10,780 --> 00:09:12,660 LORNA GIBSON: That's perfect for everybody. 214 00:09:12,660 --> 00:09:13,160 OK. 215 00:09:13,160 --> 00:09:15,420 The second part is the plateau stress 216 00:09:15,420 --> 00:09:18,930 for loading in the x2 direction. 217 00:09:18,930 --> 00:09:20,530 So that's that. 218 00:09:20,530 --> 00:09:22,220 For loading in the x2 direction, it 219 00:09:22,220 --> 00:09:26,830 could either be an elastic buckling collapse 220 00:09:26,830 --> 00:09:29,294 stress, or a plastic buckling collapse stress. 221 00:09:29,294 --> 00:09:30,960 So I'm going to calculate both, and then 222 00:09:30,960 --> 00:09:34,250 whichever one is lower, that's the one it would be. 223 00:09:34,250 --> 00:09:35,330 So let's see here. 224 00:09:35,330 --> 00:09:39,880 So that's going to be-- I'm missing my formulas again. 225 00:09:39,880 --> 00:09:40,440 Here we are. 226 00:09:40,440 --> 00:09:41,920 So here's the buckling one. 227 00:09:41,920 --> 00:09:50,200 It's n squared pi squared over 24 times t cubed over lh 228 00:09:50,200 --> 00:09:53,860 squared times 1 over cos theta. 229 00:09:53,860 --> 00:09:57,960 So you put 0.806 squared in here. 230 00:10:01,832 --> 00:10:06,860 Pi squared 24. 231 00:10:06,860 --> 00:10:07,690 So here you go. 232 00:10:07,690 --> 00:10:14,650 This is t over l cubed times h over l squared. 233 00:10:14,650 --> 00:10:19,590 So that would be 0.05 cubed. 234 00:10:19,590 --> 00:10:28,180 And that would be 1 over 2 squared, and then 1 over 0.707, 235 00:10:28,180 --> 00:10:29,550 and then whatever that equals. 236 00:10:29,550 --> 00:10:30,225 OK? 237 00:10:30,225 --> 00:10:31,430 Are we good with that? 238 00:10:36,380 --> 00:10:41,615 And then you'd want to calculate sigma star to plastic. 239 00:10:45,080 --> 00:10:52,570 And that's equal to sigma ys, t over l squared, 240 00:10:52,570 --> 00:10:59,850 and 1 over 2 cos squared theta. 241 00:10:59,850 --> 00:11:00,410 All right? 242 00:11:00,410 --> 00:11:04,910 So then sigma ys was 880 mega pascals. 243 00:11:07,600 --> 00:11:11,545 And t over l was our 0.05. 244 00:11:11,545 --> 00:11:16,780 And that's 1 over 2 times 0.707 squared, 245 00:11:16,780 --> 00:11:20,540 and then whatever that equals. 246 00:11:20,540 --> 00:11:21,820 OK? 247 00:11:21,820 --> 00:11:24,620 And then whichever one of those would be less 248 00:11:24,620 --> 00:11:26,910 is the plateau stress. 249 00:11:26,910 --> 00:11:28,325 So I don't-- yeah? 250 00:11:28,325 --> 00:11:30,366 AUDIENCE: So I know-- I think I made this mistake 251 00:11:30,366 --> 00:11:34,950 in the problem set, but here, because it's titanium, 252 00:11:34,950 --> 00:11:36,607 we don't consider it brittle. 253 00:11:36,607 --> 00:11:37,440 LORNA GIBSON: Right. 254 00:11:37,440 --> 00:11:37,870 Right. 255 00:11:37,870 --> 00:11:38,620 I mean, you could. 256 00:11:38,620 --> 00:11:40,130 But you don't need to. 257 00:11:40,130 --> 00:11:42,070 AUDIENCE: That was something with ceramics? 258 00:11:42,070 --> 00:11:44,850 LORNA GIBSON: Yeah, or if it was a glass, or ceramic, or maybe 259 00:11:44,850 --> 00:11:45,770 an epoxy. 260 00:11:45,770 --> 00:11:47,430 Something that was brittle. 261 00:11:47,430 --> 00:11:49,400 And besides which, if you look at the question, 262 00:11:49,400 --> 00:11:53,070 I only give you a yield strength and a solid modulus. 263 00:11:53,070 --> 00:11:54,570 So to get the brittle thing, I would 264 00:11:54,570 --> 00:11:56,025 have to give you a fracture strength. 265 00:11:56,025 --> 00:11:57,858 AUDIENCE: That's true with all the problems. 266 00:11:59,392 --> 00:12:01,350 LORNA GIBSON: I usually give you what you need. 267 00:12:01,350 --> 00:12:02,550 Especially on the test, I'm going 268 00:12:02,550 --> 00:12:03,410 to give you what you need. 269 00:12:03,410 --> 00:12:05,220 You're not going to be looking things up. 270 00:12:05,220 --> 00:12:06,340 OK? 271 00:12:06,340 --> 00:12:08,370 So we're good so far? 272 00:12:08,370 --> 00:12:12,360 OK, let me go back to the question. 273 00:12:12,360 --> 00:12:18,530 So then the third one is the out of plane Young's modulus 274 00:12:18,530 --> 00:12:20,810 in the x3 direction. 275 00:12:20,810 --> 00:12:24,350 And that's just going to be Es times the relative density. 276 00:12:24,350 --> 00:12:27,470 And the relative density-- let's see. 277 00:12:27,470 --> 00:12:31,940 So it's Es times rho star over rho s. 278 00:12:31,940 --> 00:12:39,236 So that's 110,000 mega pascals. 279 00:12:39,236 --> 00:12:41,110 And then there's also the very first equation 280 00:12:41,110 --> 00:12:42,720 on this is the relative density. 281 00:12:42,720 --> 00:12:51,920 So it's t/l times h/l plus 2 divided by 2 cos 282 00:12:51,920 --> 00:12:56,350 theta h/l plus sine theta. 283 00:12:56,350 --> 00:12:59,170 So I'm not going to substitute everything in, OK? 284 00:12:59,170 --> 00:13:00,310 So that was it. 285 00:13:00,310 --> 00:13:05,000 It was very plug and chug. 286 00:13:05,000 --> 00:13:05,640 Are you good? 287 00:13:05,640 --> 00:13:09,402 AUDIENCE: Can I ask a question about a specific question. 288 00:13:12,049 --> 00:13:14,340 LORNA GIBSON: I'm going to go through the rest of them. 289 00:13:14,340 --> 00:13:15,410 She wants me to do the whole test. 290 00:13:15,410 --> 00:13:16,640 You want me to do the whole test, right? 291 00:13:16,640 --> 00:13:18,999 AUDIENCE: That would be great, but if other people 292 00:13:18,999 --> 00:13:20,290 have other questions it's fine. 293 00:13:20,290 --> 00:13:21,090 LORNA GIBSON: Why don't I do the whole test. 294 00:13:21,090 --> 00:13:23,213 And then there's going to be time, I think. 295 00:13:23,213 --> 00:13:24,129 AUDIENCE: [INAUDIBLE]. 296 00:13:35,652 --> 00:13:38,260 LORNA GIBSON: Let me do the test from last unit. 297 00:13:38,260 --> 00:13:40,585 OK, so that's number one. 298 00:13:40,585 --> 00:13:42,870 Or that's one and two. 299 00:13:42,870 --> 00:13:50,800 And then three is, a closed cell elastomeric polyethylene 300 00:13:50,800 --> 00:13:54,210 foam has a relative density of 0.05 301 00:13:54,210 --> 00:13:57,869 and a volume fraction of solid in the edges of 0.6. 302 00:13:57,869 --> 00:13:59,785 They give you the Young's modulus of the solid 303 00:13:59,785 --> 00:14:01,464 is 0.2 gigapascals. 304 00:14:01,464 --> 00:14:02,880 The pressure within the cell walls 305 00:14:02,880 --> 00:14:05,360 is atmospheric, 0.1 mega pascals. 306 00:14:05,360 --> 00:14:08,170 And the Poisson's ratio of the foam is 0.3. 307 00:14:08,170 --> 00:14:10,650 And you're asked to get the Young's modulus of the foam, 308 00:14:10,650 --> 00:14:12,320 the compressive plateau stress. 309 00:14:12,320 --> 00:14:14,450 And then there's a question about why 310 00:14:14,450 --> 00:14:16,870 does the Young's modulus depend on the solid modulus 311 00:14:16,870 --> 00:14:20,330 and relative density, while the Poisson's ratio does not. 312 00:14:20,330 --> 00:14:23,725 So let me go through, then. 313 00:14:28,240 --> 00:14:31,200 So the first one is, what's e star. 314 00:14:31,200 --> 00:14:37,690 And we're told relative density is 0.05. 315 00:14:37,690 --> 00:14:41,096 The volume fraction in the edges is 0.6. 316 00:14:41,096 --> 00:14:42,860 So remember, that was what we called phi. 317 00:14:42,860 --> 00:14:45,610 So phi's 0.6. 318 00:14:45,610 --> 00:14:54,140 The Young's modulus of the solid is 0.2 gigapascals. 319 00:14:54,140 --> 00:15:00,130 The initial pressure within the cells is 0.1 mega pascal. 320 00:15:00,130 --> 00:15:08,410 and Poisson's ratio for the foam is 0.3. 321 00:15:08,410 --> 00:15:09,874 And it's a closed cell. 322 00:15:14,122 --> 00:15:15,580 So if you remember-- oh, let's see. 323 00:15:15,580 --> 00:15:16,955 I think back here, was I supposed 324 00:15:16,955 --> 00:15:18,774 to-- I was supposed to say something 325 00:15:18,774 --> 00:15:20,190 about the mechanism of deformation 326 00:15:20,190 --> 00:15:21,440 and failure for the first one. 327 00:15:21,440 --> 00:15:24,630 So the mechanism of deformation in the modulus in the 2 328 00:15:24,630 --> 00:15:26,510 direction is bending. 329 00:15:26,510 --> 00:15:28,591 The mechanism of failure here is buckling. 330 00:15:28,591 --> 00:15:30,340 The mechanism of failure here is yielding. 331 00:15:30,340 --> 00:15:34,700 And the mechanism of deformation here was axial deformation, OK? 332 00:15:34,700 --> 00:15:36,400 So I forgot to say that. 333 00:15:36,400 --> 00:15:38,030 OK, let me go back here. 334 00:15:38,030 --> 00:15:40,110 So for this one, if it's a closed cell foam, 335 00:15:40,110 --> 00:15:42,280 remember there were three terms to the modulus. 336 00:15:42,280 --> 00:15:44,710 There was one from bending of the edges, one 337 00:15:44,710 --> 00:15:46,790 from stretching of the faces, and one 338 00:15:46,790 --> 00:15:48,960 from the gas contribution if you've 339 00:15:48,960 --> 00:15:50,840 got gas inside the cells. 340 00:15:50,840 --> 00:15:53,756 So again, I'm going to have to peek at the equation. 341 00:15:56,790 --> 00:15:57,290 Foams. 342 00:15:57,290 --> 00:15:59,125 Here we go, foams. 343 00:15:59,125 --> 00:15:59,625 OK. 344 00:16:18,330 --> 00:16:19,740 And then this gas one. 345 00:16:28,115 --> 00:16:29,240 Get rid of that. 346 00:16:38,570 --> 00:16:40,540 OK, so this one is just the same kind of thing. 347 00:16:40,540 --> 00:16:42,760 It's just plug and chug. 348 00:16:42,760 --> 00:16:44,413 Do I need to put all the numbers in? 349 00:16:44,413 --> 00:16:47,144 AUDIENCE: No. 350 00:16:47,144 --> 00:16:49,450 I guess I'm a little bit confused 351 00:16:49,450 --> 00:16:53,108 why you need the pressure term. 352 00:16:53,108 --> 00:16:56,730 Because you talked about faces bursting. 353 00:16:56,730 --> 00:17:01,692 LORNA GIBSON: Ah, so if it's the modulus, remember the modulus-- 354 00:17:01,692 --> 00:17:03,150 so the question is, why do you need 355 00:17:03,150 --> 00:17:06,240 to worry about the pressure because I talked 356 00:17:06,240 --> 00:17:08,390 about the faces bursting. 357 00:17:08,390 --> 00:17:10,380 And remember, the stress strain curve 358 00:17:10,380 --> 00:17:12,599 looks something like that. 359 00:17:12,599 --> 00:17:18,079 Maybe the slope of the curve is a little bit higher over here. 360 00:17:18,079 --> 00:17:20,040 But this is the modulus down here, right? 361 00:17:20,040 --> 00:17:23,609 The modulus is related to the initial stress strain 362 00:17:23,609 --> 00:17:24,920 relationship. 363 00:17:24,920 --> 00:17:27,800 And initially, they're not going to burst. 364 00:17:27,800 --> 00:17:31,080 You'd have to load it up to some amount of stress 365 00:17:31,080 --> 00:17:33,460 before the faces burst, right? 366 00:17:33,460 --> 00:17:36,434 So when you're down here, the faces 367 00:17:36,434 --> 00:17:38,850 certainly down at the beginning, they're not burst, right? 368 00:17:38,850 --> 00:17:41,400 You have to get some stress before they're going to burst. 369 00:17:41,400 --> 00:17:44,750 And in some materials, when you get up around here, 370 00:17:44,750 --> 00:17:48,480 around the plateau stress-- let's just call that sigma 371 00:17:48,480 --> 00:17:51,340 star-- then they might burst. 372 00:17:51,340 --> 00:17:53,970 And then the pressure term would disappear 373 00:17:53,970 --> 00:17:56,890 and the face term would disappear. 374 00:17:56,890 --> 00:18:01,150 So if I don't tell you to ignore them, if I don't say 375 00:18:01,150 --> 00:18:04,720 they're going to burst, or I don't say they're negligible, 376 00:18:04,720 --> 00:18:05,930 I would calculate them. 377 00:18:05,930 --> 00:18:07,640 And then if they're small, then you say, 378 00:18:07,640 --> 00:18:09,440 well, they're negligible. 379 00:18:09,440 --> 00:18:10,854 OK? 380 00:18:10,854 --> 00:18:11,770 Are we good with that? 381 00:18:11,770 --> 00:18:13,650 You're good? 382 00:18:13,650 --> 00:18:14,590 You're good? 383 00:18:14,590 --> 00:18:17,670 Sardar, you don't need to be here. 384 00:18:17,670 --> 00:18:20,320 But you can stay if you want, but you don't need to be here. 385 00:18:20,320 --> 00:18:23,120 OK, are you good? 386 00:18:23,120 --> 00:18:25,130 Everybody else good? 387 00:18:25,130 --> 00:18:26,180 OK. 388 00:18:26,180 --> 00:18:32,450 That was A. B, what's the compressive plateau 389 00:18:32,450 --> 00:18:35,680 stress of the foam. 390 00:18:35,680 --> 00:18:38,660 So here, they want to know what sigma star is. 391 00:18:38,660 --> 00:18:40,730 You're told it's elastomeric. 392 00:18:40,730 --> 00:18:42,610 So if it's elastomeric, it's like a rubber. 393 00:18:42,610 --> 00:18:43,692 It's rubbery. 394 00:18:43,692 --> 00:18:45,400 So if it's rubbery, it's going to buckle. 395 00:18:45,400 --> 00:18:46,400 It's not going to yield. 396 00:18:46,400 --> 00:18:47,940 It's not going to be brittle. 397 00:18:47,940 --> 00:18:53,600 So you can just calculate the elastic stress here. 398 00:18:53,600 --> 00:19:01,512 And if we flip over to our handy dandy list of equations-- blah, 399 00:19:01,512 --> 00:19:02,220 blah, blah, blah. 400 00:19:02,220 --> 00:19:02,896 Oh, pooh. 401 00:19:06,270 --> 00:19:10,300 I'm realizing-- yeah, so I don't have 402 00:19:10,300 --> 00:19:14,770 the term here for the faces or for the gas, 403 00:19:14,770 --> 00:19:17,233 so here we could assume it's going to rupture. 404 00:19:17,233 --> 00:19:18,816 So let's assume it's going to rupture. 405 00:19:27,179 --> 00:19:28,970 So if we assume that it's going to rupture, 406 00:19:28,970 --> 00:19:33,670 it would be just like the open celled foams. 407 00:19:33,670 --> 00:19:39,955 And then you can just use that. 408 00:19:43,530 --> 00:19:47,310 And that's been found to work fairly well for the open celled 409 00:19:47,310 --> 00:19:48,912 and the closed cell foams. 410 00:19:48,912 --> 00:19:54,340 AUDIENCE: And we assume that faces rupture because-- 411 00:19:54,340 --> 00:19:55,860 LORNA GIBSON: Well, to be honest, 412 00:19:55,860 --> 00:19:59,154 I can't remember if last year I got to this point in the test 413 00:19:59,154 --> 00:20:01,070 and somebody said, we don't have the equation, 414 00:20:01,070 --> 00:20:03,530 and I gave them the equation with the other terms, 415 00:20:03,530 --> 00:20:06,410 or if we just assumed that the faces ruptured. 416 00:20:06,410 --> 00:20:08,040 I can't remember. 417 00:20:08,040 --> 00:20:11,820 To be honest, I think probably for elastomeric foam, 418 00:20:11,820 --> 00:20:15,162 you could assume that they'd probably don't rupture 419 00:20:15,162 --> 00:20:16,453 unless they're very, very thin. 420 00:20:19,810 --> 00:20:23,265 AUDIENCE: So what kind of foams do they typically rupture in? 421 00:20:23,265 --> 00:20:25,390 LORNA GIBSON: So certainly if you had a metal foam, 422 00:20:25,390 --> 00:20:28,390 they'd probably rupture. 423 00:20:28,390 --> 00:20:35,310 If you had, say, a polymer foam that was more rigid, 424 00:20:35,310 --> 00:20:38,520 like a rigid polyurethane. 425 00:20:38,520 --> 00:20:40,005 So polyurethanes can be flexible, 426 00:20:40,005 --> 00:20:41,880 which means they're made out of an elastomer, 427 00:20:41,880 --> 00:20:43,410 or they can be rigid. 428 00:20:43,410 --> 00:20:48,336 And the foams that are typically used for insulation, 429 00:20:48,336 --> 00:20:49,710 thermal insulation, are typically 430 00:20:49,710 --> 00:20:52,550 closed celled polyurethane foams. 431 00:20:52,550 --> 00:20:55,200 And those typically have very thin faces, 432 00:20:55,200 --> 00:20:59,420 and they would rupture. 433 00:20:59,420 --> 00:21:00,100 Yeah? 434 00:21:00,100 --> 00:21:03,040 AUDIENCE: Are you looking for the Young's modulus 435 00:21:03,040 --> 00:21:04,385 in this problem? 436 00:21:04,385 --> 00:21:05,760 LORNA GIBSON: The Young's modulus 437 00:21:05,760 --> 00:21:07,040 was the first part, right? 438 00:21:07,040 --> 00:21:08,824 So part A was the Young's modulus. 439 00:21:08,824 --> 00:21:09,490 AUDIENCE: In B-- 440 00:21:09,490 --> 00:21:12,614 LORNA GIBSON: In B is the collapse stress, the compressor 441 00:21:12,614 --> 00:21:13,114 strength. 442 00:21:13,114 --> 00:21:19,100 AUDIENCE: So I think in my notes, I think it has this. 443 00:21:19,100 --> 00:21:21,000 If the-- 444 00:21:21,000 --> 00:21:23,470 LORNA GIBSON: Right, if p0 is bigger than-- so what she's 445 00:21:23,470 --> 00:21:26,630 showing me in her notes is I had a little note in the class, 446 00:21:26,630 --> 00:21:29,610 in the lecture, that if the initial pressure in the cells 447 00:21:29,610 --> 00:21:32,280 is greater than atmospheric, then the cell walls are 448 00:21:32,280 --> 00:21:35,390 pre-stressed and you have to overcome that in the buckling. 449 00:21:35,390 --> 00:21:36,640 AUDIENCE: Is that atmospheric? 450 00:21:36,640 --> 00:21:37,700 LORNA GIBSON: That is atmospheric pressure. 451 00:21:37,700 --> 00:21:39,684 AUDIENCE: So you don't need the [INAUDIBLE]. 452 00:21:39,684 --> 00:21:40,392 LORNA GIBSON: No. 453 00:21:40,392 --> 00:21:41,840 Yeah. 454 00:21:41,840 --> 00:21:43,010 OK? 455 00:21:43,010 --> 00:21:44,880 OK. 456 00:21:44,880 --> 00:21:47,300 Yeah, you don't know that that's atmospheric. 457 00:21:47,300 --> 00:21:51,240 So do you ever do things in PSI? 458 00:21:51,240 --> 00:21:53,040 No, you don't. 459 00:21:53,040 --> 00:21:56,200 Because when I was a student a long time ago, 460 00:21:56,200 --> 00:21:58,450 the thing I remember learning was atmospheric pressure 461 00:21:58,450 --> 00:22:01,400 is 14.7 PSI, more or less. 462 00:22:01,400 --> 00:22:03,870 And the conversion between mega pascals and PSI 463 00:22:03,870 --> 00:22:06,710 is there's more or less 145 PSI to a mega pascal, 464 00:22:06,710 --> 00:22:11,120 so the atmospheric pressure is about 0.1 mega pascals. 465 00:22:11,120 --> 00:22:11,620 Yeah? 466 00:22:11,620 --> 00:22:14,266 AUDIENCE: I don't know if this is a silly question, 467 00:22:14,266 --> 00:22:17,230 but for part B, how do we get from the Young's 468 00:22:17,230 --> 00:22:18,105 modulus of the foam-- 469 00:22:18,105 --> 00:22:19,979 LORNA GIBSON: Oh, sorry, sorry, sorry, sorry. 470 00:22:19,979 --> 00:22:21,660 I put the wrong thing down here. 471 00:22:21,660 --> 00:22:24,740 Sorry, my mistake. 472 00:22:24,740 --> 00:22:26,768 OK, now you happy? 473 00:22:26,768 --> 00:22:27,267 Sorry. 474 00:22:29,950 --> 00:22:32,252 OK, shall I move on to the next part? 475 00:22:32,252 --> 00:22:32,960 Another question? 476 00:22:32,960 --> 00:22:35,174 AUDIENCE: Well, I had a question about number two, 477 00:22:35,174 --> 00:22:36,900 but maybe we can come back to that-- 478 00:22:36,900 --> 00:22:38,441 LORNA GIBSON: OK, let me finish this, 479 00:22:38,441 --> 00:22:40,110 and then we'll go back to number two. 480 00:22:40,110 --> 00:22:43,080 So this one here, the part C is why 481 00:22:43,080 --> 00:22:45,436 does the Young's modulus foam depend 482 00:22:45,436 --> 00:22:47,310 on the solid modulus and the relative density 483 00:22:47,310 --> 00:22:49,470 while the Poisson's ratio does not. 484 00:22:49,470 --> 00:22:53,400 So when I write the equation for the Young's modulus, 485 00:22:53,400 --> 00:22:55,820 the solid modulus comes into it, the relative density 486 00:22:55,820 --> 00:22:56,800 comes into it. 487 00:22:56,800 --> 00:22:58,910 And remember when we had the Poisson's ratio, 488 00:22:58,910 --> 00:23:02,140 it's just a constant that depends on the cell geometry. 489 00:23:02,140 --> 00:23:08,710 So here's C, nu star just as a constant. 490 00:23:08,710 --> 00:23:11,223 And that constant just depends on the cell geometry. 491 00:23:16,310 --> 00:23:18,750 So if you think of the Poisson's ratio, 492 00:23:18,750 --> 00:23:20,790 it's the ratio of two strains, right? 493 00:23:20,790 --> 00:23:24,730 So say I have my foam here. 494 00:23:24,730 --> 00:23:26,710 So say that's just a block of foam. 495 00:23:26,710 --> 00:23:28,600 Little cells in it here. 496 00:23:28,600 --> 00:23:29,800 Little cells. 497 00:23:29,800 --> 00:23:32,480 And say I press on it this way here. 498 00:23:32,480 --> 00:23:34,504 And let's call this the one direction 499 00:23:34,504 --> 00:23:35,420 and the two direction. 500 00:23:37,940 --> 00:23:41,380 If I press it in the two direction, 501 00:23:41,380 --> 00:23:45,460 the Poisson's ratio is then just nu would be-- 502 00:23:45,460 --> 00:23:47,550 let's see, this would be 2, 1. 503 00:23:47,550 --> 00:23:49,640 It'd be the strain in the one direction 504 00:23:49,640 --> 00:23:51,430 over the strain in the two direction. 505 00:23:51,430 --> 00:23:54,250 So it's the ratio of two strains, right? 506 00:23:54,250 --> 00:23:58,780 And if you think of our model for the elastic behavior 507 00:23:58,780 --> 00:24:03,240 of the foam, each of those strains 508 00:24:03,240 --> 00:24:06,270 is going to be related to some bending deformation in the cell 509 00:24:06,270 --> 00:24:08,560 walls or the cell struts. 510 00:24:08,560 --> 00:24:11,720 And this strain here is going to be-- 511 00:24:11,720 --> 00:24:13,740 let me make this proportional. 512 00:24:13,740 --> 00:24:17,215 It's going to be proportional to a delta over l. 513 00:24:17,215 --> 00:24:19,090 And this one here is going to be proportional 514 00:24:19,090 --> 00:24:21,060 to the delta over l. 515 00:24:21,060 --> 00:24:25,100 So this might be delta in the one direction, 516 00:24:25,100 --> 00:24:27,260 and this will be delta in the two direction. 517 00:24:27,260 --> 00:24:28,760 But those two things are both going 518 00:24:28,760 --> 00:24:33,360 to be related to the bending deflection of the beams, right? 519 00:24:33,360 --> 00:24:36,060 And since both of those deltas are 520 00:24:36,060 --> 00:24:39,304 related to the bending deflection of the beams, 521 00:24:39,304 --> 00:24:40,720 we could write them-- if you want, 522 00:24:40,720 --> 00:24:46,690 I could write that as f l cubed over E of the solid times t 523 00:24:46,690 --> 00:24:48,110 to the fourth. 524 00:24:48,110 --> 00:24:50,600 And then that's times 1 over l. 525 00:24:50,600 --> 00:24:56,270 And then this thing here is also f l cubed over E of the solid t 526 00:24:56,270 --> 00:24:59,000 to the fourth 1 over l. 527 00:24:59,000 --> 00:25:02,592 So everything cancels out except the geometrical constant. 528 00:25:02,592 --> 00:25:04,550 And if you remember when we did the honeycombs, 529 00:25:04,550 --> 00:25:06,040 it looked exactly the same. 530 00:25:06,040 --> 00:25:10,530 When we looked at the Poisson's ratio of the honeycombs, 531 00:25:10,530 --> 00:25:12,830 we had the strain in one direction 532 00:25:12,830 --> 00:25:14,600 over the strain in another direction. 533 00:25:14,600 --> 00:25:16,170 And each of those strains was related 534 00:25:16,170 --> 00:25:17,810 to some component of delta. 535 00:25:17,810 --> 00:25:20,640 There was a delta 1 and a delta 2. 536 00:25:20,640 --> 00:25:23,130 But the delta 1 might be delta sine theta 537 00:25:23,130 --> 00:25:25,440 and delta 2 was delta cos theta. 538 00:25:25,440 --> 00:25:29,680 So if the deltas are the same, then it all just cancels out. 539 00:25:29,680 --> 00:25:32,598 And all you're left with is a geometrical constant. 540 00:25:32,598 --> 00:25:33,590 OK? 541 00:25:33,590 --> 00:25:37,150 Do you get physically why that is? 542 00:25:37,150 --> 00:25:39,130 AUDIENCE: So I have a question about this. 543 00:25:39,130 --> 00:25:42,595 Because we're given most of the equations that we need, 544 00:25:42,595 --> 00:25:45,210 is it only in conceptual questions 545 00:25:45,210 --> 00:25:48,434 that we should know how we actually derived that version? 546 00:25:48,434 --> 00:25:49,850 LORNA GIBSON: I'm not going to ask 547 00:25:49,850 --> 00:25:53,015 you to derive that equation for a closed cell foam. 548 00:25:53,015 --> 00:25:54,390 AUDIENCE: I meant this last part. 549 00:25:54,390 --> 00:25:58,720 LORNA GIBSON: Well yeah, you should be able to explain that. 550 00:25:58,720 --> 00:26:00,490 But I mean just at this level. 551 00:26:00,490 --> 00:26:04,070 Nothing very mathematically involved. 552 00:26:04,070 --> 00:26:05,870 AUDIENCE: OK, cool. 553 00:26:05,870 --> 00:26:07,570 LORNA GIBSON: OK, are we good? 554 00:26:07,570 --> 00:26:11,472 So that was the end of the test for the undergraduates, OK? 555 00:26:11,472 --> 00:26:12,930 And then for the graduate students, 556 00:26:12,930 --> 00:26:15,210 just like the problem sets, I just have an extra question. 557 00:26:15,210 --> 00:26:16,750 And that's what I did this year, too. 558 00:26:16,750 --> 00:26:18,791 So the graduate students have one extra question. 559 00:26:18,791 --> 00:26:19,477 So you and you. 560 00:26:19,477 --> 00:26:20,935 Is anybody else a graduate student? 561 00:26:20,935 --> 00:26:23,350 I think it's just the two of you. 562 00:26:23,350 --> 00:26:25,140 You're post post-graduate. 563 00:26:25,140 --> 00:26:26,020 OK. 564 00:26:26,020 --> 00:26:28,700 OK. 565 00:26:28,700 --> 00:26:29,607 So let's see. 566 00:26:33,010 --> 00:26:35,410 So this one says, the performance-- 567 00:26:35,410 --> 00:26:38,504 so this is on the performance indices which I told you 568 00:26:38,504 --> 00:26:41,170 you didn't need to know for this test, partly because, remember, 569 00:26:41,170 --> 00:26:42,128 we missed two lectures. 570 00:26:42,128 --> 00:26:45,014 We're not exactly on the same spot as we were last year. 571 00:26:45,014 --> 00:26:46,180 But I can do it if you want. 572 00:26:46,180 --> 00:26:49,454 Do you want me to do it, or should we do other questions? 573 00:26:49,454 --> 00:26:54,174 AUDIENCE: What do the grad students think? 574 00:26:54,174 --> 00:26:54,965 LORNA GIBSON: Sure. 575 00:26:54,965 --> 00:26:55,430 OK. 576 00:26:55,430 --> 00:26:57,138 So the question is, the performance index 577 00:26:57,138 --> 00:26:59,920 to minimize the mass of a beam of a given bending stiffness, 578 00:26:59,920 --> 00:27:02,547 length and square cross-section is e to the one half over rho. 579 00:27:02,547 --> 00:27:04,630 So you remember, we derived that e to the one half 580 00:27:04,630 --> 00:27:06,024 over rho in class. 581 00:27:06,024 --> 00:27:08,440 In the section on wood, we saw that this performance index 582 00:27:08,440 --> 00:27:11,230 for wood is higher than that for the solid cell wall 583 00:27:11,230 --> 00:27:12,186 material in wood. 584 00:27:12,186 --> 00:27:13,060 Do you remember that? 585 00:27:13,060 --> 00:27:18,190 The e to the one half over rho for the wood was, I think, 586 00:27:18,190 --> 00:27:23,890 rho s over rho star to the one half times Es 587 00:27:23,890 --> 00:27:26,560 to the one half over rho s. 588 00:27:26,560 --> 00:27:29,100 So explain why wood has a higher value of e 589 00:27:29,100 --> 00:27:32,500 to the one half over rho than the solid cell wall material. 590 00:27:32,500 --> 00:27:35,880 And then part B is, suggest a design for an engineering 591 00:27:35,880 --> 00:27:38,663 material based on wood that has high values of e 592 00:27:38,663 --> 00:27:41,310 to the one half over rho. 593 00:27:41,310 --> 00:27:54,400 So one way to explain it is to say that if you're looking at e 594 00:27:54,400 --> 00:28:05,140 to the one half over rho, you can say for wood, E over Es 595 00:28:05,140 --> 00:28:08,250 is equal to rho star over rho s for loading 596 00:28:08,250 --> 00:28:09,820 in the axial direction. 597 00:28:09,820 --> 00:28:20,730 So this will be for loading actually along the grain. 598 00:28:20,730 --> 00:28:23,030 And that's what we were looking at. 599 00:28:23,030 --> 00:28:26,340 So that's what I'm talking about here. 600 00:28:26,340 --> 00:28:50,160 So I think-- let me just see if this is right. 601 00:28:50,160 --> 00:28:50,680 Yeah. 602 00:28:50,680 --> 00:28:54,320 So this equation here is exactly the same as that equation 603 00:28:54,320 --> 00:28:55,630 there, right? 604 00:28:55,630 --> 00:28:58,190 And this is basically saying that this is the performance 605 00:28:58,190 --> 00:29:01,130 index for the wood. 606 00:29:01,130 --> 00:29:04,150 This is the performance index for the solid. 607 00:29:04,150 --> 00:29:06,920 And this factor here is bigger than 1, 608 00:29:06,920 --> 00:29:09,990 because the solid density is higher than the wood density. 609 00:29:09,990 --> 00:29:11,150 OK? 610 00:29:11,150 --> 00:29:15,690 So really, all you have to do is say that for the wood, 611 00:29:15,690 --> 00:29:18,440 the modulus in the longitudinal or the axial direction 612 00:29:18,440 --> 00:29:22,310 along with grain varies linearly with the relative density. 613 00:29:22,310 --> 00:29:24,110 And it probably would be a good idea 614 00:29:24,110 --> 00:29:29,947 to say that this is a result of the cell walls deforming 615 00:29:29,947 --> 00:29:30,446 axially. 616 00:29:33,070 --> 00:29:36,540 So when you take the cells, if you think of the wood cells 617 00:29:36,540 --> 00:29:42,760 as being something like that and you're loading it this way on, 618 00:29:42,760 --> 00:29:45,770 the cells just actually shorten, and the modulus 619 00:29:45,770 --> 00:29:49,560 depends on the-- it just is the volume fraction of solid times 620 00:29:49,560 --> 00:29:51,760 the modulus of the solid. 621 00:29:51,760 --> 00:29:53,260 And that's where this comes from. 622 00:29:53,260 --> 00:29:57,050 And once you have this, that basically gives you that. 623 00:29:57,050 --> 00:29:58,050 OK? 624 00:29:58,050 --> 00:30:00,500 Are we good with that? 625 00:30:00,500 --> 00:30:01,680 So that's why it's higher. 626 00:30:05,770 --> 00:30:07,680 Another way to look at it as sort 627 00:30:07,680 --> 00:30:10,017 of more of a hand-wavy argument is 628 00:30:10,017 --> 00:30:11,850 that if you have a certain amount of solid-- 629 00:30:11,850 --> 00:30:13,960 so say you have a certain mass of solid. 630 00:30:13,960 --> 00:30:16,751 If it's solid, it takes up a certain cross-sectional area. 631 00:30:16,751 --> 00:30:18,750 So say that your beam's a certain length, that's 632 00:30:18,750 --> 00:30:20,624 going to have a certain cross-sectional area. 633 00:30:20,624 --> 00:30:23,560 And if you have wood, if you have a cellular material, 634 00:30:23,560 --> 00:30:25,170 if you have the same mass, you're 635 00:30:25,170 --> 00:30:28,360 essentially making the dimensions of that piece 636 00:30:28,360 --> 00:30:29,160 bigger. 637 00:30:29,160 --> 00:30:32,420 So you're moving the material further away. 638 00:30:32,420 --> 00:30:34,320 And as you're making it bigger, you're 639 00:30:34,320 --> 00:30:36,030 increasing the moment of inertia. 640 00:30:36,030 --> 00:30:40,890 And so you're increasing the bending resistance of it. 641 00:30:40,890 --> 00:30:45,090 That's another, more hand-waving way to talk about it. 642 00:30:45,090 --> 00:30:48,970 And then the second part is to suggest 643 00:30:48,970 --> 00:30:51,020 a design for an engineering material 644 00:30:51,020 --> 00:30:53,240 based on wood that would have high values of e 645 00:30:53,240 --> 00:30:54,847 to the one half over rho. 646 00:30:54,847 --> 00:30:57,180 So remember when we looked at those material performance 647 00:30:57,180 --> 00:31:00,340 charts, we said that wood was similar to engineering fiber 648 00:31:00,340 --> 00:31:01,460 composites. 649 00:31:01,460 --> 00:31:03,540 But those data for fiber composites 650 00:31:03,540 --> 00:31:06,700 are assuming that it's solid, the fiber composite's a solid. 651 00:31:06,700 --> 00:31:08,880 So if you could take fiber composites 652 00:31:08,880 --> 00:31:10,940 and make little tubes of fiber composites 653 00:31:10,940 --> 00:31:13,580 and assemble the tubes together so that it was like wood, 654 00:31:13,580 --> 00:31:15,860 would get something that would be even higher. 655 00:31:15,860 --> 00:31:24,090 So if you could make, say, a fiber composite honeycomb 656 00:31:24,090 --> 00:31:33,820 material, and you'd want to have the fibers aligned 657 00:31:33,820 --> 00:31:51,330 along the prism axis of the honeycomb, 658 00:31:51,330 --> 00:31:53,140 then you would get higher values. 659 00:31:53,140 --> 00:31:57,870 It would be the same-- it'd be this sort of argument again, 660 00:31:57,870 --> 00:31:59,720 but now with a fiber composite. 661 00:31:59,720 --> 00:32:07,360 So you'd want-- if this was your fiber composite like this, 662 00:32:07,360 --> 00:32:09,500 you'd want the fibers-- well, in wood, 663 00:32:09,500 --> 00:32:11,000 they're at a little bit of an angle. 664 00:32:11,000 --> 00:32:14,170 But say they were lined up like that. 665 00:32:14,170 --> 00:32:15,670 You'd want them something like that, 666 00:32:15,670 --> 00:32:21,280 then loading it that way on, right? 667 00:32:21,280 --> 00:32:24,120 And if one way to think about those charts 668 00:32:24,120 --> 00:32:26,520 is if you-- say we had a plot. 669 00:32:29,470 --> 00:32:33,880 And say this was log of the modulus 670 00:32:33,880 --> 00:32:35,860 and that was log of the density. 671 00:32:35,860 --> 00:32:37,730 And I think I'll just draw the envelope. 672 00:32:37,730 --> 00:32:40,140 So foams were somewhere down here, 673 00:32:40,140 --> 00:32:47,720 metals were somewhere over here, and with elastomers we're 674 00:32:47,720 --> 00:32:48,760 somewhere in here. 675 00:32:52,740 --> 00:32:56,257 I think ceramics were up here. 676 00:32:56,257 --> 00:32:58,590 And then I can't remember exactly where composites were, 677 00:32:58,590 --> 00:33:01,650 but composites were around about here. 678 00:33:01,650 --> 00:33:04,270 I'll just say FRC for fiber reinforced composites. 679 00:33:04,270 --> 00:33:08,480 And I think woods were kind of in here. 680 00:33:08,480 --> 00:33:09,606 Something like that. 681 00:33:09,606 --> 00:33:11,480 And then we had our performance index, right? 682 00:33:11,480 --> 00:33:13,440 So remember, there was a performance index, 683 00:33:13,440 --> 00:33:15,530 something like that. 684 00:33:15,530 --> 00:33:20,320 And that slope of that was e to the one half over rho. 685 00:33:20,320 --> 00:33:22,710 So every point on that line had the same value of e 686 00:33:22,710 --> 00:33:23,862 to the one half over rho. 687 00:33:23,862 --> 00:33:25,820 And essentially, if you had the fiber composite 688 00:33:25,820 --> 00:33:27,470 and you made a honeycomb out of it, 689 00:33:27,470 --> 00:33:30,550 you would be taking the data from here 690 00:33:30,550 --> 00:33:32,410 and shifting them out that way. 691 00:33:32,410 --> 00:33:34,000 You'd be pushing them out over here, 692 00:33:34,000 --> 00:33:37,750 so you'd get a higher value of that performance index. 693 00:33:37,750 --> 00:33:39,250 OK? 694 00:33:39,250 --> 00:33:41,450 So that's the test from last year. 695 00:33:41,450 --> 00:33:42,410 That's the end-- yeah? 696 00:33:42,410 --> 00:33:44,815 AUDIENCE: So for along right here 697 00:33:44,815 --> 00:33:49,624 or different it would be cubed. 698 00:33:49,624 --> 00:33:52,040 LORNA GIBSON: Yeah, so the thing about the honeycombs is-- 699 00:33:52,040 --> 00:33:52,790 AUDIENCE: The opposite. 700 00:33:52,790 --> 00:33:53,623 LORNA GIBSON: Right. 701 00:33:53,623 --> 00:33:54,960 AUDIENCE: [INAUDIBLE]. 702 00:33:54,960 --> 00:33:56,826 LORNA GIBSON: It'd be worse, that's true. 703 00:33:56,826 --> 00:33:58,200 So the thing about the honeycombs 704 00:33:58,200 --> 00:34:01,326 is they're very stiff in the axial direction, 705 00:34:01,326 --> 00:34:03,200 but you pay for that in the other directions. 706 00:34:03,200 --> 00:34:05,030 And it's the same for wood. 707 00:34:05,030 --> 00:34:08,739 So the wood is very good when you load it along the grain, 708 00:34:08,739 --> 00:34:10,840 but you pay for it the other way. 709 00:34:10,840 --> 00:34:13,429 But if you think of from the tree's point of view-- 710 00:34:13,429 --> 00:34:14,730 if you're a tree. 711 00:34:14,730 --> 00:34:16,460 So here's my little tree. 712 00:34:16,460 --> 00:34:19,440 So here, say we have a tree trunk, and we have some branch. 713 00:34:19,440 --> 00:34:22,909 Branch over here, branches, tree. 714 00:34:22,909 --> 00:34:25,830 So the grain is lined up this way. 715 00:34:25,830 --> 00:34:28,699 And then when there's a branch, the grain 716 00:34:28,699 --> 00:34:30,540 turns around and goes that way, right? 717 00:34:30,540 --> 00:34:32,473 So if you think of the tree as a whole, 718 00:34:32,473 --> 00:34:35,650 the whole tree blows in the wind like this. 719 00:34:35,650 --> 00:34:39,630 So it's like a column like this, and everything's 720 00:34:39,630 --> 00:34:40,799 lined up that way. 721 00:34:40,799 --> 00:34:42,090 And you're loading it this way. 722 00:34:42,090 --> 00:34:45,219 So that is the stiff direction, right? 723 00:34:45,219 --> 00:34:47,639 And if you're a branch, the branches 724 00:34:47,639 --> 00:34:50,239 are more loaded by gravity. 725 00:34:50,239 --> 00:34:52,070 So they're loaded that way. 726 00:34:52,070 --> 00:34:55,520 And then because the fibers, the grain turns around, 727 00:34:55,520 --> 00:34:59,950 they're also oriented in the good direction. 728 00:34:59,950 --> 00:35:03,440 So from the tree's point of view, it's optimized things. 729 00:35:03,440 --> 00:35:06,540 Then you remember when I talked about the old wooden sailing 730 00:35:06,540 --> 00:35:10,980 ships, when they made the old wooden sailing 731 00:35:10,980 --> 00:35:15,100 ships, if this was the deck here and that was the haul there, 732 00:35:15,100 --> 00:35:18,480 they would get pieces of wood to fit in here 733 00:35:18,480 --> 00:35:20,560 that were called the knees. 734 00:35:20,560 --> 00:35:21,990 That was the knee. 735 00:35:21,990 --> 00:35:24,050 And they would try to get a piece that 736 00:35:24,050 --> 00:35:25,710 was from a branch like this. 737 00:35:25,710 --> 00:35:29,890 And they would try to match the curve of that joint 738 00:35:29,890 --> 00:35:32,450 with the branch with the curve that they needed in here so 739 00:35:32,450 --> 00:35:35,780 that the grain followed the pattern of what 740 00:35:35,780 --> 00:35:38,630 they needed for the boat. 741 00:35:38,630 --> 00:35:39,164 OK. 742 00:35:39,164 --> 00:35:39,830 Other questions? 743 00:35:42,442 --> 00:35:44,843 AUDIENCE: I'm not quite sure what 744 00:35:44,843 --> 00:35:47,890 the difference between tangential versus ray here. 745 00:35:47,890 --> 00:35:49,000 LORNA GIBSON: Oh, OK. 746 00:35:49,000 --> 00:35:50,210 So in the wood, you mean? 747 00:35:50,210 --> 00:35:50,793 AUDIENCE: Yes. 748 00:35:50,793 --> 00:35:52,040 LORNA GIBSON: OK. 749 00:35:52,040 --> 00:35:53,720 So can I rub this stuff off? 750 00:35:53,720 --> 00:35:54,375 We're happy? 751 00:36:23,050 --> 00:36:23,890 Let's see. 752 00:36:23,890 --> 00:36:24,500 Say again? 753 00:36:24,500 --> 00:36:26,160 So tangential and radial. 754 00:36:26,160 --> 00:36:27,020 OK. 755 00:36:27,020 --> 00:36:32,970 So say the wood cells look something like this. 756 00:36:32,970 --> 00:36:42,910 So these would be the fiber cells or the tracheids 757 00:36:42,910 --> 00:36:48,400 And then the rays typically are more rectangular cells. 758 00:36:48,400 --> 00:36:51,510 So they might look something like that. 759 00:36:51,510 --> 00:36:55,909 And then they would be some more fibers or tracheids, 760 00:36:55,909 --> 00:36:57,950 depending on if it was a soft wood or a hardwood. 761 00:36:57,950 --> 00:37:05,892 So these would be either fibers or tracheids 762 00:37:05,892 --> 00:37:07,940 in a hardwood or a soft wood. 763 00:37:07,940 --> 00:37:10,260 And then these would be the ray cells in here. 764 00:37:10,260 --> 00:37:12,009 So they have a different structure. 765 00:37:12,009 --> 00:37:13,050 They just look different. 766 00:37:13,050 --> 00:37:13,805 They're different shape. 767 00:37:13,805 --> 00:37:14,430 AUDIENCE: This is the top? 768 00:37:14,430 --> 00:37:15,171 From the top? 769 00:37:15,171 --> 00:37:17,420 LORNA GIBSON: Yeah, this is looking from the top down. 770 00:37:17,420 --> 00:37:23,450 And then if you think of the tree-- 771 00:37:23,450 --> 00:37:25,450 so the tree's going to have growth rings, right? 772 00:37:25,450 --> 00:37:27,699 So the growth rings are going to look-- obviously 773 00:37:27,699 --> 00:37:29,490 I'm not making perfect circles, but you get 774 00:37:29,490 --> 00:37:31,840 the idea-- something like that. 775 00:37:31,840 --> 00:37:33,810 And then the rays go this way. 776 00:37:33,810 --> 00:37:36,050 They go radially. 777 00:37:36,050 --> 00:37:38,020 OK? 778 00:37:38,020 --> 00:37:42,170 So this would be the radial direction, 779 00:37:42,170 --> 00:37:44,470 and then those are the rays. 780 00:37:44,470 --> 00:37:45,453 Are we good? 781 00:37:45,453 --> 00:37:49,330 AUDIENCE: So which way's the tangential? 782 00:37:49,330 --> 00:37:52,570 LORNA GIBSON: So the tangential would be this way on, OK? 783 00:37:52,570 --> 00:37:54,980 So if I loaded this way like that, 784 00:37:54,980 --> 00:37:58,735 that would be loading it in the tangential direction. 785 00:37:58,735 --> 00:38:00,110 And if I loaded it this way, that 786 00:38:00,110 --> 00:38:01,943 would be loading it in the radial direction. 787 00:38:01,943 --> 00:38:04,538 The length of the rays runs in the radial direction. 788 00:38:04,538 --> 00:38:06,365 The length this way on. 789 00:38:10,900 --> 00:38:13,760 So this thing here corresponds to one 790 00:38:13,760 --> 00:38:15,390 of these lines I've drawn here. 791 00:38:18,630 --> 00:38:24,160 And then these guys here are the stuff in between here. 792 00:38:24,160 --> 00:38:29,890 AUDIENCE: How do you know what the tangential, the Young's 793 00:38:29,890 --> 00:38:34,120 modulus and stiffness is? 794 00:38:34,120 --> 00:38:36,930 LORNA GIBSON: So say we were loading it tangentially, 795 00:38:36,930 --> 00:38:40,140 we're loading it like that. 796 00:38:40,140 --> 00:38:42,820 Then-- 797 00:38:42,820 --> 00:38:44,830 AUDIENCE: If you have a tree, how do you 798 00:38:44,830 --> 00:38:46,450 apply a tangential load on it? 799 00:38:46,450 --> 00:38:48,700 LORNA GIBSON: Oh, well it's not the whole tree, right? 800 00:38:48,700 --> 00:38:51,620 So say we have a piece of wood that we cut out like this. 801 00:38:51,620 --> 00:38:53,290 So say I have that. 802 00:38:53,290 --> 00:38:56,820 And if I loaded it this way on, I'd be loading it tangentially. 803 00:38:56,820 --> 00:38:58,350 The tree's big, right? 804 00:38:58,350 --> 00:39:00,351 So I'm not talking about loading the whole tree, 805 00:39:00,351 --> 00:39:02,683 I'm talking about taking a piece of wood out of the tree 806 00:39:02,683 --> 00:39:03,588 and loading it. 807 00:39:03,588 --> 00:39:06,565 AUDIENCE: OK, so you can't really load tangentially 808 00:39:06,565 --> 00:39:09,475 for the entire trunk. 809 00:39:09,475 --> 00:39:11,750 LORNA GIBSON: No, I'm talking about taking a piece out 810 00:39:11,750 --> 00:39:13,664 and loading that piece. 811 00:39:13,664 --> 00:39:15,562 AUDIENCE: How about a ray here? 812 00:39:15,562 --> 00:39:16,270 Do you take the-- 813 00:39:16,270 --> 00:39:17,150 LORNA GIBSON: So the same thing. 814 00:39:17,150 --> 00:39:19,390 You'd take-- say this was the piece of wood 815 00:39:19,390 --> 00:39:21,710 that you were looking at. 816 00:39:21,710 --> 00:39:24,640 Now you would just load it this way on. 817 00:39:24,640 --> 00:39:25,280 OK? 818 00:39:25,280 --> 00:39:30,030 I think-- I brought my thing because I have the slides. 819 00:39:30,030 --> 00:39:31,870 Let me see if I can find-- I think there 820 00:39:31,870 --> 00:39:33,415 was a slide that showed this. 821 00:39:43,310 --> 00:39:44,010 OK. 822 00:39:44,010 --> 00:39:46,626 That was Furry Fridays. 823 00:39:46,626 --> 00:39:47,750 That was the wood sculptor. 824 00:39:47,750 --> 00:39:50,490 Here we go. 825 00:39:50,490 --> 00:39:53,296 OK, so imagine that that cube is your piece of wood 826 00:39:53,296 --> 00:39:54,420 that you're loading, right? 827 00:39:54,420 --> 00:39:57,819 So imagine this is the-- you cut a little piece out. 828 00:39:57,819 --> 00:39:59,110 Then you're loading it tangent. 829 00:39:59,110 --> 00:40:00,000 Can you see, then? 830 00:40:00,000 --> 00:40:02,225 You can load it-- you're not loading the whole tree. 831 00:40:02,225 --> 00:40:04,725 AUDIENCE: Yeah, I was thinking about loading the entire tree 832 00:40:04,725 --> 00:40:06,950 and then applying the tangential load on it. 833 00:40:06,950 --> 00:40:09,276 LORNA GIBSON: Yeah, because then-- I see the problem. 834 00:40:09,276 --> 00:40:10,900 AUDIENCE: It's going to give us shears. 835 00:40:10,900 --> 00:40:11,897 LORNA GIBSON: Yeah. 836 00:40:11,897 --> 00:40:13,605 Yeah, because I could say, well, if I was 837 00:40:13,605 --> 00:40:14,896 trying to load the whole thing. 838 00:40:14,896 --> 00:40:17,120 Say I was loading it from here to there. 839 00:40:17,120 --> 00:40:20,349 Well if you look at it one way, it looks tangential. 840 00:40:20,349 --> 00:40:22,390 If you look at it the other way, it looks radial. 841 00:40:22,390 --> 00:40:24,260 So think of cutting a piece out, because that 842 00:40:24,260 --> 00:40:24,940 is what you're going to do. 843 00:40:24,940 --> 00:40:26,580 You're going to cut a piece out. 844 00:40:26,580 --> 00:40:28,060 OK? 845 00:40:28,060 --> 00:40:29,172 All right. 846 00:40:29,172 --> 00:40:30,255 Are there other questions? 847 00:40:34,110 --> 00:40:34,610 Yes? 848 00:40:34,610 --> 00:40:35,062 AUDIENCE: With that formula sheet, 849 00:40:35,062 --> 00:40:37,774 do you only give that formula for the honeycombs, 850 00:40:37,774 --> 00:40:39,077 or also for the foams? 851 00:40:39,077 --> 00:40:41,660 LORNA GIBSON: I'm going to give you-- if you look at, I think, 852 00:40:41,660 --> 00:40:43,300 problem set 2, I gave you a sheet that 853 00:40:43,300 --> 00:40:47,260 had three pages of equations. 854 00:40:47,260 --> 00:40:49,570 And it looked exactly like this. 855 00:40:49,570 --> 00:40:51,720 So there was one saying, properties 856 00:40:51,720 --> 00:40:54,554 of two dimensional cellular solids-- honeycombs. 857 00:40:54,554 --> 00:40:56,470 There was a whole thing of in plane properties 858 00:40:56,470 --> 00:40:58,240 and out of plane properties. 859 00:40:58,240 --> 00:40:59,730 That was one page. 860 00:40:59,730 --> 00:41:05,070 The next page was properties of regular hexagonal honeycombs. 861 00:41:05,070 --> 00:41:08,780 And then the next page was properties 862 00:41:08,780 --> 00:41:10,656 of three dimensional cellular solids foams. 863 00:41:10,656 --> 00:41:11,655 AUDIENCE: OK, excellent. 864 00:41:11,655 --> 00:41:12,440 Thank you. 865 00:41:12,440 --> 00:41:13,320 LORNA GIBSON: OK? 866 00:41:13,320 --> 00:41:14,700 Like I just said, I think there's 867 00:41:14,700 --> 00:41:16,575 maybe one or two equations missing from this. 868 00:41:16,575 --> 00:41:19,033 But if it was something you needed, I would give it to you. 869 00:41:19,033 --> 00:41:20,230 I would give it to you. 870 00:41:20,230 --> 00:41:22,370 OK? 871 00:41:22,370 --> 00:41:24,370 AUDIENCE: So I should have scrolled down for it? 872 00:41:24,370 --> 00:41:25,310 LORNA GIBSON: What? 873 00:41:25,310 --> 00:41:26,380 You should have scrolled? 874 00:41:26,380 --> 00:41:27,687 So you're like me. 875 00:41:27,687 --> 00:41:29,020 This happens to me all the time. 876 00:41:29,020 --> 00:41:29,950 I have some website. 877 00:41:29,950 --> 00:41:30,700 I'm looking at it. 878 00:41:30,700 --> 00:41:31,310 I'm like, OK. 879 00:41:31,310 --> 00:41:31,810 I got it. 880 00:41:31,810 --> 00:41:32,680 I think I've got everything. 881 00:41:32,680 --> 00:41:35,130 And then I realized I'm supposed to-- I missed something 882 00:41:35,130 --> 00:41:36,558 because I was supposed to scroll down. 883 00:41:36,558 --> 00:41:37,474 AUDIENCE: Problem set two was only the honeycombs. 884 00:41:37,474 --> 00:41:38,390 That's why. 885 00:41:38,390 --> 00:41:39,320 LORNA GIBSON: Well, I think that was probably 886 00:41:39,320 --> 00:41:41,866 all we covered was the honeycombs on that problem set. 887 00:41:41,866 --> 00:41:43,740 AUDIENCE: So that's why I only got that part. 888 00:41:43,740 --> 00:41:44,524 LORNA GIBSON: OK. 889 00:41:44,524 --> 00:41:45,190 All right, yeah. 890 00:41:45,190 --> 00:41:48,950 So I'm going to give you, this will be attached to the test. 891 00:41:48,950 --> 00:41:49,850 OK? 892 00:41:49,850 --> 00:41:53,740 So I think on the test that I posted it was attached, 893 00:41:53,740 --> 00:41:54,910 wasn't it? 894 00:41:54,910 --> 00:41:55,547 Yeah. 895 00:41:55,547 --> 00:41:57,088 [INAUDIBLE], did you have a question? 896 00:41:57,088 --> 00:42:01,516 AUDIENCE: For 2 part D, I don't think we went over that. 897 00:42:01,516 --> 00:42:07,150 I was confused as to how-- is it just 898 00:42:07,150 --> 00:42:12,880 that equilateral triangular cells always have-- is always 899 00:42:12,880 --> 00:42:14,671 truss behavior? 900 00:42:14,671 --> 00:42:15,670 LORNA GIBSON: Let's see. 901 00:42:15,670 --> 00:42:16,503 AUDIENCE: Oh, sorry. 902 00:42:16,503 --> 00:42:18,560 It's the 2014 test. 903 00:42:18,560 --> 00:42:20,620 LORNA GIBSON: 2014-- oh, sorry. 904 00:42:20,620 --> 00:42:21,400 I missed a part. 905 00:42:21,400 --> 00:42:23,990 Sorry. 906 00:42:23,990 --> 00:42:25,760 Yeah, so it says, the same titanium alloy 907 00:42:25,760 --> 00:42:28,844 is used to make a honeycomb with equilateral triangular cells. 908 00:42:28,844 --> 00:42:30,510 And what is the in plane Young's modulus 909 00:42:30,510 --> 00:42:34,400 for loading in the x 2 direction of the triangular honeycomb? 910 00:42:34,400 --> 00:42:50,950 So this is-- say you have cells that look like that now. 911 00:42:50,950 --> 00:42:56,100 And that's x 1. 912 00:42:56,100 --> 00:42:57,123 That's x 2. 913 00:43:00,540 --> 00:43:01,800 OK. 914 00:43:01,800 --> 00:43:07,110 So the Young's modulus for this-- 915 00:43:07,110 --> 00:43:09,517 so I happen to remember the formula. 916 00:43:09,517 --> 00:43:11,100 So I guess I'm thinking you might have 917 00:43:11,100 --> 00:43:13,330 put this on your cheat sheets. 918 00:43:13,330 --> 00:43:18,960 It's 1.15 times Es times that, times the relative density. 919 00:43:18,960 --> 00:43:20,790 So I'm trying to remember. 920 00:43:20,790 --> 00:43:23,180 Do I have that on here? 921 00:43:23,180 --> 00:43:25,291 I don't have it on here. 922 00:43:25,291 --> 00:43:27,255 AUDIENCE: Are we just supposed to know that? 923 00:43:31,183 --> 00:43:32,660 Are we just supposed to-- 924 00:43:32,660 --> 00:43:34,076 LORNA GIBSON: Well, I guess what I 925 00:43:34,076 --> 00:43:37,580 would hope that you would know is maybe not the constant, 926 00:43:37,580 --> 00:43:39,710 but that it should go as Es and linearly 927 00:43:39,710 --> 00:43:41,540 with the relative density. 928 00:43:41,540 --> 00:43:44,330 Because it's a truss and because it deforms axially. 929 00:43:44,330 --> 00:43:47,880 I don't really expect that you would remember the 1.15. 930 00:43:47,880 --> 00:43:49,116 Yeah? 931 00:43:49,116 --> 00:43:52,792 AUDIENCE: So does that mean for all the foams, of which there 932 00:43:52,792 --> 00:43:55,950 were like 10 constants, we don't need to write all down, 933 00:43:55,950 --> 00:43:57,860 like what C1 equals-- 934 00:43:57,860 --> 00:44:01,860 LORNA GIBSON: I think that's what this thing gives you. 935 00:44:01,860 --> 00:44:02,370 Let's see. 936 00:44:02,370 --> 00:44:04,240 It gives you all the Cs. 937 00:44:04,240 --> 00:44:07,930 All right, then I'll make sure I give you the Cs. 938 00:44:07,930 --> 00:44:11,086 I'll make sure I give you the Cs. 939 00:44:11,086 --> 00:44:13,460 But who's got a pen so I can write that down to make sure 940 00:44:13,460 --> 00:44:14,165 that I do that? 941 00:44:18,090 --> 00:44:19,580 Does somebody got a piece of paper. 942 00:44:19,580 --> 00:44:20,940 Or I could write it down here. 943 00:44:20,940 --> 00:44:21,900 Oh, here we go. 944 00:44:21,900 --> 00:44:24,070 I can write it down this little sticky thing here. 945 00:44:37,390 --> 00:44:38,755 OK. 946 00:44:38,755 --> 00:44:41,050 So I'll stick that on there so I remember to do that. 947 00:44:41,050 --> 00:44:45,066 AUDIENCE: Will you also be writing what Cs? 948 00:44:45,066 --> 00:44:47,442 Because you have, in your equation, C1, C2-- 949 00:44:47,442 --> 00:44:48,900 LORNA GIBSON: Well, I think I would 950 00:44:48,900 --> 00:44:51,890 say-- say I asked you for, I don't know, 951 00:44:51,890 --> 00:44:57,050 the yield stress for a foam in compression 952 00:44:57,050 --> 00:44:58,830 that yields plastically. 953 00:44:58,830 --> 00:45:03,006 I would say, the constant for that is 0.3. 954 00:45:03,006 --> 00:45:04,330 I wouldn't give you C2. 955 00:45:04,330 --> 00:45:06,270 I wouldn't do it by numbers, I would tell you 956 00:45:06,270 --> 00:45:09,030 what the number was for the thing you needed, 957 00:45:09,030 --> 00:45:10,810 because I mean the way the numbers are, 958 00:45:10,810 --> 00:45:11,970 the only reason they're numbered is 959 00:45:11,970 --> 00:45:13,928 because that's the number they are in the book. 960 00:45:13,928 --> 00:45:16,240 They're just ordered sequentially in the book. 961 00:45:16,240 --> 00:45:18,510 But I don't expect you to remember which-- 962 00:45:18,510 --> 00:45:21,140 it's C6, or C5 or something. 963 00:45:21,140 --> 00:45:23,430 So anyone else? 964 00:45:26,430 --> 00:45:28,418 AUDIENCE: Can you explain the difference 965 00:45:28,418 --> 00:45:35,235 between uniaxial yield and plastic buckling? 966 00:45:35,235 --> 00:45:36,110 LORNA GIBSON: Oh, OK. 967 00:45:45,860 --> 00:45:54,930 So if you have something and it fails by uniaxial yield-- 968 00:45:54,930 --> 00:46:01,540 so say you have a honeycomb like this, 969 00:46:01,540 --> 00:46:04,190 and you're loading it this way on, OK? 970 00:46:04,190 --> 00:46:06,320 So if you're loading it that way on, 971 00:46:06,320 --> 00:46:11,530 these walls of the honeycomb are just axially deforming, 972 00:46:11,530 --> 00:46:12,170 initially. 973 00:46:12,170 --> 00:46:12,669 Right? 974 00:46:12,669 --> 00:46:16,790 So the elastic behaviors, they just axially deform. 975 00:46:16,790 --> 00:46:22,290 So it works out that if these cell walls are very thick, then 976 00:46:22,290 --> 00:46:26,200 you can reach a yield stress before any buckling occurs. 977 00:46:26,200 --> 00:46:30,860 And then the strength would just be that yield stress 978 00:46:30,860 --> 00:46:35,780 of the cell wall material times the relative density. 979 00:46:35,780 --> 00:46:36,780 OK? 980 00:46:36,780 --> 00:46:39,299 But the cell walls have to be thick for that to happen. 981 00:46:39,299 --> 00:46:41,340 So then imagine that the cell walls aren't thick. 982 00:46:41,340 --> 00:46:44,630 Imagine that the cell walls are thin. 983 00:46:44,630 --> 00:46:49,781 So say I have the same honeycomb like this. 984 00:46:57,200 --> 00:47:00,840 If the walls are thin, and say the solid material 985 00:47:00,840 --> 00:47:05,400 itself-- so this is for the solid-- it 986 00:47:05,400 --> 00:47:07,490 has some stress strain curve. 987 00:47:07,490 --> 00:47:10,400 And it may have a linear elastic part, and then 988 00:47:10,400 --> 00:47:13,060 a yield thing like that. 989 00:47:13,060 --> 00:47:14,850 So say this is the yield strength here. 990 00:47:18,110 --> 00:47:22,480 Say we compress that this way on the same thing in the three 991 00:47:22,480 --> 00:47:23,860 direction. 992 00:47:23,860 --> 00:47:28,180 Then if a material's got a yield point, 993 00:47:28,180 --> 00:47:30,310 there can be an interaction between plastic 994 00:47:30,310 --> 00:47:32,190 yielding and elastic buckling. 995 00:47:32,190 --> 00:47:34,210 And you can get plastic buckling. 996 00:47:34,210 --> 00:47:36,190 And the plastic buckling, you're going 997 00:47:36,190 --> 00:47:38,860 to get the wrinkles that go along 998 00:47:38,860 --> 00:47:40,700 the length of it this way. 999 00:47:40,700 --> 00:47:43,150 remember I showed you that tube that kind of collapsed 1000 00:47:43,150 --> 00:47:45,440 and folded up kind of thing? 1001 00:47:45,440 --> 00:47:46,840 That's plastic buckling. 1002 00:47:46,840 --> 00:47:47,970 OK? 1003 00:47:47,970 --> 00:47:53,910 And typically, people use what's called the tangential modulus 1004 00:47:53,910 --> 00:47:59,830 to calculate the buckling stress for plastic buckling. 1005 00:47:59,830 --> 00:48:03,900 And the tangential modulus would be something related 1006 00:48:03,900 --> 00:48:05,190 to the tangent over there. 1007 00:48:05,190 --> 00:48:08,890 I don't expect you to be able to derive 1008 00:48:08,890 --> 00:48:11,590 plastic buckling equations. 1009 00:48:11,590 --> 00:48:16,930 But the plastic buckling-- you know what 1010 00:48:16,930 --> 00:48:19,810 elastic buckling is, right? 1011 00:48:19,810 --> 00:48:20,580 Yeah. 1012 00:48:20,580 --> 00:48:21,420 Yeah. 1013 00:48:21,420 --> 00:48:24,550 So one way to think about plastic buckling 1014 00:48:24,550 --> 00:48:31,290 is, if you have-- and I'm trying to remember. 1015 00:48:31,290 --> 00:48:33,365 This is called the slenderness ratio. 1016 00:48:37,040 --> 00:48:39,110 And I'm trying to remember, is that l over r? 1017 00:48:39,110 --> 00:48:42,300 Imagine you had just a circular cross section 1018 00:48:42,300 --> 00:48:43,460 and you had a length, l. 1019 00:48:43,460 --> 00:48:48,960 So you have a column here, and it's got a length, l, 1020 00:48:48,960 --> 00:48:50,610 and it's got a radius, r. 1021 00:48:50,610 --> 00:48:52,470 Like that, OK? 1022 00:48:52,470 --> 00:48:55,060 So the longer it gets, the more slender it is, 1023 00:48:55,060 --> 00:48:57,420 the higher the slenderness ratio is. 1024 00:48:57,420 --> 00:49:04,070 And this, I think, is some sort of stress. 1025 00:49:04,070 --> 00:49:06,270 If the slenderness ratio of just a single column 1026 00:49:06,270 --> 00:49:08,600 is short-- if it's stubby, if you 1027 00:49:08,600 --> 00:49:11,599 had a column that looked kind of like that, 1028 00:49:11,599 --> 00:49:12,640 It's not going to buckle. 1029 00:49:12,640 --> 00:49:14,180 It's going to yield. 1030 00:49:14,180 --> 00:49:18,872 And so if you compress that, it would just yield. 1031 00:49:18,872 --> 00:49:21,205 And it's just going to yield at the yield stress, right? 1032 00:49:21,205 --> 00:49:24,480 It's just going to yield at sigma y of the solid, whatever 1033 00:49:24,480 --> 00:49:25,650 the solid is. 1034 00:49:25,650 --> 00:49:30,840 If you have a long column, it would buckle elastically 1035 00:49:30,840 --> 00:49:32,770 by an Euler buckling. 1036 00:49:32,770 --> 00:49:35,394 And Euler buckling-- let's see. 1037 00:49:35,394 --> 00:49:36,810 I'm going to run out of room here. 1038 00:49:40,090 --> 00:49:43,770 If you think of it in terms of a stress instead of a load, 1039 00:49:43,770 --> 00:49:48,840 it's going to be in squared pi squared Es i. 1040 00:49:48,840 --> 00:49:51,970 Let's say i goes as r to the fourth. 1041 00:49:51,970 --> 00:49:56,420 And this is going to be l squared r squared. 1042 00:49:56,420 --> 00:49:56,920 Right? 1043 00:49:56,920 --> 00:49:58,350 This is going to be a pi in here. 1044 00:49:58,350 --> 00:49:59,766 There's going to be a pie in here. 1045 00:49:59,766 --> 00:50:03,180 I might have lost a factor of 4, but it's 1046 00:50:03,180 --> 00:50:07,650 going to be-- let me make this proportional, OK? 1047 00:50:07,650 --> 00:50:10,160 So the slenderness ratio, there's 1048 00:50:10,160 --> 00:50:12,650 going to be an l over r squared term here. 1049 00:50:12,650 --> 00:50:17,600 So I could cancel out the four there and put a squared. 1050 00:50:17,600 --> 00:50:29,940 So sigma Euler is going to go as Es times r over l squared. 1051 00:50:29,940 --> 00:50:31,660 Like that. 1052 00:50:31,660 --> 00:50:36,209 And so this is the Euler buckling stress here, OK? 1053 00:50:36,209 --> 00:50:37,250 So this would be elastic. 1054 00:50:42,090 --> 00:50:44,020 And right here at this little corner, 1055 00:50:44,020 --> 00:50:48,340 it turns out life isn't quite that mathematically exact. 1056 00:50:48,340 --> 00:50:50,090 If you're near that corner, it's not 1057 00:50:50,090 --> 00:50:51,920 like here it's buckling elastically, 1058 00:50:51,920 --> 00:50:53,760 and here, it's buckling plastically. 1059 00:50:53,760 --> 00:50:56,290 What happens is, if you looked at data, 1060 00:50:56,290 --> 00:51:01,490 data might do something like that. 1061 00:51:01,490 --> 00:51:04,770 So the sum interaction between the elastic and the plastic. 1062 00:51:04,770 --> 00:51:07,774 And that's kind of what's going on with this thing here. 1063 00:51:07,774 --> 00:51:08,690 Does that makes sense? 1064 00:51:08,690 --> 00:51:16,461 AUDIENCE: Plastic buckling can-- OK, 1065 00:51:16,461 --> 00:51:19,000 so if you unload plastic buckling, 1066 00:51:19,000 --> 00:51:21,851 you get some of the elastic part back? 1067 00:51:21,851 --> 00:51:23,850 LORNA GIBSON: You're not going to get much back. 1068 00:51:23,850 --> 00:51:24,725 AUDIENCE: You're not? 1069 00:51:24,725 --> 00:51:31,600 LORNA GIBSON: No, because once-- to get the plastic buckling, 1070 00:51:31,600 --> 00:51:35,680 you're very close to this. 1071 00:51:35,680 --> 00:51:37,740 By the time you get that deformation, 1072 00:51:37,740 --> 00:51:39,580 you've got locally, it's yielded. 1073 00:51:39,580 --> 00:51:43,430 It's not all elastic everywhere. 1074 00:51:43,430 --> 00:51:45,870 It's going to yield in places. 1075 00:51:45,870 --> 00:51:51,950 And once it starts yielding, it's-- if you think of these 1076 00:51:51,950 --> 00:51:54,560 buckles forming, it's not like you're at one spot on this 1077 00:51:54,560 --> 00:51:56,960 curve throughout the whole thing. 1078 00:51:56,960 --> 00:52:00,390 Some of it's more deformed, and some of it's less deformed. 1079 00:52:00,390 --> 00:52:04,880 Let me pull up those plastically buckled columns, those tubes. 1080 00:52:08,520 --> 00:52:10,920 Get rid of that one. 1081 00:52:10,920 --> 00:52:14,740 Let me try and remind myself where they were. 1082 00:52:14,740 --> 00:52:18,300 I think-- honeycombs, I want honeycombs. 1083 00:52:18,300 --> 00:52:21,062 Out of plane, that's what I want. 1084 00:52:21,062 --> 00:52:24,450 It was this thing here. 1085 00:52:24,450 --> 00:52:26,830 So you see when you have-- that's just one tube, 1086 00:52:26,830 --> 00:52:29,900 but the whole honeycomb would-- imagine that you have 1087 00:52:29,900 --> 00:52:31,530 groups of tubes put together. 1088 00:52:31,530 --> 00:52:35,420 They would have to fail in some compatible way. 1089 00:52:35,420 --> 00:52:38,662 But the deformation and the stresses 1090 00:52:38,662 --> 00:52:40,870 are not going to be uniform through this whole thing, 1091 00:52:40,870 --> 00:52:41,060 right? 1092 00:52:41,060 --> 00:52:42,730 One part of it's going to be at one stress, 1093 00:52:42,730 --> 00:52:44,920 and something else is going to be at another stress. 1094 00:52:44,920 --> 00:52:46,900 So parts of it are going to yield plastically, 1095 00:52:46,900 --> 00:52:50,290 and you're not going to recover that. 1096 00:52:50,290 --> 00:52:50,790 OK? 1097 00:52:50,790 --> 00:52:54,570 So in fact, they use these sorts of things for energy absorption 1098 00:52:54,570 --> 00:52:58,490 devices, like in cars and things like that. 1099 00:52:58,490 --> 00:53:02,100 To absorb the energy from the impact. 1100 00:53:02,100 --> 00:53:02,760 More questions? 1101 00:53:06,610 --> 00:53:08,770 Let him have a turn. 1102 00:53:08,770 --> 00:53:13,740 AUDIENCE: Sorry, I have a question about I think 2013. 1103 00:53:13,740 --> 00:53:14,978 LORNA GIBSON: 2013, OK. 1104 00:53:14,978 --> 00:53:16,144 AUDIENCE: The last question. 1105 00:53:18,672 --> 00:53:19,630 It's about the plastic. 1106 00:53:24,170 --> 00:53:25,190 LORNA GIBSON: 2013. 1107 00:53:25,190 --> 00:53:26,870 Let me rub some of this stuff off. 1108 00:53:58,887 --> 00:53:59,387 OK. 1109 00:54:02,010 --> 00:54:03,890 Here we go. 1110 00:54:03,890 --> 00:54:06,770 OK. 1111 00:54:06,770 --> 00:54:09,850 Oh, we haven't covered this at all. 1112 00:54:09,850 --> 00:54:13,204 So the last question, this one here? 1113 00:54:13,204 --> 00:54:13,870 AUDIENCE: Right. 1114 00:54:13,870 --> 00:54:14,920 LORNA GIBSON: Yeah, so this question's 1115 00:54:14,920 --> 00:54:15,890 on energy absorption. 1116 00:54:15,890 --> 00:54:17,400 We haven't got there yet. 1117 00:54:17,400 --> 00:54:20,997 AUDIENCE: How about-- OK. 1118 00:54:20,997 --> 00:54:22,580 LORNA GIBSON: And the third question's 1119 00:54:22,580 --> 00:54:23,500 on sandwich structure. 1120 00:54:23,500 --> 00:54:27,060 So when I taught the course in 2013, 1121 00:54:27,060 --> 00:54:29,440 I did the topics in a different order. 1122 00:54:29,440 --> 00:54:31,145 So I did honeycombs, and I did foams. 1123 00:54:31,145 --> 00:54:32,770 And then I think I did sandwich panels, 1124 00:54:32,770 --> 00:54:34,270 and I did energy absorption. 1125 00:54:34,270 --> 00:54:37,480 And I left the stuff on the wood and the cork to the end. 1126 00:54:37,480 --> 00:54:38,480 So we haven't done that. 1127 00:54:38,480 --> 00:54:42,550 So don't panic if you haven't-- if you can't do that. 1128 00:54:42,550 --> 00:54:44,064 OK? 1129 00:54:44,064 --> 00:54:44,980 You should have known. 1130 00:54:44,980 --> 00:54:47,521 Come on, you should have known that if it talked about things 1131 00:54:47,521 --> 00:54:50,310 we haven't covered yet, I'm not going to give it on the test. 1132 00:54:50,310 --> 00:54:51,600 OK, what else? 1133 00:54:51,600 --> 00:54:54,220 AUDIENCE: Can you explain more plastic hinges? 1134 00:54:54,220 --> 00:54:57,420 LORNA GIBSON: Plastic hinge, OK. 1135 00:54:57,420 --> 00:55:03,413 So let's just say we have a beam in bending, OK? 1136 00:55:06,420 --> 00:55:10,740 And say it just has a load p in the middle, all right? 1137 00:55:10,740 --> 00:55:12,020 Are we good? 1138 00:55:12,020 --> 00:55:18,230 So this load in the middle, then this reaction is p over 2, 1139 00:55:18,230 --> 00:55:20,540 and that reaction is p over 2. 1140 00:55:20,540 --> 00:55:25,770 And if I drew the sheer force diagram, it'd go p over 2 up, 1141 00:55:25,770 --> 00:55:30,370 we go over, go p over 2 down, like 2 p over 2 down. 1142 00:55:30,370 --> 00:55:33,120 Over and back up. 1143 00:55:33,120 --> 00:55:34,820 OK? 1144 00:55:34,820 --> 00:55:38,980 And then if I drew the bending moment diagram, 1145 00:55:38,980 --> 00:55:41,792 it would go up and down like that. 1146 00:55:41,792 --> 00:55:42,750 And that would be zero. 1147 00:55:42,750 --> 00:55:43,708 And that would be zero. 1148 00:55:43,708 --> 00:55:46,350 And this would be PL over 4. 1149 00:55:49,660 --> 00:55:50,190 OK? 1150 00:55:50,190 --> 00:55:59,260 Are we OK with that? 1151 00:55:59,260 --> 00:56:01,726 We haven't got to the end of the answer, but-- 1152 00:56:01,726 --> 00:56:03,110 AUDIENCE: I have a question. 1153 00:56:03,110 --> 00:56:04,570 We're not expected to-- 1154 00:56:04,570 --> 00:56:05,810 LORNA GIBSON: No, no, you don't need to do this. 1155 00:56:05,810 --> 00:56:07,226 I'm just trying to explain it now. 1156 00:56:07,226 --> 00:56:10,650 You don't need to retain that information, 1157 00:56:10,650 --> 00:56:11,480 for heaven's sakes. 1158 00:56:11,480 --> 00:56:12,720 No. 1159 00:56:12,720 --> 00:56:15,860 Come on, I'm so disappointed. 1160 00:56:15,860 --> 00:56:18,222 AUDIENCE: For the moment, is it positive 1161 00:56:18,222 --> 00:56:20,430 for the counterclockwise turns? 1162 00:56:20,430 --> 00:56:22,929 LORNA GIBSON: Oh, so you remember for the beam bending, 1163 00:56:22,929 --> 00:56:24,220 there's a different convention. 1164 00:56:24,220 --> 00:56:26,774 It's positive if it's tension on the bottom. 1165 00:56:26,774 --> 00:56:28,190 Are you in mechanical engineering? 1166 00:56:28,190 --> 00:56:29,804 I though you were in mechanical engineering. 1167 00:56:29,804 --> 00:56:31,190 AUDIENCE: No, I take physics. 1168 00:56:31,190 --> 00:56:32,565 LORNA GIBSON: Oh, you do physics. 1169 00:56:34,120 --> 00:56:36,850 All I really want to say is the moment's maximum in the middle, 1170 00:56:36,850 --> 00:56:37,350 OK? 1171 00:56:37,350 --> 00:56:41,730 So let me just say the moment's maximum in the middle, OK? 1172 00:56:41,730 --> 00:56:43,600 So then let's look at the cross section. 1173 00:56:50,360 --> 00:56:52,220 So say I look at a cross section here. 1174 00:56:52,220 --> 00:56:56,470 Let's just make it rectangular to make it easy for me to draw. 1175 00:56:56,470 --> 00:56:58,760 So it has width, b, and a height, h. 1176 00:56:58,760 --> 00:56:59,350 OK? 1177 00:56:59,350 --> 00:57:03,150 So this would be h on this picture over here. 1178 00:57:03,150 --> 00:57:07,290 And remember, the neutral axis goes through the middle 1179 00:57:07,290 --> 00:57:09,450 on the cross-section here. 1180 00:57:09,450 --> 00:57:12,140 And so one half of the beam is in tension, 1181 00:57:12,140 --> 00:57:14,290 and the other half of the beam is in compression. 1182 00:57:14,290 --> 00:57:17,380 So for this situation here, this half of the bean 1183 00:57:17,380 --> 00:57:20,740 is going to see compression, and that half of the beam 1184 00:57:20,740 --> 00:57:24,190 is going to see tension, OK? 1185 00:57:24,190 --> 00:57:25,494 Are we happy with that? 1186 00:57:25,494 --> 00:57:26,410 We're happy with that. 1187 00:57:26,410 --> 00:57:27,420 OK. 1188 00:57:27,420 --> 00:57:29,530 Now let me draw the stress distribution. 1189 00:57:36,940 --> 00:57:43,370 So if it's linear elastic and it hasn't yielded yet, 1190 00:57:43,370 --> 00:57:45,830 the stress distribution is going to look like this. 1191 00:57:45,830 --> 00:57:48,180 So this is h again. 1192 00:57:48,180 --> 00:57:49,770 That's the height. 1193 00:57:49,770 --> 00:57:51,430 And now b is into the board. 1194 00:57:55,775 --> 00:57:58,140 And I'm plotting the stress this way. 1195 00:57:58,140 --> 00:58:00,020 So this thing here is my neutral axis. 1196 00:58:02,530 --> 00:58:03,660 It has no stress. 1197 00:58:03,660 --> 00:58:05,750 Remember, there was one plane that has no stress, 1198 00:58:05,750 --> 00:58:07,270 and for a rectangular cross-section, 1199 00:58:07,270 --> 00:58:09,565 it goes through the middle of the cross-section. 1200 00:58:09,565 --> 00:58:12,820 It goes through the centroid. 1201 00:58:12,820 --> 00:58:14,280 Is this ringing a bell? 1202 00:58:14,280 --> 00:58:16,880 I'm hoping this is ringing a bell. 1203 00:58:16,880 --> 00:58:18,270 Come on. 1204 00:58:18,270 --> 00:58:19,970 I know we did this in 302, too. 1205 00:58:19,970 --> 00:58:21,596 I know we did. 1206 00:58:21,596 --> 00:58:22,096 OK. 1207 00:58:24,760 --> 00:58:27,030 OK, so this is all linear elastic, right? 1208 00:58:33,510 --> 00:58:36,860 So at some point-- so I'm going to get to the plastic hinge. 1209 00:58:36,860 --> 00:58:40,300 At some point, If you keep loading it 1210 00:58:40,300 --> 00:58:42,070 and p gets bigger and bigger, the moment 1211 00:58:42,070 --> 00:58:44,530 gets bigger and bigger, the stress gets bigger and bigger 1212 00:58:44,530 --> 00:58:47,480 remember, the equation here for the stress 1213 00:58:47,480 --> 00:58:50,580 is equal to My over i. 1214 00:58:50,580 --> 00:58:52,650 This moment, the maximum moment's 1215 00:58:52,650 --> 00:58:55,380 going to be this moment here. 1216 00:58:55,380 --> 00:58:58,760 The maximum y is going to be h over 2, the distance 1217 00:58:58,760 --> 00:59:00,140 from the neutral axis. 1218 00:59:00,140 --> 00:59:03,020 And i is going to be bh cubed over 12 1219 00:59:03,020 --> 00:59:06,060 for the rectangular section. 1220 00:59:06,060 --> 00:59:08,020 So if I keep loading it up, at some point, 1221 00:59:08,020 --> 00:59:11,013 the maximum stress is going to equal the yield stress. 1222 00:59:20,530 --> 00:59:21,080 right? 1223 00:59:21,080 --> 00:59:22,840 And in our cellular things, is going 1224 00:59:22,840 --> 00:59:24,930 to equal the yield stress of the solid. 1225 00:59:24,930 --> 00:59:30,080 So our beam is one of our edges in the foam, 1226 00:59:30,080 --> 00:59:32,400 or struts in the honeycomb. 1227 00:59:35,220 --> 00:59:37,710 So at some point, is going to equal the yield 1228 00:59:37,710 --> 00:59:39,290 strength of the solid. 1229 00:59:39,290 --> 00:59:42,150 So let me draw the stress distribution again, 1230 00:59:42,150 --> 00:59:43,610 where we start to have plasticity. 1231 00:59:46,130 --> 00:59:50,319 So here, the stress is equal the yield stress. 1232 00:59:50,319 --> 00:59:52,610 And it's going to equal the yield stress at the bottom, 1233 00:59:52,610 --> 00:59:54,849 too, because it's all symmetric, right? 1234 00:59:54,849 --> 00:59:57,390 And the neutral axis is still going to be in the middle here. 1235 00:59:57,390 --> 00:59:59,650 So it's going to 0 down there. 1236 00:59:59,650 --> 01:00:03,780 So initially, when it's just barely reached the yield stress 1237 01:00:03,780 --> 01:00:08,140 at the outer part of the beam, then this stress distribution 1238 01:00:08,140 --> 01:00:10,490 would still be linear in between. 1239 01:00:10,490 --> 01:00:15,120 But once you start to load it more than that, 1240 01:00:15,120 --> 01:00:18,660 then the plastic region starts to seep in 1241 01:00:18,660 --> 01:00:20,690 from the outside inwards. 1242 01:00:20,690 --> 01:00:26,410 And what we do here is we assume that the solid is 1243 01:00:26,410 --> 01:00:28,160 elastic perfectly plastic. 1244 01:00:33,710 --> 01:00:36,560 And if you remember, when we said things were perfectly 1245 01:00:36,560 --> 01:00:40,607 plastic, or if they were elastic perfectly plastic, 1246 01:00:40,607 --> 01:00:41,440 they look like that. 1247 01:00:41,440 --> 01:00:45,090 The stress strain curve for the solid, I'm assuming, 1248 01:00:45,090 --> 01:00:46,235 looks like that. 1249 01:00:46,235 --> 01:00:48,110 So I'm assuming the yield stress in the solid 1250 01:00:48,110 --> 01:00:49,470 is just a constant. 1251 01:00:49,470 --> 01:00:52,630 That if I strain it more, there's no work hardening. 1252 01:00:52,630 --> 01:00:53,914 I'm neglecting work hardening. 1253 01:00:53,914 --> 01:00:55,413 AUDIENCE: You said inward that the-- 1254 01:00:58,042 --> 01:00:59,000 LORNA GIBSON: In board? 1255 01:00:59,000 --> 01:01:02,996 AUDIENCE: Inward, you said something like-- 1256 01:01:02,996 --> 01:01:03,870 LORNA GIBSON: Inward. 1257 01:01:03,870 --> 01:01:06,960 So this is-- let's see. 1258 01:01:06,960 --> 01:01:09,050 I didn't bring a bean with me today. 1259 01:01:09,050 --> 01:01:09,750 No beam. 1260 01:01:09,750 --> 01:01:11,000 Do we have anything beam-like? 1261 01:01:11,000 --> 01:01:12,470 Ah, here we have a beam-like thing. 1262 01:01:12,470 --> 01:01:13,330 OK. 1263 01:01:13,330 --> 01:01:17,720 So say this is my beam, and I'm loading it this way on, OK? 1264 01:01:17,720 --> 01:01:20,760 And this is b, and that's h. 1265 01:01:20,760 --> 01:01:26,490 So this picture here is looking at it that way on, OK? 1266 01:01:26,490 --> 01:01:30,250 And this picture here, I've drawn h, 1267 01:01:30,250 --> 01:01:34,950 but now I'm just looking at the stress distribution across h. 1268 01:01:34,950 --> 01:01:38,190 And b is into the board. 1269 01:01:38,190 --> 01:01:39,130 Is that OK? 1270 01:01:39,130 --> 01:01:41,400 Does that answer your question? 1271 01:01:41,400 --> 01:01:43,740 AUDIENCE: No, I mean for the plastic. 1272 01:01:43,740 --> 01:01:45,810 LORNA GIBSON: For this part? 1273 01:01:45,810 --> 01:01:47,470 OK. 1274 01:01:47,470 --> 01:01:48,270 I'm working up. 1275 01:01:48,270 --> 01:01:50,600 I haven't finished it yet. 1276 01:01:50,600 --> 01:01:52,470 So this is the same kind of view as here. 1277 01:01:52,470 --> 01:01:53,554 I drew it a little bigger. 1278 01:01:53,554 --> 01:01:55,178 It shouldn't have draw bigger, I should 1279 01:01:55,178 --> 01:01:56,510 have drawn it the same height. 1280 01:01:56,510 --> 01:01:57,620 But it's the same thing. 1281 01:01:57,620 --> 01:01:58,120 OK? 1282 01:02:00,790 --> 01:02:02,550 So you'll buy that at some point, 1283 01:02:02,550 --> 01:02:04,640 we reach the yield strength here. 1284 01:02:04,640 --> 01:02:07,160 And if I keep loading it up and I 1285 01:02:07,160 --> 01:02:09,760 assume that the solid is perfectly plastic, 1286 01:02:09,760 --> 01:02:15,580 that there's no work hardening, then the stress distribution 1287 01:02:15,580 --> 01:02:16,940 would look like this. 1288 01:02:16,940 --> 01:02:18,150 OK? 1289 01:02:18,150 --> 01:02:23,230 And then if it yields more, then it's going to look like that. 1290 01:02:23,230 --> 01:02:26,580 And if it yields more, eventually I'm going to get 1291 01:02:26,580 --> 01:02:30,940 to the stage here, where it's-- let me redraw this. 1292 01:02:30,940 --> 01:02:34,950 That would be-- you get the idea, OK? 1293 01:02:34,950 --> 01:02:37,400 This will go over here, and down here, and like that. 1294 01:02:37,400 --> 01:02:38,751 OK? 1295 01:02:38,751 --> 01:02:39,250 OK. 1296 01:02:42,140 --> 01:02:45,690 So are we happy with this stress distribution 1297 01:02:45,690 --> 01:02:47,100 across the cross-section? 1298 01:02:47,100 --> 01:02:47,840 Yeah, OK. 1299 01:02:47,840 --> 01:02:50,170 So that's when it forms the plastic hinge. 1300 01:02:50,170 --> 01:02:57,170 So when it forms a plastic hinge, 1301 01:02:57,170 --> 01:02:59,790 the stress distribution looks like this. 1302 01:03:04,471 --> 01:03:06,220 So these are supposed to be the same size. 1303 01:03:06,220 --> 01:03:07,170 They're not quite. 1304 01:03:16,620 --> 01:03:17,390 Let's see. 1305 01:03:17,390 --> 01:03:20,560 So one of the things I talked about 1306 01:03:20,560 --> 01:03:23,230 was the plastic moment that kind of characterized 1307 01:03:23,230 --> 01:03:25,580 that plastic hinge. 1308 01:03:25,580 --> 01:03:28,680 And the plastic moment is just the internal amount of moment 1309 01:03:28,680 --> 01:03:32,580 that the beam can withstand when it's yielded completely 1310 01:03:32,580 --> 01:03:33,970 across the whole cross-section. 1311 01:03:33,970 --> 01:03:36,040 So when we're at this point here. 1312 01:03:36,040 --> 01:03:40,080 So you calculate that by saying, that stress 1313 01:03:40,080 --> 01:03:42,650 there is equivalent to a force. 1314 01:03:42,650 --> 01:03:45,350 That stress there is equivalent to a force. 1315 01:03:45,350 --> 01:03:49,230 And you get the moment by multiplying those forces 1316 01:03:49,230 --> 01:03:51,020 times that distance there. 1317 01:03:51,020 --> 01:03:51,660 OK? 1318 01:03:51,660 --> 01:03:53,993 Because you think of those two forces as being a couple, 1319 01:03:53,993 --> 01:03:58,810 and the moments the force times the distance between them. 1320 01:03:58,810 --> 01:04:03,120 So the plastic moment was sigma ys. 1321 01:04:03,120 --> 01:04:10,800 And say we're talking about our honeycomb or foam or something. 1322 01:04:10,800 --> 01:04:12,990 That was our cell wall thickness t. 1323 01:04:12,990 --> 01:04:15,320 So let me call it t instead of h for the foams 1324 01:04:15,320 --> 01:04:17,290 and the honeycombs. 1325 01:04:17,290 --> 01:04:23,870 So this force here is going to be the stress time the area 1326 01:04:23,870 --> 01:04:25,980 over which it acts. 1327 01:04:25,980 --> 01:04:28,960 And let's say we look at it for a honeycomb. 1328 01:04:33,700 --> 01:04:37,710 Then I've got the stress is acting over this distance 1329 01:04:37,710 --> 01:04:41,400 here, and then times the depth into the page, right? 1330 01:04:41,400 --> 01:04:44,360 And if it's the honeycomb, that depth into the page 1331 01:04:44,360 --> 01:04:47,140 is just b, OK? 1332 01:04:47,140 --> 01:04:51,170 And then this moment arm here is just t over 2 1333 01:04:51,170 --> 01:04:53,540 as well, because that's t over 4, and that's t over 4. 1334 01:04:53,540 --> 01:04:57,030 So it's t over 2 again. 1335 01:04:57,030 --> 01:05:06,290 So it's sigma ys bt squared over 4 for the honeycomb. 1336 01:05:06,290 --> 01:05:08,015 And say we have an open cell foam. 1337 01:05:12,830 --> 01:05:17,180 The edges aren't of thickness b, they're of thickness t. 1338 01:05:17,180 --> 01:05:24,660 So then it's m p is just sigma ys t cubed over 4. 1339 01:05:27,400 --> 01:05:29,950 Are we happy? 1340 01:05:29,950 --> 01:05:32,770 And really, physically what that is is it 1341 01:05:32,770 --> 01:05:36,339 means that the beam can't hold any more force. 1342 01:05:36,339 --> 01:05:37,880 You can't apply any more force to it. 1343 01:05:37,880 --> 01:05:40,840 It's just going to rotate like this once you've 1344 01:05:40,840 --> 01:05:42,730 gotten to that plastic moment. 1345 01:05:42,730 --> 01:05:46,050 That's why it's called a hinge, because it just 1346 01:05:46,050 --> 01:05:48,480 can rotate like hinge rotates. 1347 01:05:48,480 --> 01:05:50,020 Like a door hinge, OK? 1348 01:05:52,906 --> 01:05:54,830 AUDIENCE: How does it rotate? 1349 01:05:54,830 --> 01:06:00,220 LORNA GIBSON: Well, where's my original picture. 1350 01:06:00,220 --> 01:06:04,170 So if this was the beam, when you form the plastic hinge, 1351 01:06:04,170 --> 01:06:05,670 your beam would just look like that. 1352 01:06:05,670 --> 01:06:09,610 And this would be your hinge point. 1353 01:06:09,610 --> 01:06:11,630 I'm a civil engineer originally. 1354 01:06:11,630 --> 01:06:15,280 We try to avoid this. 1355 01:06:15,280 --> 01:06:17,680 So that's why, in the foam and in the honeycomb, that's 1356 01:06:17,680 --> 01:06:22,220 when it fails is when you get that plastic hinge forming. 1357 01:06:22,220 --> 01:06:24,080 OK? 1358 01:06:24,080 --> 01:06:24,580 All right. 1359 01:06:24,580 --> 01:06:26,571 We have a few more minutes. 1360 01:06:26,571 --> 01:06:28,910 AUDIENCE: That kind of looks like plastic buckling. 1361 01:06:28,910 --> 01:06:33,570 LORNA GIBSON: Well yeah, it's not buckling, but it's plastic, 1362 01:06:33,570 --> 01:06:34,460 yeah. 1363 01:06:34,460 --> 01:06:35,085 It's permanent. 1364 01:06:37,650 --> 01:06:38,205 Anyone else? 1365 01:06:42,940 --> 01:06:44,109 OK. 1366 01:06:44,109 --> 01:06:44,900 No other questions? 1367 01:06:48,990 --> 01:06:51,300 Should we call it a day? 1368 01:06:51,300 --> 01:06:54,030 Is that helpful? 1369 01:06:54,030 --> 01:06:55,910 All right, then. 1370 01:06:55,910 --> 01:06:56,790 It's what I do. 1371 01:06:56,790 --> 01:06:57,350 Come on. 1372 01:06:57,350 --> 01:06:58,432 It's what I do. 1373 01:07:01,880 --> 01:07:02,680 All right. 1374 01:07:02,680 --> 01:07:05,350 So I'll see you Wednesday. 1375 01:07:05,350 --> 01:07:09,570 And my plan is to grade the tests before spring break, 1376 01:07:09,570 --> 01:07:12,180 so I shall have it back to you.