Mechanics for engineering students: Statics | Dániel Csíkos | Skillshare

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Mechanics for engineering students: Statics

teacher avatar Dániel Csíkos, Mechanical engineer

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Taught by industry leaders & working professionals
Topics include illustration, design, photography, and more

Lessons in This Class

33 Lessons (2h 24m)
    • 1. Welcome to the course!

    • 2. Introduction of vectors

    • 3. Scalar products of vectors

    • 4. Cross products of vectors

    • 5. Mechanical model

    • 6. Forces, distributed loads

    • 7. Moments, couples

    • 8. Axioms of statics

    • 9. Static moment

    • 10. Centroid of bodies

    • 11. Moment of inertia

    • 12. Moment of inertia - Basic example

    • 13. Reduction of loads

    • 14. Simplest resultant, central line

    • 15. System of loads – Basic example

    • 16. Equilibrium of two or three forces

    • 17. Constraints

    • 18. Calculation of reaction forces - Basic example

    • 19. Separation of structures

    • 20. Principle of superposition

    • 21. Structures with three pins – Basic example

    • 22. Trusses

    • 23. Method of joints

    • 24. Method of joints – Basic example

    • 25. Method of section

    • 26. Method of section – Basic example

    • 27. Stress resultants

    • 28. Stress resultant functions with basic examples

    • 29. Stress resultants of curved beams with basic example

    • 30. Coulomb friction

    • 31. Self locking with basic example

    • 32. Belt friction

    • 33. Summary

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About This Class

This mechanics course has mainly been created for students currently learning statics or related mechanical subjects at college/university. In this course, you are going to find everything that you need to know about statics!

Objective of the Course

The main objective of the course is to help you be able to solve static mechanical problems. You are going to be able to determine stress resultant functions as you are going to be able to calculate reaction forces and inner forces in statically determined systems.

By taking this statics course, you are going to get the basic knowledge to learn Strength of Materials which is the subject where you learn how to investigate or design a structure considering mechanical effects.

What will I learn?

  • Basic concepts
  • Mathematical background
  • Determination of reaction forces and inner forces in complex mechanical structures
  • How to analyze trusses
  • How to handle problems including friction

What do I need to know to start the course?

You don't need any specific prior knowledge. The course starts from scratch and take you through the topics with detailed explanations and examples.

How to make the most of this course?

There are several practice problems that you can solve by yourself. I suggest solving those problems after watching the lectures of each topic!

Meet Your Teacher

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Dániel Csíkos

Mechanical engineer


Welcome! :)

I'm a mechanical engineer and online entrepreneur. I've got my MSc level degree in mechanical engineering at Budapest University of Technology and Economics and I'm enthusiastic about sharing my knowledge and my love of engineering sciences.

I've been teaching and helping mechanical engineering students as a private tutor in various subjects for over 3 years. Therefore I not only know how to understand a topic as a student but also how to make it understandable for others. I passed most of my subjects with an excellent mark and I hope I can help you to reach the desired mark or objective for yourself by ease, too. I offer guidance to understanding either if you are a student or if you want to... See full profile

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1. Welcome to the course!: welcome to my cart of statics. It is very nice to see you here in this first video. Let me tell you about the course. The course contents of six topics of statics which are more or less be it on each other. Therefore, I suggest going in chronological order Off course. If you already know something, you can just skip a lecture on take one, which is more interesting for you First, there is a mathematical summary. If you order did Novak garage obra, just skip it. If you do not know it, then please take it to be confident in it. It has you to concentrate on mechanics later. The first mechanical topic contents the basic concept on definitions on which you can be a done. Then I tell you how to handle such system of forces that you can see on the figure. The next step is to calculate reaction forces. If you learn this part well, you can take a step forward. The analyzes off dresses is a more complex task that you are going to do with confidence. If you can calculate your kind of reaction and in our forces you can go on and learn the most important part of statics distress. Resultant. If you want to design anything, you must learn this topic well, Don't worry, I will be there to help you. The last topic is the known idea constraints. I tell you about the sticking friction on the bath friction to write in your special knowledge in statics in every mechanic a topic I summarized everything that you need to know. Then I show you a basic example. If you understood everything, then you can solve a sample problem for which I always give you a very detailed solution. I hope that everything is going to be crystal clear at the end off the sample problem. And you can move on to the next topic. I wish you good luck. And I hope you will enjoy this journey off learning with me. 2. Introduction of vectors: Hi there. Let's start our journey by reviewing the basic mathematical background off statics. This minded sounds so exciting, but I promise that we do it briefly, and then you don't have to worry about mathematics. You can just concentrate on mechanics basically for statics, or you have to know its factory algebra. What is a factor in mechanical sounds? Most of the quantities are actors such as forces, moments, velocities and so on. They can all be defined either in a dramatic away or on algebraic manner. Jim Ethically a factor is defined by its line of action. It's like on its orientation as an example. Force factor is denoted by F. It is underlined to indicate that it is in fact, a factor. It's line of action is smart with the dash line. It depends on me how long the factor actually is. Therefore, I also need a scare to relate this length to the magnitude of director. For example, I could choose that one. Newton is one centimeter on my drawing, the orientation off the factor is simply marked by the arrowhead. In the end of the factor for calculations, it is better to defend director in an algebraic form. Usually a cat is in coordinate system Misused aesthetics in three dimensions. The catalyse and coordinate system uses three coordinates X y and set on windows three axes . Unit factors are refined. I espera two x j s para. Why on que esperar it'd set? These are called the base factors and they are perpendicular to each other in case of a cat is in coordinate system in an algebraic form. I has unit length in X direction on zero light in violence. E jay has unit like in my direction and zero in the other two directions and K is respectfully the same in the Z direction. Therefore, all of actors can be described by an addition in which the magnitude in X direction is more deployed by the unit Vector I. The magnitude invite direction is multiplied by the unit factor J and the magnitude inside direction is multiplied by the unit factor k. The result is a battering three D factors have been our mathematical behavior. They are easy to work with. One of the basic operations is addition. Vectors can be added by adding the factor components a scholar's. This can also be done chair medically by adjusting the starting point off bond actor to the other actors, and this can be done with any number of factors. The resultant is a straight factor from the starting point of the first factor to the arrow had off the last factor. Other simple operation. This multiplication off factor by a scholar Number C algebraic early. The components of the factor are multiplied by this guy number C j o Matic elite. This multiplication equals to an allegation for a scar greater down from and shortening for a skeleton. Smaller, damp on. There are several more operations that can be down with the actor. For example, they can be multiplied by an other factor by calculating a scholar product or across poor attacked. That is all you need for statics. So stay tuned on. I will tell you more about them in the next lectures 3. Scalar products of vectors: one of the most important factor operations is taking the so called scaler or don't productive to actors. So let me show you how to do it. Joe medically scale our product is the light off the first factor times the length of the second factor in the first factors direction. The light is denoted in the creation by taking the absolute value of the tractor. You can see from the definition that if two actors are perpendicular off, it's not the degrees and the skill a product becomes zero. On the other hand, in case of parallel factors off zero and the skill her productive equals to the product defacto nights Marjorie Kelly the factors shall be multiplied. In the previous lecture you learned the factor equals to its component in X direction. Times I, plus its components, invite Times J plus its component in that timescale. Both factors on the same format on your terms in the brackets shabby multiplied with advance In the other bracket. The terms I times, I J Times J and K Time escape become one all of their terms. For example, I Times J become zero does Finally, the scale of product is defined as the sum off the productive components off the two factors in the same direction. Now you know how to calculate scale projects. But you may wonder, right is good for you when you do back to Raja, This operation appears quite a lot of times, but one specific application is calculation off bhakta projections To calculate the projection off of actor to another actor, it is necessary to looked in the direction on which the projection shall be made. This can be done by taking the north off factor in the direction The norm is calculated by dividing the factor by its absolute value, which means it's like this results in a unit factor in the direction of the factor. I didn't this unit director with you Afghan then back doors get a projection is reached if it is no deployed by this unit. Factor in my example after is multiplied by U F one, which is the unit factor in the direction of F bomb factor. The result is a skill number which means the blanked off after two in the direction off F bomb. If I multiply this line by the unit factor again, I obtained a factor, which is the parallel component off F two with respect to F month, This calculation is frequently needed Impact or German three. 4. Cross products of vectors: the most important factor. Operation in statics is taking cross product off two factors. This occurs in most of the calculations, so let's see how it's done. The curse Productive two factors mean another actor, Joe Methodically. This operation can only be illustrated in three dimensions. F one and F do other two factors for reached across productive steak the area of the parallelogram that the factor span is equal to the amplitude off the result of the cross product. This can be calculated as the absolute value off F one times absolute value off F two times SIGN offer. The result of the cross product is a factor. Its direction is determined by the right. Andrew F. One is your tamp after is your forefinger down? The resulting director f is your middle finger only the order off the fingers is important . So if your forefinger is taken to be half month, your middle finger is after on your Tom is off. This results in the same direction for half algebraic Ali. You can take a cross product based on the knowledge acquired in the last couple of seconds . I fried the factors up by using their three components on the base factors of the cartage and coordinate system simply I multiply or members inside the brackets. Members like the curse productive I've actor with itself are zero Cruz productive backdoors I and J results in a cave actor according to the right hand rule as ice para rabbit access X J s parapet access by on the result is perpendicular to them, which means it's pointing toward access that however, the cross product of J and I, which is the opposite order of the two terms results in the factor pointing tow opposite direction according to the right hander. By following the rules, the cross productive F bomb and F two becomes f Mumbai times have twos at times I minus F ones. At times after wife times I etcetera. You can see how the result can be written as a factor. There is a simple short got to calculate the cross product. When I want to calculate the first line of the result, I simply cover the first line on the left side of the creation. Now I can forget that it is there and I concentrate on the remaining two drugs from the first factor. I take the element in the next throat, which is half, Mom. Why? And I multiply it with the element across it. Namely after is that then I say abstract the product off the two other elements f onset and after. Why? If you want to calculate the second line, you can cover the second line on the left hand side and you can do the same. It is easy at. Imagine if you ride the first throw off the left hand side below the third row. Then you can multiply F ones that with after X as it is across it, and you can see obstruct F one X times after that for the third line. The method is the same as I mentioned in the beginning. Off this section, trust products appear a lot in statics, so you will be able to see application examples if you go on on fudge. The upcoming lectures 5. Mechanical model: in mechanics. We use models to simplify off problems. Therefore, you do not have to deal with every little aspect off a given problem. You can concentrate on the big picture. For example, in statics, we care about reaction forces or stresses Athans. But we do not care about the formations. They will be introduced calculated instrument of materials, therefore, or bodies in statics. A corset are to be rigid aesthetics. The bodies are permanently arrest. So there is always actually Librium that can be problems in three dimensions, but lots of times it is enough to investigate the problem. In two dimensions in three D, the rigid bodies have six degrees of freedom and they could move according to any degree or freedom into the disc. Reduce is to only treating These are freedom for a rigid body. The model is built up by several typical amounts. There are constraints that prevent the body to move according to its degrees of freedom. Therefore, they reduce the number of degrees of freedom. These constraints can be pins, sliders at cetera. I will tell you more about them later. The other very important mother elements are the loads. These loads can be forces moments or couples for a structure. It is the most important off course besides fulfilling it's purpose. Too bad the stress is caused by the loads to keep up, and you can learn more about floats in the upcoming lectures. 6. Forces, distributed loads: loads are very important in the mechanical models, so let's talk about them. The first loads you can learn about our forces on distributed loads forces director quantities, so they are either defend dramatically by their line of action, magnitude and orientation. Or they can be different Algebraic alley by the three components in three dimensions or by to force components in two dimensions. The force and the body is applied in a given point. That is why it is also called concentrated force in mechanical modeling. The force is fixed to its line of action. It doesn't matter in which point it is acting along the same line because its effect is the same. The statement will be more clear. After you learned how to calculate load reduction. A lot can be distributed on the line on an area or volume. The most important case for us in statics is the parallel system off loads. You can imagine that lot off small forces are acting parallel to each other given line area or for you load can be distributed on online. In this case, the load is measured as a force intensity. On its unit is Newton over meters the intensity can change according to bomb perimeter Iman Dimension. If a load is distributed on an area, you can call it pressure. So, yes, the atmospheric pressure is also in this category. The unit off pressure is Newton over meter squared or Paskah. The pressure can change according to two perimeters in two dimensions. A load distributed on a volume escort specific rate on its unit is Newton over meters Cube . This is a three dimensional measure so it can change according to three para meters. It would be hard to make a figure about specific weight is it can be a different value in the every point off a body. Four dimensions would be needed for proper plotting. You can think about it. Analogous lee to the previous cases. You can imagine the differences between distributed and concentrated forces through some everyday examples. If you get stuck with your car and someone has to tell your car to a mechanic with a rope, the rope is exerting a force on your car as it is connected in the valley. Fine point. If you want to slice of bread with a sharp knife, you've produced a load in Nevada defiant line does. The knife is exerting a force distributed longer line. Another proper example is ice skating as your body weight is distributed on two body find lines. If you simply book or you jump in the temple in, you accept a force distributed on an area. The differences within the intensity offloading on this area were determined which way you move after you made the contact. Gravity is the best example for a force distributed on the volume because you are experiencing it right now. Note that in reality there is no concentrated force. It would always be a distributed force, but if you go far enough from the effect, you can consider it as a concentrated force. As it doesn't matter how you love the system, you only care about the effect that the load causes. Distributed force can easily be substituted by a concentrated force, which is acting in the center off action off the distributed load. Let's take a force distributed on the line as example. The center of action is the dramatic center off the pressure intensity graph, which is thrown over the line, advised it can be calculated by Intergraph or Sums. I will tell you more about calculating the center off a given object in the lecture. But I introduced the Seine tradeoff bodies. The concentrated force, which which you consented to. The distributed load is calculated by an integral over the land on which the load actually acts. For two D and three d case, the calculations are similar. You only have to do additional in the grass. 7. Moments, couples: moments and couples are just as important mechanical loads US forces. So let me tell you more about them. Forces are responsible for dinner displacement, but they also cause rotating affect outside of their line of faction. This rotating effect is a moment the moment off a force can be calculated in a given point , for example, in the origin off the coordinate system. To do that, or you need to know, is the force factor on the position matter off any point. Located on the line faction of the given Force Doctors product is the moment about the origin. The unit off moment is Nuta meter. The moment off. A force can also be calculated about an axis. This can be done by firstly calculating the moment with respect to a point on the Axis than this moment shall be projected on the access by multiplying with its unit factor. Cuppers are similar to movements, but yet they shall be treated in a different manner. The name couple comes from the fact that it is caused by force capital. The two forces of the couple are parallel to each other. They have the same magnitude on different orientation. Therefore, the forces are 20 0 But as their line of action is power but not coincident, they cause a result. On moment, the magnitude of the moment of the capital equals to the product of the force amplitude on the distance between the forces. The couple acts on the plane in which the forces are located. Concentrated couple is caused by a couple contending infinitely latch forces that are infinitely close to each other. It has affinity effect, which is purely rotational. 8. Axioms of statics: there are five important axioms in statics. Mainly, they are important is understanding them perhaps your lot in solving problems in statics. So let me guide you through the five axioms. You are probably family orbit. Newton start low. For every action, there is an equal and opposite reaction aesthetics. There are lots of cases when you have to divide the structure into parts. That's what this axiom becomes handy. And you've even know that one part of the structure act on the other part, with the same force as the other acts back but in the opposite direction. Two forces can only be an actual Librium if they have the same line affection, the same magnitude and opposing orientation. This axiom were happy to determine the action forces. Every violence refers to the active eyes off loadings. It doesn't matter if I calculate the effect of two separate forces or I consider the resultant off the two forces, and I calculate the effect with death threats after this means that the system of forces can be substituted by that its atoms. The first axiom is the actually beom off a rigid body, the actual Librium, off a rigid body does not change by putting on or taking off system of floats which are in actual Librium. This axiom is the key to solve problems. By using superposition, I will introduce you to the Stepney and then you can use it as a powerful toe to solve problems and statics. The last axiom is the principal off solidification or the firm about bodies can be substituted with a rigid body in actual Librium. This axiom is a huge simplification for us. We can know the body, it will undergo deformation and then we can substitute it with the rigid body. In this way, we can calculate with rigid bodies in statics, which is much easier than calculating with the for mobile bodies. These axioms are behind the calculations that you can do to actually solve a problem in statics. 9. Static moment: static moment is a factor. Quantity. You can use it in order to obtain the sentry off anybody. Let me show you first how to calculate the static moment off a point like us. It can be done by multiplying the mass with the position factor off the point In the given Cartesian coordinate system, the result is a factor. The islands off the factor described how far the point is from the X by and south access. Now you know how to calculate the static moment off a point like mass. So that's ghost everybody. The body is beard up from infinite number of points. The static moment off the body is the some off the aesthetic moment off its points. The point. Have a position Vector Air I and the Small Data Am I Mass. Instead of summing, an integral can be taken on infinitely small muscle a month off the body. The static moment in the sentry off the rigid body is zero. This information can be used to derive information which can be used to calculate the position matter off the center of gravity. The opposition factor off the sentry can be calculated based on the mass and position factor off the small mass elements. The product of the position factor on the mass is calculated for every element they are. Some on the sum is divided by the sum off the small masses. This means the fraction off the static moment on the toe Thomas. This can also be calculated as an integral, but the sound form is more important in practical applications. 10. Centroid of bodies: in the previous lecture, I showed you how to calculate the centered of body stay erratic Alley. So let me also show you how to do it in practice. First, let me record that the position actor of the century is the fraction of the static moment on the total mass. In practice, you just have to calculate the sums in the fraction. Let me show you an example. The triangle is made out off wires in the X Y plane. You can see it's Joe Matthew on the figure. The cross section on the density off the virus are identical. Where is the san? Trade off this body. First, let's divide the triangle into parts and determine the centered of the parts. The coordinates off Stephen R. Zero Buber to zero in case of C two, they are a over 200 in case of C three, they are a over to be over two and zero. It was very easy to find the centuries off lines much easier than finding the center of the whole structure. Now we know the required position factors, but we still need the mass. For every part. The mass can be calculated as density times, William. But the volume is the cross section times the land, the density and the cross section is the same. For every part that length differs on Lee M one equals two row times capital. A time Speed M two equals two row times capital times a finally M three equals two row times capital a times the square root off a scratch plus be scrapped. In this case, the length is calculated according to protect your s Is he around? Now you can substitute everything back to the creation of the position. Matter of the century. This leads to a Roman times. I'm fun plus Aer two times and two plus air three times emptory over a mom plus M two plus emptory the German three off the three parts are known, so you can actually do the substitution as the density and the crew section at the same for every part, these perimeters dropped out of the equation. So one shabby, divided with daughter light off the elements, Then it is multiplied with the times in the bracket. If you have the numerical values of the parameters, you can easily open the solution 11. Moment of inertia: the moment of inertia, also known. A second moment of area describes how the point of the area are distributed with respect to a given access. It is an important property of the cross section of beams, and it is used frequently in strength of materials to describe the effect of banding. The second moment of area can be defined similarly to the static moment. Instead of a body, you should imagine the cross section off the body in a plane, for example, in the X Y plane. The moment of inertia is not a factor, but the metrics. But please don't care. With that respect, let me show you how to calculate the components off the moment of inertia. The moment of inertia according to access acts, is the integral off Y squared over the area. This means that you take a very small area inside the cross section. You take the square off. It's why coordinate Onda, then you multiply it with the small area. You do this to every point and you some the results. The moment of inertia according to access why is respectfully the same. But instead, off y squared X crowd is integrated. The product moment off area is the second moment of area with respect to prepare off axes for X is X and why it can be calculated as the Intergraph off X times by over the area. There are some significant geometries for which it is verse. Knowing the moment of inertia, you often have to deal with rectangle across sections. The moment of inertia is Val known. If the axes are going through the central, read off the rectangle and they are parallel with sides of the rectangle. In this case, I X is eight times be cubed over to evolve. I wy is a cubed times be over 12th and you can see that the dimension perpendicular to the axis is cubed on the other. Dimension is linear. I X Y is zero, and it will always be zero if one of the axes is a C matter access in case of exes, which are not going through the central aid, the moment of inertia can be calculated by the panel access to your am, also known as shine rt around. We would like to know the moment of inertia with respect to X. Come on, vie coma for starters. I know the moment of inertia according to Access X and why the barrel access to your, um lets us connect the central little access to the panel access so I can calculate I X coma as I X plus eight times that of Vice Squad, where I X is the moment of inertia but respected the central axis. X Capital A is the area off the cross section, which is now a Times B. That of I is the distance between the two axes in the creation. I've I coma is I've I Plus Capital eight times that X crowd in the same manner I x by coma is i x y plus eight times that x times that of I a circle is another very important germ, a tree to its central. It'll axes X and why the moment of inertia is the same i X It was, I've I on. They boost equal to R to the power of four times pi over for or you can just say D to the power of four times Spy over 64 where the is the diameter on R is the radius of the circle . The product moment of Area I X Y is zero as the circle is symmetric two. It's Entrada relaxes 12. Moment of inertia - Basic example: let me show you how you can calculate the moment of inertia off a cross section. The moment of inertia off the cross section shall be calculated with respect to the central axis off the whole coast section. The geometry is given, but we also have to determine the location off the center of mass A equals to 20 minute waiters. B is 50 millimeters. See is 70 millimeters and the is 10 millimeters. We know the geometry. Let's look for the san trade off this to the structure. We need the central eat off the group section as the central axis are going through that point on, we need the moment of inertia with respect to do's axes. The easiest way to calculate the century is calculating the sentry off parts and calculating the result. And after that, the German Attari should be split into two parts. It's easier to work with two rectangles as the central laid off a rectangle can easily be determined that the Vatican part off the cross section B, the first rectangle on the horizontal that'd be the second rectangle. The red line along the Y axis is very separated. Two parts the sentry off the first part can easily be calculated. We know that the right bottom corner is at the origin from the harder century is to the negative side off the X axis, actually at minus a work too, in the white direction. The century is at B over to to use the formula off the result and central, we need the masses. Actually, we can use areas as we are working with a two D structure, you can either just use area or use mass by technically choosing the size of the third dimension. SST. The mass of the first part is dance. It'd times a times B times sti. Here the density is considered to be the same at every point, and also the thickness D is also the same everywhere. The actual area is a times B. The second part of the structure is the horizontal part. It's a rectangle of its sides, see, and the the century is at sea, or to the over to the area off the rectangle is see Time's Day. This is more deployed by the thickness on the density which are going to drop out eventually. The sentry of the cross section is calculated by the quotient off to sums. The denominator contains the product off the masses and the coordinates of the Seine traits off each part. The denominator safely contents that some off the masses, which equals the whole mass. Off the cross section, we can substitute and see that the density and thickness rated drop out in case of a cross section area off each part on the coordinates of the sentry. Off each part is more deployment and herded in the nominator by the denominator becomes the total area off the cross section. The coordinates campus substituted. We know everything. We just have to calculate a tractor compliment. First we do the multiplication on. Then we add the components. Rove iro, the center. It of the cross section is at 8.53 millimeters, 16.8 millimeters at the X Y plane. Now we can concentrate on calculating the moments of inertia. The moment of inertia off special German threats is known. It's best to use them. So far, we have work. By dividing the cross section into two parts. The division works perfectly. We know the moment of inertia with respect to the central axis off any rectangle. That's right, these up first, and then we can calculate the result. In a moment of inertia, we can start with the Vatican part. We're going to calculate the moment of inertia to the own sent riddle. Access off the first rectangle. So the moment of inertia is not calculated. Two X is X or y, but to some exes that are parallel to X and y the moment of inertia with respect to the central axis parallel to X becomes a times B cubed over 12. The science parallel to the access is linear in the formula, and the science perpendicular to the access is cute. In the formula, the rules are reversed if we brought the moment of inertia to the other. Central access, which is parallel to buy the moment of inertia, is a cute times B word for the product. Moment of area becomes zero as we use symmetry axis off the rectangle. The same person do. It goes for the horizontal part of the structure. See, Time's the cubed over 12 is the moment of inertia with respect to the centered of access but allowed to X as C s para two X and the is perpendicular to see cubed times. The over 12 is the moment of inertia with respect to the other central access as the roles are switched, the product moment off area is also zero in this case. Now the moment of inertia off each part shall be reduced to the central axis off the whole process. Action by the half, off the parallel access t around the distance off each part central access on the whole cross sections central access are needed and not just the difference between the coordinates is needed, but was so direction is needed. So the distances that we are about to calculate have signed the centered off the hawker section on The sentries off the part are unknown, so we can easy to calculate the distances. But be aware that actually the component off a backdoor unaided to need the distance factor which starts from the central read off apart and points to the center off mass off the whole cross section. To get this factor the coordinates off the first part Synthroid is subtracted from the coordinates off the centered off the hawker a section by substituting the two factors we get the result in direction X. The distance is 18.5 millimeters in direction. By distance, it's minus 8.2 millimeters. The same goes with the other part, the coordinates off the center it off. The second part is subtracted from the coordinates of the center of the Electra section. The distance factors points to the common century. The distance factor becomes minus 26.5 millimeters and 11.8 millimeters on the X Y plane. Now we can apply the barrel access T around to each moment of inertia, and then we can some the moments of inertia off each part. We can only do this some nation with the moment of inertia that are calculated with respect to a common access. They also need the areas for calculations, so I brought them up to Area one equals two. A Times B Area two equals two C times. Let's check the result on moment of inertia with respect to the central axis that is parallel to X. The moment of inertia of part is written a scribe bracket. Each of them is calculated by adding the moment of inertia with respect to their own access andan. Other term this other term is the product off the area off the part and the distance off the to access, among which we use the parallel access to your, um in case off axis, parallel to X, the distance in direction. Why is used? These are the distances that we calculated previously in case off access para. Why the distance in Direction X is used in the product moment of area. Both distances are used. The science off the distances only become important in case of the product moment off area . Just work as I showed you before and you get appropriate sign we can substitute as you already know every measure that it's nated. These are the moments of inertia with respected central access off the cross section. 13. Reduction of loads: a structure can be loaded by a system of floats which consists off forces distributed forces on capitals to describe how the system acts on a given point of the structure. It is essential to calculate the resultant off the loads. This is called the reduction off loads become mother other loads of the rigid body with forces and coppers. You have learned about four type off loads. The distributed forces can be substituted by forces. The moments are caused by the forces, so both of these loads are included in forces. Concentrated couples cannot be substituted by forces. So to make a proper model, we need two types of floats forces on concentrated capitals. We can substitute all loads in a given point with force on the concentrated couple. Now, let this point be the origin of the coordinate system. Oh, the result and force can be calculated as some off the forces. The resultant concentrated couple equals to the some off the concentrated couples, plus the some off the moments forces with respect to a given point in your case to the origin. The moment of the force with respected origin is calculated as the trust product off the position matter off the point of action on the force factor. All together we got a factor pair, which describes the loading in point. Oh, we can do this calculation to any other point. The forces that is the same in your case is the concentrated capital is different. For example, we could calculate the concentrated capital in point A It equals two m o loose air A oh, cross here m o on f at the resultant is calculated in point. Oh, so as the system of floats substituted with these two loads, they are only loads acting on the system. It is important that factor a o is pointing towards 00.0, from point A aesthetics we always have. Actually, Bree um so it is an important loading case. There's actually be, um if both the resultant force on a couple are serial factors. In this case, the factor pair is zero actor for all other points to this is easy to see from the previously introduced appellation. The first factor remains zero actor The capital in a equals to the capital in O plus the cross redact zero plus across product off something and zero becomes zero off course 24 systems are active element. If you can add the same force and copper to them on both of them. We have been actually Priam, simply if the resultant are the same. In a point the system of floats are active on, there are several loading cases which are easy to categorize after the load reduction. 2.0, if both the force and the couple are zero, there is actually Librium. This case is general aesthetics. If the forces not zero but the couple is the resultant is a force. If the forces zero. But the couple is not. The resultant is a concentrated couple. If neither the force nor the couple zero, there are two cases. If there's kind of productive zero, the resultant is a force. If the scaler productive 02 actors at weapon the killer. The force can't goes a moment perpendicular to itself, so you can find a point where the moment of the force equals to the concentrated capital. So, in fact, bomb force is enough to produce this type of floating. If the scholar product is not zero, we have a general case where the simplest resultant is a force on the couple. You can find the point by the force and the couple were be parallel. This is called the simplest resultant, and you can learn more about it in the next lecture. 14. Simplest resultant, central line: Now that you know how to calculate the resultant off a system of floats, let me introduce a special case. The simplest trees, after in case of the simplest ways after and the resultant force factor at the resultant concentrated capital of actor are parallel to each other. Let's assume that bees a point where the statement is true we carried use the system of floats to be here. The force and the couple are far lower. This can be checked inside. The the cross productive to para of actors is always zero. The position matter of B can easily be determined a show you the short derivation off the formula. Let's consider the reduction formula for the concentrated capper from O to be down, we take the cross product off the force and the recreation. The position matter can be considered as the some off two components, one Isparta out to the force and the other is perpendicular to the force. The cross product off the parallel component on the force zero Also the left hand side is zero. The term with the too close for dots can be reviewed him according to a mathematical formula. F Times F becomes F squared on this killer productive F and the mathematical opposition matter zero simply because they are perpendicular to each other, become right out the remaining part of the creation. By rearranging the terms, we get a formula for the pay particular part of the opposition factor of B. This position matter points from B to go. But if O is the origin off the coordinate system, it is better to know the factor pointing from O to be. We only need to change the sand on the right hand side if we want to change the direction off the factor. The center line is the line on which the resultant is the simplest. Resultant. It's a creation can be written based on the formula we obtain for Point B on, based on the fact that the center line is parallel to the resultant force. The second statement is true because if we go outside of that line than the moment of the resultant force with cause a moment which is perpendicular to the fourth factor in that case, the resultant cover could not be prouder with the force. So the formula for the center line reads as air London equal to Air Obi perpendicular plus Lambda Times E f. Here. Lambda is a perimeter on E f. Is the unit tractor in the direction off the force? 15. System of loads – Basic example: practice. The task is to reduce the system off loads to the origin. The system off loads consists off two forces on the couple, the loads unknown and they're acting. Point is also known as the science off recuperate. Actually them, the result is going to contain to the there is a result of force and the resultant moment with respect to the origin. These can be calculated separately. Let's start with the resultant force. The resultant force equals to the some of the forces acting on a structure. If we substitute the forces with their some acting on the origin, the same force effect is applied because substitute the forces as vectors and we can hurt the factor components one by bomb. The result and force factor equals one minus 57 new tones. Now we need to calculate the resultant moment This is caused by the couples and also by the forces. The resultant moment around the origin is calculated by adding the some off the couples on the some off the moments off the force is with respect to the origin. Now we have one couple. So the first sound equals two m von on. We have two forces. So the second, some contents do terms. If we calculate the moment with respect to the origin, we need the position factor pointing from the origin to the point of action off a force down. We calculate the cross product off the position factor on the force to get the moment. Watch out at this point air bomb and air to must point to the forces from the origin and the order in which we do the cross product is our times off and not half times are. Let's determine the position factors if we just call it the coordinates off the points off . Action off the forces. We have the proper actors. The point of action off F one is located on the X axis. So the Y and Z component off caravan is definitely 01 in the axe. Access the distance off after one on the origin is a The point of action off after is located within the X Y plane. The Vertex off the coup Boyd is at the coordinates a B zero. These factors are going to work for us. Let me at something though the forces do not actually have a specificity point where they act, they have a line which they act, so any point off that line can be used as a point of action. Now we use these points as they are the easiest to find, but you can use any of the points that are located on the action line off the force. Now we have, with the necessary data, to calculate the resultant moment with respect to the origin that substituting to the recreation, there are two crisper. That's that we need to calculate. I have shown you how to handle such more applications, but let me get you to the calculations only. Let's see the first prosper backed. We can calculate the first component of the result by using the second and third drove the factors. The second term in the first factor is zero. This is multiplied by the third component off the second factor, which is four down. The south track the product off the third component of the first factor, and the second component off the second director. This is zero times minus do the second component off. The result comes similarly from the third and for throw off the factors the third component off The first factor is multiplied by the first component of the second director. This is zero times bomb, then the first component of the first factor is multiplied by the third component off the second rector. This gives us 0.4 times for that. We need to subtract from the other product. Finally, the third component of the result comes from the first and second row off. The factors 0.4 is multiplied by minus 20 is multiplied by far. Now we can do mood applications and subtractions to get the result zero minus 1.6 minus 1.8 is the result. We still have one more cross product the same matter This applied every job is calculated by deleting the row from the original two actors and multiplying the terms that are not in the same position. In the remaining factors. Just try to memorize the formula. It is very useful in three D mechanical problems. No justice statics. The calculations can be done on. We find that the result of movement with respect to the origin is minor. 0.2 du pont du minus one measured in Newton meters the complete solution is a factor. Pair in the origin. There's a resultant force marked by F and the resultant moment marked by M. O. These are acting in the origin, and they have the same effect as the original system off loads that contained a one after A and AMP. If you continue with the calculations, you can just think off one force and one moment as well. Loads are just reduced to this bomb force and the moment you can absolutely forget about everything else. 16. Equilibrium of two or three forces: the actual Librium off two forces on the actual Abramoff three forces are both very important. In lots of cases, the reaction forces can be calculated based on these two special cases. The actually room off to forces has already been discussed. As it is one of the axioms off statics two forces are actually Bree. Um if their line affection is the same, their magnitude is the same and the orientation is opposed. Therefore, F two equals to mind the staff one. The equilibrium of three forces is a bit more complicated. The resultant force on the resultant concentrated copper must be zero. The following conditions are required to reached. It's go first. They must be in the same plane. Second, there is a common intersection off the lines of action. If this is too and they are in the same plane, the resultant concentrated couple is zero in the intersection. If it is zero in the intersection and the forces at Kent is also zero than the couple is zero everywhere. So one part of the task is done. We still have to ensure that the forces are tent zero. The third condition is the following the factors must produce a closed factor triangle. Finally, the first condition is that the direction off the factors must be continuous in the factor tying girl. In this way, we can go around from the start here the end on We arrive at the same point that Ford the forces at 20 0 If the factors for a closed factor polygon with continues directions that resultant will always be zero. 17. Constraints: reaction forces occur in constraints, so it's time to get to know them. Constraints are constraining the degrees of freedom in the three dimensional space. Abadi has six degrees of freedom. It can move along the three axis, so that can be displacement in X y and that directions. The body can also be rotated around X Y and said that actions in a two dimensional plane body has three degrees of freedom. It can have a displacement in X and y directions. If I choose the X Y plane in the X I plane, the body can be rotated around access. If you to do rotate around any other access, it would move out off plane. Now that you know how body can move, let's see the common constrains. A fixed support fixes one point off the body. This point cannot move on. The body cannot rotate around this point, so the fixed constraint prevents the body to move according to any off its degree of freedom. In three D, This ministry displacements and three rotations into the the two displacements on the only vie rotation is constrained. A being joined fixes one point off the body, but the body can rotate around the pin. Therefore the displacements are constrained in treaty, this means street displacements on then to D this means to displacement. But all life support means that the body can roll over the super parallel to it. Naturally, the body cannot go through the support, so it constants the displacement perpendicular to itself. The role of support doesn't constant any off the rotational degrees of freedom. So this is all boost in three D On to the one displacement is constrained the van which is perpendicular to the rural life support. All of these constraints are models. So when I tell you that we have a fixed support or we have a being joined in reality, they might not look like the models that we grow physically up in joint on the fixed support can be very similar. The differences only that the pin joint loves rotations on the fix. Support doesn't alone rotations. So maybe if I fast in a screw tighter opinion joint can become a fixed support in the modeling sense from all this. Please only remember that we use more does. So if you have a really life problem, you should always start with finding the prepare model we cannot directly calculate with the constraints, but each constants can be substituted with forces on couples. These are the reaction forces and reaction couples. You need a force in every direction in which there is a constant goes by the given support and the same goes for couples and constant rotations. Let's see an example. A beam is connected to a role of support in a onto a pin joint in B. We can consider this to be a two dimensional problem. The free body diagram can be drawn first before we always need a coordinate system. So I decided to use a Cartesian coordinate system in which X points toward my right and why points upward? The roller support only constants the movement perpendicular to itself. This means direction. Why in the present case, so there is a reaction force in point A in the direction off, why I did not eat with a by There is a pin jointing, be it constants or displacements but loves rotation. So I have to force into dimensions. Be X is in the X direction on Beav. I is in the vie direction on the free body diagram. You can choose the orientation off the forces on the calculations. We give you the rial off orientation in statics we need the body in equilibrium so or degrees of freedom. Shabby, constrained, statically determinate cases at the best. In this case, we have constraints for all degrees of freedom, but we don't have any more. So the number off constraints refers to the number off creations into the three. Actually, bloomy creations can be written part body in three D. We have 60 creations. If we have more constraints, it means more unknowns in this creations. For example, if there are two pin joints on von Beam, there are four unknowns into the but you only have three equations. This problem could not be easy dissolved, but there is a solution For that case, however, you need to learn standoff materials to understand that solution. Now I keep it simple, and you will have statically determinate problems. 18. Calculation of reaction forces - Basic example: let me show you a basic example to clear how the solution mattered. Looks like for determining the reaction forces. The beam is connected to a roll off support in a onto a pain joint in B two forces act on the beam. F one is 100 new tones after is 50 new tons Afghan acts in the middle of the beam and F two acts in point A. So the loading and the German three is well known. We need to determine the reaction forces in A and B. First of all, we should always through a free body diagram. The first step is to choose a coordinate system. Now I choose to work in the X Y plane where X points right words on by points upwards. The loads F one on after can be copied from the problem sheet. Then we should put the reaction forces on the free body diagram in the pain. Joint displacement is constant in both directions, so there can be a B X force component in the X direction Onda be by force component in the vie direction. The orientation off the factors is up to me depending on the reality I feel get a positive or a negative result. The rollers support only constraints, the displacement perpendicular to itself now in divi direction, so there can only be on a vie component off the reaction force in a one way to solve the problem is through writing up the actually broom creations. There's actually Librium, so the some of the forces is zero and the resultant couple off the system floats zero in any point. So let's start writing up creations into the We have three of them. The some of the forces zero in direction. Axe in the action X There is B X and F two votes with positive orientation, So be X plus after we close to zero. The some off the forces is also zero in the direction by in that direction, a Buy and Levi are pointing in the positive direction on F one points in the negative direction. So a by plus B by minus F month equals +20 The result and concentrated couple can be written up to any point, as you have learned in the lecture about reduction of floats, I choose point a so the some off the concentrated couples and the moments with respect to point a zero. There is no concentrated couple, so we only have to deal with the moments of the forces. A by B X and F two are going to point a so the arms off these forces are zero only be vie on F. One have moments with respect to point a the positive Z direction. It's pointing outwards of the plane. Place your Tom in the direction off X so to the right, Place your forefinger in the direction of why so up towards straight and your middle finger to the tour direction, it points towards yourself outwards of the plane. If positive Z is pointing in that direction, the positive direction off the moment can be determined by adjusting your Tom in that direction. Disappoint your tamp towards yourself, then your other fingers show. Which way is the positive movement and rotation? Now it is counter clock rice on the X Y plane. So be by times l is a positive moment with respect to a on F one times l over to its negative moment. Now we have the equations. We just have to solve them. In this case, it is quite simple. From the first immigration we get be X three X equals two minus after. So it is minus 50 Newtons from the 30 creation we can determine Beav. I believe I equals two F 1/2. So it is 50 new dams From the second recreation, we can determine a vie. It equals two Afghan minus b. Why? So it is also 50 new terms. The results can be written in factor for month now You should write the factor components in the global coordinate system. This would be important if I said that a vie, for example, points downwards. Then I would have got a buy equals two minus 50 as the result. But the Eva actor would be equal to zero in X on minors. A vie in divi direction, so minus minus 50 would be 50 again. A graphical solution can also be done. A graphical solution always require scare. A choose 10 U turns to be vast centimeter. The solution mattered is based on the equilibrium off three forces, so you cannot do it. If there's a concentrated couple, you can see very more forces on the free body diagram. But there is always three forces. Now Von Force is in a wound. Force is in be on to forces acting on the beam. But the resultant off these forces can be determined. So only Von Force remains in the problem. To graphically get the resultant, you have to search for the intersection off the lines of action off the two forces. In that point, the forces can be added according to the parallelogram low off addition. Finally, we only have bomb force F 12 as a load. The next step is to solve the actual Librium for three forces. Three forces can only be in actually boom If they are involved Plane which is surely through now they must have a common intersection point off their lines affection. This is what we have to ensure. Now we know f 12 and we know the line affection off A. It is perpendicular to the roller support. So it must be Vatican. The two known line of actions are crossing each other. At the beginning, I had no idea about the land affection of B, but now I know that it must go through that intersection point and off course. It goes through B because it x there so I can. Through the line of action of B two, the forces must form a closed factor Chai anger So I could be a friend to to the intersection Point and I drew a line parallel to the line of action off B to the end off the new F want to factor. Now the closed batter triangle is forming. But it is also important that the directions off the factors must be connected continuously . Now we got the result. With the half off the scale we can get numerical is that, Believe me, it is the same as the vans that we got with the calculations. 19. Separation of structures: So far, you have seen pretty simple constructions, which are easy to analyze. If you face a more complex problem, you need some tricks to analyze it. For example, to calculate the reaction force is one of these streaks is the separation off structures. Let's see a structure with three pins. The structure is fixed to the environment by to pin joints, and there is an additional inner pin joint which is connecting the two beams for simplicity . There are only two forces acting on this structure. I can draw the free body diagram and now you can probably see by the structure is problematic. There are four reaction forces to for each being joined. However, into the I can only use three creations based on this free body diagram to calculate reaction forces. So I have no chance in determining the reaction forces. That is why I have to use a trick. Now. I assure you the separation off structures and you can learn other trick in the next lecture, that one. We use the principal off superposition aesthetics. Everything is in actual Librium. So if I split the structure into multiple parts, every section were being actually Bree um, off course. If I split the structure into parts, I have to do the same as in case of substituting the constants with forces. If I'd even vom part, I have to consider its action on the other part, and also the other part has a reaction on the first part. This is basically Newton stirred low For every action. There is an equal and opposite reaction. Let's use these rules and solve the problem. Let's split the structure into parts in point. See, for the first part, I have reaction forces in point A and in our reaction forces in point C, I can choose the directions for every force. I've you get the properties that after numerical calculations for the second part, the direction off the reaction forces in C cannot be choosen according to the action and reaction low. They must be opposite to the Vance acting on the first part. This is important to remember Now you have six unknowns, but you can use six e creations three for each free body diagram. You can see that this problem aesthetically determined you can actually solve it because of the addition up in joint note that I used the pin joint to split the structure into two parts. If you split the structure anywhere else, that would be to reaction forces and also a couple that you have to consider the Pingeot and cannot carry moments. That's why that are only two forces acting in it. You will understand this better after you have learned to calculate the stress. Resultant so I don't go into details. Now. If you understood the separation off structures, you can join me in the next lecture. But I will tell you about the principle off superposition. And after that you can solve this problem Numerical e in two ways. 20. Principle of superposition: the principle of superposition gives away to solve complex problems in statics. Let's check out the problem that we already discussed in the previous lecture. It is a structure with three pins from which to is connected to the environment. And fun is an inner pain joint. We can throw the free body diagram. The problem is that we have four unknowns for three actually blew me creations. We need to produce more recreations. This could be done by separation off the two beams in point C. But now I show you another trick. You need a little bit of terror to know. First, the principle of superposition states that if there are more loads acting on the body, the effect off a single load, for example the effort off a force is the same as if it would be the only load. Acting on the body on this effect can be added to the effect off the other loads to obtain their superposition. This means linearity in the mathematical sense. But you only have to know that you can load in steps, take only the Afghan force. The reaction force is a coma. Then take only the F to force. The reaction force is a double a comma if you add the effect of forces f Fun and F two. The reaction force is the sum off a comma and a double comma. This can be done with any number off loads. You should also be family are be to force members. These are based on the actual Librium off two forces. That name represents the fact that only two forces are acting on the member off the structure. As an example, you can see a beam to forces acting on its doyennes two forces on actual Librium if their line of action is the same. So the forces are acting parallel to the beam. The magnitude off A and C are the same, and the orientations are the opposite to each other. We can go back to the original problem that's consider the loads individually and make 23 buddy diagrams. First, I take off F two and F one remains the only load in this case. The beam between Point A and C becomes a to force member, as there is one reaction Force acting in A, which is off course, has a component in X and y directions, and there is an other inner reaction force acting in. See as the direction off a comma is parallel to the beams direction. We get an additional germ ethical equation, which connects a X coma and ive I coma. Besides that, we have three, actually Brima creations. So what together four recreations for four unknowns for reaction forces can be calculated in the next step. I take off F Bomb and put F two on the structure. In this case, the beam between B and C is a two first member, so I haven't additional dramatic equation for the reaction force in Point B. I can solve this problem, too. Then I can add the results for the two cases and obtained the reaction forces. 21. Structures with three pins – Basic example: forces in structures with treatments in practice, the structures in In the terroristic. A lecture is investigated. The geometry on the loads are given sufficiently. We must determine the reaction force is occurring in A and B we can through a free body diagram. As the structure is connected to the surroundings in A and B, there are reaction forces in each Been there are two Reaction force compliments. Well together. There are four reaction force components, but as always, we only have three actually remake rations. The trick is needed to get more recreations. We have learned to tricks the separation off structures and the principal off superposition . Now I'm going to show you the solution with both matters. Let's start by the separation off structures. We can split the structure into two parts at sea. It is important to split the structure at the Turpin as there is only an inner force. Inside that been, there is no inner moment. So by splitting the structure, we get to new unknown force confidence. All together, there are six condoms, but we have two times three actually bream equations. Now we can determine all unknowns based on these creations. When it split the structure. You must throw the inner forces at point C. It is important to consider the action reaction connection. The forced components on the two drawings are pointing to opposite directions. Besides this, you can freely choose the dye actions. So now C X points right words on the left, throwing on left words and the right drawing, but it could be in the opposite way. Let's go like the actually blew immigrations off the first part of the structure along the direction x a X plus e x plus after two equals zero along the direction. Why a By plus C right equals +20 These are the actual Umbria off the forces. We still need an equation, including moments. Let's consider the actual Abramoff moments everything around point a some M a equals to see why. Times a minus. C x Times C minus. After times, the the arms of the forces can be seen on the original structure of throwing. Let's continue with the actually bloom off the second part. Along dine action X b x minus E X equals zero along direction by B Y minus. C right minus half Want the question zero as the turn recreation we can use the actually bloom off moments, for example, acting around point be some m B equals to see right times B plus C exam C plus F one times B minus e. You can see that all of the forces are acting positive. Lee around. Be on all of them would turn the structure counter clockwise. What, Together we have six decorations that we can use to calculate the six. Al knows, But let me remind you that everything is in, actually, really even the original structure Is that actually, Bree, um, you can also used the actually human creations off the original structure. So if you want, you can use in under three big rations, but you do not have to Ride is up. This is just an opportunity along the X direction a X plus B x plus after requested zero based on the original structures. Three by the diagram along Divide direction a vie plus B by minus F. One interested zero, and for examples, the moment around A reads as be right times a plus B minus a fun times a plus e minders after times t This equals +202 As you can see a X a by and be X goes to a suit that arm zero b writers. The system counter clockwise around a by the other two forces after in the system in the opposite direction. All together, there are nine decorations. You can use any six off them. A show you one possible solution from the creation nine b y equals two for ridicule. A new terms from Equation eight a bi equals 20 from the creation to see right in close to zero from equation three c x equals two minus a 20.89 killer Newton's from equation one a X equals two minus 11.11 killing U turns and finally, from a creation seven b x d cross the minus 8.89 Continue tells you can also use the 1st 6 situations, and you would get the same result. I've read the solution, in fact, or four months. All of the components are known, but be careful about the time actions. Now I throw everything in Positive act survived that action on the free body diagram, so everything is substituted in the global factor as we got it. If you drove something in a negative direction, you should consider its negative sign when you substitute it into the factor. Former. Let's see the alternative solution based on the principle off superposition according to the principle of superposition, we can loot the structure and steps. We get the same result either by calculating the effect of forces separately and heading down or calculating the effects together. So first we calculate the reaction force is caused by Afghan, and then we at the reaction force is caused by after. The result gives the same reaction force that is actually occurring in the system. Barbosa's F one and F to act on the first free body diagram only after one act on the second free body diagram only after two x for simplicity. I marked the reactions differently. In the first case, I used one comma markings. In the second case, I used to come on markings. Let's start with the case than Onley. Afghan acts on the structure. In this case, the left county lever becomes the two first member as forces are only acting at its end points in the pens, there can only be an axial force in the to force member, so the force in a must be paralleled with the canti lever. So there is a dramatic connection between the components of the reaction force. A comma, a X coma over a bike. Amount equals two a were see according to the geometry. This ratio ensures that a common factor is parallel with canti lever. This is an additional recreation, so this on the three actually gloomy creations provide four creations or together. These are enough to determine the four unknown reaction force Confidence in direction X e X comma plus B x Kama equals to zero in direction by If I come a plus B Reich amount minus F one requested zero the some off the moments around a zero. So we've I come a times a plus. B minus F one times a plus. B equals +20 We can simply solve the system off decorations. The creation four gives us be right Come on, which equals two territory killing your toes. A by kama equals to eight killing Newton's from the third creation a X comma across the 4.44 kill a new towns from the first day creation and finally be X comma averse to minus 41 44 killer new tons from the second creation. These are the reaction forces, if only after one acts on the structure. Let's see the second loading case when only half to act on the structure. This time, the canti lever on the right hand side is a to force member, as there are only two forces acting it on it in the tube Ian's. This means that be to come on back to respond aloud with the canti lever. The germ. A tree gives us an in creation between the confidence off B two comma. The X component over the vie component Big First, minus B oversee. I assume that those components point to the positive direction. However, the gentle math trick constraint says that one of them is going to be negative. For example, if B X two comma is positive, be by to come up must be negative to get the proper direction. This is vie Vinnie the minus sign. We also need the actually human creations. For this loading case, a X to come a plus B x to command plus f do equals zero a by two Demopoulos be writing commodity crust. Zero on Were I to come on times a plus B minus after times the equals zero, according to the actually come off moments around a we can solve the creations be vie to come out equals to eight Kill a new terms According to the Forced, the creation from the Joe Matic immigration BX to Khama becomes minus 4.44 killing Utahans a bye to Mama First U minus eight Killer Newtown's from the third creation and finally, a X to come out equals two minus 15.56 killing Utahans. These are the reaction forces that only have to act as a load. We must some the results off the loading cases will of the force. Confidence must bay some separately. It is very important to consider the direction off each confident. I assumed everything to be pointing in the positive axe or via direction, so I only need to add the components. If you drove something in the negative X survive direction on the free body diagram, then you must take it into account with the negative sign. Now we have to provide the global factors, according to the X my coordinate system. It doesn't matter how you drew the forces on the free body diagram. Be careful of and you have to sound the results. As you can see approximately go the same results with the two solution matters This slight differences only come from rounding. 22. Trusses: trusses are important structures, both in civil engineering and mechanical engineering. So let me introduce their basic properties. By knowing these properties, you will be able to analyze trusses by calculating the inner forces. You are surrounded by trusses. For example. There are lots of bridges which are trusses in mechanical modeling. Or you can also think of the steel structure off the skyscrapers. But maybe the most famous truss structure is the Eiffel Tower. The mechanical model off a trance bridge can be taken as an example. You can see that there is a pin joint on the roller support under the structure. If you look around in your neighborhood, you can probably find a bridge on which you can observe that it is in fact, connected to a pin joint. The pain joint is needed to lower rotations, so the affirmations caused by the heat do not stress the bridge too much. You can also observe that the bridge has delegation are months which followed this place month and can be modelled by roller supports. Now that you surely know buttresses are, let's see its properties. Considering the germ a three you should know to properties. Firstly, trusses consist off straight rigid beams. Secondly, these beams are connected through idea being joints at their aunt. Considering the loads, you should know they only act at the pain joints, therefore or kind of distributed forces are neglected. For example, gravity is neglected as the loads can only after the pain joins or beams become to force members. There can only be to forces acting on each beam, one at one end and the other at the other. And these forces can be just reaction forces in the pen joins, or they can be the some off reaction forces and loads anyway. That can only be to off them, as the beams are to first members, and they are straight or in our forces are longer to deny in the beams. By knowing these properties, you can start to analyze trusses, but first joined me and I will tell you about the method of joints and the method of sections. These are the two best matters to open in her forces in trusses 23. Method of joints: let me show you a very effective mattered for opening all in our forces. In trusses. Let's consider trans bridge. You can see it's mechanical model. First, you can usually open the reaction of forces in A and B, just as you learned in the previous lectures. If you cannot open the reaction force is simply or you want to know the inner force in any of the beams, the method of joints becomes handy. As you already know, everything is in actual Librium aesthetics. In a trust, you can see multiple up in joints on beams, and all of them are in actual Librium. This fact is putting news. The method of joints use that the pain joints are actually Bree um, and you can draw a free body diagram off, each being joined. Then you can use the equilibrium E creations to obtain the forces acting on the pain joint . If you separate the structure that our forces that each beam except on the pain joint. And also there are reaction forces that are excited on the beams by the pin joint. According to Newton's Third Low, these forces are equally magnitude, but they have opposing directions each beam is a two point member, so there can only be longer to deny forces inside them. Let me assume that these forces are positive if they are pointing outwards from the beam. Therefore, a positive force means tension in the given beam and the negative force means compression. The force acting on a pin joint is opposed to this force. So if the positive direction is outwards from the beam, its reaction force is pointing outwards from the pain joint. In this manner, you can draw a free body diagram for example, in point C but your forces me intention for the respective beams. Off course loads like F one has to be in the free body. Diagram to you can do the same for each pin and then you can solve the actually blew me creations into D. You can only use two of these actually Bria me creations. The some off forces are zero in the two directions. It is also true that the result and moment in the pain zero, but this is automatically full feared as the arm of the moment is zero for each force acting on the pain joint. Therefore, if you can't It's practical to pick up in joint, in which only two forces are unknown, because then you can determine them with ease and you can continue with the next being joint. By doing this, you can open all in our forces in trusses. 24. Method of joints – Basic example: let me show you how you can use the method of joints In practice. Trust is investigated, the structure is connected to it. Surroundings by a pin in A and Earl a Super and be there are two. First is acting on the trust. We must determine the reaction forces and then the inner forces. By the help of the method of joints, the Juma three and the loads are given. I might every pin by a letter and a number or the county lovers for easier reference before doing anything else. We must determine the reaction forces. To do that, we need the free body diagram on the actual IBRA. Immigration's a pin is connecting the structure to the surroundings in a so there is a reaction force competent in X and y directions. There's a runner support in be so only a force competent perpendicular to the ruler. Support arises so there's only of Attica force. Competent and be, the constraints are substituted by the reaction force is still there is a pin in A and B as actually pins are connecting the county lovers in the trust. In case of the free body diagram, we only substitute the constraints that connect the trust to the surroundings. These constraints were both connected to pence off the trust. Now let's check the free body diagram the rectory Unknown Reaction Force components. This can easily be calculated by using the three actually beom creations. There could be more force competence. In that case, you can use the method of joints to get additionally creations based on the actually boom off any joint. You can use those additionally creations to express the reaction forces. Now it's enough to consider the actual Abramoff the trust in Direction X, a X plus after equals +20 in direction by A i i plus B by plus F one in close to zero. We need one more recreation from actually room off moments around the point. At this point, be a several forces go to a and do not have a moment with respective a a x a by and have to go through a so their arm off moment zero. We only have no zero moments from Levi, and after me, right rotates the system counterclockwise around a. We consider this direction to be positive after my rotates in the same direction, so that has a positive moment to the absolute moment is a plus B for beer I and A for half long a x the first minus after based on the first day creation. Therefore, a X equals to nine ist lenticular in you toes. The Tory creation does us that be by close to 4.76 killing Dems. Finally, the second recreation Let us calculate a vibe which equals minus 14.76. Given domes, the reaction forces can revert back to her four month and from now we must cap later in our forces off all of the county lovers Well, we know the reaction forces the method of joints, beards on the actually room off the joints. Everyone off them is at Akron Librium, so we can draw several free body diagrams that contained the inner forces. Actually, bream off moments is obviously threw off. Yet for every joint, as the forces acting on a pin are going to that same point, therefore we have to we creations with which we can capitalize the inner forces at every joint. The actual Abramoff forces can only be used. Therefore, if we can, we must choose a point in which to connecting in our forces. On look at the structure Pill A and lengthy is connected to only two county lovers. The solution should start with one of them. If we start with been a, we can calculate the inner forces in county lover bomb on three. Then we can also use Pindi in our forces. Off Canti Lever four and seven can be calculated after that. Every remaining joint is sufficient as only two in our forces are unknown in B, C and E. So let's start by stating that pin A is that actually Agrium. Let's through a free body diagram. The trust is connected to its surroundings through point A. So the reaction force components, a acts and a vie act on the This is how the surroundings are acting on the structure. They exert force on the pin in a on then the pin distributes the force between the two connecting county lovers. The two connecting county lovers also act underpin the cantilevers off trust are to force members as only their endpoints are loaded. The inner forces act acsi Ali in the to force members so the inner forces must be drawn parallel to the county lovers. I assume that the two first members our intention. So the inner forces point out farts off a county lever. The reaction force off such inner force points outwards from the pin here I drove s bomb which is the force that can't lever from exerts on the joint a on as three which is the force that can't lover tree exact on joint A Always make the inner forces point outwards of the pin In this manner, if you get a positive result, you know that the inner force mints tension for the candid lover And if you get a negative result, you know that the inner force means compression for the canti lever. Let's jump to the actually blew me creations. However, before that we need to know geometric data namely, the angle between the forces from the gel material trance you can see the tangent offer equals to see over a. This means that offer equals to 50.2 victories. Now we can use the equally human creations in direction X. The forces are actually beom a X plus s mom plus Estrich time school sign offer equals +20 in direction by a bi minus s three times Sign off equals 20 We can calculate asthma and history as three comes from the second equation. And then we get a small from the first migration history. It was to a by over signed off. It is minus 19 point. Do kill a new terms. The negative result means that there's a compression within Can't celebratory s one equals to 32 point curricular Newtown's, which is a positive results, so can't deliver fund is intention we can continue with the next joint. Pindi is also at actually bloom and there are only two undone in our forces that act on this been as three is already know. So only the inner force of Canti lever four and seven. It's the we calculated justice Before I assumed that the inner forces find outwards off the pin. There's also load which act in the F one acts as it is seen in the original figure or so the previously calculated offer angle appears in the German tree. The force actually bloomin direction X means that s seven minus a street times goes an awfully close to zero in direction. My F bomb plus s four plus F three times. Sign off for a 1st 0 s seven equals two minus for of point curricular Newtown's based on the first day creation on them. As for the 1st 24.75 continue tones Based on the second recreation, we can go on and choose any of the remaining joints The work it that's use being e which is also at actually Really there are only two Canty lovers that are connected to pin me so we can calculate in our forces even before calculating any other force within the trust after is a low that active e. Besides that, inner forces off Canti Lever two and six are drawn on a free body diagram as forces pointing outwards of the been in direction X after minus s two e cost zero in direction of I as six is simply zero. There's actually no force in Cantel ever six. It seems like there is no point having this canti lever in the structure at all. But in statics becomes the regent bodies. In reality, there are some informations, so actually there is a function of this canti lever to as to the question after two Based on the first day creation so panicky Lannoo turns off Stangel act meeting canti lever to there is bomb unknown in our force the von Waiting Canti lever five I choose to calculated by the half off the actual Librium off PNC To use the rations, we need an angle that can be calculated based on the geometry beater e close to Arctic chancy over be By knowing this angle, we can calculate us five either from the force actively, um, of the action ex or by indirection X as two plus s five times cause and beat Ah minus s mom in close to zero in direction by minus as four minus us five times Signed Vita equals zero for example, as five equals demise as for over San Mita, which means minus 6.44 killing your does this result can be checked by using the other recreation or so you can check your work by using the actually bloom of joint be as that has no being used at all. We have already got all of the results. At this point, the remaining joints are just good to check your results 25. Method of section: if you have a trust and you only need to know the inner force in just some off the beams, not in all of them. You may be better off choosing the method off section instead of the method of joints. We can take the same trust bridge example as before. Now, I'm only interested in calculating the force acting in the beam which connect see on the so why should I calculate all other forces? I can find the result I need directly. I just have to use the method of section as its name indicates section off. The whole structure is taken into account. To do that, I have to cut the structure into two parts with a straight line. Now I do it across beam 45 and 10. Wherever I cut the structure, I leave on part and I only keep the other part for further investigation. The part which is taken away has an effect on the remaining part. In every being which is cut, there are enough forces. Therefore, the free body diagram off the remaining part contents these inner forces, that is all we need to do to consider the effect of the other half of the structure, and this can be done with both parts. As you can see, the inner forces in the beams are longer to denounce, as the beams are to force members in trusses. That's how you can easily road in our forces on the free body diagram. There are some rules off cutting the structure. I have to cut the beam, which is under investigation so it's in Our force could be calculated. I can cut maximum off three beams as this lead to three unknowns. If the reaction forces and loads are already known, these three unknowns can be determined by using the three actually be a me creations in case off a two D problem. Maximum to cut beams are connected to the same being joints. If or three were connected to vamping joint, the actually bloomy creation for moments in the pain joint would automatically be full field. Thus, I could not use three actually blew me creations for calculation. By keeping these rules, you can use the method of Section two easily obtain the inner force in a given beam, even in case of a very complicated structure 26. Method of section – Basic example: I show you how you can use the mattered off section in practice. This problem has already been investigated by the help off the method of joints. This time, only the inner force in canti lever five months be calculated on top off the reaction forces. In such cases, it's verse using the method of section instead of the method of joints. The method of section can give a direct solution by the method of joints. Require several steps to calculate the inner force in county never five, the same data is used. This before the reaction forces can be calculated based on the free body diagram and actually human creations. The steps are the same as you have seen it earlier. Now we can just use the previous results and concentrate on the second part of the problem . We must determine the inner force off Canti Lever five. In case of the method off section, we use a straight line to cut the structure into two parts. Wherever we cut the structure, we cut some country lovers in which, in our forces act, the inner forces are acting from one part of the structure on the other. So we can choose one of the two parts and consider it. Actually, Librium, the inner forces carried the effect off the donated part. There are three actually bloom equations in a plane, so we can just express three cardinals. If the reaction forces are ordered in. Now we can calculate tree in our forces by the help of the mattered off section. If we need the inner force in canti lever five, we must cut that 12 main inner force appear we can cut maximum to further county lovers. If we can't three cantilevers Mystere have to make sure that they are not joining the same pin. If three county lovers joined the same thing, the result moment off them with respect, that bill is zero. So we lose the actually bloom off moments, and we can just used to actually view me creations. So we use a straight line. We must cut county never five and maximum two other Canty lovers. Why we make sure that two off the tree can't camp delivers can join to the same thing. The only suitable way is to cut through Canty lovers 25 and seven. There is no other way to cut Canti Lever five and maximum to other county lovers. After the cat, it's up to you to choose which side of the structure you investigate. Now I keep the right side of the trust with the roller suffered and be on the after forcing e we must, through the inner forces on the cut structure along canti lever to 57 The tree in our forces substitute the effect off the dilated part. I assume every inner force to be Tash. Um So the inner forces point outwards from the county lovers. So positive result means tension on the negative result means compression. The structure is at actually live Liam. The tree usually creations can be used to determine as to us fight on Osama, though only as five months be calculated. So it's enough to write an equation. For that, we need a better angle which can be expressed based on the German. Then we can really calculate us five in direction by the force. Actually, beom only contents us five as an unknown B y plus s five times sine theta equals +20 as five equals the minus b y over sign metta, which means minus 6.46. Killing your dollars. That's it. We have got the result. That we need. Worst case scenario is that we have to solve three creations to find the result. It's way faster than the matter of joints. If just one in our forces needed off course, the method of joints gave the same result For the inner force. There is only a slight difference Beauty surrounding Gary roars, so actually you can use both methods choose as huge. 27. Stress resultants: stress. Resultant splay huge Dolin engineering. The structure is designed or check based on the occuring stresses. If the stress maximum exceeds the year stress or the ultimate tens, I strapped the structure of your be damaged or destroyed. There can also be lost off stability, for example, battling which can lead to damage. If you can determine the stresses, Earth and factions, you are going to be able to beard on your statics knowledge and become a good engineer. So this topic is very important. Let's see the details taken arbitrary beam and cut it into two parts with an area summer this area must repair. Can they collected the center line of the being? There are forces and couples acting on the beam on those parts, or together the resultant in Syria as there is actually agreement statics. Let's reduce the system off loads to the center of the Catholic or of section for both sides. This results in the force and the couple for both sides which have the same magnitude and line affection but opposite orientation. This maybe family are to you as we have already used this step in the previous topics. The in fact off forces and couples on one side can be substituted with the singing of force on the couple. These are the forces Athens, which are the loads transferred within the body in a given cross section. In practice, you only have to know that the stress resultant in a given crow section are loves. Reduced to death cross section, there are four types off stressors Athans. These are the components off the resultant force, and the resultant couple let me define this components and is the normal actor off the cross section. T is a factor back and ignore. For example, if an is aligned with coordinate axes aligned with coordinate by the first stresses are turned is the normal force. It is the component of the fourth resultant, which is perpendicular did a cross section. The other component off the fourth resultant, is called sheer force. It is perpendicular to the normal force. The torch in or twisting moment is the component of the resultant couple, which is normal to the cross section, and the bending woman is packed and vehicular to the torch in a moment. The San CA mansions are important in case of stresses happens in the following. I use the Sachem rations, which are usually used by mechanical engineers. I cut the big summer. The normal force is positive if it points outwards off the material. In other words, if there's tension in the given intrasection, if I start to ride a faction of the normal force from the left hand off the body, the normal force is positive. If it points left words, if I go from the right and of the body that it is positive if it's pointing to the right, this is how you shall imagine the arrows that are drawn. The figures Don't worry if it is not clear yet. You will surely understand this in the next lecture. But I show you some examples. The sheer force is positive if it points upwards. If I start in the left hand, it is positive if it points downwards, if I go from the right and the bending moment is positive if it bans the beam downwards. This definition is not the same for every engineering training. But don't worry. If you learnt it differently in school, just be consistent and always use the same definition for engineering purposes. The sign doesn't matter. Just the real direction off the stresses which will be used instructive materials later. Now you know how I think off the directions like give yourself the torch in a moment is positive if it points outwards of the materia just like a case of the normal force. Join me in the next lecture, but I show you some examples. 28. Stress resultant functions with basic examples: let me show you how to determine the stresses out and functions in practice. Just take a quick look at the Sankoh mansions as a reminder. The arrows showed the positive directions. You shall apply the balm on the left if you have writing up the function from the left and you shall apply the one on the right. If you have writing up the function from the right, let's check out the first example. There is a canti lever with the force F acting on its free and toe right. The stresses Artan functions. Usually the reaction forces have to be calculated. But now I can do it from the free and without calculating the reaction forces, we can define a perimeter which is measured from the free. And this perimeter s is greater or equal to zero. But it is definitely smaller than El, the land of the canti lever. It can't equal to the land because there is a change in the loading at that point, as the reaction forces are acting at the beatin and there is only a normal force. So all other stresses are tons become zero. We have to divide the length off the country lever into as many parts as money have different loadings. Now the loading is the same along the whole length, so one function is enough. If we would have force in the middle of the county lever, it would change the loading. Then I would have to segments in the first fund s would go from zero to AL over two and in the second s would go from Delaware to L. Now let's determine the function of the normal force. I measured the perimeter from the left, so the positive direction off the normal force would be to the left. The force points to the right so into the material, which means a negative effect. Therefore, the function of the Norma force equals two minus half. Let's see the second example there's a county lover with distributed load again. We can measure the perimeter from the free and to avoid calculating the reaction forces. There is no normal force and there is no torch in a torque, so these functions become zero. However, there is cheering on banding, so we have to think about those functions. What I do is basically to take the point, which is at the distance as from the free and the forces are turned on Capitol resultant shall be calculated there. So I'm in the point which has the distance as from the free and I've ride the functions from the right and off the canti lever. So I used the corresponding signed conventions and I forget everything which is located to the left. From this point, let me consider that I'm at the position shown with the red line. I only calculate with the loads which are located to the right from this red line. The result and force at that section is P Times ass, and it is pointing upwards. Considering the San Conventions, this is a negative sheer force. Thus the sheer force function equals to mind the speed times us the bending moment off. This resultant force can also be calculated. We still only think about the part which is located on the right off the red line. The magnitude off the force which can replace the distributed load, equals to P times us. If I replace the distributed force, I do it with the force which act at the center off the distributed force. Now it is the middle off the given part, so its distance from the red line is s over to. Therefore, the bending moment equals two minor speed times as times as over two minus. Because this distributed force bans the canti lever upwards on the magnitude is determined as the magnitude off concentrated force times the arm off the force. With respect to the red line, you can always ride the stresses that and functions like this. The third example contents to forces one at the free end off the country lover and fun at a given point. As the loading is not the same along the whole county lever land, the stresses, art and functions shall be determined in two steps. In the 1st 1 the perimeter s goes from zero to a. It is measured from the free end. So I don't calculate the reaction forces. We can again throw a red line, and we can imagine that we are reducing the loads to the point marked with the red line as the perimeter is measured from the left and only the loads on the left side off the red line are in interest. You must forget that there's anything on its right side. The problem looks the same as it was for the first example. There's a normal force which points into the material, so it means compression. Therefore, the Norma force equals two minus F one. The other fortunes out and functions are zero on the segment off the country lover, Let's move to the second step. Perimeter s goes from a to l. We can do red line again Now it is after after you. So now both forces are to the left from the point marked with the red line compared to the previous segment, the Norma force is unchanged. It equals two minus F one. There is still no torch in a moment, but there is a sheer force on the bending moment. After causes both effect, it is pointing downwards. This means negative sheer force. As the left arrow off, the Sankoh mention counts as we ride the functions from the left that, for the sheer force in the segment, equals two minus. After the force is pointing downwards, which causes the left and off the country lever to ban downwards. This means a positive banding moment. Now we just have to determine its magnitude as usual. The bending moment equals to the force times the arm off the force that for the bending moment equals two F two times s minus a. The arm of the force must be calculated between the red line and the line of action off F two. The distance between the canti lever and on the red line is us. And the distance between the canti lever and and F two is a that difference. Give us the arm length, which is s minus a. You can see that if you take the derivative off the bending moment according to perimeter s , you almost get the sheer force with my sign conventions. If you've read the functions from the left side, the derivative off, the bending moment is minus one times the sheer force. If you ride the functions from the right, the derivative off the bending moment is the sheer force. This fact is useful for checking your results 29. Stress resultants of curved beams with basic example: so far, I have told you about straight beams, but not all the beam substrate. Some of them are curved, but despite this, you must be able to handle them. So let's talk about the stresses. Athens off curved beams. First of all, let's talk about the signed commissions. They are basically the same as in case of the straight beams, but it never hurts to revise it. The normal force is positive if it's pointing outwards off the materia. So if it is a chance, I force the sheer force is positive. If I've ride the function from the left hand and the force points upwards, or if I've read the function from the right hand and the fourth points downwards in case of the bending moment, we have a slight difference compared to the previous case. The banding moment is positive if it increases the curvature off the beam and it is negative if it straightens the beam. So in this case you shall not think about the global directions. Just look at the shape of the beam and comfort the direction off the banding effect. To that shape, the torch in our moment is still positive. If it's pointing outwards of the material. Let's see an example. There's a curve to be with the beard and the fourth act on its free, and in this case, it is not practical to use X or y coordinate as a perimeter. Instead of that, you should choose an angle. I choose angle fee, which is measured from the free and free cambuur I from zero to pi over two we can talk to in the stress result and functions with the hop off this perimeter. There is no torture that whole. So the torch in a moment 1st 0 the other ones are more difficult. But let's do it step by step. First, let's just the over the forces attend. We are investigating at the point, which is described with perimeter. Fire at the perimeter is measured from the right. The force half is always on the right of 50 snow, it is considered. The forces included, the force resultant it feels to have in that point, the resultant force can be divided into two complements. The normal force is in the normal direction. Off the cross section. The fear force is perpendicular to the normal force. This way we can obtain a function for both the normal and the sheer force. The normal force is pointing outwards from the material, so it is surely a positive stressors. Afternoons the Norma force becomes halftime signed five According to the jail. My three off the right and go to try and go. The sheer force is pointing upwards as I fried the function, starting from the right end of the beam. Therefore, the sheer force is negative, so it becomes minus halftime school sand fire According to the gentleman tree, the last step is to determine the banding movement. The force is known, so we only need to determine the arm off the force with respect to the cross section. Describe but angle Phi. Let's see a new figure. The red line indicates the arm off the force. There is a right and go try anger, little hip attenders. Which land is our? So the arm off the force is our times. Sci fi. The fourth straightens the beams. It decreases the courage. Therefore, it is a negative bonding moment. Therefore, the bending moment becomes minus half times at times sci fi. It is still true that the derivative off depending. Movement is connected to the sheer force. It should differentiate the sign. Five becomes school sad five. But there is an additional multiplication with our I don't want to go into details. It comes from the fact that we choose an angle Asieh coordinate. If you understood this example on the sample problem, you are just ready to take a step in your mechanics. Studies and love stank of materials. In the next section, you can learn about known idea constraints, which basically means the effect of friction. That section violence your knowledge in statics and prepares you for taking the step towards the moving objects. The feared off dynamics. 30. Coulomb friction: The most important non idea constraint is the Coolum friction. It appears in most of the fundamental boosting aesthetics or dynamics. Sometimes the kuna friction can be neglected, but sometimes it's not. So let me show you what the Coolum friction is and how to handle problems where it occurs. Coolum friction occurs at rough Surface Let's let the body with force half this body is on a rough surface. We can calculate the reactions on the surface as it is a constraint. So let's draw a free body diagram besides stuff that can be gravity if we are in a gravitational field. So I added that to the free body diagram to it acts at the center of the body. In reality, there is a distributed force which is accepted on the body by the ground. This distributed force keeps the body on the surface of Article E as the body is on the surface and cannot go through or so this distributed force acts against forced F. This part is the fictional component. This distributed force is highly dependent on the exact German three off the surface, but we do not want to go into details. Therefore, let's substitute the distributed force with a concentrated force. K. This force has two components. The horizontal oven is the friction which I marked with us and the Vatican. One is the normal force, which I marked with. Um, let's concentrate on this force. You can see the batter decomposition on the figure. Okay. Equals s plus on. I introduced the anger row to describe the connection off the friction and the normal force . Real note. Is the limit off this angle. Please start to imagine this problem in treaty. There is a cone which is described Withrow Note as row note is the half off the apex angle off this cone. If the reaction force is in the cold, then there can be actually Librium. Therefore, if low is smaller or equal to row note there is actual Librium. Limiting friction occurs if ro equals the row. Note the friction force us is maximal. In this case, you can see that the condition can be defined for sticking by the half of the forces. If the absolute value off us over on which equals two tension throw is smaller than tension . Throw note and there is equilibrium done. The body sticks to the surface. If you do the calculations and you get the absolute value of s over, um, it's greater than this value. There is sliding that doesn't belong to statics. Then the problem becomes a dynamic of problem aesthetics. We just check if there is really sticking or not, or we can check the limit Case of friction Tangent Row note equals to me a note and it is called The coefficient of friction, we note belongs to sticking. There is another coefficient of friction, which belongs to sliding friction that is usually marked with me instead of me. Note. In reality, the sticking limit friction is usually larger than the sliding friction. This is why it is better if you don't start sliding on the road with your car because it is pretty hard to reach the sticking point again. But now you can forget about dynamics. Just concentrate on statics and the sticking friction. Let me take an additional remark. You can see that in the definition, the absolute value off s over and can be a maximum off tangent row note. The tension function doesn't have a maximum, so actually me note can be greater than fun, and the friction force can also be greater than the normal force off course. Usually it is much smaller, but the coefficient of friction depends on the two materials that are connected. And if they are really rough, that can be such micro heels on the surface, which makes it very hard to move an object parallel to the surface. This could increase the coefficient, and they do increase it for race cars. 31. Self locking with basic example: South looking is an interesting and important mechanical fundamental. To understand it, a show you an example. This example will also be a good practice on calculating with friction at the case off limiting friction. You can see the problem that I'm about to. So for you there is a body on the roof slope which has a given anger with respect to the horizontal plane. The rough surface is a frictional coefficient. Off me. Note gravity acts on the body verticality as usual. Besides that force F acts on the body, What is the maximum? A minimum off force F for which the body stays in actual Librium. In those cases, we have limiting friction on the calculate according to death, This is the only state where we have an expletive connection between the friction on the normal force. However, that connection doesn't contain science for the forces. I know that the normal force must be perpendicular to the surface and it is pointing upwards as the body is supported on the surface. But I don't really know much about the friction. Yet The friction is definitely parallel to the surface. According to the definition on its orientation Depends on the forces which act on the system. I must get the proper orientation off the friction force on the free body diagram to have the correct results for the fourth F First, let's see the case where we look for the minimum value off the force. But what do I mean by a minimum? If the force would be too small, the body we just go down on the slope so the body would move to the left as you could have heard, the friction always act against the motion in case of sliding friction. The limiting friction case is similar to that, but it isn't statics. If it would move to the left, the friction force must point right Words Now I know everything to drew a free body diagram . There are force F at its minimum value, a gravitational force, a normal force and the Friction force which aids half men to keep the body in actual Librium, I use a coordinate system in which X coordinate its parallel with the slope. I could have chosen anything. I just felt that this is the most comfortable for me. Now you can ride, actually bloom recreations based on the free body diagram. There are three equally room recreations. However, I cannot really calculate with the actual Abramoff moments as I don't know where the forces act. Sure, I could know where f acts I know of RG act, and I also know the line affection off the friction. But I don't know where an act it could act at any point off the bottom surface of the body . I could calculate the line of action off an in fact, from the equilibrium off moments. Let me l Oh, just one more remark the land affection off and changes. So the reaction force prevents the body from turning over the actually broom in direction. X reads as S plus F Men Times Co Sign offer minus M times G Time sign. Offer it for +20 as G equals two. Sometimes G. This is the first in creation the equilibrium in die action by reads as an minus half meantime, Sana fa minus Sometimes G Times CO sign off are equal to zero. This is the second equation. I don't know the value off us, um, and F men. So I have three our nose, but just to e creations and needed to recreation. This comes from the fact that the force is minimal if the friction is as large as possible . So there is a limiting friction. Therefore, s equals theme. You note times and this is the turkey creation these decorations can be. So for F men, the solution is M times G time signed off a minus. Marino Times Co. Sign offer over cosign offer plus Marino Time sign offer. This can be calculated easily as we know the perimeters. If San offer minus Marino Times Co sign offer smaller than zero than the minimum force is also smaller than zero. It is not a huge problem. It only means that there is no need for a force. The actual Librium would exist. Even if you put the body downwards with the force, let's see the other case for the maximum force. In this case, If the force would be slightly larger, the body would start moving to the right, in this case only wanting changes. Compared to the previous case, the friction points left words to prevent the body from moving to the right. Now I can throw the free body diagram. You can see the difference Action X against the maximum force to keep. Actually, Bree um, the actual Librium in direction X reads as minus as plus F marks Times cause and offer my news. M times G time signed off equal to zero The equilibrium in direction Why reads as n minus F Marks Times sign offer minus M times G Times Co Sign offer equal to zero as there is limiting friction again s equals to me. Note times. Um, therefore, we have the three creations for the three unknowns, and we can solve these creations to find half max. The solution is M times G Time sign offer plus Marie note times cosign offer over cause and offer minus Marino time sign offer if a course and offer minus minal times sign offer is smaller than zero than F marks would also be smaller than zero. It sounds weird, isn't it? Off course it is weird because this is just not well, it Physically. I've wanted to move the body to the right and I got the results that I need the force pointing to the left to move my body to the right. It sounds ridiculous. In fact, my last sentence is just not true because these south looking the body of your never moved to the right. The coefficient of friction is just too large to love movement with this geo matter. Mathematical e re note is greater than contention offer Let me talk just a bit about south looking. It is a very useful thing. There are several cases, very use it. Maybe even without noticing. This is the only reason why you can climb a leather without falling to the ground. This helps a skewed to function. Screws are designed so they would look independently of the acting force. You can only faster or lose an askew by applying a concentrated capital on it, which is a very special loading. You cannot just pull out a screw, and it is because of self looking. As an engineer, you should just keep south looking in mind. It might help you to construct something at a point in your career 32. Belt friction: as its name indicates, bad friction is the friction off about one another body. This other body is usually a cylinder, for example, a capstan. From this example, you can see that a very popular, really Bert application off the bath friction is in sailing, but the rope is wound around the capstan. This helps you to hold against pretty huge forces pretty easily. Let me introduce the effect in details. Firstly, it is practical, if you know the mechanical definition off the rope in mechanic, a sense the rope is infinitely 10 such that its length is much greater than its cross section. The rope is perfectly flexible. That can only be tension inside the rope. It cannot carry shearing or banding. It would just deform if there is bending moment acting on it. Also, if that would be a compression, the rope would just collapse. It doesn't hold against compression. Just think off, Rheal Life ropes. The rope doesn't along gate. As we used every very statics, there is no deformation. This is needed to soak in the bath. Friction in creation. I do not show you how to prove the Bath Friction creation, which is also called That's capstan creation. I just introduce you to the results. Let's see the mood. At first, there's a cylinder rand, which there's a rope. The rope is always marked with the dash line. That is why you see a dash line at one and the extension for Steve on and on the other end there stands reinforced E two. If Steven is greater than T two, the rope would start moving into di action off Stephen. However, there's a friction between the rope and the cylinder, which prevents the rope from moving. That is why you can see a lot off small forces acting on the surface off the cylinder, the exact moment with respect to the center of the cylinder in the same orientation Esti Tu does. Therefore, the friction has t two and ellos equilibrium. Therefore, if you put a huge rate on the rope, you can hold it with your bare hands. If the friction is large. Enough offer is the anger, which shows the total anger swept by the rope around the cylinder. It is measured in radiance. It can be as large as you want if you turn the rope around the ceiling. There multiple Times offer is two pi greater for every turn. The bad friction equation states that the larger force Steven over the smaller force. T two is smaller or equal to E at mute Note. Times offer. There's equality in case of limiting friction. If you have a huge load which causes divan force, you can elevate this load if you use a rope to allow yourself to hold the load with a much smaller force. Me by the ceiling there can be rotated by a motor so the load rate is elevated. This is a very special topic and statics so you might not need it during your carrier at all. But I believe that it is good to know as many tricks as possible. So if you ever need SAT solution now, you know that you must look at the captain equation. 33. Summary: Congratulations. You have reached the end of the scores. I hope that you learned a lot. If something is still unclear, please go back and revise that part of the course. If you still have some questions, do not hesitate to reach out and ask them. I'm at your service. I want you to understand everything properly. You have come a long way in statics. The most important is to have racks elit knowledge of calculating reaction forces in our forces and stress resultant. If you continue your mechanic studies, these are going to be essential for you. I hope that you like to discourse on more importantly, that you liked statics. The next step in mechanics is either toe words learning strength off materials or dynamics . I believe you got everything to take that step.