Okay everybody we're gonna get started here. I'm still admitting students. Into this meeting remind me a few minutes once everybody's in to make that more general announcement about. How to login properly I am as you guys probably know I'm gonna be recording these I'm also recording the chat window, so if you ask a question in the chat window, and for some reason I miss it during the lecture which I hope not to do but if I do you I'll follow up with you after the meeting is over. 

Turn on my camera here, so you can see my smiling face. Now we'll go ahead and get started with the lecture a couple of announcements first as you are probably aware of the first homework set was posted. Something that's kind of notable this year with homework. I'm trying something new something. 

I've never tried before we're gonna do it for this first homework set. I anticipate doing it throughout the semester, although I can't be sure. I guess it depends a little bit on how well it goes this first time basically what I'm using is some software that helps me to standardize how I'm grading prevents me from having to memorize how many points I'm taking off as deductions for various mistakes that occur that can be error prone. 

It allows me to standardize the comments that I give to you guys as students, so I'm looking forward to trying that. I think it'll be helpful in fact this this time around it will be the TA as Andrew who will be using that software first before I do but either way the the point of the software is to standardize the format to the greatest extent possible make things as fair and efficient as possible in the in the way that grades are evaluated and submitted. 

In order to accomplish that it's actually quite helpful to have a standard format that you submitted in as well, so I am requiring that you submit in a specific format if you see if you've downloaded and begun the homeworks that you already know what I'm talking about if you haven't I'll look carefully at the instructions when you receive them basically what I'm asking you to do is to print the problem set and then make sure that your answer is contained in a specific area on the sheet and that will help when when the grading is occurring for me to be able to find. 

Your answer as quickly as possible and and helps to standardize the process now. I understand that that is a bit more work for you potentially I do value your time more than you may realize. I don't want you to have to do extra work. I think in this case the small increment of additional work actually pays great dividends for everybody and also allows me to save a lot of time for myself, so I appreciate that you're gonna be willing to do that work for me. 

I do ask that you do it, of course. Do your best if you need additional sheets of paper in order to complete the work if I didn't leave you enough room to do the work then attach those at the end of your homework set rather than in the middle if you need to if you don't have a printer and a scanner available to you at home then this time around I'm gonna we're gonna be open-minded we're gonna try our best it's certainly possible to use a phone camera the PDF type of app these days to take pictures have the camera. 

On your phone assemble it into a PDF and then you can upload that PDF directly as a submission. I'm okay with you attempting to do that, please do try to follow the format that is required and really the format in this case is that the answer needs to be towards the bottom of the page there's a defined box if you print it if you haven't printed it yourself and I'd ask you to create your own box that you know loosely mirrors the box that I've created and try and make sure that your answer lives in that place so you do have my permission. 

And I've been asked by a couple of students you do have my permission to try that we'll do it this time around if everything goes well if you guys are doing a good job putting together those packets that PDF then likely will be able to continue that if things don't go well for some reason then I might have to be a little bit more strict next time, you know, again, it's not about me being strict about the format it's rather about just making the format work well and so I'm I'm just hopeful that that's gonna work really well for all of us. 

So reminder, of course that that will be due on Thursday, you'll submit it to e-learning prior to the lecture start the lecture starts at 2:30 the e-learning drop box closes at 2:30 sharp and don't forget that there is no late submission of homework, so make sure to get it up there on time. 

I see a couple of questions have come in Ethan has said in person senior design got out late, so that's probably why part of the people are still logging on okay thanks Ethan for letting me know. I appreciate that. Caden says is there any issue with digitally writing on the document? 

I have no issue with you digitally writing on the document if you want to mark up a PDF that's perfectly acceptable. I would prefer that you not type equations in an unformatted format, you know, it is possible of course to use. Software when you're you know, mathematical manipulation software and or for example matlab has a symbolic manipulation tool, you're not allowed to just submit the the output of the mathematical tool if it's in a non-human digestible format. 

I won't accept that but if you want to type something over an image on a PDF in order to make it more legible that would be great anything that helps me read. And helps increase the clarity is probably acceptable and anything that detracts from the clarity makes it more difficult that's of course the opposite going to be unacceptable. 

Okay, I'm not seeing any other questions on that at the moment so I'll go ahead and move on. Today's lecture is largely going to be an introduction to aircraft structures from a historical perspective sort of how did we get where we are now? We're not going to go back thousands of years in structural. 

Analysis technique but we will go back to about the era of the Wright brothers that's an uh, 1903 actually just slightly before that and bring in those perspectives and then we'll talk about some modern topics as well in this process. So before we actually get into the historical perspective on structural mechanics, I want to introduce you to a or give you an overview of aerospace engineering what I'm going to call the subdisciplines of aerospace engineering. 

What do I mean by that? If you look at if you classify expertise within aerospace engineering, usually the way people view that expertise is to fall into one of several broad categories. The, It occurs to me now just let me pause for a moment to my sharing my screen, can you see my screen? 

I mean, let me double check that my screen is being appropriately shared. I apologize if it has not been. 

It looks like I was not sharing my screen, don't worry you haven't missed anything yet. There we go. That's better. So those those subdisciplines of of aerospace engineering are sort of broadly classified into these. One would be aerodynamics perhaps the most famous of our disciplines. There are propulsion systems. 

Of course that's how we generate the forces necessary to propel our aircraft and spacecraft through the atmosphere and in space. There's the overall subdiscipline of control systems which show you guys are all familiar with. A rather complex, but outside of the scope of this particular class. A relatively new sub-discipline of expertise is one that has come in to the forefront in the last say two decades that's called system engineering. 

That's instead of looking at individual technical subdisciplines, if you look at the whole system as one sort of optimization global optimization to view and so system engineering is is relatively new as a as a sub-discipline of aerospace. And engineer. During its relatively new as an area of expertise or study within our program but now it has achieved essentially equal standing as these other four in terms of broad scope and applicability and you know recognize technical expertise. 

And the last but of course not the least is the structural mechanics sub-discipline, and that's what this course is all about as well as at least a half dozen to a dozen courses in the graduate curriculum. What is structural mechanics? Well, instructional mechanics, we focus on static and dynamic equilibrium. 

In this class, we're going to focus primarily on static equilibrium structures also includes the concept of strength of materials and how materials fail. We have some sub expertise in static stability and control. Now this is different from stability in the control systems or stability in a vehicle sense like an aerodynamic sense for an aircraft. 

We have concepts of static stability the most prominent of which is buckling if you have ever seen a panel buckle on an aircraft or if you've seen, you know, even something as simple as a piece of spaghetti break as you push on it along its axe. Because it it bows into a curved piece of spaghetti before ultimately failing that concept falls into the general a subdomain of stability and control and then of course we have dynamic stability, this is the concept of air elasticity this is how fluids and structures interact. 

We of course want to have dynamic stability in our aircraft so we don't get something like a wing divergence which would cause a structural failure and then lastly again within this general area of structural mechanics is the concept of integrated structural and other functions. So, for example, you might have a fuel tank that's made of a porous foam that porous foam, of course is where the fuel could be contained and then perhaps a metallic foam is really what I'm thinking of. 

Here and then that can be quite structurally rigid it can also be used for cooling the aircraft if you've got external skins of the aircraft that are let's say forward facing and therefore, you know, in a in a mock scenario getting very hot you can actually use fuel to cool the structure of the airframe if you've got a well-integrated structure that perhaps as a multifunction structure. 

So those are some examples that we'll see in this class. 

Now, let me mention some challenges associated with structural mechanics. These are important challenges things that we as structural engineers need to meet. Number one, of course is safety. You may or may not know but travel. By commercial airline is one of the safest forms of travel that exists on the planet. 

It is no accident that the aerospace industry has done suction such an excellent job in safety. It is very rare that we have a structural failure that leads to a catastrophic failure and loss of a flight vehicle. Of course, it's not unheard of it happens. It's happening much less now than it has in past years, but we need to design our structures to be safe. 

Now, what is safe mean? In the aerospace industry. That means amongst other things likely that your structure needs to have some redundant load carrying capacity. So that not so that there's no single source of failure that can cause a catastrophic loss of the vehicle. Now, that's not possible for all structures, but it is in fact required for many of our structures as well as many other, you know, functional devices within an aircraft that don't fall under the structures sub-discipline. 

We want to have no failure. Now we want to draw a distinction between local and global failures we'll do that later in the semester but what does that mean we we don't want failures that occur that cause catastrophic loss of the vehicle. However, if we have a failure of a structural component that because of redundancies allows us to still safely operate that vehicle down to the ground for a repair or replacement of that part. 

That is also considered to be a safe aircraft. Now, we can make our aircraft even safer than we do however there are challenges associated with doing so the most important of those is that instruction mechanics, we're not allowed not really permitted to overdesign our system. We can't put in more structure than it's actually necessary and of course that's because weight is equal to cost weight is the enemy of an aircraft the heavier the aircraft the poorer the fuel economy is the poor the performances the longer the takeoff distance. 

The faster the takeoff speed the longer the landing speed the faster the landing speed the more drag all of those things come directly as a result of weight. So if you over-design a structure in this discipline, you you really are being quite detrimental to the overall performance of that vehicle and perhaps can make it an infeasible design or one that is not competitive. 

There's very little room for error and in aerospace structures we we have to get it right the first time what do I mean by that failure can be fatal, you know, if we have a flight test of vehicle that ultimately fails in it, it causes a fatality the whole world knows about it, it becomes news very quickly redesign of that vehicle to fix any issues that have arisen can be extremely costly for a large company like Boeing. 

It has a huge, you know penalty in public confidence has a huge impact on their financial bottom line for a smaller company, like what some of you may end up working for when you graduate, you know, a redesign could be costly enough that the the company doesn't even survive by the way. 

I shouldn't imply that you wouldn't work for a large company we have many of our graduates who are not who are now working for large aerospace firms. 

So what's typically assigned to you as a structural engineer knows aerospace, you know, I'm giving a few simplistic examples. But nevertheless it gives you an idea of what you might expect you as a designer rarely as a structural designer. I should say rarely have the ability to impact the overall surface shape of the vehicle the external skins of the vehicle are largely designed by aerodynamicists and they are then dictating what their needs are down to your level as a structural engineer in in the work in the modern world where we have a system engineer who works with the aerodynamic system. 

Engineers and tries to optimize the overall vehicle as opposed to any one particular component the structural engineer may have a little bit of influence in that sense but you know largely the external skins of an aircraft our our dictated by our colleagues on the aerodynamic side and so we have to live within the volume that they allow us as we design our structure. 

The second thing that we're typically given is a set of loads that we must be able to carry again these could be a loads in a number of forums some examples of loads that are fairly typical are the aerodynamic loads those are the lift loads the drag loads also the inertial loads that those are the loads associated with rapid changes of direction in some sort of high-speed maneuver, we also of course need to think about similar inertial loads that may be classified in other ways, for example landing gear loads when you come in, A for a landing often it's assumed to be a controlled crash for from the structural engineers perspective you meet a certain obligation to carry a load factor relative to the weight of the aircraft on the structure and so those things are what are typically given. 

Was required of us as structural engineers we need we need to design a structure that of course is safe and safe means no failures that's a given but perhaps even before we get to the point of no failures we need to meet certain stiffness requirements, so the wing deflection can only be a certain level in order to maintain our aerodynamic performance things like the fuselage can't deform too much or we get you know, some sort of air wellastic loop that occurs that causes, you know, Uncontrollable flight so we need to be able to meet certain stiffness requirements even something as simple as making sure the wing tips don't strike the ground when you fill the aircraft with fuel that there's a famous aircraft which at some point will discuss designed by Bertrand where that was a particular issue, so again, we'll we'll get back to that at some point in the future, but these are the type of requirements that we need to meet we need to meet a certain stiffness now stiffness is generally considered to be the relationship between the load and the over. 

All deflection of the structure. We may also be required to deliver a set of materials, so when we when we design our aircraft structure, we're specifying the geometry within the volume that we're permitted and we also need to specify the materials that are being used and those materials are often very much dependent on the flight envelope of the aircraft if it's a supersonic aircraft we made need to have external skins that can manage much higher temperatures if we're in another area where temperatures are really critical would be in an engine structure. 

Let's say we're in the nozzle of of a jet engine obviously we need to live within a very high temperature environment higher temperature isn't the only challenge of course low temperatures are also quite relevant to a flight vehicle if you're up at altitude the temperature can be about minus 40 and so you know, we need to be able to manage those temperature extremes we need to specify materials that are able to survive the cyclic loads that occur. 

From a fatigue perspective. Such that over the service life of the vehicle there's no fatigue crack propagation that causes a catastrophic failure and so these are the things that are sort of the predominant things that we need to deliver is a is a certain stiffness with the geometry associated with it and a certain set of material selections in order to meet our requirements. 

Now of course, we're not living in a constraint-free world so we have many constraints the first and and foremost constraint in any aerospace vehicle, of course that is is that we need to have a fairly optimal structure so that it can be as lightweight as possible for all the reasons that I alluded to previously. 

We need to have sufficient interior spaces for example to contain fuel for example to contain passengers and payload we need to meet manufacturing constraints, of course the things that we design need to be able to be manufactured that includes things like the ability to you know, even fit a fuselage in an autoclave is one particular example of a of a manufacturing constraint, you know, one of the reasons that fuselages. 

Are often assembled from barrels, ie this the length of the fuselage is is cut up into smaller sections which are eventually joined together is because of things like manufacturing and transportation constraints, we need to be able to manufacture it and then get it to the assembly plant, of course cost is always a constraint. 

Cost the more costly something is the fewer people will buy it of it the less profit is available to be made for obvious reasons and then of course the last and and perhaps unsung of all the constraints is the maintenance constraint. Why do I say maintenance is such a constraint many of the aircraft that are flying today are decades older than anybody that is present here today virtually, you know, the aircraft that I fly. 

I'm a pilot I fly an aircraft that was made in 1963, the only way that in aircraft manufactured in 1963 can still be flying in 2020 is through maintenance and so we need to be able to design the structure that is is is well maintained. Now the average age of the general aviation fleet is well over 40 years. 

The commercial fleet, of course is newer but nevertheless the length of time that a commercial airliner is in service is measured in decades many decades often the flight vehicle that you designed today will outlive your professional career, if not outlive you as an individual. I'm gonna pause for a moment and look and see if there are any questions. 

So Caden asks a question thanks Caden for the question is the center of gravity of the craft typically given of course the center of gravity of the aircraft is something that needs to be computed and tracked very carefully where I should say really the center of gravity envelope of the aircraft because typically in any airframe the center of gravity is is changing continuously from a structural engineers perspective. 

We need to I guess the answer to that is yes, we need to make sure that our structures allow the aircraft to live within its CG envelope that includes all the passenger and payload loading so at the end of the day we need to make sure that our weight distributions are appropriate so what a great question Caden next time. 

I teach this class. I'm gonna include a bullet point about that, so great thanks for the question. 

I think that looks pretty good for questions for the moment. So again, what are some of the loads that are specified? I mentioned these earlier I should have gone ahead and flip to this slide. Many of the loads are aerodynamic. I explicitly had mentioned lift and drag previously. Now, we can't forget pitching moments. 

So these are not only the pitching moment of the entire aircraft which needs to be controlled typically through a downforce on the horizontal tail balancing the overall pitching moment to the aircraft. But also these this includes pitching moments specifically along a lifting surface. So obviously a asymmetric airflow will generate a pitching moment as it generates lift and that pitching moment generates a torque and that torque needs to be carried throughout the entirety of the wing into the airframe reacted at the fuselage and then ultimately carry to that tail. 

So those are some examples of the aerodynamic loads. The thrust load, of course is another very large point source. Of load in an aircraft. So this will be typically assumed to be a point source from the perspective of the flight vehicle the structural engineer. If you're far field from that point source, you can treat it as a point source. 

If you're actually looking at the hard mounting points of the engine on the aircraft, then you're going to consider it perhaps as a more distributed load, but the thrust loads are some of the most substantial loads on an aircraft. We're talking about aircraft engines on a modern aircraft can be. 

You know upwards of a hundred thousand pounds of thrust and more on a lift I should say on a launching vehicle you could potentially be talking about millions of pounds of thrust when you aggregate all the rocket nozzles and rockets together. I mentioned the inertial loading that's the landing of the dynamic maneuvers. 

I didn't mention gust but that you know gusts ultimately manifest themselves as aerodynamic loads in maneuvers, and ultimately become what amounts to. An inertial load because the aircraft is changing direction quickly. So you you not only have the aerodynamic load associated with the you know, change in aerodynamic forces on the wing but you then also get the loads associated with the movement of the mass and so in some cases even though the aerodynamic loads would be fine for the wing we might have internal loads that could be problematic for internal load paths, you know, for example, you could have a, Overhead bin fail as a result of the inertial loads because there's too much mass in the bin or the the bin itself doesn't have sufficient structural load carrying capacity. 

The last type of flight load that I haven't mentioned about but is quite prominent and important to us as aerospace structural engineers is vibrations vibration ultimately leads to fatigue and fatigue can lead to failure. So we don't cover vibration much in this class but nevertheless it's something that an aerospace structural engineer needs to have a pretty good concept of and be able to design around. 

So as I had mentioned a couple of times now weight drives geometry. You know, we need to have the lightest possible structures that we can design. Safely and this is really been true since the Dawn of Flight goes. In fact to evolution we can go back much further than the Wright brothers and recognize that that flight vehicles that is birds have even evolutionarily speaking recognized to the importance of weight in their own designs if you look at the bones of birds. 

They take on many characteristics of the structures that we're now designing as aerospace engineers that bones are hollow. They tend to have a foamy interior in this case, it would be called. Cancellous bone to make sure that they can carry some compression strength and so this has been true for for millennia since the Donna Flight obviously also true for us as modern aerospace engineers. 

So the requirement to have a lightweight structure drives us to designing very structurally efficient aircraft. 

What does that mean? What is structurally efficient mean? Well, you know, obviously since we're talking about weight we need to be able to carry as much load as we can at the lowest weight and in order to accomplish that we have to be very careful about how we manage the load paths in a flight vehicle. 

Local loads are transferred around the structure of the aircraft and ultimately our necessary to carry the payload. So, for example. Aerodynamic loads at the wind skins in order to carry you through the air as a passenger in an aircraft or as a pilot in the aircraft the wing skins are generating lift. 

Now those those skins as they generate lift the skins themselves tend to be very thin membranous or shell-like structures and so they're not particularly structurally sound in and of themselves. They form part of a of a system of structures that ultimately carries the load. So typically what happens is the wind skins very quickly offload the pressure onto a stringer. 

The stringer passes the load onto a rib and the rib passes the load onto a spar. So wing skin stringers ribs bars the spars then carry the load into the fuselage and then you can go further, of course the fuselage carries the load through the floor into your seat and then from your seat to your person. 

And so this principle of load transfer and load path management is one that's quite critical to us as structural engineers. The wing skins can be very thin and ultimately still be able to pass their load through to your to your body by offloading them onto structures that are purpose-built to carry different types of loads. 

It's a pressure load at the skin at the stringer, it becomes a local bending load which is passed to a rib which is again now a local bending load ultimately into the spars which becomes a global bending load and then into the fuselage. 

So I had a question in the chat from Steve Steve asked is vibration technically a cyclical load. Now, that's a great question. Steve thanks for asking. So the vibration in and of itself, you know, you're asking a bit more of a semantic question that that then a practical one so the vibration itself often will be not of great concern to a structural engineer, however what can occur is as something vibrates, of course, it's deforming it's deformed deforming quite quickly. 

And so when it's deforming it's straining and when you have strain, of course, you have stress. And so that vibration is ultimately manifesting itself as a strain and stress cyclical load on the material. So, you know, whether you're calling this a load depends on who you're talking to but for us as a structural engineer, we need to be able to carry those cyclic loads those vibration loads, we need to keep those loads low enough such that they don't induce a fatigue failure which in very simplistic terms is often keeping it before below of fatigue limit if we had something like a structural steel or perhaps some of the aluminum alloys. 

But. In a more complex sense when we start to talk about composite materials, they have no fatigue stress limit and so now it's a cyclical count of how many cycles to failure. So we need to design our structures such that the loads occur at loads that are small enough at stresses and strains that are small enough. 

That that structure can carry those you know, essentially and infinite number of time many millions or billions of times. 

Okay, let's divert for just a few minutes here. And talk about what the principal structural elements in an aircraft are. What when I say structural elements what I mean is that we typically classify our aircraft structures into types of structures that are more easily analyzed and you'll see what I mean in just a minute. 

Here's an example of that. 

Axial members, carry extensional and compressive loads. So when I say an axial load carrying member what I'm describing is something that predominantly has its loads in one direction. And it's typically carried along the axis of a geometry that's pretty well defined in that same direction. You can also call them columns. 

In fact, in architecture axial load carrying members are typically called columns less frequently that they're called columns in an aircraft structure but nevertheless it is another name that is commonly used. So here for example, we have the landing gear. Of an aircraft. This one happens to be a supersonic aircraft, although I don't recall what the aircraft is specifically you can see the load that is carried through that landing gear at least a substantial portion of that 80% of that is columnar. 

So it's carrying a predominantly axial load. Now, it's also carrying a bit of bending load and will get back to that in just a minute but if it's predominately carrying an axial load, we'll call an axial member. You'll see why we categorize it in this way as we get a little bit deeper into our analysis techniques, particularly our engineering analysis techniques such as beam theory, which is going to come in the middle third of the class. 

Well, the next classification of aircraft structures is something that we would call a bending member. Of any member, of course curious a bending moment. Well, what's a bending moment bending moment is something that is predominantly causes bending it's a force at a distance. So it is a torque in some sense. 

Torx or moments moments are torques, but in this case when we orient our structure. And the load relative to that structure, we now think of that structure as something that's causing a bending deformation. These are a subset of these bending members are called beams in fact the largest subset of these bending members are called beams and an example of a beam is the example that you're looking at here in this case, it's an aircraft inside the aircraft wing is a spar and that spar is we call it a spar an aerospace we would call it a beam in a more general engineering setting. 

And of course both of those would be classified as bending members. So forces acting at a distance carrying bending loads spending moments. That's what a beam is that's what a bending member is the most famous beam is probably an IBM you've seen those previously obviously that's what the photo is here on the left hand side, we don't have too many eye beams in aerospace but we have many beams that sure similar characteristics to the eye beam and we'll get into that later in the semester. 

Torsion members. Carry twisting moments or twerks. Torsion members now again, what's a torque a torque is mathematically similar to a bending moment, it's a force acting on a distance now what distinguishes bending moments from torx is the direction that is occurring relative to the geometric structure that makes up the member that carries it. 

So if I were to go back one slide for a moment bending members carry bending loads, these are loads that have their forces oriented such that it causes in this case we have an XYZ coordinate system with z coming out of the plane, so a bending moment is a force and distance combination that causes torque about the z axis in this image, so that's the out of plane axis in this image. 

Now contrasting that with torque even though it's mathematically the same torx we generally consider to be causing that bending moment or that I should say that force times distance that force couple. Around an axis aligned with the principal geometric feature so in this case if we had the same x y z coordinate system where the x axis is the x is, the the axis along the length of the geometry it would be a torque about that x axis, so that's the difference between torques and bending moments mathematically the same physically different with respect to the geometry and those things that carry these torques are called a torsion. 

Member. Is we often though not always call them shafts so a subset are called shafts it's toward the fronts time this year usually that's in July it happens to be right now. September in the covet era, so I thought I'd use this bicycle as my example this this is a bicycle that drives the the the drive wheel with a shaft rather than with a chain so again what you're doing is you're cranking on that shaft you're creating a torque the torque then of course is what is carried. 

Back to the main axle of that wheel and then drives the wheel and accelerates the bike in the aerospace world, of course, we have many shafts perhaps the most predominant of those would be like, for example a crankshaft in an engine or the shaft that carries. You know, all the the blades in a turbine engine. 

There are other structural members that carry torsion as well that wouldn't be classified as shafts, for example the fuselage carries torsion the wings carry torsion those aerodynamic moments manifest themselves as torsion about the long axis of the wing and again in those circumstances we're not typically calling them a shaft we would then just call it a torsion member. 

Or a torque box, we'll get that language clarified and why we call it different things as we go through the semester. 

Another type of structural element in aerospace and and of course more broadly in all of engineering is something called the sheer panel. Now what does a sheer panel a sheer panel? I'll give you a sort of dry description, it's a thin sheet of material used to carry in-plane shear load. 

Or in the aerospace we would off often call it a skin, so we would have the wing skins. Or the wing panels again geometrically, what is the the predominant feature of these is that they are thin in one direction relative to the other two directions and from a load perspective it carries sheer load in the plane of that thin skin and in that circumstance it's called the shear panel. 

Now we have sheer panels, of course throughout the entirety of the aircraft structure the fuselage the reason we have thin skins on a few slash, of course the external arrow dynamic loads but also because they're really quite phenomenally efficient at carrying torx. I will tell you more about that as we get later in the semester but I think that is not as intuitive as this example that I'm showing on this slide. 

I don't know if you've thought much about it previously but your house or your apartment is also built structurally in some ways quite similarly to an aircraft in that you have sheer panels in your house or sheer panels in your apartment. We call it drywall most of the time in modern construction. 

If you think about what's going on if you if you put these studs vertically. The studs can carry the axial loads, just fine. You put a weight up there. The the most of that load is transferred through the stud from the top surface to the bottom surface. However, what those studs can't carry in the absence of having something like drywall on that wall if you got a lateral force something pushing from the side. 

What would happen is that that would manifest itself as a bending moment that's very difficult to carry from the perspective of the stud particularly at the top and bottom where it attaches to the headers. And so you could quite easily get failures you what amount to hinges at the top and bottom surface. 

So a wall will not have strength to carry this lateral load until you put this sheer panel on it in this case it would be drywall. So that's a simplistic example. Relatively easy and intuitive to understand but there is much similarity between that concept in the concept of an aircraft wing skin as its carrying the the torque the aerodynamic moments down the wing. 

I'm gonna pause for a moment and just check on questions. You know, you guys are asking most of your questions. You know in the chat box and that's fine. I would also encourage you if you if you could more easily ask it, you know by unmuting yourself turning your mic on and and asking it verbally I'd be happy to take those questions as well a little bit more interactive in some ways that way and and so if it's a real brief question asking it the chat's fine but feel free to interrupt me. 

If this is confusing to you in any way and ask a full-length question, 

Another type of structure that a structural engineer would be interested in would be what we call a beam on an elastic foundation. Now, we don't see these quite as commonly in aerospace engineering, although there are some sort of specialized structures that this would apply to in an aircraft or a spacecraft. 

The example that would most likely apply. In an aerospace structure would be if you have to use adhesive to bond one structure to another particularly a beam-like structure to a more rigid structure. So this this happens periodically in aerospace engineering particularly with modern composite manufacturing techniques, but perhaps the most intuitive example is not that example in aerospace, but rather the example that I have shown here on the screen. 

That is the tracks of a train. Now the tracks of a trainer are made out of steel. And of course they they feel quite rigid when you are walking on them if you've ever walked on train tracks, but of course trainsway many hundreds of thousands of pounds and so if you watch closely as a train rolls down the track, you actually see the track deforming quite well. 

Now the elastic foundation in this context is actually the ground and it's the sub it's the subtract surface. So the beam that is the train track carries the load and ultimately distributes forces sort of spreads those point forces out over larger area, but then those forces still need to be carried into the ground and the ground acts behind the beam to elastically minimize or mitigate the deformation of the beam itself. 

And so we can in this context treat this as a combination of a beam plus a spring-like element behind it. So I'm already beginning to use the language that we use instructional mechanics to to classify things geometrically, so I've already referenced a beam a beam is generally something that's much longer in one direction than it is in another direction. 

And the elastic foundation in this case is comparatively large. 

Another type of aerospace structure, you should be familiar with is a membrane membrane, excuse me, a membranous structure. We have a lot of membrane structures and aerospace things like, you know here obviously this example, the airship. What it's what amounts to a balloon with internal pressureization and well, so what's characteristic of a membrane? 

A membrane is something that's thin and one dimension relative to the other dimensions. And perhaps even more critically what differentiates a membrane from what we'll see in a few minutes that's a plate or a shell a membrane is so thin that it cannot carry any axial compressive load if you press on a membrane. 

That plane that it is thin it will obviously buckle and fail. So you can't if you took a sheet of aluminum foil and tried to use it as a compression load carrying device, it would fail immediately. It can however do quite a good job of carrying tension. So a membranous structure is thin and one dimension relative to the others. 

It can carry tension in that plane, but it cannot carry compression in that plane. So lots of membranes in aerospace. You if if it were particularly thin you could even consider wind skins to be membranes, you know, you think about the Wright brothers they made their wing skins out of out of a cloth called Muslim. 

And so if you if you can imagine trying to compress cloth in plain, you know, obviously it's not going to carry load. So you can have membranous wings. You can have a hot air balloons, you can have parachutes the the sort of one-dimension. Al analogy of a membrane is a rope. 

You've probably heard the words you can't push a rope. So, that's what a membrane is in aerospace. 

I mentioned briefly plates and shells. So plates and shells similar to a membrane they are relatively thin and one direction compared to the other two directions. So aircraft skins are often considered to be shells ship holes or obviously going to be shells pressure vessels storage tanks are some examples of things that are that would be classified as plates or shelves. 

Now, what's the difference between a plate and a membranous structure and that's the ability to carry in plane bending loads and in some cases carrying some modest degree of axial load? So plates and shells carry bending loads much like beams, you know, a beam is sort of a one-dimensional bending member a plate or a shell is a two-dimensional bending member. 

It can carry bending moments about two different axes. So the difference between plates and beams is in the fact that it has two dimensions that are much larger than the third a beam has one dimension that's much larger than the other two but in both cases they can carry bending loads and to some degree axial loads. 

Whether to classify this is a difference type of structure is sort of something that I guess falls to a individual preference, but you can also have curved beams. In this case, it looks an awful lot like a spring. A spring of course as an energy storage mechanism, it can be used to have a force that's you know, loosely up a proportionate force to an amount of deflection that occurs you guys have all seen springs before but if you look carefully at what a spring is and what it is geometrically speaking it has one dimension that is longer than the other two dimensions it happens to be oriented in a helical coordinate system. 

Such that along the helix it's longer than it is in the other dimensions but nevertheless it in many ways can be treated quite similarly to how we would treat a beam from an engineer's perspective curious bending loads one one particular complexity of springs and curve beams that would not necessarily be present in a typical engineering beam is that if you deform it to a large enough extent then you're actually closing two surfaces on each other. 

And you get a very strong nonlinearity in their response so it it's it becomes very stiff very quickly once those surfaces come in contact with each other so they are treated somewhat differently than a typical beam but nevertheless you can see how they're easily classified as beams. 

So those are the predominant classifications of structures within the the aeros structure subdiscipline beams plate shells membranes axial columns in in reality most structures in an aircraft ultimately are some combination of those idealizations. So everything that I described was an idealization of a geometric structure and a load combination and as you can see in these images most of our arrow structures are some combination of those idealized structures on the left, we have a fuselage the fuselage has stressed skins in it, so it has membranous or shell-like structures which are reinforced with stringers, as you can see the stringers carry load in the launch tunal direction of the aircraft. 

Or of the fuselage in this case those stringers themselves locally are treated like beams and they are reinforced by what are shown here as formers those formers now are ensuring that the fuselage maintains its shape globally so the stringers make sure it maintains its shape locally so that the wing skins don't deform too much the formers make sure that the stringers don't deform too much and so therefore make sure that the fuselage maintains it's overall shape the former's themselves can be treated. 

As beams with point attachments and then you can go through and look at things like bulkheads and and other structural aspects of the fuselage. On the right hand side, we see a schematic of a wing. Again the same story. What do we have? We have wing skins which are either going to be membranous membranes, or they could be shells depending on how much load they can carry and what style of low they're able to carry. 

We have spars which can be treated as. Beams. We have ribs which can be locally treated as beams. All that same load path discussion that we started about 15 to 20 minutes ago is now manifest in this schematic of the wing and you can see how we I can idealize that wing as a combination of these subsets. 

Okay, I see we have a question from Nicole Nicole's question is what exactly is a fuselage, sorry if it's a dumb question no it's not a dumb question, absolutely not. My major is mechanical. So, thanks Nicole for the question the fuselage. Let's see here, that would be this here on the left the fuselage is the. 

Structure of the overall structure that well it serves many functions, first of all, it contains the payload so it contains the seats the floor. It also is a load carrying member so it carries load from the tail up to the main wing and from the main wing back to the tail. 

It's the overall structure, it's the main body if you will of the fuselage. I'm sorry of the aircraft. So the fuselage is the main body of the aircraft It's the part that everybody lives inside as opposed to the wing which of course attaches to the fuselage here at the wing attachment point and yeah, so that's what the fuselages. 

Thanks for asking that question Nicole I know particularly for those who may have come from a different discipline or maybe have haven't always been super excited about aerospace and I've only really gotten interested in it recently if I'm using terminology that you're unfamiliar with absolutely ask me questions really, there is no such thing as The dumb question. 

I don't like people saying that my question is done, you know, it's it's not a dumb question. It's a great question. If you're in an interview and you know, you don't know what it is, then you might be in trouble. So don't get through this class without learning what it is. 

So thanks for the question. So Joel, oh Joel answer the question for her. He said think main body of the aircraft, it's the tube you put people and cargo in. David thanks for echoing that's not a dumb question. Joel thank you as well, okay, so great thanks. I mentioned the load paths earlier this now can you know, we're gonna return to that concept throughout the class, you know, here's an example a number of figures from the textbook figure one nine top left is an idealization of a box beam structure, so what's a box beam structure, well the box beam structure has as sort of as shown here has a thin skin. 

That forms a box and then it locally has what amount to sparse in it with spark cap so in an arrows we'll look here in the middle in an aerospace wing-like structure if we look at a wing so here's overall the best picture of what the wing structure looks like again, this is a simplified wing single cell effectively for our torque box. 

The spar is this portion here. 

Which is characterized by having concentrated areas in the cross section and these are called spark caps. The spark caps for reasons, which you will learn soon carry the predominant bending load, so it's material that's that's pushed far from the neutral axis of the beam. So that it can carry large loads at low stresses. 

Now the spar web is I'm gonna try and highlight the spar web here now. I recognize I don't have a very large photo to show you the spar web is that part that I just kind of showed there and that acts to keep the that actually carries the sheer load in the beam will get into much more detail on that later in the semester and then the whole thing forms this box. 

Which is kind of schematically shown over here on the left hand side. Okay now the box ultimately. Now combines multiple structures, so you have a front bar that's a it's like a beam that's carrying bending loads you have a rear spar that's a beam that's carrying bending loads, but you also have the box which is these thin skins which you'll soon learn carry torsion loads. 

Okay so torsions now along the axis of that wing structure. Now these stringers are these localized reinforcements. Which I've circled. Again localized reinforcements what they do is they make sure that the thin wing skin. Doesn't deform too much and then as you move into the wing, you would have. 

A rib that eventually sort of collects the loads that those stringers have carried and then it transfers that load inwards so let me see if I can draw a rib. 

It looks like it's no longer letting me edit the the chalk that I've added here, but so as you as the stringer carries the load towards the rib or in this case towards the frame as it's described in the fuselage section, then that frame carries carries the load in the XY plane over to the beam structure that is the spar that then ultimately carries it into the fuselage so again wing skin, so the distributed load picked up by the stringer carried to the, Rib from the rib it's carried over to the fuselage. 

I'm sorry over to the spar from the spar down to the fuselage and then from the fuselage, of course, you can imagine it goes further into the floor and into your seat. 

Here's a photograph showing a modern aircraft fuselage and it really does show that that you know these conceptual idealizations and these photos that you see in your textbook. These figures that you see in your textbook actually are reflective of what's going on in the real world, this is an outer skin. 

This part here which makes up the fuselage. Outer skin. These. Are the. Stringers that are carrying it into in this case it's we're calling it a frame so what's the frame the frame is like a it's very much like a rib and a wing but instead it's in a few slides why we call it something else to be honest. 

I don't really know it's just what we call it and if if this frame instead we're a complete circle or we're used to prevent things from moving from one down the down the length of the fuselage, so you could put what amounts to a door there then we would call it. 

A bulkhead so a frame in a bulkhead are similar now they both maintain the overall shape of the fuselage, but one's a lot one allows something to move through the fuselage and the other actually blocks it it's a separator to keep cargo on one side or the other. Anyway, so this is one barrel of the 787 fuselage it comes from the autoclave like this and then it's assembled with multiple other barrels before ultimately the floor is assembled inside so that photo that we just saw there is a figure one four. 

I'm sorry one point one four the fuselage down at the bottom. 

Okay, so we have about 20 minutes left at least in this lecture. I'm going to divert a little bit, you know, I've I've now introduced you to some overall structural concepts some idealizations some language that we've used instructional mechanics particularly aerospace structural mechanics, but I'm going to now kind of step back from that a little bit and give you some overall perspective on on a structures and mass from a bigger perspective now these examples actually came. 

From a lecture by a guy named dr Jim justifer he is an engineer by training but then ultimately went on to be an orthopedic surgeon and he he wrote a paper who actually we wrote a paper together in the past. Where we were comparing, you know, giving some perspective on structural mechanics and and orthopedics. 

So these slides are largely from him and I'm sharing them with you because I think they're really insightful so here's a question that we can start this partial portion of the lecture with a flea the size of a man could jump about how high. So is it gonna be one meter? 

10 meters a hundred meters. 

Now why is that question relevant? It's it's relevant because of scaling laws it's relevant because of scaling laws and really it comes down to a something that was identified perhaps first by a Galileo Galilei. You of course know him because of his telescope and his contributions to astronomy but that's not the only contribution that he made and he he actually found this problem that we're now discussing to be quite an interesting problem, that is the problem of scaling in mechanics. 

So he Galileo he was one among the earliest to document the sizing problem the scaling problem, this is a quote from him who does not know that a horse falling from a height of three or four cubits will break bones while a dog falling from the same height or a cat from a height of enter eight or ten cubits will suffer no injury. 

And just as small animals are proportionally stronger and more robust than the larger soul also are smaller plants and for some reason this is cut off the slide but it continues similarly to that. 

This is no longer Galileo, but. Another of his contemporaries. I'm certain you both know that an oak 200 cubits high would not be able to sustain its own branches if they were distributed as a tree of ordinary size and that nature cannot produce a horse's largest 20 ordinary horses or a giant 10 times taller than an ordinary man unless by miracle or by greatly altering the proportions. 

Now. What does that so? I don't know if you can see it here there well, oh focus on the tree. Here's another another of the contemporaries. You can drop a mouse down a thousand yard mine shaft and an arriving on the bottom, it'll get a slight shock and walk away a rat is killed a man is broken and a horse splashes again, so what do we observing with all of these comments made by those who are centuries our predecessors? 

They are observing the scaling problem. Roughly called the cube squared problem. What does that mean? If you increase the size of a cube in its linear dimension by 10x. You are not increasing its volume by 10x. Rather you're actually increasing its volume by 10 cubed. Now that's interesting but what we really need to observe here is that the cross-sectional area of that and again this is Galileo thinking about animals and animal bones you recall that photo of an animal bone that I put right alongside Galileo's photo. 

The load carrying capacity. Of a bone. Scales approximately as it's cross-sectional area. Which goes essentially as a characteristic length squared. Whereas the weight or mass of that same body. Instead scales approximately proportionally to that characteristic length to the cube. So length cubed. 

What does that mean that means the relative strength of something decreases inversely with length scale? 

That's why mice are comparatively stronger than rats compared to really stronger than humans comparatively stronger than horses. 

It's the length cube length squared cubed law. Now, if you were to plot this now these are all relative terms. And we take something and we look at its relative strength and then as we scale its length scale up. Following this law, it's relative strength decreases. So by the time something is 10 times larger, it only has about 10 percent of the relative strength. 

Now this isn't perfect. This isn't perfect and in fact we'll highlight some of the differences in just a minute, but conceptually this is a really key thought that we need to instill in our brains. This sizing law impacts us as engineers and and particularly us as aerospace engineers. We need to be aware that we're sort of up against it when it comes to scaling things to be larger. 

Just for kicks here. I'm going to show you some properties of human bone. So human bone actually is quite interestingly. Asymmetric. Asymmetric meaning that it is stronger in compression than it is intention. In some ways stronger in bending than it is intention. As you can see the strength of bone is impacted by your age. 

So it starts out in the neighborhood of a hundred and fifteen to 120 megapascals as you age. As you get osteoporosis as your bone mineral content changes the strength of bone actually changes. Ultimately as you get to be someone who would be classified as elderly in your seventies to eighties or beyond your strength of your bone now is on the order of seventy percent of what it might have been. 

It actually gets worse than that because not only does your strength of your bone decrease but also the wall thickness of your bone decreases so the overall strength of your bones is even more adversely affected. Okay, it's perhaps notable that the ultimate strain to failure that is well, you will talk about strain extensively. 

I'm sure your conceptually aware of what that is delta. L over L change in length over length. That is comparatively more consistent. Than the strength, although it's it also falls off over time. So children's bones quite elastic can take higher strains than adult bones. 

Here's a chart just for interest. You don't need to get into all the details of it here, but essentially what we find is that rats humans cows have similar bone strengths. So turns out that in bone were much less consistent than we have in terms of reported strengths for any particular metallic alloys engineered materials tend to have much narrower. 

You know tolerances on the on the yield strength or ultimate tensile strength of a material. So the numbers that are quoted here in this particular plot are not necessarily identical to those on that prior table but the point is is that humans horses cows rats have strengths that are all sort of on the same order of magnitude, they're there in many ways quite similar. 

So we're talking 200. Megapascals as the approximate number for bending strength of a bone. Rats all the way up through cows. So here 200 megapascals kind of falls in this regime here. That's what this. This chart is now if we're looking at this along the x axis here, we have the body mass of a variety of animals and their peak bone strength. 

So if you look at this it's kind of interesting chipmunks, they're peak bone strength is in the neighborhood of 50. You know, small kangaroo rats again small ground squirrels there in the neighborhood of a hundred prairie dogs in the neighborhood of a 50 again dogs, goats horses buffalo elephants. 

Note that they're all pretty much in they achieve the same peak bone stress. Now, I when I say peaks bone stress, this is based on their activity level. Versus the strength of bone. So in sort of nominal activities chipmunks, prairie dogs dogs horses and elephants all load their bones at approximately the same stress levels. 

And they have a similar safety margins with regard to their overall peak bone strength. So with scaling eventually an animal's bones will break, you know that scaling law it's area versus volume. You can't go arbitrarily large but gate and activity level do matter as due to some degree geometry and when I say geometry if we're talking about hollow structures, hollow structures are a little bit different than solid structures. 

So the length, Versus area versus volume characteristics are a little bit different if you've got something that's hollow. Anyway, so bottom line animals know their own strength and a larger animals tend to move much slower elephants are very slow movers. If you've ever noticed, they stand quite vertical their legs legs are very straight up and down carrying as much of that load directly axially as they can. 

They don't have much bending in their legs because they don't want to create bending loads. They don't want very high dynamic loads. So David asks are those test results from live bones or dead bones. Those are from a dead bone. It's in a laboratory setting. They do generally it's it's typically fresh or fresh frozen tissue. 

Typically now that's not to say that people haven't made attempts to test strengths of live bones, but there's animal cruelty issues associated with that that's kind of a complex topic but what what's being reported here? Is is fresh tissue but not live. Now, of course just in the the activity level in the these are really you should consider these to be estimated peak bone stresses for activity levels. 

No one's putting strain gauges in chipmunks or at least if they are and and it has been done, you know, they're they're doing it in such a way that they are actually influencing the activity. So it's these are best estimates of stress during activity for these animals. Now, why did I take you on that little side journey? 

It's because there are many analogies to this length area volume problem. Instructional mechanics. Historically our building structures have similar limitations. You know you think about the Egyptians and and their architecture they were building with stone and so there was this the same length area of volume concept that they were battling when they were designing their structures. 

We've got to be quite efficient in our structural engineering, so then the question is does this principle apply in aerospace? And the answer is a little bit complicated. It's well yes, of course. Because length area and volume do scale. Differently ones squared ones cubed. And yet at the same time we have to be a little bit cautious and recognize that it's not quite that simple. 

So I'm going to say yes and no. Now, we're not going to talk about aerodynamics very much but it turns out that aerodynamic scaling laws actually want us to increase the size of our aircraft to gain efficiency. You know you it's no secret that that most commercial airliners for long haul are quite large. 

There's very good reasons for it. It's aerodynamic reasons the and and ultimately it really has to do with weight fractions of payload versus weight fractions of. Structure overall. Now, I have to be careful as we say this. So larger aircraft tend to be more efficiency longer distances require exponential fuel increases, so there's some complexity there. 

But structural weight in an aircraft scales it somewhat less than length cubed. Okay, that's pretty critical somewhat less than length cubed. Now why? Because in aerospace engineering often the volume, that's contained. Is rather empty. Okay, it's not like a block that solid where we get this direct length area volume criteria. 

Instead what we get is an open space inside the fuselage that now becomes empty space or in the wing that becomes empty space. However, it's not as simple. So in that case, you're now talking about the weight scaling with the area of the skin of the aircraft in some sense. 

Okay. So now it's a length versus area problem. Well, I have to step back from that statement just a little bit why because we rarely leave aircraft empty when we can put something useful in them and so the payload now becomes something that you know, sort of the volume of payload and the the weight of the payload scales with the volume of the payload goes as the length cubed. 

So it's somewhat less than length cubed it's not directly length cubed but it's somewhat more. Than area. So we ultimately bottom line we still need to be thinking in those terms. 

Now it just as a somewhat tangential comment materials themselves structures can have size effects and flaws which dictate complex outcomes larger isn't definitely better. Composite materials are node known to have some size effects. 

I will also mention that arrow structures. Also tend to limit activity based on size much like animals elephants move more slowly mice are much more agile and move more quickly. If you compare the maneuver of a Boeing 747 to something like an extra 300 which is an aerobatic aircraft or an RC aircraft, obviously the activity the maneuvers that that aircraft conduct are very much dependent on the size of the aircraft. 

Now, I see that it is almost four o'clock, so we've run a little bit over. I appreciate you sticking with me through to the end. Happy to take any questions that you may have otherwise we'll pick up again here on Thursday. 

Hi professor. I'm sure you I'm doing well. Who is who is this again? Jeremy Bayak, hi Jeremy. 

Yeah. 

Well, I when the cargo you're potentially scaling. To be then the, Density of like the wing structure. Because you're not scaling this thing material. Yeah, so you have to keep in mind that this length area volume law sort of makes this assumption that you have the same material and of course we're talking really about different materials, so you need to generalize that concept to be more of a overall view rather than something that's quite specific and would apply to everything however it's it's nonetheless still quite relevant so think for example about the fuel that we need to carry. 

Right the density of that fuel times its volume ultimately is its mass. And that very much directly scales with volume. Right now there are other things that scale instead a little bit differently so for example in order to carry the loads of a large aircraft in comparison to a small aircraft you're going to see that later in the semester you're going to see that the bending stiffness. 

Goes as the area moment of inertia and the area moment of inertia goes so it actually goes as something that's the integral of y squared d a which ends up being you know, something that depending on whether you're you know, I'll use a circular cross-section as an example, so if you had a circular cross-section in bending the bending rigidity or stiffness of that, which is represented by its material modulus. 

E times its area moment of inertia i in that case the area moment of inertia goes as the fourth order the the the fourth power of the radius. So it's stiffness goes as the fourth power of the radius it's weight goes as the square of the radius. I either cross-sectional area and so there are other sort of underlying. 

Scaling laws that drive us to have large structures, but ultimately still end up being limited by this length area volume argument. You know, because there are portions of the airplane that are in you know, you think about so what I just described which goes as the fourth order of the length that's for something in bending but something in axial compression goes as the area so that goes as the length squared so the landing gear are going to be governed by length squared whereas the wings are going to be governed by length to the fourth. 

So there are these underlying scaling mechanisms. You need to keep in the back of your head and I've given you this amp example mostly to conceptually introduce you to the concept rather than to be very explicit about saying that this always applies in all circumstances. 

You're welcome. 

Okay, I'm not seeing any more a questions in the chat and I'm not hearing anybody chime in with verbal questions, so I'm gonna go ahead and wrap up and log off of the meeting. Office hours Thursday 10 o'clock if you have questions, of course by email or phone and feel free to contact Xander Sereni as well the TA with any questions you may have I look forward to seeing you guys on Thursday.