WEBVTT 1 00:00:00.120 --> 00:00:03.080 Imagine for a second that you are standing at the 2 00:00:03.200 --> 00:00:07.679 absolute base of a massive modern skyscraper, or you know, 3 00:00:08.000 --> 00:00:12.160 maybe you're looking out across a sprawling, continuous steel bridge 4 00:00:12.199 --> 00:00:12.839 over a river. 5 00:00:13.080 --> 00:00:17.000 Yeah, those really massive structures where you just feel tiny exactly. 6 00:00:17.120 --> 00:00:19.760 You see the concrete, you see the steel, but what 7 00:00:19.839 --> 00:00:23.559 you probably rarely think about is the like the invisible 8 00:00:23.679 --> 00:00:25.800 war going on inside those materials. 9 00:00:25.879 --> 00:00:29.079 Oh. Absolutely, there is a whole lot of invisible drama happening. 10 00:00:28.760 --> 00:00:32.479 In there, right, Like, how do civil engineers actually calculate 11 00:00:32.560 --> 00:00:35.600 the precise internal forces to make sure the whole thing 12 00:00:35.640 --> 00:00:37.960 doesn't just, you know, collapse under its own weight. 13 00:00:38.079 --> 00:00:40.600 It's not a trivial thing. You've got bending moments trying 14 00:00:40.600 --> 00:00:43.880 to snap beams in half, sheer forces trying to slice them, 15 00:00:44.240 --> 00:00:47.679 and axial loads just squishing columns. 16 00:00:47.359 --> 00:00:50.000 Straight down, squishing them with immense weight. Yeah, And that's 17 00:00:50.000 --> 00:00:52.759 why today we're doing a deep dive into a really 18 00:00:52.799 --> 00:00:57.039 foundational text for civil engineers. We're looking at structural analysis, 19 00:00:57.079 --> 00:01:01.640 the set, fourth edition by ss Pabacatti. Classic, really it is, 20 00:01:01.840 --> 00:01:05.120 and our mission today is to unpack the mathematical toolkit 21 00:01:05.239 --> 00:01:09.400 that structural engineers use to analyze what they call indeterminate structures. 22 00:01:09.599 --> 00:01:11.640 Right, And for those of you listening who are say 23 00:01:12.120 --> 00:01:15.439 engineering students or young professionals or even just ambitious self 24 00:01:15.439 --> 00:01:17.680 taught folks, you probably know that term. 25 00:01:17.840 --> 00:01:21.159 Yeah, we're talking about structures where the basic equations of 26 00:01:21.239 --> 00:01:25.079 static equilibrium just well, they leave you with too many unknowns. 27 00:01:25.359 --> 00:01:28.719 That is the core dilemma right there. Because in a simpler, 28 00:01:29.120 --> 00:01:34.519 statically determinate structure like a seesaw or a basic bridge, 29 00:01:34.879 --> 00:01:38.480 resting on just two simple supports, you can figure out 30 00:01:38.519 --> 00:01:40.079 all the forces pretty. 31 00:01:39.799 --> 00:01:42.760 Easily because you just use the basic rules. Right, Some 32 00:01:42.840 --> 00:01:47.519 of vertical forces, horizontal forces, rotational moments all equal zero. 33 00:01:47.640 --> 00:01:51.159 Exactly three equations three unknowns. But the second you add 34 00:01:51.200 --> 00:01:53.840 a third support to that bridge, or you know, you 35 00:01:53.920 --> 00:01:56.560 weld the joints of a steel frame together so it's continuous, 36 00:01:56.920 --> 00:01:59.239 those three equations are suddenly mathematically useless. 37 00:01:59.319 --> 00:02:01.799 You end up with like six or ten unknowns. 38 00:02:02.000 --> 00:02:04.920 Right. You need a much deeper, far more sophisticated toolkit 39 00:02:04.920 --> 00:02:05.359 to solve it. 40 00:02:05.640 --> 00:02:08.080 And we want to really get into the technical weeds 41 00:02:08.120 --> 00:02:10.639 of that toolkit today. We want to understand the mechanics, 42 00:02:10.960 --> 00:02:14.879 keeping the science clear but grounded in practical applications because 43 00:02:14.919 --> 00:02:19.199 I mean, before we had supercomputers and fancy structural analysis software. 44 00:02:19.560 --> 00:02:20.719 How did they do it. 45 00:02:20.719 --> 00:02:24.599 It's a great question because solving a fifty story skyscraper 46 00:02:24.639 --> 00:02:26.560 by hand sounds completely impossible. 47 00:02:26.599 --> 00:02:27.680 It sounds like a nightmare. 48 00:02:27.800 --> 00:02:30.560 It was a monumental task, for sure. But the first 49 00:02:30.719 --> 00:02:35.120 major breakthrough was actually a shift in perspective. Engineers realized 50 00:02:35.159 --> 00:02:37.800 that if the static forces alone can't give you the answer, 51 00:02:38.199 --> 00:02:40.840 you have to look at how the structure physically moves. 52 00:02:40.599 --> 00:02:42.840 Wait moves like swaying. 53 00:02:42.639 --> 00:02:45.960 Well, specifically how it bends and rotates at its joints 54 00:02:46.080 --> 00:02:48.759 under a load. And this brings us to the slope 55 00:02:48.800 --> 00:02:52.039 deflection method. Okay, yeah, this was finalized in a really 56 00:02:52.080 --> 00:02:54.879 polished form by Professor G. A. Maney at the University 57 00:02:54.879 --> 00:02:57.159 of Minnesota way back in nineteen fifteen. 58 00:02:57.280 --> 00:02:59.599 Nineteen fifteen, so we are firmly in the era of 59 00:02:59.719 --> 00:03:02.560 slow rules and just massive sheets of grid paper. 60 00:03:02.840 --> 00:03:07.439 Oh yeah, serious tedious hand calculations. And the absolute genius 61 00:03:07.439 --> 00:03:10.719 amomoities approach was that he treated the slopes and the 62 00:03:10.759 --> 00:03:14.039 deflections of the joints as the fundamental unknown. 63 00:03:13.759 --> 00:03:15.199 Instead of the forces themselves. 64 00:03:15.319 --> 00:03:19.319 Exactly. You essentially set up a series of simultaneous equations 65 00:03:19.439 --> 00:03:22.719 based on the equilibrium conditions of those specific joints. 66 00:03:22.960 --> 00:03:25.039 So if you can solve for how much a joint 67 00:03:25.120 --> 00:03:29.400 rotates and how much it deflects or moves, you can 68 00:03:29.479 --> 00:03:33.520 plug those geometric values back into his equations define the 69 00:03:33.520 --> 00:03:36.400 internal end moments of every connected member you've got. 70 00:03:36.400 --> 00:03:38.840 It's basically a two step process. First you calculate the 71 00:03:38.879 --> 00:03:41.639 physical movement of the structure skeleton, and then you use 72 00:03:41.680 --> 00:03:44.199 that movement to back calculate the invisible forces. 73 00:03:44.280 --> 00:03:46.360 Okay, but to make this math work by hand in 74 00:03:46.439 --> 00:03:49.919 nineteen fifteen, MANI had to make some pretty bold assumptions, right, 75 00:03:50.000 --> 00:03:53.039 because looking through Bavacatti's text, the first big one is 76 00:03:53.039 --> 00:03:54.800 that all joints are perfectly rigid. 77 00:03:54.960 --> 00:03:58.000 Right. The assumption there is that the angle between any 78 00:03:58.000 --> 00:04:01.199 two members meeting at a joint does not change even 79 00:04:01.240 --> 00:04:04.159 after the entire structure deforms under a massive load. 80 00:04:04.400 --> 00:04:08.360 I always picture that like taking two heavy wooden rulers 81 00:04:08.400 --> 00:04:11.520 and super gluing them together at a perfect ninety degree angle. 82 00:04:11.599 --> 00:04:13.159 That's a really good way to visualize it. 83 00:04:13.280 --> 00:04:15.800 Yeah, so if I press down hard on the end 84 00:04:15.800 --> 00:04:18.360 of one of those rulers, the rulers themselves will bend 85 00:04:18.360 --> 00:04:22.160 into curves. But that glued corner itself, the joint, it 86 00:04:22.439 --> 00:04:26.360 stays locked at exactly ninety degrees. It just rotates as 87 00:04:26.399 --> 00:04:28.639 a single rigid unit in space. 88 00:04:28.800 --> 00:04:32.240 That's highly accurate. Yeah, the internal angle stays rigid. But 89 00:04:32.360 --> 00:04:35.360 many these other primary assumptions are the ones that they 90 00:04:35.399 --> 00:04:38.519 often raise eyebrows for first year structural students. 91 00:04:38.720 --> 00:04:40.959 Let me guess ignoring certain deformations. 92 00:04:41.120 --> 00:04:44.639 Yep, the slope deflection method dictates that both axial deformations 93 00:04:44.680 --> 00:04:47.279 and sheer deformations are entirely neglected. 94 00:04:47.399 --> 00:04:49.920 Okay, I have to pause and challenge that. Are we 95 00:04:50.079 --> 00:04:53.600 just like ignoring fundamental physics to make the math easier. 96 00:04:53.720 --> 00:04:54.720 It sounds like it, I know. 97 00:04:54.800 --> 00:04:57.279 Because if you put a massive multi ton load on 98 00:04:57.480 --> 00:05:00.759 a vertical concrete column, that column is going to compress, 99 00:05:00.800 --> 00:05:03.920 It's going to swish down acxially. How is it fundamentally 100 00:05:03.959 --> 00:05:05.639 safe to just pretend that doesn't happen. 101 00:05:05.920 --> 00:05:08.800 It is a very logical concern. But if we look 102 00:05:08.839 --> 00:05:12.959 at the actual structural behavior of rigid jointed frames, we 103 00:05:13.079 --> 00:05:16.439 have to consider the relative magnitude of these deformations. 104 00:05:16.560 --> 00:05:18.360 You mean, how big they are compared to each other. 105 00:05:18.480 --> 00:05:22.639 Exactly the distortion caused by axial forces, so that column 106 00:05:22.680 --> 00:05:27.319 physically compressing, and the distortion caused by sheer forces, they 107 00:05:27.319 --> 00:05:30.759 are just infinitesimally small when you compare them to the 108 00:05:30.800 --> 00:05:33.600 distortion caused by flexure or bending, So it's just. 109 00:05:33.639 --> 00:05:35.720 A matter of scale. Then the bending is just that 110 00:05:35.800 --> 00:05:37.279 much more prominent, profoundly. 111 00:05:37.319 --> 00:05:41.040 So Yeah, when a steel or concrete frame bends, the 112 00:05:41.079 --> 00:05:45.399 displacement from that flexual rotation completely dwarfs the tiny, tiny 113 00:05:45.439 --> 00:05:48.000 fraction of a millimeter it compresses axially. 114 00:05:48.240 --> 00:05:48.759 Oh wow. 115 00:05:48.879 --> 00:05:52.759 Okay, so by ignoring the axial and sheer deformations, many 116 00:05:52.839 --> 00:05:57.240 drastically simplified the mathematical complexity. He reduced the unknowns without 117 00:05:57.240 --> 00:05:58.879 sacrificing any practical safety. 118 00:05:58.959 --> 00:06:01.959 I suppose it's like, uh, calculating the total weight of 119 00:06:01.959 --> 00:06:04.079 a fully loaded freight truck and deciding you don't need 120 00:06:04.079 --> 00:06:05.879 to factor in a speck of dust on the windshield. 121 00:06:05.959 --> 00:06:07.399 That captures the reasoning perfectly. 122 00:06:07.480 --> 00:06:10.279 Yeah, it's mathematically there in the real world, but it 123 00:06:10.319 --> 00:06:12.439 has zero bearing on the suspension design. 124 00:06:12.800 --> 00:06:16.920 Right now, Even with those assumptions, analyzing a frame requires 125 00:06:17.199 --> 00:06:21.399 setting up joint equilibrium equations, and you rely on a 126 00:06:21.439 --> 00:06:25.279 core property called flexial rigidity, which is denoted as EI. 127 00:06:25.879 --> 00:06:28.759 Let's define EI really quickly for our listeners who might 128 00:06:28.800 --> 00:06:29.639 be a bit rusty. 129 00:06:29.800 --> 00:06:34.199 Sure, so E stands for the modulus of elasticity, that is, 130 00:06:34.240 --> 00:06:35.879 the material property. 131 00:06:35.560 --> 00:06:37.519 Like how stiff the actual material is. 132 00:06:37.600 --> 00:06:40.959 Exactly, Steel has a much higher E than wood, for example. 133 00:06:41.399 --> 00:06:43.680 And then I stands for the moment of inertia, which 134 00:06:43.680 --> 00:06:45.920 is entirely about the cross sectional geometry. 135 00:06:46.040 --> 00:06:48.879 Right, because an ibeam shape resists bending much bisch than 136 00:06:48.959 --> 00:06:51.199 just a solid square block of the same weight. 137 00:06:51.120 --> 00:06:54.759 Spot on, so EI together represents the overall resistance to bending. 138 00:06:55.279 --> 00:06:57.759 Use that EI, along with the fixed end moments in 139 00:06:57.839 --> 00:07:00.800 rotations to write out your equilibrit equations. 140 00:07:01.040 --> 00:07:03.439 But I mean, even with all these simplifications, if you 141 00:07:03.600 --> 00:07:07.279 have a structure with more than three unknown joints, you 142 00:07:07.319 --> 00:07:11.759 are suddenly staring down a barrel of incredibly complex simultaneous equations. 143 00:07:11.800 --> 00:07:16.079 You really are, And doing that by hand becomes profoundly 144 00:07:16.160 --> 00:07:19.759 tedious and honestly highly prone to arithmetic errors. 145 00:07:19.879 --> 00:07:22.600 Oh, I bet one drop negative sign and the whole 146 00:07:22.639 --> 00:07:23.319 thing is ruined. 147 00:07:23.439 --> 00:07:27.120 A single negative sign ruins the entire analysis. Yeah, but 148 00:07:27.279 --> 00:07:30.759 it is crucial for modern engineers to realize that Manny's 149 00:07:30.759 --> 00:07:34.959 slope deflection method is the direct foundational ancestor of the 150 00:07:35.000 --> 00:07:37.040 stiffness matrix method, which. 151 00:07:36.839 --> 00:07:39.560 Is the math running under the hood of almost all 152 00:07:39.600 --> 00:07:42.720 structural analysis software today. Right, the computer is just doing 153 00:07:42.759 --> 00:07:46.000 matrix algebra to solve thousands of those equations in a 154 00:07:46.000 --> 00:07:47.240 fraction of a second. 155 00:07:47.000 --> 00:07:49.839 Exactly right. But back in the nineteen twenties and thirties, 156 00:07:50.120 --> 00:07:53.439 engineers didn't have matrixs running on microchips, and structures were 157 00:07:53.439 --> 00:07:54.720 getting substantially bigger. 158 00:07:54.959 --> 00:07:57.319 Right, we're talking about the dawn of the true skyscraper 159 00:07:57.360 --> 00:07:57.839 era here. 160 00:07:58.000 --> 00:08:02.279 Yeah, skyscraper's in massive continuous bridges. Solving endless equations by 161 00:08:02.319 --> 00:08:06.879 hand was exhausting. The industry desperately needed smarter more forgiving 162 00:08:06.920 --> 00:08:08.800 mathematical workarounds. 163 00:08:08.240 --> 00:08:10.600 Which brings us to the brilliance of engineers like Gasper 164 00:08:10.639 --> 00:08:13.879 Khani and Hardy Cross. Let's look at Kanyie's method of 165 00:08:13.959 --> 00:08:18.959 rotation contribution first, How did he bypass that simultaneous equation nightmare? 166 00:08:19.240 --> 00:08:23.319 So Gasper Khani, who is a German engineer, he developed 167 00:08:23.319 --> 00:08:28.399 this brilliant iterative distribution procedure. It's actually still based directly 168 00:08:28.439 --> 00:08:30.240 on Manny's slope deflection equations. 169 00:08:30.279 --> 00:08:31.959 Okay, so same foundation game. 170 00:08:31.800 --> 00:08:34.840 Foundation, But instead of trying to solve all the equations 171 00:08:34.879 --> 00:08:38.919 at once in a massive, interconnected matrix, you distribute the 172 00:08:38.960 --> 00:08:43.759 bending moments joint by joint, over and over again in cycles. 173 00:08:44.120 --> 00:08:47.679 And according to Paviacatti's book, Khanyie's method has this one 174 00:08:47.960 --> 00:08:51.440 absolute superpower that made it the go to method for 175 00:08:51.600 --> 00:08:54.960 multi story frames. It is fundamentally self correcting, it. 176 00:08:54.879 --> 00:08:58.360 Is, and that is its absolute greatest practical advantage. 177 00:08:58.399 --> 00:09:00.639 I really need you to break down the Maccarex of this, 178 00:09:00.759 --> 00:09:03.799 because how can math just, you know, forgive a mistake. 179 00:09:03.919 --> 00:09:04.799 It sounds like magic. 180 00:09:04.840 --> 00:09:07.000 I know it really does. Because if I'm doing long 181 00:09:07.039 --> 00:09:09.159 division by hand and I carry the wrong number, my 182 00:09:09.240 --> 00:09:12.840 final answer is permanently wrong. How does Connie's method naturally 183 00:09:12.919 --> 00:09:14.080 fix a human error? 184 00:09:14.159 --> 00:09:17.840 You have to think about the mechanism of iterative converging mathematics. 185 00:09:18.399 --> 00:09:21.600 Think of it like balancing a complex set of scales 186 00:09:21.639 --> 00:09:24.879 that are all tied together in a normal simultaneous equation. 187 00:09:25.320 --> 00:09:28.720 If you write one coefficient wrong, the final result is 188 00:09:28.720 --> 00:09:33.360 skewed forever. But Kanyie's method works by applying rotation factors 189 00:09:33.559 --> 00:09:36.480 and distributing moments in a continuous loop. You move from 190 00:09:36.559 --> 00:09:39.159 joint a to B to C and then back to A. 191 00:09:39.519 --> 00:09:42.399 So you are constantly recalculating the balance right. 192 00:09:42.480 --> 00:09:46.279 Every single cycle is a forced rebalancing act. Let's say 193 00:09:46.320 --> 00:09:49.240 you make a math error cycle two. You accidentally calculate 194 00:09:49.279 --> 00:09:52.000 a moment of fifty at joint A instead of forty. 195 00:09:51.799 --> 00:09:53.120 So you've messed up the scale. 196 00:09:53.240 --> 00:09:56.720 You've essentially just pushed one of those scales out of equilibrium. 197 00:09:56.879 --> 00:09:59.519 But in cycle three, the math forces you to look 198 00:09:59.519 --> 00:10:02.679 at joint A again in relation to his neighbors. The 199 00:10:02.720 --> 00:10:06.159 formula basically sees that joint A is now mathematically out 200 00:10:06.159 --> 00:10:07.080 of balance. 201 00:10:06.679 --> 00:10:09.600 With joint B OH, and then it automatically distributes that 202 00:10:09.639 --> 00:10:12.639 excess moment away based on the relative stiffness of the members. 203 00:10:12.799 --> 00:10:16.320 Exactly because you keep looping, that initial error gets diluted 204 00:10:16.399 --> 00:10:17.000 down to zero. 205 00:10:17.200 --> 00:10:18.840 It naturally irons out the mistake. 206 00:10:19.120 --> 00:10:21.600 That is amazing it is it might take you six 207 00:10:21.639 --> 00:10:24.000 cycles to converge on the true answer instead of five, 208 00:10:24.360 --> 00:10:27.120 but you will absolutely arrive at the mathematically correct. 209 00:10:26.879 --> 00:10:30.360 Answer that provides such an immense safety net. I can 210 00:10:30.399 --> 00:10:32.919 only imagine the peace of mind that gave a practicing 211 00:10:32.960 --> 00:10:37.879 engineer in like nineteen fifty knowing a two am calculator 212 00:10:38.000 --> 00:10:40.399 error wouldn't result in a structural failure. 213 00:10:40.639 --> 00:10:44.480 Seriously, But Canee wasn't the only one finding clever backdoors 214 00:10:44.519 --> 00:10:47.879 into complex math. We also have Professor Hardi Cross and 215 00:10:47.919 --> 00:10:51.159 his column analogy method, which he introduced in nineteen thirty two. 216 00:10:51.279 --> 00:10:53.919 Hardi Cross, he's kind of a titan in the field, right, oh, a. 217 00:10:53.879 --> 00:10:57.919 Total titan, and his column analogy is just a beautiful 218 00:10:57.960 --> 00:11:03.159 piece of mathematical abstraction. He noticed identical mathematical structures between 219 00:11:03.200 --> 00:11:06.120 the equations used to find moments in a continuous fixed 220 00:11:06.159 --> 00:11:09.159 beam and the equations used to find stresses in an 221 00:11:09.200 --> 00:11:10.559 eccentrically loaded column. 222 00:11:10.600 --> 00:11:12.960 Okay, let's paint a picture of an eccentrically loaded column 223 00:11:13.000 --> 00:11:13.600 for our listeners. 224 00:11:13.679 --> 00:11:14.159 Good idea. 225 00:11:14.200 --> 00:11:16.480 That's a vertical column where the heavy load on top 226 00:11:16.559 --> 00:11:19.279 isn't perfectly centered in the middle. Right, it's offset to 227 00:11:19.320 --> 00:11:22.120 the edge, which means the load is pushing straight down, 228 00:11:22.200 --> 00:11:24.960 but it's also acting like a lever trying to twist 229 00:11:25.039 --> 00:11:26.519 or bend the column at the same time. 230 00:11:26.720 --> 00:11:30.720 That dual action is the key. In an eccentrically loaded column. 231 00:11:31.000 --> 00:11:34.720 The total stress is calculated by taking the direct axial stress, 232 00:11:35.240 --> 00:11:37.399 so the load divided by the area and adding the 233 00:11:37.399 --> 00:11:42.039 bending stress cross realized the mathematical format of that column 234 00:11:42.039 --> 00:11:46.480 formula is structurally identical to the formula for finding indeterminate 235 00:11:46.559 --> 00:11:47.840 moments in a fixed beam. 236 00:11:48.000 --> 00:11:49.360 That's wild. 237 00:11:49.600 --> 00:11:53.279 So he proposed mapping the one D beam problem into 238 00:11:53.279 --> 00:11:54.440 a two D column space. 239 00:11:54.639 --> 00:11:56.600 Okay, walk me through how you actually do that. You 240 00:11:56.679 --> 00:11:58.200 just create an imaginary column. 241 00:11:58.240 --> 00:12:01.039 You do. You create what he called in an analogous column. 242 00:12:01.159 --> 00:12:04.200 You draw a short imaginary column whose length equals the 243 00:12:04.240 --> 00:12:07.399 span of your real beam and whose width is equal 244 00:12:07.440 --> 00:12:11.120 to one over ei one divided by the flexial rigidity 245 00:12:11.120 --> 00:12:12.039 we talked about earlier. 246 00:12:12.080 --> 00:12:14.559 Oh okay, So if the real beam is very stiff, 247 00:12:14.639 --> 00:12:17.519 meaning a high EI, the width of your imaginary column 248 00:12:17.519 --> 00:12:18.440 becomes very narrow. 249 00:12:18.480 --> 00:12:21.559 Correct. You then take the free bending moment diagram of 250 00:12:21.600 --> 00:12:24.320 your real beam, and you treat that diagram as the 251 00:12:24.320 --> 00:12:28.039 physical load pressing down on your analogous column. Yeah. If 252 00:12:28.080 --> 00:12:31.399 you calculate the resulting stress on that imaginary column, the 253 00:12:31.480 --> 00:12:35.000 number you get is exactly equal to the indeterminate bending 254 00:12:35.039 --> 00:12:36.360 moment on the real beam. 255 00:12:36.559 --> 00:12:39.720 That is just mind bending. It's like translating a tremendously 256 00:12:39.759 --> 00:12:43.840 difficult calculus problem into a straightforward geometry and area problem. 257 00:12:43.879 --> 00:12:47.279 It really is, and Pavicatti notes this is uniquely powerful 258 00:12:47.320 --> 00:12:49.840 for beams with variable cross. 259 00:12:49.600 --> 00:12:52.519 Sections, like if a bridge girder gets much thicker at 260 00:12:52.559 --> 00:12:54.320 the supports and thinner in the middle. 261 00:12:54.440 --> 00:12:59.399 Exactly analyzing that with pure slope deflection is incredibly difficult 262 00:12:59.519 --> 00:13:02.960 because the the moment of inertia is constantly changing. But 263 00:13:03.039 --> 00:13:04.519 in column analogy, what do you do? 264 00:13:04.679 --> 00:13:07.559 I guess you simply change the width of your imaginary 265 00:13:07.600 --> 00:13:11.559 column to match the changing one over ei along the span. 266 00:13:11.759 --> 00:13:14.960 That's it. It turns a nightmare of integral calculus into 267 00:13:15.000 --> 00:13:17.679 calculating the area of a slightly weird shaped column. 268 00:13:17.720 --> 00:13:20.559 Okay, so Connie and Cross gave engineers the tools to 269 00:13:20.600 --> 00:13:24.799 solve complex continuous static frames, but a bridge isn't a 270 00:13:24.799 --> 00:13:28.840 static environment. What happens when a multi ton freight train 271 00:13:29.279 --> 00:13:33.240 rolls across a continuous steel bridge because the structural math 272 00:13:33.360 --> 00:13:36.440 must completely change when the load is constantly moving. You 273 00:13:36.440 --> 00:13:39.519 can't just pause time and recalculate an entire slope deflection 274 00:13:39.639 --> 00:13:42.759 matrix for every single inch the train rolls forward. 275 00:13:42.919 --> 00:13:46.519 You definitely can't. The computational load would be infinite. And 276 00:13:46.519 --> 00:13:49.600 this is where the industry turns to influence line diagrams 277 00:13:49.679 --> 00:13:53.279 or ILDs, and specifically the Miller Breslau principle. 278 00:13:53.559 --> 00:13:55.519 Okay, I read this section of the source material a 279 00:13:55.519 --> 00:13:57.679 few times because it feels a bit like a magic trick. 280 00:13:57.879 --> 00:14:00.639 It often feels that way to students. Yeah. An influence 281 00:14:00.720 --> 00:14:03.399 line diagram is simply a graph, but instead of showing 282 00:14:03.440 --> 00:14:06.600 how forces exist under a stationary load, it shows how 283 00:14:06.639 --> 00:14:10.679 a specific internal force, say the sheer force the midpoint 284 00:14:10.720 --> 00:14:13.879 of a bridge, varies as a single unit load moves 285 00:14:13.879 --> 00:14:15.279 from one end of the bridge to the other. 286 00:14:15.639 --> 00:14:18.639 Right and for determinate simple structures, these lines are. 287 00:14:18.519 --> 00:14:24.799 Straight correct, but for indeterminate continuous structures they are complex curves, and. 288 00:14:24.879 --> 00:14:27.279 To figure out what those curves look like, you use 289 00:14:27.320 --> 00:14:30.879 the Miller Breslau principle. The principle states that if you 290 00:14:30.919 --> 00:14:35.279 take the structure physically, remove the restraint for the specific 291 00:14:35.320 --> 00:14:38.720 stress you were looking for, and then introduce a small deformation, 292 00:14:39.480 --> 00:14:43.200 the physical shape the structure takes actually represents the influence 293 00:14:43.240 --> 00:14:44.559 line diagram itself. 294 00:14:44.600 --> 00:14:45.320 That is the essence of it. 295 00:14:45.399 --> 00:14:47.600 Yet I need to visualize this to make sure I 296 00:14:47.679 --> 00:14:52.960 understand the mechanics I'm picturing. Like a physical scale model 297 00:14:53.240 --> 00:14:56.600 of a continuous bridge beam made out of flexible rubber 298 00:14:56.879 --> 00:14:58.000 resting on three supports. 299 00:14:58.080 --> 00:14:59.039 Okay, great visual. 300 00:14:59.240 --> 00:15:01.000 Let's say I want an I know the influence line 301 00:15:01.000 --> 00:15:04.279 for the vertical reaction force at that middle support. According 302 00:15:04.320 --> 00:15:06.720 to the principle, I would physically cut the rubber beam 303 00:15:06.799 --> 00:15:09.639 free from that middle support and then physically push the 304 00:15:09.679 --> 00:15:12.919 rubber straight up at that spot. And the physical wavy 305 00:15:12.960 --> 00:15:15.679 shape that the rubber beam curves into, like bending up 306 00:15:15.679 --> 00:15:18.159 in the middle and dipping down toward the ends. That 307 00:15:18.200 --> 00:15:22.039 physical shape is literally a structural graph of the forces. 308 00:15:22.519 --> 00:15:25.720 It is. The deflected shape of the release structure is 309 00:15:25.759 --> 00:15:30.919 the influence line for that reaction. It's an incredible experimental approach. Historically, 310 00:15:30.960 --> 00:15:34.320 engineers would actually build physical spline models to form them, 311 00:15:34.639 --> 00:15:37.519 and literally trace the curves onto paper to get their 312 00:15:37.559 --> 00:15:38.279 influence lines. 313 00:15:38.399 --> 00:15:41.039 Okay, but I have to push back here. Tracing a 314 00:15:41.120 --> 00:15:43.799 rubber curve sounds like a great visual aid for a 315 00:15:43.879 --> 00:15:49.039 university classroom. But does this physical, deformed shape trick actually 316 00:15:49.080 --> 00:15:52.799 give engineers the exact, quantifiable numbers needed to build a 317 00:15:52.879 --> 00:15:56.120 real bridge. I just can't imagine submitting a hand traced 318 00:15:56.159 --> 00:15:57.960 line to a city building department. 319 00:15:58.679 --> 00:16:01.360 No, you wouldn't, and that a critical distinction to make. 320 00:16:02.080 --> 00:16:04.639 While it works experimentally, it is first and foremost a 321 00:16:04.720 --> 00:16:05.919 rigorous analytical tool. 322 00:16:06.000 --> 00:16:07.840 Okay, so how do you get the exact numbers. 323 00:16:07.960 --> 00:16:10.159 To get the exact numbers, you don't use rubber models. 324 00:16:10.279 --> 00:16:14.679 You apply the principle mathematically. You mathematically remove the restraint, 325 00:16:14.759 --> 00:16:17.639 apply a unit load a force of exactly one in 326 00:16:17.679 --> 00:16:20.440 the direction of the released restraint, and then you calculate 327 00:16:20.440 --> 00:16:22.879 the precise deflections of the beam at various points. 328 00:16:23.240 --> 00:16:25.960 And how do you calculate those precise deflections without getting 329 00:16:25.960 --> 00:16:27.200 bogged down in terrible math? 330 00:16:27.240 --> 00:16:30.279 Again? Engineers commonly use the conjugate beam. 331 00:16:30.039 --> 00:16:33.519 Method, which is another one of those brilliant mathematical workarounds. Right, 332 00:16:34.039 --> 00:16:35.879 let's break down how that works mechanically. 333 00:16:36.200 --> 00:16:39.759 It's fantastic. The conjugate beam method is a mathematical trick 334 00:16:40.080 --> 00:16:42.320 where you swap out the physical loads on a beam 335 00:16:42.600 --> 00:16:45.600 for the bending moments themselves. Okay, you set up a 336 00:16:45.600 --> 00:16:48.600 fictional conjugate beam and you load it with the m 337 00:16:48.720 --> 00:16:52.679 over Ei diagram of the real beam. By simply calculating 338 00:16:52.720 --> 00:16:56.080 the static sheer and static bending moment on this imaginary beam, 339 00:16:56.440 --> 00:16:59.600 the resulting numbers give you the exact slope and exact 340 00:16:59.639 --> 00:17:01.320 deflection of your real beam. 341 00:17:01.600 --> 00:17:05.480 So it turns complex calculus back into basic statics exactly. 342 00:17:05.519 --> 00:17:07.839 So, by using the conjugate BEM method on a structure 343 00:17:07.920 --> 00:17:10.640 released according to the Miller Preslau principle, you get the 344 00:17:10.680 --> 00:17:15.119 exact ordinance, the precise decimal values of your influence line. 345 00:17:15.200 --> 00:17:19.000 It gives you a quantifiable, exact diagram, allowing you to 346 00:17:19.039 --> 00:17:22.359 perfectly design a bridge to handle the moving stresses of 347 00:17:22.400 --> 00:17:23.039 a freight train. 348 00:17:23.240 --> 00:17:24.480 Correct. Incredible. 349 00:17:24.960 --> 00:17:28.200 Okay, we have covered continuous beams, we've solved moving loads, 350 00:17:28.680 --> 00:17:30.839 but we really have to talk about scaling up to 351 00:17:31.079 --> 00:17:32.440 massive proportions. 352 00:17:32.759 --> 00:17:35.799 Ah yes, multi story frames. 353 00:17:35.680 --> 00:17:38.759 Right, what happens when you were tasked with analyzing a 354 00:17:38.799 --> 00:17:43.599 fifty story commercial skyscraper Because the degree of indeterminacy in 355 00:17:43.640 --> 00:17:47.680 a grid that massive is just staggering. Every single joint 356 00:17:47.680 --> 00:17:50.759 where a beam meets a column introduces multiple unknowns. 357 00:17:51.279 --> 00:17:55.079 This is where theory hits reality really hard. Exact solutions 358 00:17:55.160 --> 00:17:58.519 using slope deflection for a fifty story frame are completely 359 00:17:58.640 --> 00:18:02.640 ruled out for manual calculs. A human being simply cannot 360 00:18:02.680 --> 00:18:02.960 do it. 361 00:18:03.079 --> 00:18:05.759 So how did they build safe skyscrapers in the nineteen sixties, 362 00:18:05.799 --> 00:18:06.920 before computers took over. 363 00:18:07.119 --> 00:18:11.079 For quick practical solutions, structural design engineers rely on approximate methods, 364 00:18:11.480 --> 00:18:14.400 and the most prominent one for handling vertical gravity loads, 365 00:18:14.640 --> 00:18:17.119 meaning the dead weight of the concrete itself plus the 366 00:18:17.160 --> 00:18:20.160 live weight of the people, desks, and equipment, is the 367 00:18:20.160 --> 00:18:21.559 substitute frame method. 368 00:18:21.640 --> 00:18:24.079 The premise of the substitute frame method is wild to me. 369 00:18:24.240 --> 00:18:27.039 It basically assumes that the bending moments transferred from one 370 00:18:27.079 --> 00:18:31.240 floor of a skyscraper to another floor are basically negligible. Yes, Therefore, 371 00:18:31.279 --> 00:18:33.759 you don't calculate the whole building at once. You analyze 372 00:18:33.759 --> 00:18:35.839 the skyscraper entirely, floor by floor. 373 00:18:36.079 --> 00:18:39.000 Right, you isolate a substitute frame. You take only the 374 00:18:39.039 --> 00:18:42.319 specific floor beam you are interested in, the columns directly 375 00:18:42.319 --> 00:18:45.400 attached above it, and the columns directly attached below it, 376 00:18:45.599 --> 00:18:48.160 just that tiny section. You assume the far ends of 377 00:18:48.200 --> 00:18:50.559 those columns are fixed, yeah, and you analyze just that 378 00:18:50.680 --> 00:18:51.839 isolated cross section. 379 00:18:52.359 --> 00:18:55.519 My analogy for this is like looking at a single 380 00:18:55.640 --> 00:18:59.480 rung on a massive towering scaffolding ladder. You want to 381 00:18:59.519 --> 00:19:01.680 know if number twenty is going to buckle under the 382 00:19:01.680 --> 00:19:05.240 weight of a worker. The substitute frame method basically says 383 00:19:05.359 --> 00:19:08.279 only look at Rung twenty and the immediate vertical pikes 384 00:19:08.279 --> 00:19:10.039 connecting Rung nineteen and twenty one. 385 00:19:10.480 --> 00:19:12.359 That's a highly accurate way to conceptualize it. 386 00:19:12.480 --> 00:19:16.359 Yeah, you completely ignore the heavy equipment sitting way down 387 00:19:16.400 --> 00:19:17.079 on Rung three. 388 00:19:17.359 --> 00:19:19.039 You isolate the local geometry. 389 00:19:19.400 --> 00:19:23.519 But is that genuinely safe to just completely ignore the 390 00:19:23.559 --> 00:19:26.880 rest of a massive, heavy building When doing these calculations, 391 00:19:27.200 --> 00:19:29.400 it feels like we are ignoring a massive amount of 392 00:19:29.440 --> 00:19:30.559 structural reality. 393 00:19:30.920 --> 00:19:34.200 I know it sounds risky, but it is safe, And 394 00:19:34.240 --> 00:19:38.480 it comes down to understanding how flexial loads physically distribute 395 00:19:38.480 --> 00:19:41.359 through a grit. When you place a heavy vertical load 396 00:19:41.440 --> 00:19:44.640 on a beam on the twentieth floor, it induces a 397 00:19:44.720 --> 00:19:48.599 massive bending moment in that specific beam, and it transfers 398 00:19:48.640 --> 00:19:51.960 a portion of that rotational moment into the columns directly 399 00:19:52.000 --> 00:19:52.640 attached to it. 400 00:19:53.079 --> 00:19:55.279 The joint rotates, taking the columns with it. 401 00:19:55.480 --> 00:19:58.920