WEBVTT 1 00:00:00.120 --> 00:00:03.040 Usually when you look at a traditional building, say a 2 00:00:03.120 --> 00:00:06.440 standard brick warehouse, or you know, a classic four story 3 00:00:06.480 --> 00:00:10.480 apartment block, you can almost intuitively sense how it's standing up. 4 00:00:10.880 --> 00:00:13.560 Gravity pushes down and the thick walls hold it up. 5 00:00:13.640 --> 00:00:16.000 It makes perfect physical sense to the human eye. 6 00:00:16.199 --> 00:00:18.839 Yeah, it's that basic load bearing logic humans have been 7 00:00:18.920 --> 00:00:22.559 using for thousands of years, just you know, stack heavy 8 00:00:22.600 --> 00:00:24.359 things on top of other heavy things. 9 00:00:24.440 --> 00:00:26.960 Right, But then you walk into a modern city center 10 00:00:27.039 --> 00:00:32.320 and look up and you see these wild, sweeping architectural marvels, 11 00:00:32.799 --> 00:00:37.399 massive cant livered stadium roofs just hanging in mid air. 12 00:00:37.479 --> 00:00:41.560 Oh, absolutely twisted steel frames, glass structures leaning at angles 13 00:00:41.600 --> 00:00:45.000 that seem to like actively defy gravity exactly. 14 00:00:45.079 --> 00:00:47.759 And when you look at those, your intuition just short circuits. 15 00:00:47.840 --> 00:00:50.880 You can't help but wonder, how do we actually prove 16 00:00:50.920 --> 00:00:53.320 a design that complex isn't going to just snap in 17 00:00:53.359 --> 00:00:55.280 the wind before we even pour the foundation. 18 00:00:55.560 --> 00:00:58.960 Well, with geometries, that wild human intuition is pretty much 19 00:00:59.079 --> 00:01:01.880 entirely useless. You can no longer rely on paper and 20 00:01:01.920 --> 00:01:05.760 pencil math because you aren't just calculating gravity pushing down anymore. 21 00:01:05.879 --> 00:01:07.359 Right, It's way more complicated than that. 22 00:01:07.640 --> 00:01:11.879 Oh yeah, you are dealing with this staggering three dimensional 23 00:01:11.959 --> 00:01:15.840 web of interconnected forces. A wind gust hitting the roof 24 00:01:16.239 --> 00:01:21.239 transfers twisting forces down through thousands of interconnected diagonal beams 25 00:01:21.480 --> 00:01:23.159 all the way to the bedrock. 26 00:01:23.159 --> 00:01:25.480 Which brings us to our mission. For this deep dive. 27 00:01:26.079 --> 00:01:30.200 We are exploring doctor srinavas on chunder Securin's highly comprehensive 28 00:01:30.239 --> 00:01:34.879 textbook Advance Structural Analysis. With Matt lab, our goal today 29 00:01:35.079 --> 00:01:38.480 is to demystify computational structural analysis for you. 30 00:01:38.760 --> 00:01:41.400 That's right, because if you are a civil engineering student 31 00:01:41.439 --> 00:01:44.959 currently banging your head against matrix equations, or self taught code, 32 00:01:45.000 --> 00:01:46.920 or wanting to understand structural physics, or. 33 00:01:46.840 --> 00:01:48.799 Even a young professional try and figure out what your 34 00:01:48.799 --> 00:01:51.760 design software is actually doing behind the scenes, this deep 35 00:01:51.799 --> 00:01:52.959 dive is built for you. 36 00:01:53.159 --> 00:01:56.079 We're going to bridge that gap between classical structural physics, 37 00:01:56.400 --> 00:01:59.239 the fundamental principles you learn on a chalkboard, and the 38 00:01:59.280 --> 00:02:03.040 modern matrix based computer coding that actually calculates the safety 39 00:02:03.040 --> 00:02:03.920 of our modern world. 40 00:02:04.239 --> 00:02:08.719 Okay, let's untack this core conflict of structural analysis. If 41 00:02:08.800 --> 00:02:11.400 you want to code a structure into a program like 42 00:02:11.479 --> 00:02:14.759 Matt Lab, you first have to formulate the problem mathematically, 43 00:02:15.120 --> 00:02:16.759 and the book points out that we have to tackle 44 00:02:16.800 --> 00:02:19.080 something called structural indeterminacy. 45 00:02:19.520 --> 00:02:21.639 To give that some context, let's look at the baseline, 46 00:02:21.639 --> 00:02:26.439 which is statically determinate structures. These are relatively simple. Imagine 47 00:02:26.479 --> 00:02:29.680 a wooden plank resting on two concrete blocks, like a 48 00:02:29.719 --> 00:02:30.560 basic footbridge. 49 00:02:30.680 --> 00:02:31.680 Okay, I'm picturing it. 50 00:02:31.919 --> 00:02:34.840 Because there are only two supports. You can calculate the 51 00:02:34.919 --> 00:02:39.840 forces using only the standard equations of static equilibrium, meaning 52 00:02:41.159 --> 00:02:44.520 the sum of all vertical forces must equal zero, or 53 00:02:44.520 --> 00:02:45.400 the bridge moves. 54 00:02:45.639 --> 00:02:48.280 But modern structures aren't just planks on two blocks. 55 00:02:48.400 --> 00:02:52.280 Far from it. Modern structures are almost always statically indeterminate. 56 00:02:52.800 --> 00:02:54.879 Imagine that same footbridge, but now you add a third 57 00:02:54.879 --> 00:02:56.000 concrete block right in the middle. 58 00:02:56.120 --> 00:02:58.840 Oh right, So suddenly you have more supports than are 59 00:02:58.879 --> 00:03:01.759 strictly mathematically necessary to keep the plank from falling. 60 00:03:02.120 --> 00:03:06.199 Exactly. It makes the bridge significantly safer and stiffer, but 61 00:03:06.319 --> 00:03:10.560 it completely breaks those basic static equations. There are just 62 00:03:10.680 --> 00:03:14.400 too many unknown support forces and not enough simple equations 63 00:03:14.439 --> 00:03:15.159 to solve for them. 64 00:03:15.240 --> 00:03:17.840 So to solve an indeterminate structure, the book says we 65 00:03:17.919 --> 00:03:21.439 have to choose between two classical grouping methods. We have 66 00:03:21.599 --> 00:03:24.360 the flexibility method and the stiffness. 67 00:03:23.879 --> 00:03:25.759 Method, right, the two big heavyweights. 68 00:03:25.840 --> 00:03:29.039 Yeah. And if you are trying to visualize this, think 69 00:03:29.080 --> 00:03:32.719 of testing the stability of a physical object. The flexibility 70 00:03:32.759 --> 00:03:35.879 method is like taking a complex object and measuring exactly 71 00:03:35.960 --> 00:03:38.400 how hard you have to push or pull on different 72 00:03:38.400 --> 00:03:41.520 parts to get it to move. You are dealing in forces, right. 73 00:03:42.000 --> 00:03:44.680 But the stiffness method takes completely different approach. It's like 74 00:03:45.439 --> 00:03:49.719 placing tiny digital tracking dots on all the structural joints 75 00:03:50.080 --> 00:03:53.520 and measuring exactly how far those dots physically shift or 76 00:03:53.520 --> 00:03:54.479 twist under pressure. 77 00:03:54.879 --> 00:03:57.919 That tracking dot visualization is great. It gets right to 78 00:03:57.960 --> 00:04:01.800 the heart of what the text calls static versus kinematic indeterminacy. 79 00:04:01.879 --> 00:04:05.400 The flexibility method deals with static indeterminacy. 80 00:04:04.840 --> 00:04:08.719 Meaning the unknowns you are hunting for are the forces right, 81 00:04:09.199 --> 00:04:13.039 the sheer force, axial force, bending moment exactly? 82 00:04:13.080 --> 00:04:15.199 To solve it, an engineer has to look at the 83 00:04:15.199 --> 00:04:19.639 structure and creatively decide which of those redundant supports to 84 00:04:19.680 --> 00:04:23.399 temporarily release in their math to make the structure solvable, 85 00:04:23.800 --> 00:04:25.560 and then add the forces back in later. 86 00:04:25.600 --> 00:04:27.480 See I can see where this is going. Software doesn't 87 00:04:27.480 --> 00:04:29.839 really do creative very well. If I'm writing a matt 88 00:04:29.920 --> 00:04:32.040 Lab script, I don't want the program to have to 89 00:04:32.079 --> 00:04:34.680 guess which beam to temporarily ignore. 90 00:04:34.600 --> 00:04:38.759 What's fascinating here is exactly that point. The flexibility method 91 00:04:38.879 --> 00:04:43.399 requires human intuition. It's highly specific to the unique geometry 92 00:04:43.399 --> 00:04:44.439 of whatever building you. 93 00:04:44.319 --> 00:04:46.759 Were looking at, so it's terrible for automation. 94 00:04:46.959 --> 00:04:49.920 Terrible. The stiffness method, on the other hand, deals with 95 00:04:50.040 --> 00:04:53.959 kinematic indeterminacy. Here, the unknowns aren't the forces, they are 96 00:04:53.959 --> 00:04:56.800 the joint displacements. You're looking for how much the joints 97 00:04:56.920 --> 00:04:58.399 rotate and translate, so. 98 00:04:58.839 --> 00:05:02.319 You mathematically love the entire structure down to make it rigid, 99 00:05:02.399 --> 00:05:05.399 and then calculate how much those joints would naturally move 100 00:05:05.480 --> 00:05:06.040 under the load. 101 00:05:06.319 --> 00:05:11.079 Yes, both methods are mathematically valid in pure physics, but 102 00:05:11.120 --> 00:05:14.519 the stiffness method is the undisputed champion for computer programs 103 00:05:14.560 --> 00:05:19.199 like matt Lab. It is generic, highly repetitive, and completely 104 00:05:19.199 --> 00:05:21.600 independent of the building's specific shape. 105 00:05:21.319 --> 00:05:23.160 Which is exactly what a computer needs. 106 00:05:23.319 --> 00:05:27.680 Right, you identify the degrees of freedom, the possible directions 107 00:05:27.680 --> 00:05:30.319 a joint could potentially move, and it results in a 108 00:05:30.360 --> 00:05:35.120 massive but incredibly predictable set of linear equations. It is 109 00:05:35.199 --> 00:05:38.240 tailor made for matrix operations. You just set up the 110 00:05:38.240 --> 00:05:41.000 loops and the computer does the brute force calculation. 111 00:05:41.519 --> 00:05:44.079 So by choosing the stiffness method, we are adopting the 112 00:05:44.160 --> 00:05:47.120 language the computer inherently speaks. But the book makes a 113 00:05:47.120 --> 00:05:50.360 hard stop here. Before we start typing matrices into our code, 114 00:05:50.399 --> 00:05:52.920 we have to establish the physical boundaries that make this 115 00:05:53.000 --> 00:05:54.759 math valid in the real world. 116 00:05:54.639 --> 00:05:57.439 Because a mathematical model is useless if it doesn't reflect 117 00:05:57.439 --> 00:06:02.319 physical reality. The author outlines three foundational assumptions, basically the 118 00:06:02.360 --> 00:06:05.199 three golden rules of your mathematical model that must be 119 00:06:05.319 --> 00:06:07.199 satisfied before you can trust any output. 120 00:06:07.480 --> 00:06:10.759 Let's run through them. Rule number one, a linear relationship 121 00:06:10.839 --> 00:06:15.480 exists between an applied load and the resulting displacement, Meaning 122 00:06:15.680 --> 00:06:18.759 if you push a beam twice as hard, it bends 123 00:06:18.839 --> 00:06:20.600 twice as far, which makes. 124 00:06:20.439 --> 00:06:24.600 The principle of superposition valid. You can calculate the windload 125 00:06:24.639 --> 00:06:27.319 on a building, then calculate the gravity load of the 126 00:06:27.319 --> 00:06:29.800 snow on the roof separately, and just add the two 127 00:06:29.920 --> 00:06:30.600 results together. 128 00:06:31.279 --> 00:06:34.160 Simple enough. Rule number two is that the material must 129 00:06:34.240 --> 00:06:38.120 obey Hook's law. This means the structural material is not 130 00:06:38.199 --> 00:06:39.959 stressed beyond its elastic limit. 131 00:06:40.160 --> 00:06:42.920 Right, if you step on a steel beam, it bends slightly. 132 00:06:43.399 --> 00:06:45.319 When you step off, it acts like a very stiff 133 00:06:45.360 --> 00:06:48.639 spring and snaps perfectly back to its original straight shape. 134 00:06:48.680 --> 00:06:50.360 But if you overload it to the point that the 135 00:06:50.360 --> 00:06:54.319 steel permanently deforms, meaning it yields, Hook's law goes out 136 00:06:54.360 --> 00:06:55.639 the window right completely. 137 00:06:56.079 --> 00:06:59.600 Your stiffness matrix is suddenly outputting garbage because the physical 138 00:06:59.639 --> 00:07:01.759 nature the material has fundamentally changed. 139 00:07:02.040 --> 00:07:04.360 That brings us to rule number three. And I have 140 00:07:04.439 --> 00:07:06.040 to be honest. As I was reading this section, I 141 00:07:06.079 --> 00:07:10.079 found myself thoroughly confused. The text states that the equations 142 00:07:10.120 --> 00:07:13.720 of static equilibrium are developed using the geometry of the 143 00:07:13.920 --> 00:07:15.000 undeflected model. 144 00:07:15.120 --> 00:07:15.319 Ah. 145 00:07:15.399 --> 00:07:19.480 Yes, it says the changes in shape are considered negligible. Wait, 146 00:07:19.600 --> 00:07:21.600 if the whole point of the stiffness method is to 147 00:07:21.639 --> 00:07:25.319 place those digital tracking dots to measure displacement to find 148 00:07:25.360 --> 00:07:28.839 out exactly how much the joints move, why are we 149 00:07:29.000 --> 00:07:33.600 mathematically pretending the geometry is undeflected. Isn't that a massive contradiction? 150 00:07:34.040 --> 00:07:36.839 This raises an important question, and it really sounds like 151 00:07:36.879 --> 00:07:40.439 a contradiction. It's a major stumbling block for students when 152 00:07:40.439 --> 00:07:44.079 they first transition from theory to coding. To untangle it, 153 00:07:44.319 --> 00:07:47.000 you have to think about the physical scale of civil engineering. 154 00:07:47.319 --> 00:07:49.360 Okay, put me in a real world scenario. Let's say 155 00:07:49.360 --> 00:07:52.279 we have a massive steel beam in a commercial building. 156 00:07:52.399 --> 00:07:54.959 Sure, let's say that beam is five thousand millimeters long. 157 00:07:55.360 --> 00:07:57.360 Under a heavy floor load, that beam is going to 158 00:07:57.399 --> 00:08:00.480 bend and the joints at the ends will displace. We 159 00:08:00.600 --> 00:08:04.639 absolutely want to calculate that displacement. However, for a beam 160 00:08:04.720 --> 00:08:08.399 that massive, the actual downward movement might only be ten millimeters. 161 00:08:08.759 --> 00:08:11.360 Oh wow, okay, ten millimeters of movement on a five 162 00:08:11.399 --> 00:08:14.480 thousand millimeter beam. It's a tiny fraction, exactly. 163 00:08:14.800 --> 00:08:20.000 The micro movements are infinitesimally small relative to the overall geometry. 164 00:08:20.240 --> 00:08:22.360 If we tried to update the geometry in our mat 165 00:08:22.480 --> 00:08:26.639 Lab code for every single millimeter the beam deflected, constantly 166 00:08:26.639 --> 00:08:29.319 reclculating the new angles of the joints as they sagged, 167 00:08:29.879 --> 00:08:32.120 the math would become nonlinear. 168 00:08:31.600 --> 00:08:33.279 And we don't want nonlinear maths. 169 00:08:33.360 --> 00:08:37.159 We really don't. The equations would constantly be changing themselves. 170 00:08:37.720 --> 00:08:41.279 Nonlinear math is an iterative nightmare to solve, taking massive 171 00:08:41.279 --> 00:08:45.519 amounts of computational power by using the undeflected original shape. 172 00:08:45.519 --> 00:08:49.159 To write our baseline equilibrium equations, we acknowledge that the 173 00:08:49.200 --> 00:08:52.919 angles haven't drastically changed enough to alter the fundamental physics 174 00:08:52.919 --> 00:08:53.480 of the loads. 175 00:08:53.639 --> 00:08:56.879 It keeps the matrix math linear and actually solvable, so 176 00:08:56.919 --> 00:08:59.600 we aren't ignoring the movement. We're just making a pragmatic 177 00:08:59.639 --> 00:09:03.320 engineering assumption. We assume a ten millimeter SAG doesn't turn 178 00:09:03.320 --> 00:09:05.759 a horizontal beam into a diagonal one. Right. 179 00:09:05.960 --> 00:09:08.559 It is the only way to make commercial structural analysis 180 00:09:08.600 --> 00:09:12.480 possible without requiring a supercomputer for every small office building. 181 00:09:12.759 --> 00:09:17.000 That makes perfect sense. So with our stiffness methods selected 182 00:09:17.080 --> 00:09:20.039 and our three golden rules firmly in place, we can 183 00:09:20.080 --> 00:09:22.919 finally do what this book does best, translate a physical 184 00:09:22.919 --> 00:09:26.840 structural beam into a digital matrix. Let's start with planar 185 00:09:26.960 --> 00:09:28.120 orthogonal structures. 186 00:09:28.399 --> 00:09:31.200 Just a standard two dimensional grid where all the columns 187 00:09:31.200 --> 00:09:34.000 and beams meet at right angles, the training wheels of 188 00:09:34.000 --> 00:09:35.200 structural analysis. 189 00:09:35.279 --> 00:09:38.279 Right. But if I'm coding this, how does Matt lab 190 00:09:38.360 --> 00:09:41.360 notes looking at a thick concrete column versus you know, 191 00:09:41.600 --> 00:09:45.720 a flimsy wooden beam. A matrix is just a box 192 00:09:45.759 --> 00:09:48.840 of numbers. The text explains that we capture the physical, 193 00:09:48.960 --> 00:09:52.519 tangible properties of those materials and distill them into what's 194 00:09:52.559 --> 00:09:54.399 called a local stiffness matrix. 195 00:09:54.600 --> 00:09:56.799 Yeah, you can think of the local stiffness matrix as 196 00:09:56.799 --> 00:10:00.960 the personality profile of that specific structural member. It dictates 197 00:10:01.000 --> 00:10:04.600 exactly how stubbornly that particular beam will resist being pushed, bent, 198 00:10:04.759 --> 00:10:05.320 or stretched. 199 00:10:05.440 --> 00:10:08.120 Here's where it's really interesting, because that profile is built 200 00:10:08.159 --> 00:10:12.159 from specific physical traits. First is E, the modulus of elasticity, 201 00:10:12.320 --> 00:10:15.519 which represents the raw material itself, like steel has a 202 00:10:15.519 --> 00:10:17.559 different E than concrete exactly. 203 00:10:17.799 --> 00:10:21.519 And second is I, the moment of inertia, which represents 204 00:10:21.559 --> 00:10:24.440 the cross sectional shape of the beam, and I beam 205 00:10:24.600 --> 00:10:27.320 resists bending very differently than the solid square tube. 206 00:10:27.320 --> 00:10:30.320 So you multiply E and I together and you get 207 00:10:30.399 --> 00:10:33.639 the flexoral rigidity, basically the beam's resistance to bending. 208 00:10:33.759 --> 00:10:36.320 You also factor in the cross sectional area A and 209 00:10:36.360 --> 00:10:40.080 the total length L to account for axial deformation. That 210 00:10:40.120 --> 00:10:43.080 tells the computer how much the beam stretches or squishes 211 00:10:43.080 --> 00:10:44.600 along its length when compressed. 212 00:10:45.039 --> 00:10:47.840 The book breaks down how to generate this local matrix, 213 00:10:47.879 --> 00:10:51.600 and it spends a lot of time calculating specific rotational coefficients. 214 00:10:52.039 --> 00:10:55.279 It derives values like four Ei over L and two 215 00:10:55.360 --> 00:10:58.480 Ei over L I understand, the ei and L are 216 00:10:58.480 --> 00:11:01.000 our physical properties. But where do the four and the 217 00:11:01.000 --> 00:11:03.399 two come from? Why are those numbers so important? 218 00:11:03.679 --> 00:11:06.320 Well, this touch is on a really elegant physical phenomenon 219 00:11:06.360 --> 00:11:09.559 called the carryover moment. I mean, imagine you are holding 220 00:11:09.639 --> 00:11:12.000 a physical ruler in front of you with both hands. 221 00:11:12.080 --> 00:11:12.879 Okay, got it. 222 00:11:13.039 --> 00:11:15.440 If you forcefully twist your right wrist to bend the 223 00:11:15.519 --> 00:11:17.799 right end of the ruler, what happens to your left hand. 224 00:11:17.879 --> 00:11:19.879 I'd feel the ruler trying to twist my left hand too. 225 00:11:20.159 --> 00:11:22.919 Exactly, even though you only applied force to one end, 226 00:11:23.200 --> 00:11:25.960 the physical stiffness of the material carries a portion of 227 00:11:25.960 --> 00:11:29.039 that twisting force over to the other end. In linear 228 00:11:29.080 --> 00:11:31.639 structural physics, if you apply a bending moment to one 229 00:11:31.720 --> 00:11:34.200 end of a beam, that's the four ei over L 230 00:11:34.399 --> 00:11:37.919 part of matrix, the beam inherently transfers exactly half of 231 00:11:37.919 --> 00:11:40.200 that force to the far end that's the two Ei 232 00:11:40.279 --> 00:11:40.720 over L. 233 00:11:41.320 --> 00:11:43.240 Ah. So it's a ratio. The four and the two 234 00:11:43.360 --> 00:11:45.919 just represent that a beam transfers fifty percent of the 235 00:11:45.919 --> 00:11:48.080 twisting force to the opposite giant. 236 00:11:48.200 --> 00:11:50.799 Yeah, it mathematically links the two ends of the beam together. 237 00:11:50.840 --> 00:11:54.080 In the code a rotation at node A automatically creates 238 00:11:54.080 --> 00:11:55.360 a predictable force at node B. 239 00:11:55.639 --> 00:11:57.600 So when you are writing a matt Lab script, you 240 00:11:57.639 --> 00:12:01.200 are essentially constructing a giant grid of these coefficients for 241 00:12:01.279 --> 00:12:04.879 every beam. But the text emphasizes that to do this accurately, 242 00:12:05.080 --> 00:12:07.639 you have to meticulously separate your degrees of freedom in 243 00:12:07.679 --> 00:12:11.639 the matrix. It categorizes unrestrained degrees of freedom labeled as 244 00:12:11.919 --> 00:12:13.840 U from restrained ones labeled ER. 245 00:12:14.320 --> 00:12:17.480 This partitioning is crucial for the math to work. An 246 00:12:17.559 --> 00:12:21.279 unrestrained degree of freedom, the U is a joint up 247 00:12:21.279 --> 00:12:23.440 in the air that is free to move and bend. 248 00:12:23.960 --> 00:12:26.679 A restrained one air is a support like a column 249 00:12:26.759 --> 00:12:29.039 bolted into a massive concrete. 250 00:12:28.559 --> 00:12:31.960 Foundation, because we know the foundation isn't going to move right. 251 00:12:32.279 --> 00:12:34.960 By mathematically separating the elements that can move from the 252 00:12:35.000 --> 00:12:38.440 elements that have zero displacement, we can shrink the massive 253 00:12:38.440 --> 00:12:43.440 matrix down into smaller, manageable sub matrices that only focus 254 00:12:43.519 --> 00:12:44.639 on the unknown movements. 255 00:12:45.080 --> 00:12:47.480 And once we've partitioned the matrix, how do we actually 256 00:12:47.480 --> 00:12:50.759 get the answers? The text talks about assembling the global 257 00:12:50.759 --> 00:12:54.799 stiffness matrix and then inverting a specific sub matrix for 258 00:12:54.879 --> 00:12:56.480 those of us who are a bit rusty on our 259 00:12:56.559 --> 00:13:00.559 linear algebra. What is physically happening when matt Lab inverts matrix? 260 00:13:00.799 --> 00:13:03.559 Think of it like basic middle school algebra. The fundamental 261 00:13:03.600 --> 00:13:06.440 equation of structural analysis is F equals k times d. 262 00:13:06.960 --> 00:13:10.879 Force equals stiffness multiplied by displacement. We know the external 263 00:13:10.879 --> 00:13:12.759 forces like the wind or the weight of the floors. 264 00:13:12.799 --> 00:13:15.559 We just built the stiffness matrix the k. What we 265 00:13:15.600 --> 00:13:17.759 want to find is the displacement D how much the 266 00:13:17.759 --> 00:13:18.480 building moves. 267 00:13:18.759 --> 00:13:22.639 So in basic algebra, to solve for D, you would 268 00:13:22.679 --> 00:13:24.159 just divide force by stiffness. 269 00:13:24.240 --> 00:13:27.639 Yes, but in matrix mathematics there is no such thing 270 00:13:27.639 --> 00:13:30.840 as standard division. You can't just hit the divide key 271 00:13:30.879 --> 00:13:34.200 on a ten thousand cell grid of numbers. Instead of dividing, 272 00:13:34.440 --> 00:13:38.000 you multiply by the inverse matrix. Inversion is the mathematical 273 00:13:38.000 --> 00:13:39.039 equivalent of division. 274 00:13:39.279 --> 00:13:40.720 Oh wow, okay. 275 00:13:40.480 --> 00:13:43.679 So when your Matlab code inverts the stiffness matrix and 276 00:13:43.759 --> 00:13:47.320 multiplies it by the force vector, it is simultaneously solving 277 00:13:47.399 --> 00:13:51.279 thousands of algebraic equations to find the exact displacement of 278 00:13:51.320 --> 00:13:52.240 every single joint. 279 00:13:52.279 --> 00:13:54.759 And once we finally know exactly how much every joint 280 00:13:54.919 --> 00:13:58.159 rotated and shifted. We plug those displacement answers back into 281 00:13:58.200 --> 00:14:01.200 a specific formula for each in individual beam to find 282 00:14:01.200 --> 00:14:04.559 the internal forces, and this involves adding something called fixed 283 00:14:04.759 --> 00:14:06.039 end moments or FEM. 284 00:14:06.360 --> 00:14:08.720 The fixed end moments are vital if you have a 285 00:14:08.720 --> 00:14:10.759 heavy piece of machinery sitting right in the middle of 286 00:14:10.759 --> 00:14:14.559 a beam. That weight creates internal stresses inside the beam 287 00:14:14.600 --> 00:14:17.919 before the joints even start to move. The FEM represents 288 00:14:17.960 --> 00:14:19.840 the forces that would exist if both ends of the 289 00:14:19.879 --> 00:14:21.799 beam were glued in place and couldn't move at all. 290 00:14:22.120 --> 00:14:24.759 So the final total force on a beam is a 291 00:14:24.759 --> 00:14:28.360 combination of the external loads sitting directly on it the FEM, 292 00:14:28.399 --> 00:14:31.600 plus the forces transferred into it by the joints twisting 293 00:14:31.600 --> 00:14:32.200 and moving at. 294 00:14:32.159 --> 00:14:35.559 The ends exactly. It is an incredibly comprehensive way to 295 00:14:35.679 --> 00:14:38.039 track every ounce of stress in a structure. 296 00:14:38.200 --> 00:14:40.120 Now, a two D grid of right angles is a 297 00:14:40.120 --> 00:14:42.320 great way to learn the ropes. But like we said 298 00:14:42.360 --> 00:14:44.759 at the start, we don't just build brick boxes anymore. 299 00:14:44.879 --> 00:14:47.559 A two D grid works for a billboard, but what 300 00:14:47.639 --> 00:14:51.360 if we are designing a stadium, dome or a truss 301 00:14:51.399 --> 00:14:55.799 bridge with hundreds of diagonal crossbraces. How does our matt 302 00:14:55.879 --> 00:14:59.200 Lab code handle members that aren't perfectly horizontal or vertical. 303 00:14:59.320 --> 00:15:01.519 That's where things a get tricky. The text tackles this 304 00:15:01.600 --> 00:15:05.559 in chapters two and four, moving into planar non orthogonal. 305 00:15:05.080 --> 00:15:08.240 Structures right, because the moment you introduce a slanted beam, 306 00:15:08.559 --> 00:15:11.519 you run into a fundamental geometric clash. Let's say you 307 00:15:11.519 --> 00:15:14.639 have a diagonal beam holding up a roof. That beam 308 00:15:14.679 --> 00:15:17.799 has its own internal local axes, let's call them by 309 00:15:17.799 --> 00:15:20.080 two and y two, which runs straight down the center 310 00:15:20.080 --> 00:15:21.600 of the beam and perpendicular to it. 311 00:15:21.879 --> 00:15:24.039 Because if you push directly down on the top of 312 00:15:24.080 --> 00:15:27.320 that slanted beam, the force isn't traveling straight into the ground, 313 00:15:27.399 --> 00:15:29.559 It's traveling diagonally down the link of the seal. 314 00:15:29.840 --> 00:15:32.960 But the building as a whole, and the global maclix 315 00:15:32.960 --> 00:15:36.279 you're building in matt Lab operates on a standard global 316 00:15:36.320 --> 00:15:40.039 axis system, just standard vertical gravity and horizontal win by 317 00:15:40.039 --> 00:15:43.000 one and why one. So you have a beam feeling 318 00:15:43.080 --> 00:15:48.240 forces diagonally, but your global matrix only understands up, down, left, 319 00:15:48.320 --> 00:15:48.799 and right. 320 00:15:49.159 --> 00:15:52.200 And to bridge that gap, the book introduces a really 321 00:15:52.240 --> 00:15:55.759 powerful tool called the transformation matrix, denoted as a T 322 00:15:56.159 --> 00:15:56.879 in brackets. 323 00:15:57.399 --> 00:15:59.600 I find this elegant because the core code we just 324 00:15:59.600 --> 00:16:03.480 discussed the local stiffness matrix with this four ei over 325 00:16:03.679 --> 00:16:06.360 l it doesn't actually change. You just add a transformation 326 00:16:06.440 --> 00:16:08.360 step before you drop it into the global model. 327 00:16:08.480 --> 00:16:12.159 Right, it relies purely on basic trigonometry by determining the 328 00:16:12.200 --> 00:16:15.000 angle of the diagonal beam let's call it data. The 329 00:16:15.039 --> 00:16:17.960 transformation matrix uses the sign and cosign of that angle. 330 00:16:18.399 --> 00:16:20.960 If a force is pushing diagonally down into the right, 331 00:16:21.279 --> 00:16:24.080 the cosine function figures out exactly what percentage of that 332 00:16:24.120 --> 00:16:26.399 force is pushing purely to the right, and the sign 333 00:16:26.440 --> 00:16:29.039 function calculates what percentage is pushing purely down. 334 00:16:29.279 --> 00:16:32.360 So you take your diagonal beams local matrix, multiply it 335 00:16:32.399 --> 00:16:36.440 by the transformation matrix, and it instantly translates those diagonal 336 00:16:36.519 --> 00:16:40.200 forces into standard vertical and horizontal components. Now it can 337 00:16:40.200 --> 00:16:43.039 speak the same language as every other beam in the structure. 338 00:16:43.159 --> 00:16:45.759 And if we connect this to the bigger picture, this 339 00:16:45.840 --> 00:16:49.720 trigonometric translation is the ultimate reason we chose the stiffness 340 00:16:49.759 --> 00:16:51.600 method over the flexibility method. 341 00:16:51.639 --> 00:16:52.840 Oh, because of the automation. 342 00:16:53.159 --> 00:16:56.480 Exactly, imagine a trust bridge with five hundred diagonal members 343 00:16:56.919 --> 00:17:00.080 all at slightly different angles, trying to manually balance all 344 00:17:00.080 --> 00:17:03.159 those diagonal forces using the flexibility method would take months, 345 00:17:03.759 --> 00:17:06.160 but in matt Lab using the stiffness method, you just 346 00:17:06.200 --> 00:17:07.279 write a simple for loop. 347 00:17:07.400 --> 00:17:10.759 The computer just grabs number one, looks at its THEATA angle, 348 00:17:11.160 --> 00:17:15.400 creates a transformation matrix, translates the local stiffness into global 349 00:17:15.480 --> 00:17:18.720 vertical and horizontal forces, and drops it into the master grid. 350 00:17:18.759 --> 00:17:20.799 Then it grabs number two and just repeats. 351 00:17:20.880 --> 00:17:24.960 It handles hundreds of diagonal beams effortlessly, which sets us 352 00:17:25.039 --> 00:17:27.720 up perfectly for when the text scales up into true 353 00:17:28.240 --> 00:17:29.720 three dimensional space frames. 354 00:17:29.960 --> 00:17:32.319 Now, jumping to three D just means we add the 355 00:17:32.440 --> 00:17:35.880 z axis for depth. Right, the textbook mentions yzx and 356 00:17:36.000 --> 00:17:38.240 zyx transformation sequences. 357 00:17:37.799 --> 00:17:41.000 Well, adding the z axis increases the variables exponentially. In 358 00:17:41.039 --> 00:17:43.240 a two D frame, a joint only has three degrees 359 00:17:43.279 --> 00:17:45.160 of freedom. It can move up and down, left to right, 360 00:17:45.200 --> 00:17:48.240 and rotate. In a three D space frame, every single 361 00:17:48.279 --> 00:17:51.119 joint has six degrees of freedom. It can translate along x, 362 00:17:51.240 --> 00:17:53.200 y and z, and it can twist around x, y 363 00:17:53.279 --> 00:17:53.519 and z. 364 00:17:53.880 --> 00:17:56.960 Oh wow, So the local stiffness matrix for a single 365 00:17:57.000 --> 00:17:59.960 beam expands from a six x six grid to two 366 00:18:00.119 --> 00:18:02.319 a massive twelve x twelve grid, which. 367 00:18:02.200 --> 00:18:05.119 Is why the order of those transformation sequences matters so much. 368 00:18:05.119 --> 00:18:07.920 In the math. You have to systematically rotate the axis 369 00:18:07.920 --> 00:18:10.680 in a specific order like y then z then x 370 00:18:11.039 --> 00:18:13.160 to align a three D beam to the global grid 371 00:18:13.559 --> 00:18:15.799 without the mathematical angles getting tangled. 372 00:18:15.519 --> 00:18:18.119 Up, and it introduces a unique physical quirk you have 373 00:18:18.160 --> 00:18:20.359 to control. In two D space, you just need an 374 00:18:20.480 --> 00:18:22.319 X and y coordinate for the start and end of 375 00:18:22.359 --> 00:18:24.200 a beam, but in three D space, knowing where the 376 00:18:24.240 --> 00:18:26.599 two ends are located isn't enough, right. 377 00:18:26.400 --> 00:18:29.279 The book introduces the spatial ci angle here. Think of 378 00:18:29.319 --> 00:18:31.799 a standard wooden ruler. If you hold it flat, you 379 00:18:31.799 --> 00:18:33.839 can easily bend it up and down, but if you 380 00:18:33.880 --> 00:18:37.200 turn it ninety degrees on its edge, it becomes incredibly stiff. 381 00:18:37.240 --> 00:18:38.359 You can barely bend it at all. 382 00:18:38.480 --> 00:18:39.440 Okay, that makes sense. 383 00:18:39.559 --> 00:18:42.559 Now, imagine an ibeam in a three D computer model. 384 00:18:43.200 --> 00:18:46.440 The computer knows the exact coordinates of node A and 385 00:18:46.519 --> 00:18:49.200 node B, but it doesn't know if that ibam is 386 00:18:49.319 --> 00:18:53.160 lying flat on its side or standing upright on its strong. 387 00:18:52.920 --> 00:18:55.839 Edge, because the beam could be rolled or twisted along 388 00:18:55.880 --> 00:18:57.319 its own axis exactly. 389 00:18:57.599 --> 00:19:00.960 That is the ci angle. It defines this facial orientation 390 00:19:01.359 --> 00:19:04.400 the role of the beam. It ensures the computer aligns 391 00:19:04.440 --> 00:19:08.079 the strong axis of the material correctly before it calculates 392 00:19:08.119 --> 00:19:09.039 the bending forces. 393 00:19:09.359 --> 00:19:12.119 Without the CI angle, the software might assume a beam 394 00:19:12.240 --> 00:19:15.079 is sitting on its strong edge when it's actually lying flat, 395 00:19:15.359 --> 00:19:18.279 which would lead to a catastrophic under calculation of the 396 00:19:18.279 --> 00:19:19.119 building safety. 397 00:19:19.240 --> 00:19:22.559 It's so much physical reality captured in sine cosin and 398 00:19:22.599 --> 00:19:23.400 a matrix grid. 399 00:19:23.599 --> 00:19:26.720 It really is. As we wrap up our deep dive 400 00:19:26.759 --> 00:19:30.759 into doctor Schendersachern's text, let's retrace the path we took today. 401 00:19:31.799 --> 00:19:35.319