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 --> 00:00:13,300 To make a donation, or view additional materials 6 00:00:13,300 --> 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 00:00:28,180 --> 00:00:30,480 LORNA GIBSON: OK, so we should probably start. 9 00:00:30,480 --> 00:00:34,130 So last time we finished up talking about energy absorption 10 00:00:34,130 --> 00:00:36,870 in foamy cellular materials. 11 00:00:36,870 --> 00:00:38,560 And today I wanted to start a new topic. 12 00:00:38,560 --> 00:00:40,630 We're going to talk about sandwich panels. 13 00:00:40,630 --> 00:00:44,140 So sandwich panels have two stiff, strong skins 14 00:00:44,140 --> 00:00:47,180 that are separated by some sort of lightweight core. 15 00:00:47,180 --> 00:00:51,110 So the skins are typically, say, a metal like aluminum, 16 00:00:51,110 --> 00:00:53,350 or some sort of fiber composite. 17 00:00:53,350 --> 00:00:56,310 And the core is usually some sort of cellular material. 18 00:00:56,310 --> 00:00:58,630 Sometimes it's an engineering honeycomb. 19 00:00:58,630 --> 00:00:59,740 Sometimes it's a foam. 20 00:00:59,740 --> 00:01:01,540 Sometimes it's balsa wood. 21 00:01:01,540 --> 00:01:03,400 And the idea is that what you're doing 22 00:01:03,400 --> 00:01:06,840 with the core is you're using a light material to separate 23 00:01:06,840 --> 00:01:09,466 the faces, and if you think about an I-beam-- 24 00:01:09,466 --> 00:01:11,340 so if you remember when we talk about bending 25 00:01:11,340 --> 00:01:14,270 and we talk about I-beams, the whole idea is that in bending, 26 00:01:14,270 --> 00:01:16,080 you want to increase the moment of inertia. 27 00:01:16,080 --> 00:01:19,040 So you want to make as much material as far 28 00:01:19,040 --> 00:01:21,610 away from the middle of the beam as possible 29 00:01:21,610 --> 00:01:23,290 to increase the moment of inertia. 30 00:01:23,290 --> 00:01:26,480 So if you think about an I-beam, you put the flanges far apart 31 00:01:26,480 --> 00:01:29,810 with the web, and that increases the moment of inertia. 32 00:01:29,810 --> 00:01:31,530 And the sandwich panels and the sandwich 33 00:01:31,530 --> 00:01:33,080 beams essentially do the same thing, 34 00:01:33,080 --> 00:01:36,490 but they're using a lightweight core instead of a web. 35 00:01:36,490 --> 00:01:38,510 And so the idea is you use a lightweight core. 36 00:01:38,510 --> 00:01:40,270 It separates the faces. 37 00:01:40,270 --> 00:01:41,975 It increases the moment of inertia. 38 00:01:41,975 --> 00:01:43,600 But you don't add a whole lot of weight 39 00:01:43,600 --> 00:01:46,330 because you've got this lightweight core in the middle. 40 00:01:46,330 --> 00:01:48,410 So I brought some examples that I'll pass around 41 00:01:48,410 --> 00:01:49,690 and we can play with. 42 00:01:49,690 --> 00:01:53,420 So these are some examples up on the screen, and some of those 43 00:01:53,420 --> 00:01:54,810 I have down here. 44 00:01:54,810 --> 00:01:57,820 So for instance, the top-- turn my little gizmo 45 00:01:57,820 --> 00:02:01,770 on-- the top left here, this is a helicopter rotor blade, 46 00:02:01,770 --> 00:02:04,160 and that has a honeycomb core in it. 47 00:02:04,160 --> 00:02:07,530 This is an aircraft flooring panel that has a honeycomb core 48 00:02:07,530 --> 00:02:09,740 and has carbon fiber faces. 49 00:02:09,740 --> 00:02:10,840 So that's this thing here. 50 00:02:10,840 --> 00:02:13,160 I'll pass that around in a minute. 51 00:02:13,160 --> 00:02:14,520 This is a downhill ski. 52 00:02:14,520 --> 00:02:18,190 This has aluminum faces and a polyurethane foam core. 53 00:02:18,190 --> 00:02:19,940 And that's the ski here. 54 00:02:19,940 --> 00:02:21,600 And it's quite common in skis now 55 00:02:21,600 --> 00:02:23,570 to have these sandwich panels. 56 00:02:23,570 --> 00:02:26,050 This is a little piece of a small sailing boat. 57 00:02:26,050 --> 00:02:30,160 It had, I think, glass fiber faces and a balsa wood core. 58 00:02:30,160 --> 00:02:31,840 And I don't know if any of you sail, 59 00:02:31,840 --> 00:02:33,680 but MIT has new sailing boats. 60 00:02:33,680 --> 00:02:34,552 Do you sail? 61 00:02:34,552 --> 00:02:36,010 AUDIENCE: I do not much these days. 62 00:02:36,010 --> 00:02:37,110 LORNA GIBSON: OK, but those little tech 63 00:02:37,110 --> 00:02:38,735 dinghies that you see out in the river, 64 00:02:38,735 --> 00:02:40,640 those have sandwich panel holes to them. 65 00:02:40,640 --> 00:02:42,610 So those are little sandwich panels. 66 00:02:42,610 --> 00:02:46,240 This is an example from a building panel. 67 00:02:46,240 --> 00:02:50,390 This has a dry wall face and a plywood face and a foam core, 68 00:02:50,390 --> 00:02:52,830 and the idea with panels for buildings 69 00:02:52,830 --> 00:02:56,370 is that usually they use a foam core because the foam has 70 00:02:56,370 --> 00:02:57,680 some thermal insulation. 71 00:02:57,680 --> 00:02:59,700 So as well as sort of separating the faces 72 00:02:59,700 --> 00:03:01,840 and having a structural role, it has 73 00:03:01,840 --> 00:03:04,640 a role in thermally insulating the building. 74 00:03:04,640 --> 00:03:06,200 The foams are a little less efficient 75 00:03:06,200 --> 00:03:07,435 than using a honeycomb core. 76 00:03:07,435 --> 00:03:10,480 So for the same weight, you get a stiffer structure 77 00:03:10,480 --> 00:03:12,370 with a honeycomb core than a foam core. 78 00:03:12,370 --> 00:03:14,850 But if you want thermal insulation as well as 79 00:03:14,850 --> 00:03:18,160 a structural requirement, then the foam cores are good. 80 00:03:18,160 --> 00:03:21,040 And these are a couple examples of sandwiches in nature. 81 00:03:21,040 --> 00:03:23,060 This is the human skull. 82 00:03:23,060 --> 00:03:26,450 And your skull is a sandwich of two dense layers 83 00:03:26,450 --> 00:03:27,950 of the compact bone, and you can see 84 00:03:27,950 --> 00:03:32,470 there's a little thin layer of the trabecular bone in between. 85 00:03:32,470 --> 00:03:35,950 So your head is like a sandwich, your skull is like a sandwich. 86 00:03:35,950 --> 00:03:38,492 And I don't know if I'll get to it next time, 87 00:03:38,492 --> 00:03:39,950 but in the next couple of lectures, 88 00:03:39,950 --> 00:03:41,740 I'm going to talk a little bit about sandwich panels 89 00:03:41,740 --> 00:03:43,380 in nature, sandwich shells in nature. 90 00:03:43,380 --> 00:03:44,620 You see this all the time. 91 00:03:44,620 --> 00:03:46,210 And this is a bird wing, here. 92 00:03:46,210 --> 00:03:48,442 And so you can see there's got the dense bone 93 00:03:48,442 --> 00:03:49,900 on the top and the bottom, and it's 94 00:03:49,900 --> 00:03:52,000 got this kind of almost trust-like structure 95 00:03:52,000 --> 00:03:52,762 in the middle. 96 00:03:52,762 --> 00:03:54,720 And obviously birds want to reduce their weight 97 00:03:54,720 --> 00:03:57,000 because they want to fly, so reducing the weight's 98 00:03:57,000 --> 00:03:58,390 very important. 99 00:03:58,390 --> 00:04:01,490 And so this is one of the ways that birds reduce their weight, 100 00:04:01,490 --> 00:04:03,954 is by having a sandwich kind of structure. 101 00:04:03,954 --> 00:04:05,370 So I have a couple of things here. 102 00:04:05,370 --> 00:04:08,320 These are the two panels at the top there. 103 00:04:08,320 --> 00:04:11,920 This is the ski, and you can yank those around. 104 00:04:15,330 --> 00:04:18,985 I also have a few panels that people at MIT have made. 105 00:04:18,985 --> 00:04:20,860 And I have the pieces that they're made from. 106 00:04:20,860 --> 00:04:23,020 So you can see how effective the sandwich thing is. 107 00:04:23,020 --> 00:04:26,470 So this was made by a guy called Dirk Moore. 108 00:04:26,470 --> 00:04:29,350 He was a graduate student in ocean engineering. 109 00:04:29,350 --> 00:04:33,252 And it has aluminum faces and a little thin aluminum core. 110 00:04:33,252 --> 00:04:35,710 So you can see, if you try and bend that with your fingers, 111 00:04:35,710 --> 00:04:38,980 you really can't bend it any noticeable amount. 112 00:04:38,980 --> 00:04:42,430 And this panel here is roughly the same thickness 113 00:04:42,430 --> 00:04:43,340 of the face on that. 114 00:04:43,340 --> 00:04:44,960 And you can see how easy it is for me 115 00:04:44,960 --> 00:04:47,270 to bend that-- very easy. 116 00:04:47,270 --> 00:04:49,470 And this is the same kind of thing as the core. 117 00:04:49,470 --> 00:04:50,970 It's thicker than that core, but you 118 00:04:50,970 --> 00:04:53,480 can see how easy it is for me to bend this, too. 119 00:04:53,480 --> 00:04:56,690 So each of the pieces is not very stiff at all. 120 00:04:56,690 --> 00:05:00,310 But when you put them all together, it's very stiff. 121 00:05:00,310 --> 00:05:02,290 So that's really the beauty of this. 122 00:05:02,290 --> 00:05:04,274 You can have lightweight components, 123 00:05:04,274 --> 00:05:06,190 and by putting them together in the right way, 124 00:05:06,190 --> 00:05:07,310 they're quite stiff. 125 00:05:07,310 --> 00:05:09,920 So here's another example here. 126 00:05:09,920 --> 00:05:12,700 This is a panel that one of my students, Kevin Chang made. 127 00:05:12,700 --> 00:05:15,137 And this has actually already been broken a little bit, 128 00:05:15,137 --> 00:05:16,970 so it's not quite as stiff as it used to be. 129 00:05:16,970 --> 00:05:19,520 And you can kind of hear, it squeaks. 130 00:05:19,520 --> 00:05:22,860 But you can feel that and see how stiff that is. 131 00:05:22,860 --> 00:05:25,720 And this is the face panel here. 132 00:05:25,720 --> 00:05:28,970 And you can see, I can bend that quite easily with my hands. 133 00:05:28,970 --> 00:05:30,230 Doodle-doot. 134 00:05:30,230 --> 00:05:31,840 And then this is the core piece here. 135 00:05:31,840 --> 00:05:34,214 And again, this is very flexible. 136 00:05:34,214 --> 00:05:36,505 So it's really about putting all those pieces together. 137 00:05:36,505 --> 00:05:39,270 So you get this sandwich construction 138 00:05:39,270 --> 00:05:42,186 and you get that effect, OK? 139 00:05:42,186 --> 00:05:44,145 [? Oop-loo. ?] 140 00:05:44,145 --> 00:05:46,370 All right, so what we're going to do 141 00:05:46,370 --> 00:05:50,610 is first of all look at the stiffness of these panels, 142 00:05:50,610 --> 00:05:52,640 calculate their deflection. 143 00:05:52,640 --> 00:05:55,220 We're going to look at the minimum weight design of them. 144 00:05:55,220 --> 00:05:57,050 So we're going to look at how for, 145 00:05:57,050 --> 00:05:59,220 say, given materials in a given span, 146 00:05:59,220 --> 00:06:02,790 how do we minimize the weight of the beam for a given stiffness? 147 00:06:02,790 --> 00:06:05,590 And then we're going to look at the stresses in the sandwich 148 00:06:05,590 --> 00:06:06,090 beams. 149 00:06:06,090 --> 00:06:08,160 So there's going to be one set of stresses in the faces, 150 00:06:08,160 --> 00:06:10,493 and a different kind of stress distribution in the core. 151 00:06:10,493 --> 00:06:12,530 So we'll look at the stress distribution. 152 00:06:12,530 --> 00:06:14,420 And then we'll talk about failure modes, 153 00:06:14,420 --> 00:06:16,510 how these things can fail, and then 154 00:06:16,510 --> 00:06:18,990 how to figure out which failure mode is dominant, which 155 00:06:18,990 --> 00:06:20,770 one occurs at the lowest load. 156 00:06:20,770 --> 00:06:22,680 And then we'll look at optimizing the design, 157 00:06:22,680 --> 00:06:26,626 minimizing the design for a certain strength and stiffness. 158 00:06:26,626 --> 00:06:29,000 So we're not going to get all that way through everything 159 00:06:29,000 --> 00:06:33,320 today, but we'll kind of make a start on that. 160 00:06:33,320 --> 00:06:33,820 OK. 161 00:06:33,820 --> 00:06:35,950 So let me start. 162 00:06:35,950 --> 00:06:41,860 So the idea here is we have two stiff, strong skins, or faces, 163 00:06:41,860 --> 00:06:43,576 separated by a lightweight core. 164 00:07:01,890 --> 00:07:05,080 And the idea is that by separating the faces, 165 00:07:05,080 --> 00:07:07,770 you increase the moment of inertia 166 00:07:07,770 --> 00:07:09,153 with little increase in weight. 167 00:07:36,800 --> 00:07:40,224 So these are particularly good if you want to resist bending, 168 00:07:40,224 --> 00:07:41,640 or if you want to resist buckling. 169 00:07:41,640 --> 00:07:43,806 Because both of those involve the moment of inertia. 170 00:08:01,640 --> 00:08:03,004 And they work like an I-beam. 171 00:08:07,380 --> 00:08:11,100 So the faces of the sandwich are like the flanges of the I-beam, 172 00:08:11,100 --> 00:08:12,480 and the core is like the web. 173 00:08:32,799 --> 00:08:36,059 And the faces are typically made of either fiber reinforced 174 00:08:36,059 --> 00:08:38,511 composites or metals. 175 00:08:42,179 --> 00:08:43,809 So typically, something like aluminum, 176 00:08:43,809 --> 00:08:45,220 usually you're trying to reduce the weight 177 00:08:45,220 --> 00:08:48,000 if you use these things, so a lightweight metal like aluminum 178 00:08:48,000 --> 00:08:50,640 is sometimes used. 179 00:08:50,640 --> 00:09:03,570 And the cores are usually honeycombs, or foams, or balsa. 180 00:09:03,570 --> 00:09:06,045 And when they use balsa wood, what they do 181 00:09:06,045 --> 00:09:08,170 is-- I brought a piece of balsa here-- what they do 182 00:09:08,170 --> 00:09:09,670 is they would take a block like this 183 00:09:09,670 --> 00:09:12,470 and chop it into pieces around here. 184 00:09:12,470 --> 00:09:15,410 And then they would lay those pieces on a cloth mat. 185 00:09:15,410 --> 00:09:17,870 So typically the pieces are maybe two inches by two inches. 186 00:09:17,870 --> 00:09:21,060 They lay them on a cloth mat, and because they're not 187 00:09:21,060 --> 00:09:24,380 one monolithic piece, they can then shape that mat 188 00:09:24,380 --> 00:09:25,580 to curved shapes. 189 00:09:25,580 --> 00:09:27,650 So it doesn't have to be just a flat panel. 190 00:09:27,650 --> 00:09:30,510 They can curve it around a curved surface if they want. 191 00:09:43,820 --> 00:09:45,990 So we'll say the honeycombs are lighter 192 00:09:45,990 --> 00:09:48,065 than the foams for a given stiffness or strength. 193 00:10:06,270 --> 00:10:08,840 But the foams provide thermal insulation as well as 194 00:10:08,840 --> 00:10:10,450 a mechanical support. 195 00:10:34,930 --> 00:10:38,260 And the overall mechanical properties of the honeycomb 196 00:10:38,260 --> 00:10:41,300 depend on the properties of each of the two parts, 197 00:10:41,300 --> 00:10:43,126 of the faces and the core, and also 198 00:10:43,126 --> 00:10:44,500 the geometry of the whole thing-- 199 00:10:44,500 --> 00:10:46,700 how thick's the core, how thick's the face, how dense is 200 00:10:46,700 --> 00:10:47,270 the core? 201 00:10:47,270 --> 00:10:48,440 That kind of thing. 202 00:11:14,800 --> 00:11:17,840 And typically, the panel has to have some required stiffness 203 00:11:17,840 --> 00:11:18,960 or strength. 204 00:11:18,960 --> 00:11:20,820 And often what you want to do is minimize 205 00:11:20,820 --> 00:11:23,170 the weight for that required stiffness or strength. 206 00:12:21,030 --> 00:12:24,650 So often these panels are used in some sort of vehicle, 207 00:12:24,650 --> 00:12:27,300 like we talked about the sailboat, or like a helicopter, 208 00:12:27,300 --> 00:12:28,950 or like an airplane. 209 00:12:28,950 --> 00:12:31,080 They're also used in like refrigerated trucks-- 210 00:12:31,080 --> 00:12:32,829 they would have a foam core because they'd 211 00:12:32,829 --> 00:12:34,560 want the thermal insulation. 212 00:12:34,560 --> 00:12:36,330 So if you were going to use it in some sort of a vehicle, 213 00:12:36,330 --> 00:12:38,079 you want to reduce the mass of the vehicle 214 00:12:38,079 --> 00:12:40,905 and you want to have the lightest panel that you can. 215 00:12:45,070 --> 00:12:45,570 Yup? 216 00:12:45,570 --> 00:12:48,740 AUDIENCE: So if you saw the base material 217 00:12:48,740 --> 00:12:53,410 and you'd have the [INAUDIBLE] sandwich 218 00:12:53,410 --> 00:12:57,100 panel, that piece [INAUDIBLE] the sandwich panel 219 00:12:57,100 --> 00:13:00,317 with something [? solid ?] in the middle? 220 00:13:00,317 --> 00:13:01,150 LORNA GIBSON: Well-- 221 00:13:01,150 --> 00:13:04,110 AUDIENCE: So, as we're getting [INAUDIBLE] aluminum piece 222 00:13:04,110 --> 00:13:06,273 that's was as thick as a sandwich panel. 223 00:13:06,273 --> 00:13:07,064 LORNA GIBSON: Yeah. 224 00:13:07,064 --> 00:13:08,680 AUDIENCE: [INAUDIBLE]. 225 00:13:08,680 --> 00:13:10,990 LORNA GIBSON: Oh, well, if you have the solid aluminum 226 00:13:10,990 --> 00:13:12,540 piece that was as thick as the sandwich, 227 00:13:12,540 --> 00:13:13,956 it's going to be stiffer, but it's 228 00:13:13,956 --> 00:13:15,440 going to be a lot, lot heavier. 229 00:13:15,440 --> 00:13:19,850 So the stiffness per unit weight would not be as good. 230 00:13:19,850 --> 00:13:20,490 OK? 231 00:13:20,490 --> 00:13:24,460 So we're going to calculate the stiffness in just one minute. 232 00:13:24,460 --> 00:13:29,620 And then we're going to look at how we minimize the weight, OK? 233 00:13:29,620 --> 00:13:30,120 OK. 234 00:13:46,920 --> 00:13:49,026 So what I'm going to do is set this up 235 00:13:49,026 --> 00:13:50,150 as kind of a general thing. 236 00:13:50,150 --> 00:13:52,110 We're just going to look at sandwich beams rather than 237 00:13:52,110 --> 00:13:53,620 plates, just because it's simpler. 238 00:13:53,620 --> 00:13:55,737 But the plates, everything we say for the beams 239 00:13:55,737 --> 00:13:57,070 basically applies to the plates. 240 00:13:57,070 --> 00:13:59,952 The equation's just a little bit more complicated. 241 00:13:59,952 --> 00:14:01,827 So we're going to start with analyzing beams. 242 00:14:25,650 --> 00:14:28,386 And I'm just going to start with a beam, 243 00:14:28,386 --> 00:14:31,210 say, in three-point bending. 244 00:14:31,210 --> 00:14:35,063 So there's my faces there. 245 00:14:35,063 --> 00:14:35,562 Boop. 246 00:14:40,880 --> 00:14:43,117 And I've made it kind of more stumpy 247 00:14:43,117 --> 00:14:45,700 than it would be in real life, just because it makes it easier 248 00:14:45,700 --> 00:14:47,500 to draw it. 249 00:14:47,500 --> 00:14:51,130 And then if I look at it the other way on, 250 00:14:51,130 --> 00:14:53,740 it would look something like that. 251 00:14:53,740 --> 00:14:57,040 So say there's some load P here. 252 00:14:57,040 --> 00:14:59,880 Say the span of the beam is l. 253 00:14:59,880 --> 00:15:03,810 Say the load's in the middle, so each of the supports just 254 00:15:03,810 --> 00:15:06,240 sees a load of P/2. 255 00:15:06,240 --> 00:15:09,780 And then let me just define some geometrical parameters here. 256 00:15:09,780 --> 00:15:13,190 I'm going to say the width of the beam is b. 257 00:15:13,190 --> 00:15:17,650 And I'm going to say the face thicknesses are each t. 258 00:15:17,650 --> 00:15:20,490 So the thickness of each face is t. 259 00:15:20,490 --> 00:15:25,200 And the thickness of the core is c, OK? 260 00:15:25,200 --> 00:15:28,020 So that's just sort of definitions. 261 00:15:28,020 --> 00:15:31,480 And I'm going to say the face has a set of properties, 262 00:15:31,480 --> 00:15:33,234 the core has a set of properties, 263 00:15:33,234 --> 00:15:35,150 and then the solid from which the core is made 264 00:15:35,150 --> 00:15:37,550 has another set of properties. 265 00:15:37,550 --> 00:15:39,480 So the face properties that we're going to use 266 00:15:39,480 --> 00:15:41,110 are a density of the face. 267 00:15:41,110 --> 00:15:42,840 We'll call that row f. 268 00:15:42,840 --> 00:15:46,760 The modulus of the face, Ef, and some sort of strength 269 00:15:46,760 --> 00:15:49,640 of the face, let's imagine it's aluminum and it yields, 270 00:15:49,640 --> 00:15:52,060 that would be sigma y of the face. 271 00:15:52,060 --> 00:15:55,080 And then the core similarly is going 272 00:15:55,080 --> 00:15:58,180 to have a density, rho star c. 273 00:15:58,180 --> 00:16:01,639 It's going to have a modulus, E star c. 274 00:16:01,639 --> 00:16:03,180 And it's going to have some strength, 275 00:16:03,180 --> 00:16:06,180 I'm going to call sigma star c. 276 00:16:06,180 --> 00:16:07,850 And then the solid from which the core 277 00:16:07,850 --> 00:16:10,540 is made is going to have a density row 278 00:16:10,540 --> 00:16:17,580 s, a modulus Es, and some strength, sigma ys, OK? 279 00:16:21,160 --> 00:16:22,700 So the core is going to be some kind 280 00:16:22,700 --> 00:16:26,090 of cellular material, a honeycomb, or a foam, or balsa. 281 00:16:26,090 --> 00:16:28,190 And typically, the modulus of the core 282 00:16:28,190 --> 00:16:30,595 is going to be a lot less than the modulus of the face. 283 00:16:34,280 --> 00:16:38,820 So I'm just going to say here that the E star c is typically 284 00:16:38,820 --> 00:16:40,720 much greater than Ef. 285 00:16:40,720 --> 00:16:42,437 And we're going to use that later on. 286 00:16:46,810 --> 00:16:49,700 So we're going to derive some equations, for example, 287 00:16:49,700 --> 00:16:54,040 for an equivalent flexural rigidity for the section, an Ei 288 00:16:54,040 --> 00:16:55,280 equivalent. 289 00:16:55,280 --> 00:16:57,180 And that has several terms. 290 00:16:57,180 --> 00:16:59,379 But if we can say the core stiffness 291 00:16:59,379 --> 00:17:00,920 is much less than the face thickness, 292 00:17:00,920 --> 00:17:04,653 and also if we can say the core-- the stiffness is less 293 00:17:04,653 --> 00:17:06,069 and also the thickness of the core 294 00:17:06,069 --> 00:17:08,089 is much greater than the thickness of the face, 295 00:17:08,089 --> 00:17:10,550 a lot of the expressions we're going to use simplify. 296 00:17:10,550 --> 00:17:13,540 So we're going to make those assumptions. 297 00:17:13,540 --> 00:17:17,480 So let me just draw the shear diagram here. 298 00:17:32,070 --> 00:17:35,120 So V is shear, so that's the shear diagram. 299 00:17:35,120 --> 00:17:37,880 We have some load P/2 at the support. 300 00:17:37,880 --> 00:17:40,660 There's no other load applied until we get to here. 301 00:17:40,660 --> 00:17:44,510 Than the shear diagram goes down by P, so we're at minus P/2. 302 00:17:44,510 --> 00:17:47,380 Then there's no load here, so this just stays constant, 303 00:17:47,380 --> 00:17:49,590 and then we go back up to 0. 304 00:17:49,590 --> 00:17:52,670 And then let me just draw bending moment diagram. 305 00:17:57,520 --> 00:17:59,550 The bending moment diagram for this 306 00:17:59,550 --> 00:18:02,637 is just going to look like a triangle. 307 00:18:02,637 --> 00:18:04,470 Remember, if we integrate the shear diagram, 308 00:18:04,470 --> 00:18:06,960 we get the bending moment diagram. 309 00:18:06,960 --> 00:18:10,730 And that maximum moment there is going to Pl over 4. 310 00:18:16,921 --> 00:18:17,420 OK. 311 00:18:21,890 --> 00:18:24,980 So initially, I'm going to calculate the deflections. 312 00:18:24,980 --> 00:18:26,980 And I don't really need those diagrams for that, 313 00:18:26,980 --> 00:18:28,450 but then I'm going to calculate the stresses, 314 00:18:28,450 --> 00:18:30,700 and I'm going to need those diagrams for the stresses. 315 00:18:30,700 --> 00:18:33,150 So just kind of keep those in mind for now. 316 00:18:33,150 --> 00:18:38,200 So to calculate the deflections, sandwich panels 317 00:18:38,200 --> 00:18:43,360 are a little bit different from homogeneous beams. 318 00:18:43,360 --> 00:18:45,750 In a sandwich panel, the core is not 319 00:18:45,750 --> 00:18:48,090 very stiff compared to the faces. 320 00:18:48,090 --> 00:18:52,930 And we've got some shear stresses acting on the thing. 321 00:18:52,930 --> 00:18:55,820 And the shear stresses are largely carried by the core. 322 00:18:55,820 --> 00:18:57,880 So the core is actually going to shear, 323 00:18:57,880 --> 00:19:00,350 and there's going to be a significant deflection 324 00:19:00,350 --> 00:19:03,210 of the core and shear as well as the overall bending 325 00:19:03,210 --> 00:19:04,020 of the whole panel. 326 00:19:04,020 --> 00:19:05,396 So you have to count for that. 327 00:19:05,396 --> 00:19:08,020 So we're going to have a bending term and a shear term-- that's 328 00:19:08,020 --> 00:19:10,850 what those two terms are there. 329 00:19:10,850 --> 00:19:20,171 So we're going to say there's a bending deflection and a shear 330 00:19:20,171 --> 00:19:20,670 deflection. 331 00:19:23,980 --> 00:19:28,890 And that shearing deflection arises 332 00:19:28,890 --> 00:19:38,470 from the core being sheared and the fact that the core, say, 333 00:19:38,470 --> 00:19:41,750 Young's modulus or also the shear modulus, 334 00:19:41,750 --> 00:19:43,920 is quite a bit less than the face modulus. 335 00:19:43,920 --> 00:19:46,420 So if you think of the core as being much more compliant 336 00:19:46,420 --> 00:19:48,000 than the face, then the core is going 337 00:19:48,000 --> 00:19:53,564 to have some deflection from that shear stress. 338 00:19:53,564 --> 00:19:55,480 OK, so we're going to start out with this term 339 00:19:55,480 --> 00:19:58,650 here, the bending term. 340 00:19:58,650 --> 00:20:04,550 And if I just had a homogeneous beam in three-point bending, 341 00:20:04,550 --> 00:20:06,270 the central deflections-- so these 342 00:20:06,270 --> 00:20:09,890 are all the central deflections I'm calculating here-- with Pl 343 00:20:09,890 --> 00:20:14,200 cubed over it turns out to be 48 is the number, and divided 344 00:20:14,200 --> 00:20:15,580 by EI. 345 00:20:15,580 --> 00:20:18,580 And because we don't have a homogeneous beam here, 346 00:20:18,580 --> 00:20:21,362 I'm going to call that equivalent EI. 347 00:20:21,362 --> 00:20:23,070 And to make it a little bit more general, 348 00:20:23,070 --> 00:20:25,440 instead of putting 48, that number, 349 00:20:25,440 --> 00:20:27,760 I'm just going to put a constant B1. 350 00:20:27,760 --> 00:20:30,350 And that B1 constant is just going 351 00:20:30,350 --> 00:20:32,670 to depend on the loading geometry. 352 00:20:32,670 --> 00:20:36,080 So any time I have a concentrated load on a beam, 353 00:20:36,080 --> 00:20:39,290 the deflection's always Pl cubed over EI, 354 00:20:39,290 --> 00:20:41,830 and then the sum number in the denominator and that number 355 00:20:41,830 --> 00:20:43,610 just depends on the loading configuration. 356 00:20:43,610 --> 00:20:45,510 So for three-point bending, it's 48. 357 00:20:45,510 --> 00:20:49,795 For the flexion of a cantilever, B1 would be 3. 358 00:20:49,795 --> 00:20:51,170 So think of that as just a number 359 00:20:51,170 --> 00:20:53,710 that you can work out for the particular loading 360 00:20:53,710 --> 00:20:55,250 configuration. 361 00:20:55,250 --> 00:21:00,702 So here we'll say B1 is just a constant that depends 362 00:21:00,702 --> 00:21:01,910 on the loading configuration. 363 00:21:13,110 --> 00:21:21,715 And I'll say, for example, for three-point bending, B1 is 48. 364 00:21:32,820 --> 00:21:35,580 For a cantilever end deflection, then B1 would be 3. 365 00:21:35,580 --> 00:21:38,665 So it's just a number. 366 00:21:38,665 --> 00:21:40,290 So the next thing we have to figure out 367 00:21:40,290 --> 00:21:41,753 is what's the EI equivalent. 368 00:21:44,980 --> 00:21:46,825 So if this was just a homogeneous beam, 369 00:21:46,825 --> 00:21:50,280 and it was rectangular, E would just be E of the material 370 00:21:50,280 --> 00:21:54,720 and I would be the width B times the height H 371 00:21:54,720 --> 00:21:57,239 cubed divided by 12. 372 00:21:57,239 --> 00:21:59,030 So here we don't quite have that because we 373 00:21:59,030 --> 00:22:00,740 have two different materials. 374 00:22:00,740 --> 00:22:03,210 So here we have to use something called the parallel axis 375 00:22:03,210 --> 00:22:05,430 theorem, which I'm hoping you may have seen somewhere 376 00:22:05,430 --> 00:22:08,100 in calculus, maybe? 377 00:22:08,100 --> 00:22:10,520 But, yeah, somebody is nodding yes. 378 00:22:10,520 --> 00:22:14,400 OK, so what we do, what we want to do 379 00:22:14,400 --> 00:22:18,130 is get the equivalent EI-- I'm going to put it back up, 380 00:22:18,130 --> 00:22:20,660 don't panic-- of this thing here, right? 381 00:22:20,660 --> 00:22:24,380 So I want-- this is the neutral axis here, 382 00:22:24,380 --> 00:22:28,930 and I want the EI about that neutral axis there. 383 00:22:28,930 --> 00:22:31,190 So, OK, you happy? 384 00:22:31,190 --> 00:22:32,530 There. 385 00:22:32,530 --> 00:22:34,610 OK, so I've got a term for the core. 386 00:22:39,350 --> 00:22:41,660 OK, the core, that is the middle of the core, right? 387 00:22:41,660 --> 00:22:46,360 So for the core, it's just going to be E of the core times 388 00:22:46,360 --> 00:22:48,660 bc cubed over 12. 389 00:22:48,660 --> 00:22:53,360 Remember, for a rectangular section, it's bh cubed over 12 390 00:22:53,360 --> 00:22:54,490 is the moment of inertia. 391 00:22:54,490 --> 00:22:58,290 And here our height for the core is just c, OK? 392 00:22:58,290 --> 00:23:02,130 And then if I took the moment of inertia 393 00:23:02,130 --> 00:23:06,660 for, say, one face about its own centroidal axis, 394 00:23:06,660 --> 00:23:13,130 I would get E of the face now times bt cubed over 12. 395 00:23:13,130 --> 00:23:15,000 So that's taking the moment of inertia 396 00:23:15,000 --> 00:23:18,460 of one face about the middle of the face. 397 00:23:18,460 --> 00:23:20,040 And I have two of those, right? 398 00:23:20,040 --> 00:23:22,110 Because I have two faces. 399 00:23:22,110 --> 00:23:24,810 And the parallel axis theorem tells you 400 00:23:24,810 --> 00:23:27,750 what the moment of inertia is going to be if you move it, 401 00:23:27,750 --> 00:23:34,330 not to the-- you don't use the centroid of the area, 402 00:23:34,330 --> 00:23:37,760 but you use some other parallel axis. 403 00:23:37,760 --> 00:23:42,230 And what that tells you to do is take the area 404 00:23:42,230 --> 00:23:46,050 that you're interested in-- so the area of the face is bt, 405 00:23:46,050 --> 00:23:49,410 and you multiply by the square of the distance between the two 406 00:23:49,410 --> 00:23:52,746 axes that you're interested in. 407 00:23:52,746 --> 00:23:53,245 Oop, yeah. 408 00:23:56,510 --> 00:23:59,088 Let me change my little brackets. 409 00:23:59,088 --> 00:23:59,587 Boop. 410 00:24:02,790 --> 00:24:05,436 So, oop-a-doop-a-doop. 411 00:24:05,436 --> 00:24:07,935 Maybe I'll stick this, make a little sketch over here again. 412 00:24:12,510 --> 00:24:18,250 OK, all right. 413 00:24:18,250 --> 00:24:22,480 So this term here, Ef bt cubed over 12, 414 00:24:22,480 --> 00:24:27,750 that would be the moment of inertia of this piece here, 415 00:24:27,750 --> 00:24:30,500 about the axis that goes through the middle of that, right? 416 00:24:30,500 --> 00:24:32,485 Its own centroidal axis. 417 00:24:32,485 --> 00:24:34,110 But what I want to do is I want to know 418 00:24:34,110 --> 00:24:37,400 what the moment of inertia of this piece is about this axis 419 00:24:37,400 --> 00:24:37,900 here. 420 00:24:37,900 --> 00:24:40,660 This is the neutral axis. 421 00:24:40,660 --> 00:24:42,675 So let's call this the centroidal axis. 422 00:24:47,720 --> 00:24:49,720 And the parallel axis theorem tells me 423 00:24:49,720 --> 00:24:52,600 what I do is I take the area of this little thing 424 00:24:52,600 --> 00:24:55,290 here, so that's the b times t, and I 425 00:24:55,290 --> 00:24:58,290 multiply by the square of the distance between those two 426 00:24:58,290 --> 00:24:59,650 axes. 427 00:24:59,650 --> 00:25:03,870 So the distance between those axes is just c plus t over 2, 428 00:25:03,870 --> 00:25:05,930 and I square it. 429 00:25:05,930 --> 00:25:07,790 And then I multiply that whole thing by 2 430 00:25:07,790 --> 00:25:10,620 because I've got two faces. 431 00:25:10,620 --> 00:25:11,417 Are we good? 432 00:25:11,417 --> 00:25:12,371 AUDIENCE: [INAUDIBLE]. 433 00:25:12,371 --> 00:25:12,848 LORNA GIBSON: Yeah? 434 00:25:12,848 --> 00:25:15,056 AUDIENCE: The center [INAUDIBLE] and the [INAUDIBLE], 435 00:25:15,056 --> 00:25:17,394 are those Ed's or Ef's? 436 00:25:17,394 --> 00:25:19,810 LORNA GIBSON: These are Ef's because this is the face now, 437 00:25:19,810 --> 00:25:21,610 right? 438 00:25:21,610 --> 00:25:24,510 So this term here is for the core. 439 00:25:24,510 --> 00:25:29,380 So here the core is E star c. 440 00:25:29,380 --> 00:25:34,090 And these two Ef's are for the face up there, OK? 441 00:25:34,090 --> 00:25:36,644 Because you have to account for the modulus 442 00:25:36,644 --> 00:25:38,560 of the material of the bit that you're getting 443 00:25:38,560 --> 00:25:40,250 the moment of inertia for. 444 00:25:40,250 --> 00:25:43,287 Are we good? 445 00:25:43,287 --> 00:25:43,786 OK. 446 00:25:59,260 --> 00:26:01,020 So now I'm just going to simplify 447 00:26:01,020 --> 00:26:02,220 these guys a little bit. 448 00:26:09,645 --> 00:26:11,900 Doodle-doodle-doodle-do-doot. 449 00:26:11,900 --> 00:26:14,320 OK? 450 00:26:14,320 --> 00:26:17,010 So I've just multiplied the twos, 451 00:26:17,010 --> 00:26:19,890 and maybe I'll just write down here this is the parallel axis 452 00:26:19,890 --> 00:26:20,389 theorem. 453 00:26:32,050 --> 00:26:32,900 Doot-doot-doot. 454 00:26:32,900 --> 00:26:33,852 Yes, sorry? 455 00:26:33,852 --> 00:26:36,212 AUDIENCE: So for the term that comes from the parallel 456 00:26:36,212 --> 00:26:39,516 axis theorem, why do we only consider Ef and not 457 00:26:39,516 --> 00:26:41,054 [INAUDIBLE]. 458 00:26:41,054 --> 00:26:42,470 LORNA GIBSON: Because I'm taking-- 459 00:26:42,470 --> 00:26:45,581 what I'm looking at-- so the very first term, this guy, 460 00:26:45,581 --> 00:26:46,080 here-- 461 00:26:46,080 --> 00:26:47,246 AUDIENCE: Yeah, [INAUDIBLE]. 462 00:26:47,246 --> 00:26:49,380 LORNA GIBSON: Accounts for this, right? 463 00:26:49,380 --> 00:26:53,506 And these two terms both account for the face. 464 00:26:53,506 --> 00:26:57,100 AUDIENCE: Oh, OK, so the face acting-- 465 00:26:57,100 --> 00:26:58,840 LORNA GIBSON: Yeah, about this axis. 466 00:26:58,840 --> 00:27:01,290 So the parallel axis theorem says 467 00:27:01,290 --> 00:27:04,970 you take the moment of inertia of your area 468 00:27:04,970 --> 00:27:08,000 about its own centroidal axis, and then you 469 00:27:08,000 --> 00:27:09,110 add this term here. 470 00:27:09,110 --> 00:27:13,910 But it's really referring to that face, OK? 471 00:27:13,910 --> 00:27:18,330 Let me scoot that down and then scoot over here. 472 00:27:49,190 --> 00:27:53,860 And this is where we get to say the modulus of the face 473 00:27:53,860 --> 00:27:56,910 is much greater than the modulus of the core. 474 00:27:56,910 --> 00:28:01,730 And also, typically c, the core thickness, 475 00:28:01,730 --> 00:28:05,770 is much greater than t, the face thickness. 476 00:28:05,770 --> 00:28:08,290 So if that's true, then it turns out 477 00:28:08,290 --> 00:28:10,710 this term is small compared to that one. 478 00:28:10,710 --> 00:28:13,440 And also this term is small compared to this one. 479 00:28:13,440 --> 00:28:15,060 And also this term, instead of having 480 00:28:15,060 --> 00:28:18,010 c plus t squared, if c is big compared to t, 481 00:28:18,010 --> 00:28:20,200 then I can just call it c squared, OK? 482 00:28:20,200 --> 00:28:22,480 So you can see here, if Ec is small, 483 00:28:22,480 --> 00:28:24,540 then this is going to be small compared to these. 484 00:28:24,540 --> 00:28:27,360 If t is small, then this guy is going to be small. 485 00:28:27,360 --> 00:28:30,240 So even though it looks ugly, many times 486 00:28:30,240 --> 00:28:32,401 we can make this simpler approximation. 487 00:28:46,010 --> 00:28:47,880 OK, so we can just approximate it 488 00:28:47,880 --> 00:28:51,255 as Ef times btc squared over 2. 489 00:28:59,470 --> 00:29:02,760 So then this bending term here, we've got everything 490 00:29:02,760 --> 00:29:05,482 we need now to get that bit there. 491 00:29:05,482 --> 00:29:07,898 So the next bit we want to get is the shearing deflection. 492 00:29:10,650 --> 00:29:12,940 So what's the shearing deflection equal to? 493 00:29:12,940 --> 00:29:15,370 So say we just thought about the core, 494 00:29:15,370 --> 00:29:17,860 and all we're interested in here is 495 00:29:17,860 --> 00:29:21,290 what's the deflection of the core and shear? 496 00:29:21,290 --> 00:29:30,725 And so say that's P/2, that's P/2, that's l/2. 497 00:29:30,725 --> 00:29:34,740 We'll say that's-- oops. 498 00:29:34,740 --> 00:29:37,990 That's our shearing deflection there. 499 00:29:37,990 --> 00:29:41,100 We can say the shear stress in the core 500 00:29:41,100 --> 00:29:46,200 is going to equal the shear modulus times the shear strain, 501 00:29:46,200 --> 00:29:49,610 so we can say P over the area of the core 502 00:29:49,610 --> 00:29:54,610 is going to be proportional to the Young's modulus times 503 00:29:54,610 --> 00:29:57,350 delta s over l. 504 00:29:57,350 --> 00:30:00,186 And let's not worry about the constant just yet. 505 00:30:03,000 --> 00:30:06,775 So delta s is going to be proportional to-- well, let 506 00:30:06,775 --> 00:30:08,900 me [? make it ?] proportional at this point. 507 00:30:08,900 --> 00:30:17,010 Delta s is going to equal Pl divided by some other constant 508 00:30:17,010 --> 00:30:19,440 that I'm going to call B2, and divided 509 00:30:19,440 --> 00:30:23,120 by the shear modulus of the core, 510 00:30:23,120 --> 00:30:27,810 and essentially the area of the core. 511 00:30:27,810 --> 00:30:29,560 And here B2 is another constant. 512 00:30:39,060 --> 00:30:41,460 So again, B2 just depends on the loading configuration. 513 00:31:00,414 --> 00:31:02,580 Yeah, this is a little bit of an approximation here, 514 00:31:02,580 --> 00:31:05,670 but I'm just going to leave it at that. 515 00:31:05,670 --> 00:31:07,870 OK, so then we have these two terms 516 00:31:07,870 --> 00:31:12,867 and we just add them up to get the final thing. 517 00:31:12,867 --> 00:31:13,700 Start another board. 518 00:31:37,480 --> 00:31:37,980 OK. 519 00:31:44,730 --> 00:31:48,160 So that would give us an equation for the deflection. 520 00:31:48,160 --> 00:31:51,580 And one thing to note here is that this shear modulus 521 00:31:51,580 --> 00:31:54,784 of the core, if the core is a foam, 522 00:31:54,784 --> 00:31:56,200 then we have an equation for that. 523 00:31:56,200 --> 00:32:00,870 We also could use an equation if it's a honeycomb. 524 00:32:00,870 --> 00:32:04,260 But I'm just going to write for foam cores. 525 00:32:04,260 --> 00:32:04,760 Whoops. 526 00:32:20,635 --> 00:32:26,654 This is for-- that will be for open-cell foam cores. 527 00:32:30,740 --> 00:32:36,830 Oops, don't want to-- and get rid of that. 528 00:32:36,830 --> 00:32:38,690 We won't update just now, thank you. 529 00:33:39,250 --> 00:33:41,720 OK, so the next thing I want to think about 530 00:33:41,720 --> 00:33:44,640 is how we would minimize the weight for a given stiffness. 531 00:33:44,640 --> 00:33:46,290 So say if we're given a stiffness, 532 00:33:46,290 --> 00:33:49,460 we're given P over delta, so I could take out the two P's 533 00:33:49,460 --> 00:33:50,090 here. 534 00:33:50,090 --> 00:33:52,260 If I divide it through by P, delta over P 535 00:33:52,260 --> 00:33:55,210 would be the compliance, P over delta would be the stiffness. 536 00:33:55,210 --> 00:33:58,570 So imagine that you're given the face and core materials, 537 00:33:58,570 --> 00:34:01,320 and you're told how long the span has to be, 538 00:34:01,320 --> 00:34:03,600 you're told how wide the beam is going to be, 539 00:34:03,600 --> 00:34:05,350 and you're told the loading configuration. 540 00:34:05,350 --> 00:34:08,179 So you know if it's three-point bending, or four-point bending, 541 00:34:08,179 --> 00:34:10,409 or a cantilever-- whatever it is. 542 00:34:10,409 --> 00:34:13,199 And you might be asked to find the core thickness, the face 543 00:34:13,199 --> 00:34:16,460 thickness, and the core density that would minimize the weight. 544 00:34:16,460 --> 00:34:18,719 So I have a little schematic here. 545 00:34:18,719 --> 00:34:20,719 I don't know if you're going be able to read it. 546 00:34:20,719 --> 00:34:22,385 So I'm going to walk through it and then 547 00:34:22,385 --> 00:34:25,610 I'll write things on the board. 548 00:34:25,610 --> 00:34:27,080 Whoops, hit the wrong button. 549 00:34:34,139 --> 00:34:36,517 OK, so we start with the weight equation here. 550 00:34:36,517 --> 00:34:38,350 The weight's obviously the sum of the weight 551 00:34:38,350 --> 00:34:41,590 of the faces, the weight of the core, so those two terms there. 552 00:34:41,590 --> 00:34:43,469 So I'll write that down in a minute. 553 00:34:43,469 --> 00:34:45,469 And then we have the stiffness constraint here. 554 00:34:45,469 --> 00:34:48,150 So this equation here is just this equation 555 00:34:48,150 --> 00:34:51,270 that I have down here on the board, OK? 556 00:34:51,270 --> 00:34:54,780 Then what you do is you solve that stiffness constraint 557 00:34:54,780 --> 00:34:57,020 for the density of the core. 558 00:34:57,020 --> 00:34:59,800 So this equation here just solves-- 559 00:34:59,800 --> 00:35:02,870 we're solving this equation here in terms of the density, 560 00:35:02,870 --> 00:35:05,980 and we get the density by substituting in this equation 561 00:35:05,980 --> 00:35:08,390 here for the shear modulus of the core. 562 00:35:08,390 --> 00:35:10,360 So you substitute that there. 563 00:35:10,360 --> 00:35:12,890 It's kind of a messy thing, but you solve 564 00:35:12,890 --> 00:35:14,660 that in terms of the density. 565 00:35:14,660 --> 00:35:16,360 Then you put that version of the density 566 00:35:16,360 --> 00:35:19,719 here in terms of this weight equation up here. 567 00:35:19,719 --> 00:35:22,010 So then you've eliminated the density out of the weight 568 00:35:22,010 --> 00:35:23,930 equation, now you've just got it in terms 569 00:35:23,930 --> 00:35:25,680 of the other variables. 570 00:35:25,680 --> 00:35:27,930 And then you take the partial derivative of the weight 571 00:35:27,930 --> 00:35:29,346 with respect to the core thickness 572 00:35:29,346 --> 00:35:32,650 c, set that equal to 0, and you take the partial derivative 573 00:35:32,650 --> 00:35:35,110 of the weight with respect to the face thickness, t, 574 00:35:35,110 --> 00:35:36,830 and you set that equal to 0. 575 00:35:36,830 --> 00:35:39,264 And that then gives you two equations and two unknowns. 576 00:35:39,264 --> 00:35:41,430 You've got the core thickness and the face thickness 577 00:35:41,430 --> 00:35:42,370 are the two unknowns. 578 00:35:42,370 --> 00:35:43,744 And you've got the two equations, 579 00:35:43,744 --> 00:35:44,875 so then you solve those. 580 00:35:44,875 --> 00:35:47,000 So the value you get for the core thickness is then 581 00:35:47,000 --> 00:35:50,240 the optimum, so it's going to be some function of the stiffness, 582 00:35:50,240 --> 00:35:51,770 the material properties you started 583 00:35:51,770 --> 00:35:53,630 with in the beam geometry. 584 00:35:53,630 --> 00:35:56,380 And similarly, you get some equation for the optimum face 585 00:35:56,380 --> 00:35:57,410 thickness, t. 586 00:35:57,410 --> 00:36:00,440 And again, it's a function of the stiffness 587 00:36:00,440 --> 00:36:02,960 and the material properties in the beam geometry. 588 00:36:02,960 --> 00:36:05,230 Then you take those two values for c and t, those two 589 00:36:05,230 --> 00:36:09,010 optimum values, and plug it back into this equation here, 590 00:36:09,010 --> 00:36:12,190 and get the optimum value of the core density. 591 00:36:12,190 --> 00:36:14,040 And so what you end up are three equations 592 00:36:14,040 --> 00:36:16,840 for the optimal values of the core thickness, the face 593 00:36:16,840 --> 00:36:19,090 thickness, and the core density in terms 594 00:36:19,090 --> 00:36:22,050 of the required stiffness, the material properties, and then 595 00:36:22,050 --> 00:36:24,430 the loading geometry. 596 00:36:24,430 --> 00:36:27,330 So I'm going to write down some more notes, because I'll 597 00:36:27,330 --> 00:36:28,650 put this on the Stellar site. 598 00:36:28,650 --> 00:36:30,754 But it's hard to read just here. 599 00:36:30,754 --> 00:36:33,420 So let me write it down and I'll also write out the equations so 600 00:36:33,420 --> 00:36:36,020 that you have the equations for calculating 601 00:36:36,020 --> 00:36:38,200 those optimum values. 602 00:36:38,200 --> 00:36:40,664 So before I do that, though, one of the interesting things 603 00:36:40,664 --> 00:36:42,580 though is if you figure out the optimal values 604 00:36:42,580 --> 00:36:43,996 of the core thickness and the face 605 00:36:43,996 --> 00:36:47,290 thickness and the core density, and you substitute it back 606 00:36:47,290 --> 00:36:49,222 into the weight, and you calculate 607 00:36:49,222 --> 00:36:51,430 this is the weight of the face relative to the weight 608 00:36:51,430 --> 00:36:54,730 of the core, no matter what the geometry is, 609 00:36:54,730 --> 00:36:56,500 and what the loading configuration is, 610 00:36:56,500 --> 00:36:58,270 the weight of the face is always a quarter 611 00:36:58,270 --> 00:36:59,640 of the weight of the core. 612 00:36:59,640 --> 00:37:01,485 So the ratio of how much material 613 00:37:01,485 --> 00:37:03,440 is in the core and the face is constant, 614 00:37:03,440 --> 00:37:07,830 regardless of the core-- of the loading configuration. 615 00:37:07,830 --> 00:37:09,580 And this is the bending deflection 616 00:37:09,580 --> 00:37:11,980 relative to the total deflection. 617 00:37:11,980 --> 00:37:13,160 It's always 1/3. 618 00:37:13,160 --> 00:37:14,660 And the shearing deflection relative 619 00:37:14,660 --> 00:37:17,040 to the total deflection is always 2/3. 620 00:37:17,040 --> 00:37:20,980 So regardless of how you set things up, 621 00:37:20,980 --> 00:37:23,150 the ratio of what weight the face 622 00:37:23,150 --> 00:37:27,500 is relative to the core and the amount of shearing and bending 623 00:37:27,500 --> 00:37:31,650 deflections is always a constant at the optimum. 624 00:37:31,650 --> 00:37:36,715 OK, so let's say we're given the face and the core materials. 625 00:37:40,500 --> 00:37:44,390 So that means we're given their material property, too. 626 00:37:44,390 --> 00:37:53,900 And say we're given the beam length and width 627 00:37:53,900 --> 00:37:55,375 and the loading configuration. 628 00:38:03,410 --> 00:38:07,490 So that means we're given those constants, B1 and B2. 629 00:38:07,490 --> 00:38:09,260 If I told you it was three-point bending, 630 00:38:09,260 --> 00:38:11,020 you would know what B1 and B2 are. 631 00:38:18,630 --> 00:38:22,570 So then what you need to do is find the core thickness, c, 632 00:38:22,570 --> 00:38:36,270 the face thickness, t, and the core density, rho c, 633 00:38:36,270 --> 00:38:39,732 to minimize the weight of the beam. 634 00:38:46,550 --> 00:38:49,195 So there's two faces, so the weight of the face 635 00:38:49,195 --> 00:38:51,900 is 2 rho f g times btl. 636 00:38:55,141 --> 00:39:01,442 And then the weight of the core is rho c g times bcl. 637 00:39:05,231 --> 00:39:06,730 So I'm going to write down the steps 638 00:39:06,730 --> 00:39:08,313 and then I'll write down the solution. 639 00:39:32,120 --> 00:39:34,230 So you solve. 640 00:39:34,230 --> 00:39:37,470 So you put this equation for the shear modulus of the core 641 00:39:37,470 --> 00:39:40,320 into here, and then you rearrange 642 00:39:40,320 --> 00:39:45,750 this equation in terms of the density of the core here. 643 00:39:45,750 --> 00:39:48,150 So you have an equation for the core density in terms 644 00:39:48,150 --> 00:40:02,080 of that stiffness, and then you solve the partial derivatives 645 00:40:02,080 --> 00:40:04,070 of the weight equation with respect 646 00:40:04,070 --> 00:40:06,615 to the core thickness, c, and put that equal to 0. 647 00:40:10,415 --> 00:40:13,420 And then the partial of dw [? over ?] dt 648 00:40:13,420 --> 00:40:14,670 and set that to 0. 649 00:40:26,247 --> 00:40:27,830 And if you do that, you can then solve 650 00:40:27,830 --> 00:40:30,550 for the optimal values of the face and core thicknesses. 651 00:40:30,550 --> 00:40:31,050 Yes? 652 00:40:31,050 --> 00:40:33,894 AUDIENCE: [INAUDIBLE] for weight, what is g? 653 00:40:33,894 --> 00:40:34,810 LORNA GIBSON: Gravity. 654 00:40:34,810 --> 00:40:35,362 AUDIENCE: OK. 655 00:40:35,362 --> 00:40:37,695 LORNA GIBSON: Just density is mass, mass times gravity-- 656 00:40:37,695 --> 00:40:38,170 weight. 657 00:40:38,170 --> 00:40:38,878 That's all it is. 658 00:41:14,680 --> 00:41:17,200 And then you've got a version of this that's 659 00:41:17,200 --> 00:41:20,100 in terms of the core density. 660 00:41:20,100 --> 00:41:23,030 You can substitute those values of the optimum face 661 00:41:23,030 --> 00:41:24,980 and core thicknesses into that equation 662 00:41:24,980 --> 00:41:26,505 and get the optimum core density. 663 00:41:31,740 --> 00:41:38,016 And then in the final equations, you get, when you do all that, 664 00:41:38,016 --> 00:41:39,890 and I'm going to make them all dimensionless, 665 00:41:39,890 --> 00:41:42,910 so this is the core thickness normalized 666 00:41:42,910 --> 00:41:49,265 by the span of the beam is equal to this thing, here. 667 00:45:11,850 --> 00:45:14,770 So you can see each of these parameters here, 668 00:45:14,770 --> 00:45:17,950 the design parameters that we're calculating the optimum of. 669 00:45:17,950 --> 00:45:21,400 I've grouped the constants B1 and B2 together 670 00:45:21,400 --> 00:45:23,120 that describe the loading configuration 671 00:45:23,120 --> 00:45:24,490 so you'd be given those. 672 00:45:24,490 --> 00:45:26,710 C2 is this constant-- oop, which I just 673 00:45:26,710 --> 00:45:31,070 rubbed off-- that relates the shear modulus of the foam core. 674 00:45:31,070 --> 00:45:32,290 So you'd be given that. 675 00:45:32,290 --> 00:45:34,970 These are the material properties of the-- you 676 00:45:34,970 --> 00:45:37,054 know, say, it's a polyurethane foam core, 677 00:45:37,054 --> 00:45:38,970 this would be the density of the polyurethane. 678 00:45:38,970 --> 00:45:40,428 Say it's aluminum faces, that would 679 00:45:40,428 --> 00:45:41,770 be the density of the aluminum. 680 00:45:41,770 --> 00:45:42,728 so you'd be given that. 681 00:45:42,728 --> 00:45:47,820 You'd be given the stiffnesses of the two 682 00:45:47,820 --> 00:45:51,850 materials, the solid from which the core is made and the face 683 00:45:51,850 --> 00:45:52,610 material. 684 00:45:52,610 --> 00:45:55,020 And then this is the stiffness here that you're given, 685 00:45:55,020 --> 00:45:57,489 just divided by the width of the beam, B. So the stiffness, 686 00:45:57,489 --> 00:46:00,030 you'd be given the width B. So you're given all those things, 687 00:46:00,030 --> 00:46:04,910 then you could calculate what that optimum design would be. 688 00:46:04,910 --> 00:46:08,490 So the next slide here just shows some experiments. 689 00:46:08,490 --> 00:46:10,300 And these were done on sandwiches 690 00:46:10,300 --> 00:46:14,250 with aluminum faces and a rigid polyurethane foam core. 691 00:46:14,250 --> 00:46:16,000 And here we knew what the relationship 692 00:46:16,000 --> 00:46:17,150 was for the shear modulus. 693 00:46:17,150 --> 00:46:18,800 We measured that. 694 00:46:18,800 --> 00:46:21,170 And what we did here was we designed the beams 695 00:46:21,170 --> 00:46:24,010 to all have the same stiffness, and they all 696 00:46:24,010 --> 00:46:27,130 had the same span in the width, B, then we 697 00:46:27,130 --> 00:46:29,702 kept one parameter at the optimum value 698 00:46:29,702 --> 00:46:30,910 and we varied the other ones. 699 00:46:30,910 --> 00:46:33,770 So here, on this beam, this set of beams here, 700 00:46:33,770 --> 00:46:35,720 the density was at the optimum. 701 00:46:35,720 --> 00:46:37,190 And we varied the core thickness, 702 00:46:37,190 --> 00:46:38,890 and we varied the face thickness, 703 00:46:38,890 --> 00:46:43,560 and the solid line was our model or our sort of optimization. 704 00:46:43,560 --> 00:46:45,925 And the little X's were the experiments. 705 00:46:45,925 --> 00:46:48,144 So you can see there's pretty good agreement there. 706 00:46:48,144 --> 00:46:50,310 Then the second set here, we kept the face thickness 707 00:46:50,310 --> 00:46:53,560 at the optimum value and we varied the core thickness, 708 00:46:53,560 --> 00:46:54,860 we varied the core density. 709 00:46:54,860 --> 00:46:59,600 So the same thing, the solid line is the sort of theory 710 00:46:59,600 --> 00:47:01,440 and the X's are the experiments. 711 00:47:01,440 --> 00:47:03,840 And here we had the core thickness of the optimum value, 712 00:47:03,840 --> 00:47:07,460 and we varied the face thickness and the core density. 713 00:47:07,460 --> 00:47:11,370 So you can kind of see how you can see this here. 714 00:47:11,370 --> 00:47:13,960 And over here, just because I forgot to say it, 715 00:47:13,960 --> 00:47:16,860 this is the stiffness per unit weight, over here, OK? 716 00:47:16,860 --> 00:47:20,280 So these are the optimum designs here, all right? 717 00:47:20,280 --> 00:47:21,940 So there was pretty good agreement 718 00:47:21,940 --> 00:47:29,095 between these calculations and what we measured on some beams. 719 00:47:29,095 --> 00:47:30,470 Do I need to write anything down? 720 00:47:30,470 --> 00:47:31,980 Do you think you've got that? 721 00:47:31,980 --> 00:47:32,160 Yeah? 722 00:47:32,160 --> 00:47:34,535 AUDIENCE: I was just going to ask, for the optimum design 723 00:47:34,535 --> 00:47:36,470 column that you have there, do those numbers 724 00:47:36,470 --> 00:47:38,480 like fall out of these equations if you do the math? 725 00:47:38,480 --> 00:47:39,646 LORNA GIBSON: They do, yeah. 726 00:47:39,646 --> 00:47:41,860 I mean, it's-- yeah, exactly. 727 00:47:41,860 --> 00:47:45,600 So if you remember the equation we had for the weight, 728 00:47:45,600 --> 00:47:53,443 so the weight is equal to 2 rho f gbtl plus the density 729 00:47:53,443 --> 00:48:00,662 of the core, bcl, so if you plug these things into there, then-- 730 00:48:00,662 --> 00:48:03,120 so this is the way to the face, that's the way to the core, 731 00:48:03,120 --> 00:48:05,280 then it drops out to be a quarter. 732 00:48:05,280 --> 00:48:06,280 So it's kind of magical. 733 00:48:06,280 --> 00:48:08,490 I mean, you have this big, long, complicated gory thing, 734 00:48:08,490 --> 00:48:10,031 and then, poof, everything disappears 735 00:48:10,031 --> 00:48:12,220 except a factor of 1/4. 736 00:48:12,220 --> 00:48:14,660 And the same for the bending deflection. 737 00:48:14,660 --> 00:48:16,290 So we had those two terms, so there 738 00:48:16,290 --> 00:48:17,960 was the bending and the shear. 739 00:48:17,960 --> 00:48:20,380 If you just calculate each of those terms 740 00:48:20,380 --> 00:48:22,020 and take the ratio of 1 over the total, 741 00:48:22,020 --> 00:48:24,230 or the one over the other, everything 742 00:48:24,230 --> 00:48:26,359 drops out except that number. 743 00:48:26,359 --> 00:48:28,400 So that's why I pointed it out, because it seemed 744 00:48:28,400 --> 00:48:30,316 kind of amazing that everything would drop out 745 00:48:30,316 --> 00:48:32,190 except for that one thing. 746 00:48:32,190 --> 00:48:34,300 OK, so then the next thing-- so that's 747 00:48:34,300 --> 00:48:36,510 the stiffness in optimizing the stiffness. 748 00:48:36,510 --> 00:48:38,101 Are we happy-ish? 749 00:48:38,101 --> 00:48:38,600 Yeah? 750 00:48:38,600 --> 00:48:39,440 OK. 751 00:48:39,440 --> 00:48:42,574 So the next thing-- oh, well, let's see. 752 00:48:42,574 --> 00:48:43,990 I don't think I need to write any. 753 00:48:43,990 --> 00:48:45,906 I think if you have that graph, I don't really 754 00:48:45,906 --> 00:48:47,690 need to write much down. 755 00:48:47,690 --> 00:48:50,860 So the next thing then is the strength of the sandwich beams. 756 00:48:50,860 --> 00:48:53,280 So let me get rid of that. 757 00:49:21,620 --> 00:49:23,600 You guys OK? 758 00:49:23,600 --> 00:49:24,490 Yeah? 759 00:49:24,490 --> 00:49:25,360 AUDIENCE: Yeah. 760 00:49:25,360 --> 00:49:27,776 LORNA GIBSON: Yeah, but you're shaking your head like this 761 00:49:27,776 --> 00:49:29,230 is very, very helpful for me. 762 00:49:29,230 --> 00:49:30,466 AUDIENCE: [INAUDIBLE]. 763 00:49:30,466 --> 00:49:32,076 LORNA GIBSON: Oh OK, that's OK. 764 00:49:32,076 --> 00:49:33,532 AUDIENCE: [INAUDIBLE]. 765 00:49:33,532 --> 00:49:35,240 LORNA GIBSON: That's OK, you can do that. 766 00:49:35,240 --> 00:49:35,781 I don't mind. 767 00:49:35,781 --> 00:49:38,830 But as long as you don't have questions for me. 768 00:49:38,830 --> 00:49:40,540 OK, and so the first step in trying 769 00:49:40,540 --> 00:49:42,041 to figure out about this strength 770 00:49:42,041 --> 00:49:44,165 is we need to figure out the stresses in the beams. 771 00:49:55,040 --> 00:49:57,680 So we need to find out about the stresses. 772 00:49:57,680 --> 00:49:59,950 And we're going to have normal stresses 773 00:49:59,950 --> 00:50:03,257 and we're going to have shear stresses. 774 00:50:03,257 --> 00:50:05,090 So I'm going to do the normal stresses first 775 00:50:05,090 --> 00:50:07,012 and then we'll do the shear stresses. 776 00:50:07,012 --> 00:50:08,470 So you do this in a way that's just 777 00:50:08,470 --> 00:50:10,303 analogous to how you figure out the stresses 778 00:50:10,303 --> 00:50:12,260 in a homogeneous beam. 779 00:50:12,260 --> 00:50:14,900 So we'll say the stresses in the face-- normally it 780 00:50:14,900 --> 00:50:17,144 would be My over I. M is the moment, 781 00:50:17,144 --> 00:50:18,810 y is the distance from the neutral axis, 782 00:50:18,810 --> 00:50:20,680 I is the moment of inertia. 783 00:50:20,680 --> 00:50:24,610 So this time, instead of having a moment of inertia, 784 00:50:24,610 --> 00:50:28,810 we have this equivalent moment of inertia. 785 00:50:28,810 --> 00:50:32,740 And we multiply by E of the face. 786 00:50:32,740 --> 00:50:35,575 So you can think of this as being the strain essentially. 787 00:50:35,575 --> 00:50:37,950 And then you multiply by E of the face to get the stress. 788 00:50:43,420 --> 00:50:48,880 The maximum distance from the neutral axis, we can call c/2. 789 00:50:48,880 --> 00:50:50,340 So that's y. 790 00:50:50,340 --> 00:50:57,840 Then EI equivalent we had Ef btc squared over 2. 791 00:50:57,840 --> 00:51:02,100 And then I have a term of Ef here. 792 00:51:02,100 --> 00:51:04,050 c squared. 793 00:51:04,050 --> 00:51:08,730 So one of the c's goes, the 2's go, the Ef's go. 794 00:51:11,745 --> 00:51:15,610 Then you just get that the normal stress in the face 795 00:51:15,610 --> 00:51:17,660 is the moment at that section divided 796 00:51:17,660 --> 00:51:19,970 by the width, b, the face thickness, t, 797 00:51:19,970 --> 00:51:21,454 the core thickness, c. 798 00:51:24,814 --> 00:51:26,530 And I can do the same kind of thing 799 00:51:26,530 --> 00:51:31,200 for the stress in the core, except now 800 00:51:31,200 --> 00:51:33,190 I multiply by the core modulus. 801 00:51:37,270 --> 00:51:55,760 So if I go through the same kind of thing, 802 00:51:55,760 --> 00:51:58,180 it's the same factor of M over btc, 803 00:51:58,180 --> 00:52:04,350 but now I multiply times E of core over E of the face. 804 00:52:04,350 --> 00:52:07,740 And since E of the core is a lot smaller than E of the face, 805 00:52:07,740 --> 00:52:09,890 typically these normal stresses in the core 806 00:52:09,890 --> 00:52:12,619 are much smaller than the normal stresses in the face. 807 00:52:34,260 --> 00:52:37,935 So the faces carry almost all of the normal stresses. 808 00:52:42,820 --> 00:52:45,380 And if you look at an I-beam, the flanges of the I-beam 809 00:52:45,380 --> 00:52:47,090 carry almost the normal stresses. 810 00:52:57,881 --> 00:52:59,380 So I want to do one more thing here. 811 00:52:59,380 --> 00:53:02,740 I want to relate the moment to some concentrated load. 812 00:53:02,740 --> 00:53:16,930 So let's say we have a beam with a concentrated load, P. 813 00:53:16,930 --> 00:53:20,795 So for example, something in three-point bending, 814 00:53:20,795 --> 00:53:22,920 typically we're interested in the maximum stresses, 815 00:53:22,920 --> 00:53:24,790 so we want the maximum moment. 816 00:53:24,790 --> 00:53:30,230 So M max is going to be P times l over some number. 817 00:53:30,230 --> 00:53:33,697 And this B3 is another constant that depends on the loading 818 00:53:33,697 --> 00:53:34,280 configuration. 819 00:53:57,380 --> 00:54:00,000 So if it was three-point bending, B3 would be 4. 820 00:54:00,000 --> 00:54:02,030 If it was a cantilever, B3 would be 1. 821 00:54:11,360 --> 00:54:13,630 So if I put those things together, 822 00:54:13,630 --> 00:54:22,310 the normal stress in the face is Pl B3 divided by btc. 823 00:54:36,440 --> 00:54:39,004 OK, so that's the normal stresses. 824 00:54:39,004 --> 00:54:40,920 And then the next thing is the shear stresses, 825 00:54:40,920 --> 00:54:43,169 and the shear stresses are going to be carried largely 826 00:54:43,169 --> 00:54:44,020 by the core. 827 00:54:44,020 --> 00:54:46,150 And if you do all the exact calculations, 828 00:54:46,150 --> 00:54:48,570 they vary parabolically through the core. 829 00:54:48,570 --> 00:54:50,550 But if we make those same approximations 830 00:54:50,550 --> 00:54:52,690 that the face is stiff compared to the core, 831 00:54:52,690 --> 00:54:54,812 and that the face is thin compared to the core, 832 00:54:54,812 --> 00:54:56,520 then you can say that the shear stress is 833 00:54:56,520 --> 00:54:57,920 just constant through the core. 834 00:55:02,810 --> 00:55:12,682 So we'll say the shear stresses vary parabolically 835 00:55:12,682 --> 00:55:13,390 through the core. 836 00:55:25,930 --> 00:55:27,820 But if the face is much stiffer than the core 837 00:55:27,820 --> 00:55:30,960 and the core is much thicker than the face, 838 00:55:30,960 --> 00:55:34,390 then you can say that the shear stress in the core 839 00:55:34,390 --> 00:55:39,130 is just equal to the sheer force over the area of the core, bc. 840 00:55:39,130 --> 00:55:42,595 So here, V is the shear force of the cross-section 841 00:55:42,595 --> 00:55:43,470 you're interested in. 842 00:55:48,695 --> 00:55:53,235 And bc is just the area of the core. 843 00:55:53,235 --> 00:55:56,270 And we could say the maximum shear force is just 844 00:55:56,270 --> 00:56:01,180 going to be V over-- actually, let's make it 845 00:56:01,180 --> 00:56:06,210 P, P over yet another constant. 846 00:56:06,210 --> 00:56:09,209 And B4 also depends on the loading configuration. 847 00:56:20,136 --> 00:56:21,510 So if I was giving you a problem, 848 00:56:21,510 --> 00:56:24,695 I would give you all these B1, B2, B3, B4's and everything. 849 00:56:36,670 --> 00:56:39,210 So the maximum shear stress in the core 850 00:56:39,210 --> 00:56:42,890 is in just the applied load, P, divided by this B4 851 00:56:42,890 --> 00:56:44,615 and divided by the area of the core. 852 00:57:25,090 --> 00:57:28,650 OK, so this next figure up here just shows 853 00:57:28,650 --> 00:57:29,840 those stress distributions. 854 00:57:29,840 --> 00:57:33,652 So here's a piece of the cross-section here. 855 00:57:33,652 --> 00:57:35,860 So there's the face thickness and the core thickness. 856 00:57:35,860 --> 00:57:39,080 You can think of that as a piece along the length, if you want. 857 00:57:39,080 --> 00:57:41,950 This is the normal stress distribution, here. 858 00:57:41,950 --> 00:57:44,110 So this is all really from saying 859 00:57:44,110 --> 00:57:46,127 plane sections remain plane. 860 00:57:46,127 --> 00:57:48,460 These are the stresses, the normal stresses in the core. 861 00:57:48,460 --> 00:57:50,040 And you can see they're a lot smaller 862 00:57:50,040 --> 00:57:53,640 in this schematic than the ones in the face. 863 00:57:53,640 --> 00:57:58,070 And then this is the parabolic stress in the core. 864 00:57:58,070 --> 00:58:02,160 And similarly, there'd be a different parabola in the face. 865 00:58:02,160 --> 00:58:03,890 And these are the approximations. 866 00:58:03,890 --> 00:58:05,500 Typically these approximations are 867 00:58:05,500 --> 00:58:07,430 made so the normal stress in the face 868 00:58:07,430 --> 00:58:08,930 is just taken as a constant. 869 00:58:08,930 --> 00:58:12,630 The normal stress in the core is often neglected. 870 00:58:12,630 --> 00:58:15,900 And here the shear stress in the core is just a constant here. 871 00:58:15,900 --> 00:58:18,150 So the two things you need to worry about 872 00:58:18,150 --> 00:58:20,260 are the normal stress for the face 873 00:58:20,260 --> 00:58:23,630 and the shear stress for the core. 874 00:58:23,630 --> 00:58:24,750 Are we good? 875 00:58:24,750 --> 00:58:25,400 We're good? 876 00:58:25,400 --> 00:58:27,600 Yeah, good-ish. 877 00:58:27,600 --> 00:58:30,430 OK, so if we want to talk about the strength of the beam, 878 00:58:30,430 --> 00:58:33,690 we now have to talk about different failure modes. 879 00:58:33,690 --> 00:58:35,950 And the next slide just shows some schematics 880 00:58:35,950 --> 00:58:37,340 of the failure modes. 881 00:58:37,340 --> 00:58:39,420 So there's different ways the beam can fail. 882 00:58:39,420 --> 00:58:40,920 Say it's in three-point bending just 883 00:58:40,920 --> 00:58:42,730 for the sake of convenience. 884 00:58:42,730 --> 00:58:45,980 One way it can fail is, say it had aluminum faces. 885 00:58:45,980 --> 00:58:47,800 This face here would be in tension, 886 00:58:47,800 --> 00:58:49,202 and the face could just yield. 887 00:58:49,202 --> 00:58:51,160 So you could just get yielding of the aluminum. 888 00:58:51,160 --> 00:58:52,360 That would be one way. 889 00:58:52,360 --> 00:58:53,860 It could be a composite face and you 890 00:58:53,860 --> 00:58:56,550 could have some sort of composite failure mode. 891 00:58:56,550 --> 00:58:59,586 You can get more complicated failure modes for composites, 892 00:58:59,586 --> 00:59:01,460 but there could be some sort of failure mode. 893 00:59:01,460 --> 00:59:04,160 This face up here is in compression, 894 00:59:04,160 --> 00:59:07,234 and if you compress that face, you 895 00:59:07,234 --> 00:59:08,900 can get something called face wrinkling. 896 00:59:08,900 --> 00:59:10,710 You get sort of a local buckling mode. 897 00:59:10,710 --> 00:59:13,900 So imagine you have the face, that you're pressing on it, 898 00:59:13,900 --> 00:59:16,730 but the core is kind of acting like an elastic foundation 899 00:59:16,730 --> 00:59:17,880 underneath it. 900 00:59:17,880 --> 00:59:20,060 And you can get this kind of local buckling, 901 00:59:20,060 --> 00:59:22,240 and that's called wrinkling. 902 00:59:22,240 --> 00:59:24,080 That's another mode of failure. 903 00:59:24,080 --> 00:59:26,100 You can also get the core failing in shear. 904 00:59:26,100 --> 00:59:30,310 So here's these two little cracks, denoting shear failure 905 00:59:30,310 --> 00:59:31,720 in the core. 906 00:59:31,720 --> 00:59:33,750 And there's a couple of other modes you can get, 907 00:59:33,750 --> 00:59:36,350 but we're going to not pay much attention to those. 908 00:59:36,350 --> 00:59:39,590 The whole thing can delaminate, and, as you might guess, 909 00:59:39,590 --> 00:59:42,640 if the whole thing delaminates, you're in deep doo-doo. 910 00:59:42,640 --> 00:59:45,670 Because, remember when I passed those samples around, 911 00:59:45,670 --> 00:59:48,400 how flexible the face was by itself and how 912 00:59:48,400 --> 00:59:49,970 flexible the core is by itself. 913 00:59:49,970 --> 00:59:51,512 If the whole thing delaminates, you 914 00:59:51,512 --> 00:59:53,636 lose that whole sandwich effect and the whole thing 915 00:59:53,636 --> 00:59:55,170 kind of falls apart. 916 00:59:55,170 --> 00:59:57,030 We're going to assume we have a perfect bond 917 00:59:57,030 --> 00:59:58,870 and that we don't have to worry about that. 918 00:59:58,870 --> 01:00:00,710 The other sort of failure mode you can get 919 01:00:00,710 --> 01:00:02,380 is called indentation. 920 01:00:02,380 --> 01:00:04,260 So imagine that you apply this load here 921 01:00:04,260 --> 01:00:06,120 over a very small area. 922 01:00:06,120 --> 01:00:07,580 The load can just transfer straight 923 01:00:07,580 --> 01:00:09,246 through the face and just kind of indent 924 01:00:09,246 --> 01:00:10,362 the core underneath it. 925 01:00:10,362 --> 01:00:12,070 We're going to assume that you distribute 926 01:00:12,070 --> 01:00:14,570 this load over a big enough area here, 927 01:00:14,570 --> 01:00:15,989 that you don't indent the core. 928 01:00:15,989 --> 01:00:18,280 So we're going to worry about these three failure modes 929 01:00:18,280 --> 01:00:21,180 here-- the face yielding, the face wrinkling, 930 01:00:21,180 --> 01:00:24,270 and the core failing and shear, OK? 931 01:00:24,270 --> 01:00:26,258 So let me just write that down. 932 01:01:17,654 --> 01:01:25,460 And then you also can have debonding or delamination, 933 01:01:25,460 --> 01:01:27,375 and we're going to assume perfect bond. 934 01:01:36,240 --> 01:01:38,335 And then you can have indentation, 935 01:01:38,335 --> 01:01:39,960 and we're going to assume the loads are 936 01:01:39,960 --> 01:01:42,168 applied over a large enough area that you don't get-- 937 01:02:20,560 --> 01:02:24,170 So you can have different modes of failure, 938 01:02:24,170 --> 01:02:28,420 and the question becomes which mode is going to be dominant? 939 01:02:28,420 --> 01:02:30,240 So whichever one occurs at the lowest load 940 01:02:30,240 --> 01:02:31,948 is going to be the dominant failure mode. 941 01:02:31,948 --> 01:02:35,510 So you'd like to know what that lowest failure mode is. 942 01:02:35,510 --> 01:02:40,630 So we want to write equations for each of these failure modes 943 01:02:40,630 --> 01:02:43,423 and then figure out which one occurs first. 944 01:02:49,970 --> 01:02:52,350 So we'll look at the face yielding here. 945 01:02:52,350 --> 01:02:54,740 And face yielding is going to occur just 946 01:02:54,740 --> 01:02:56,322 when the normal stress in the face 947 01:02:56,322 --> 01:02:58,030 is equal to the yield stress of the face. 948 01:02:58,030 --> 01:02:59,805 So this is fairly straightforward. 949 01:03:08,620 --> 01:03:12,840 So this was our equation for the stress in the face. 950 01:03:12,840 --> 01:03:16,000 And when that's equal to the face yield strength, 951 01:03:16,000 --> 01:03:17,070 then you'll get failure. 952 01:03:25,544 --> 01:03:31,680 And the face wrinkling occurs when the normal compressive 953 01:03:31,680 --> 01:03:33,890 stress in the face equals a local buckling stress. 954 01:03:57,071 --> 01:03:58,820 And people have worked that out by looking 955 01:03:58,820 --> 01:04:02,100 at what's called buckling on an elastic foundation. 956 01:04:02,100 --> 01:04:04,300 So the core acts as elastic support. 957 01:04:04,300 --> 01:04:05,829 You can think that as the face is 958 01:04:05,829 --> 01:04:07,495 trying to buckle into the core, the core 959 01:04:07,495 --> 01:04:09,500 is pushing back on the face. 960 01:04:09,500 --> 01:04:13,540 And so the core is acting like a spring that pushes back, 961 01:04:13,540 --> 01:04:16,210 and that's called an elastic foundation. 962 01:04:16,210 --> 01:04:21,410 So people have calculated this local buckling stress, 963 01:04:21,410 --> 01:04:25,090 and they found that's equal to 0.57 964 01:04:25,090 --> 01:04:30,200 times the modulus of the face to the 1/3 power 965 01:04:30,200 --> 01:04:33,670 times the modulus of the core to the 2/3 power. 966 01:04:51,990 --> 01:04:58,650 And here, if we use our model for open cell foams, 967 01:04:58,650 --> 01:05:02,580 we can say the core modulus goes as the relative density squared 968 01:05:02,580 --> 01:05:04,600 times the solid modulus. 969 01:05:04,600 --> 01:05:06,080 And so you can plug that in there. 970 01:05:48,780 --> 01:05:50,200 So then the wrinkling occurs when 971 01:05:50,200 --> 01:05:54,310 the stress in the face, the Pl over the B3 btc 972 01:05:54,310 --> 01:05:55,510 is equal to this thing here. 973 01:06:50,650 --> 01:06:52,310 OK, so one more failure mode that's 974 01:06:52,310 --> 01:07:00,920 the core shear, and that's going to occur when the shear stress 975 01:07:00,920 --> 01:07:06,280 in the core is just equal to the sheer strength of the core. 976 01:07:06,280 --> 01:07:12,260 So the shear stress is P over B4 times bc, 977 01:07:12,260 --> 01:07:17,360 and the shear strength is some constant, I think it's C11, 978 01:07:17,360 --> 01:07:21,300 times the relative density of the core to the three halves 979 01:07:21,300 --> 01:07:24,740 power times the yield strength of the solid. 980 01:07:24,740 --> 01:07:29,367 And here, this constant is about equal to 0.15, 981 01:07:29,367 --> 01:07:30,200 something like that. 982 01:07:39,310 --> 01:07:40,730 So now we have a set of equations 983 01:07:40,730 --> 01:07:42,930 for the different failure modes, and we 984 01:07:42,930 --> 01:07:45,550 could solve each of them, not in terms of a stress, 985 01:07:45,550 --> 01:07:47,660 but in terms of a load P. The load P is what's 986 01:07:47,660 --> 01:07:49,380 applied to the beam, right? 987 01:07:49,380 --> 01:07:52,150 So we could solve each of these in terms of the load, P. 988 01:07:52,150 --> 01:07:55,049 And then we can see which one occurs at the lowest load, P. 989 01:07:55,049 --> 01:07:57,090 And that's going to be the dominant failure mode. 990 01:07:57,090 --> 01:07:59,150 So one way to do it would be to, for every time 991 01:07:59,150 --> 01:08:02,110 you wanted to do this, to work out all these three equations 992 01:08:02,110 --> 01:08:03,990 and figure out which one's the lowest load. 993 01:08:03,990 --> 01:08:06,800 But there's actually something called a failure mode 994 01:08:06,800 --> 01:08:08,950 map, which we're going to talk about. 995 01:08:08,950 --> 01:08:12,534 So let me just show you it and we'll start now. 996 01:08:12,534 --> 01:08:14,200 I don't know if we'll get finished this. 997 01:08:14,200 --> 01:08:18,020 But there's a way that you can manipulate these equations 998 01:08:18,020 --> 01:08:21,740 and plot the results as this failure mode map. 999 01:08:21,740 --> 01:08:24,750 And you'll end up plotting the core density on this plot, 1000 01:08:24,750 --> 01:08:29,180 on this axis here, and the face thickness to span ratio here, 1001 01:08:29,180 --> 01:08:31,550 and so this will kind of tell you, 1002 01:08:31,550 --> 01:08:35,446 for different configurations of the beam, different designs, 1003 01:08:35,446 --> 01:08:37,529 for these ones here, the face is going to wrinkle, 1004 01:08:37,529 --> 01:08:39,570 for those ones there, the face is going to yield, 1005 01:08:39,570 --> 01:08:41,750 and for these ones here, the core is going to shear. 1006 01:08:41,750 --> 01:08:43,710 So I'm going to work through these equations, 1007 01:08:43,710 --> 01:08:45,770 but I don't think we're going to finish it today. 1008 01:08:45,770 --> 01:08:47,602 So this is just kind of where we're headed 1009 01:08:47,602 --> 01:08:48,560 is to getting this map. 1010 01:08:53,370 --> 01:08:57,460 So we'll say the dominant failure mode is the one that 1011 01:08:57,460 --> 01:08:58,890 occurs at the lowest load. 1012 01:09:21,369 --> 01:09:22,910 So the question we're going to answer 1013 01:09:22,910 --> 01:09:25,479 is how does the failure mode depend on the beam design? 1014 01:09:35,620 --> 01:09:37,330 And we're going to do this by looking 1015 01:09:37,330 --> 01:09:40,310 at the transition from one failure mode to another. 1016 01:09:59,540 --> 01:10:01,900 So at the transition from one mode to another, 1017 01:10:01,900 --> 01:10:03,980 the two modes occur at the same load. 1018 01:10:26,880 --> 01:10:28,380 So I'm going to take those equations 1019 01:10:28,380 --> 01:10:32,076 I had for each of the failure modes, 1020 01:10:32,076 --> 01:10:33,950 and instead of writing this in terms of, say, 1021 01:10:33,950 --> 01:10:35,825 the stress in the face, I'm going to write it 1022 01:10:35,825 --> 01:10:49,540 in terms of the load, P. So using 1023 01:10:49,540 --> 01:10:53,150 that first one over there, the load for face yielding, 1024 01:10:53,150 --> 01:10:55,030 I'm just rearranging that. 1025 01:10:55,030 --> 01:11:03,820 It's B3 times bc times t/l times the yield strength of the face. 1026 01:11:12,520 --> 01:11:15,280 And similarly for face wrinkling, 1027 01:11:15,280 --> 01:11:18,880 I can take this equation down here and solve it for this P 1028 01:11:18,880 --> 01:11:20,158 here, OK? 1029 01:11:50,970 --> 01:11:53,130 And then I can take that equation at the top 1030 01:11:53,130 --> 01:11:56,490 and solve that for P2 for the core shear, 1031 01:11:56,490 --> 01:12:03,210 and that's equal to C11 times B4 times bc times 1032 01:12:03,210 --> 01:12:08,110 sigma ys times-- oops, wrong thing-- 1033 01:12:08,110 --> 01:12:11,410 times the relative density to the 2/3 power. 1034 01:12:19,410 --> 01:12:19,910 OK? 1035 01:12:22,990 --> 01:12:25,660 And then the next step is to equate these guys. 1036 01:12:25,660 --> 01:12:28,410 So you get a transition from one mode to the other 1037 01:12:28,410 --> 01:12:31,125 when two of these guys are equal to each other, right? 1038 01:12:31,125 --> 01:12:33,000 So there's going to be a transition from face 1039 01:12:33,000 --> 01:12:36,590 yielding to face wrinkling when these guys are equal. 1040 01:12:36,590 --> 01:12:38,529 And I'm not going to start that because we're 1041 01:12:38,529 --> 01:12:39,570 going to run out of time. 1042 01:12:39,570 --> 01:12:43,450 But let me just say that I can pair these two up and say 1043 01:12:43,450 --> 01:12:46,370 there's a transition between those two. 1044 01:12:46,370 --> 01:12:50,180 And that transition is going to correspond to this line 1045 01:12:50,180 --> 01:12:51,510 here, OK? 1046 01:12:51,510 --> 01:12:54,460 So at this line here, that means you get face yielding 1047 01:12:54,460 --> 01:12:57,690 and face wrinkling at the same load, OK? 1048 01:12:57,690 --> 01:13:01,280 And then if I paired up-- let's see here. 1049 01:13:01,280 --> 01:13:04,030 If I paired up face wrinkling and core shear, these two guys 1050 01:13:04,030 --> 01:13:09,130 here, I'm going to get this equation here on that plot. 1051 01:13:09,130 --> 01:13:12,020 And then if I paired up these two guys here, the face 1052 01:13:12,020 --> 01:13:16,300 shielding and the core shear, I would get that line there, OK? 1053 01:13:16,300 --> 01:13:19,790 So once I have those lines, that tells me, you know, 1054 01:13:19,790 --> 01:13:23,620 anything with a lower density core and a smaller face 1055 01:13:23,620 --> 01:13:25,800 thickness is going to fail by face wrinkling. 1056 01:13:25,800 --> 01:13:27,840 Anything with a bigger density is 1057 01:13:27,840 --> 01:13:29,530 going to fail by face yielding. 1058 01:13:29,530 --> 01:13:33,320 And anything with a larger face thickness and a larger density 1059 01:13:33,320 --> 01:13:35,010 is going to fail by core shearing. 1060 01:13:35,010 --> 01:13:37,447 And so you can start to see that it-- 1061 01:13:37,447 --> 01:13:39,030 I'll work out the equations next time, 1062 01:13:39,030 --> 01:13:41,571 but you can start to see that it kind physically makes sense. 1063 01:13:41,571 --> 01:13:44,550 Intuitively, this face wrinkling, 1064 01:13:44,550 --> 01:13:48,669 it depends on the normal stress in the face, in compression. 1065 01:13:48,669 --> 01:13:50,960 So obviously the thinner the face gets, the more likely 1066 01:13:50,960 --> 01:13:52,168 that's going to be to happen. 1067 01:13:52,168 --> 01:13:54,960 So it's going to happen at this end of the diagram. 1068 01:13:54,960 --> 01:13:58,040 And it also depends on that elastic foundation, on how much 1069 01:13:58,040 --> 01:14:01,270 spring support the foundation has, right? 1070 01:14:01,270 --> 01:14:03,960 So the lower the core density, the more likely 1071 01:14:03,960 --> 01:14:05,500 that is to happen. 1072 01:14:05,500 --> 01:14:08,570 Then if you, say you have small t, so the face is going 1073 01:14:08,570 --> 01:14:12,070 to fail before the core, as you increase the core density, 1074 01:14:12,070 --> 01:14:15,030 you're making that elastic foundation stiffer and stiffer, 1075 01:14:15,030 --> 01:14:17,280 and you're making it harder for the buckling to occur. 1076 01:14:17,280 --> 01:14:19,330 It can't buckle into the elastic foundation, 1077 01:14:19,330 --> 01:14:21,480 so then you're going to push it up to the yielding. 1078 01:14:21,480 --> 01:14:24,230 And then as you make the face thickness bigger, 1079 01:14:24,230 --> 01:14:26,631 as t gets bigger, then the face isn't going to fail 1080 01:14:26,631 --> 01:14:27,880 and the core is going to fail. 1081 01:14:27,880 --> 01:14:29,720 So you can kind of see just looking 1082 01:14:29,720 --> 01:14:31,570 at the relative position of those things, 1083 01:14:31,570 --> 01:14:33,910 they all kind of make physical sense. 1084 01:14:33,910 --> 01:14:35,420 So I'm going to stop there for today 1085 01:14:35,420 --> 01:14:38,740 and I'll finish the equations for that next time. 1086 01:14:38,740 --> 01:14:40,730 And we'll also talk about how to optimize 1087 01:14:40,730 --> 01:14:42,400 for strength next time. 1088 01:14:42,400 --> 01:14:47,080 And we'll talk about a few other things on sandwich panels.