– [Voiceover] Let's learn fractions. Understanding fractions can be easy as one, two, three. Just watch this video, and you'll be using fractions with ease. It's easy to count whole things, like a pizza, a pie, or a donut. But what if we only had part of one? How would we count it then? We'd use fractions. A fraction looks like this. The bottom number is called the denominator. It shows how many equal pieces an object has been divided into. The top number is called the numerator. It shows how many parts of the denominator you are looking at. For example, a pizza is usually divided into eight slices. That means that, in a fraction, the denominator would be eight. If you normally eat one slice of pizza, then one would be the numerator. That means that you eat one eighth of a pizza. If you normally eat two slices of pizza, then two would be the numerator. The denominator would still be eight, since there are always eight slices of pizza. That would mean that you eat two eighths of a pizza. Let's try another example. This pie is divided into six pieces, and you eat four pieces of it.
If there were six pieces of pie, and you eat four, what is the denominator of your fraction? (upbeat music) That's right, six. What is the numerator? (upbeat music) Yep, it's four. That means that you ate four sixths of the pie. Let's try one more example. If you and your friend have one donut to share, then you will cut it into two pieces. If there are two pieces of donut and you eat one, what is the numerator of your fraction? (upbeat music) That's right, the numerator is one. What is the denominator? (upbeat music) Right again! It's two. When the fraction is one over two, we call that one half. Thanks for watching. (upbeat music).
The problem with the Matrix sequels is that
their target audience is a 13 year old boy who has somehow earned a PhD in philosophy. That is to say: someone who loves philosophical
conversations and also enjoys ridulously over the top action sequences. Yes! That was a bowling pin sound effect! I didn't even add that! These movies are goofy…it's okay. Embrace it! So as a 13 year old kid in 2003 who didn’t
have a philosophy degree, I hated the sequels and spent the rest of my life up until about
a month ago pretending that they DID not exist.
But then a funny thing happened. I got old. And when I popped these suckers in fifteen
years after the fact, my thoughts on the movies did a complete 180. This feels like a confession, but I had SO
much fun watching these movies. More fun than Hugo Weaving had acting in them. And if you’re thinking right now, “Hold
on, Sage. You’re running a YouTube channel mostly
about movies and you want to tell me that the Last Jedi AND the Matrix Sequels are good? Do you know what F****** platform you’re
on?” And yeah, I can list just as many problems
these movies have as anybody else can. These are flawed movies by any measure that
don’t live up to the tightly wound perfection of the original. The biggest obstacle to enjoying these movies
is that the philosophy isn’t neatly woven into the narrative. The philosophical stuff in the first film
was like a little side dish.
It didn’t dominate the rest of the story,
and the emphasis was always on the characters and the action. The philosophy in the sequels? Ooof. It’s just dropped like an anvil onto the
audience. "As I was saying, she stumbled onto a solution whereby nearly 99% of all test subjects … What are you talking about? Between every action scene we get a GIANT. PHILOSOPHICAL. LECTURE. about the nature of power, free will,
determinism, purpose, causality, belief, and systemic control.
I’ve talked about techniques to integrate
exposition into a story before, and these movies follow zero of those guidelines. As a piece of mass market entertainment, this
absolutely kills the pacing, so it’s no wonder that 13 year old me was bored. But nowadays? Listening to some guy explain philosophy for
15 minutes at a time is something I do for FUN! Like, in my free time.
So again, my relationships to the films have
changed. Where all of this just went over my head as
a child, I can now finally find a coherent through-line to the ideas in the film. My confusion with these films is epitomized by a pair of paradoxical statements made by Neo at the end of the trilogy. To get anything from the films, other than pure indulgence, I have
to make sense of these two conflicting lines… “Why Mr. Anderson? Why, why do you persist?” “Because I choose to.” You were right Smith. You were always right. It was inevitable. Because I choose to. It was inevitable.” How can we resolve this paradox? Well, first it helps to clarify how Neo actually
beats agent Smith. Back in Reloaded, Agent Smith tells us that
he has been freed from his programing as a result of being killed by Neo in the original. Usually, when a program no longer has a purpose,
it is supposed to go to a place called The Source where they are deleted.
But being freed from his programing also meant
Smith suddenly had free will. He could decide to NOT go to the Source, and
instead starts making copies of himself, telling Neo it’s inevitable that he will make a
Smith clone out of Neo too. “It’s inevitbale.” As The One, Neo is also supposed to go to
The Source. The machines are trying to get him there because
once The One reaches The Source, the Matrix is rebooted. (Man, it is sort of irritating that every
name in this universe is just a noun with the word “the” in front of it.) At the end of the final film, Neo goes to
the Machine City which is the location of the Source.
So when Smith copies himself onto Neo, he
unwittingly connects himself to the Source which deletes him. Since Smith has become so powerful in the
Matrix, this is the only way that Neo can possibly defeat him. Once Neo has made the decision to fight Smith,
it is inevitable that this would be the way he is defeated. The mechanics of the plot don’t allow another
option. So that’s one explanation for
the line about inevitibility. The philosophical answer comes by way of asking:
is Neo free to make the choice to fight in the first place? Is he free to make any of the choices he’s
making? This is where all of that talk talk talk talk
talk in the Matrix Reloaded comes in handy. At the climax of that movie, Neo confronts
The Architect, the machine program responsible for creating the Matrix, who informs him that
everything he thought he knew about the war is a lie.
The resistance has been created and guided
by the machines at every step of the way. Neo isn’t that special. He is The One, but there have been many Ones
before him, and if all goes according to plan, there will be many Ones after him. The Architect gives him a choice. If he goes through one door, the Matrix will
be rebooted and Neo will restart the Revolution with 23 other individuals.
This is the door that the machines want him
to choose, but choosing it means that Trinity will die He can, however, go through the other door
and try to save her. The Architect tells him that he will fail
to do so even if he goes through that door, and if he does choose that door, then the
entire human race will die along with them. So. The way the Architect frames this choice is
as one between Moral Duty and Love.
It’s a choice Neo has faced before: this
is the third time he’s had to make this decision in this very movie. The first one comes early on in the film. Neo wants to spend time with Trinity, but
he is confronted by a crowd of people who need him for spiritual support. No one’s lives are at risk, so in a tensionless
choice between Love and his Duty to the world, Neo chooses to help others. The second comes around the halfway point
of the film. When the Merovingian refuses to help him,
his wife Persephone, (hey look an actual character name) says she will help on one condition:
"I'll give you what you want.
But you have to give me something."
"What?"
"A kiss."
Excuse me? Now the stakes are higher: humanity’s survival
hangs in the balance, or so Neo believes, so once again he has to choose: Love or Duty. Again, he chooses to help others. So by the time he reaches this point of the
story, the Architect has every reason to believe that Neo will promptly walk through that door
to The Source and do his Job.
But Neo…doesn’t. When the stakes are raised to their highest
degree and when Trinity’s life hangs in the balance, he chooses to save her, and the
Architect tells us why: “Already I can see the chain reaction, the chemical precursors that signal the onset of an emotion designed specifically to overwhelm logic and reason. An emotion that is blinding you from the simple and obvious truth: she is going to die and there is nothing you can do to stop it." Essentially, he’s saying that Neo is a slave
to his passions, to emotion, which means that him going through that door proves that he
does not have free will. His actions are in line with what the Merovingian
was talking about earlier in the film: Causality. "You see, there is only one constant, one universe that is only real truth: Causality. Action, reaction.
Cause and effect." "Everything begins with choice." "No, wrong. Choice is an illusion created by those with power and those without." Neo was told Trinity was in danger, he had
an emotional reaction, and then he sped off to save her. Cause and effect. But I don’t think we can just simply take
the Architect at his word that Neo is blinded by emotion. In her paper on the film, “Neo’s Kantian
Choice "The Matrix Reloaded" and the Limits of the Posthuman” DANA DRAGUNOIU analyzes
Neo’s decision according to Immanuel Kant’s Categorical Imperatives, so let’s look at
the first one of those. Kant writes that we should, “Act only according
to that maxim whereby you can, at the same time, will that it should become a universal
law.” So what does that mean? Well, basically you should only do things
that you would be okay with everyone else doing all the time.
So you shouldn’t throw a plate of cookies against a wall because you wouldn’t
want to live in a world where everyone was constantly violent towards dessert. Hugo stop having so much fun! This Categorical
Imperative basically breaks the ethical dilemma Neo is in. See, the Architect is only capable of thinking
in purely mathematical terms, so it’s no surprise that he wants Neo to think like a
Utilitarian. Utilitarianism, for those who aren’t watching
The Good Place as they should be, is the idea that you should act so as to maximize utility,
or for the purposes of this example, to maximize the survival of the most number of people. So since 23 people surviving is better than
none, Neo should go to the Source. It’s basically Thanos logic. But when you think about this decision in
Kantian terms, the logic falls apart. To quote Dragunoiu, “If everyone (or, to
use the language of the trilogy, every “one”) were to sacrifice Zion and betray the Resistance
whenever the machines threatened humanity with extinction, the very notion of resistance
would be rendered absurd” Walking through this door to The Source would
only continue to perpetuate the systems of control that the machines have constructed
to maintain power over humanity.
It dooms them to endlessly repeat the cycle
of revolution and near extinction until the end of time. Under this thinking, the danger Trinity faces
is irrelevant. If Neo actually wants to fulfill his duty
to save humanity, then he needs to walk through this door no matter what. The choice isn’t between Duty and Love. Both of those values are behind the same door.
Only servitude is behind the other. Because Neo is acting on principle, doing something, doing something that is outside of his own self-interest, his decisions become unpredictable to the machines, suggesting that he does have free will.
He won't sacrifice one life to save many but instead insists on finding another solution. It’s basically Captain America logic. “We don’t trade lives.” But the fact that Trinity’s life also
hangs in the balance complicates our analysis of Neo’s motives here. Was he a slave to his passions after all, or did he
choose to do what is more morally right? What actually motivated him to go through
this door? Love or moral duty. Because the first indicates determinism, and
the second indicates free will. Luckily, the events of the third film clarify this by removing Trinity from the equation.
Instead of trying to keep Trinity out of the
line of fire like he did in Reloaded, in Revolutions he doesn’t stop her from joining him on
his suicide mission to the Machine City. Doing so demonstrates his willingness once
again to sacrifice his own self-interest, his love, in order to do what is morally required
of him. Her death along the way essentially allows
him to have clarity over his own motivations. That’s what the Oracle has been telling
him he needed to do this entire time: to understand why he makes the decisions he makes.
To know himself. "It's Latin. It means know thyself." "But if you already know, how can I make a choice?" "Because you didn't come here to make the choice, you've already made it. You're here to try to understand why you made it." Neo’s journey is, like all epic stories,
about finding a deeper understanding of self, and what Neo discovers about himself after
making all of these choices is that he always chooses to serve others over himself. I’m reminded here of a more recent story
about the nature of free-will, which argued that the principles that people hold are actually
akin to programming. In Westworld, your moral attitudes can be calculated for and are thus predetermined. but in the Matrix morality confounds the machines. The deterministic universe they've set up only caters to the common urges like hunger and pleasure that humans experience. Neo and the other resistance fighters operate outside of those bounds fighting for a higher purpose that these machines cannot account for or understand.
But at the end of the story, Neo has become self-aware about his own character. Giving him the knowledge to know that he would always make the hard choice and that he would always choose to oppose Smith. The question at the heart of the sequels isn't just about whether or not humans have free will or Neo has free will but do we have free will when it comes to resisting the systems of control put upon us by the powerful. These movies are about all the ways that those with power maintain control over a population. Controlling both sides in an eternal struggle creating the illusion that descenders are powerless. That we are caught in a loop without end. Its conclusion is that through moral courage, changing the system isn't just possible…it's well… "It was inevitable." So while I went into a rewatch of these movies
expecting to loathe every scene every step of the way, I came out of these movies exhilirated,
thrilled to finally see through what I once thought was totally unclear.
Opinions need refreshing every now and then. There are some movies whose reputations are
so seemingly universally agreed upon, good or bad, that we stop questioning them. We take it as a given. But the great thing about art is that while
it stays the same, you don’t, and when you take a look back at something you thought
you knew, it feels like waking up to a new reality when all you’ve ever known was the
illusion. I think the most exciting part of the fantasy
of the Matrix trilogy for me has always been this idea of living in a world where you can
just download a skill into your brain and instantly know how to do something. And while we sadly don’t have that technology
yet, the next best thing is Skillshare, a platform where experts in the fields of design,
writing, animation and more explain how they do what they do in accessible, well-curated
online classes.
I want to thank Skillshare for sponsoring
this episode. For writers, I recommend checking out Steve
Alcorn’s Writing Academy courses. He’s got a bunch of different classes on
topics like creating characters, writing dialogue, and structuring a story. So if you're interested in Skillshare, the first 500 people to click this link in the description will get a 2 month free subscription and access to over 20,000 classes with their premium account. Thanks for watching everyone, and a big thanks
to my Patrons on Patreon for supporting this channel.
it was a whole lot of fun watching Deadpool 2 in theaters but since it's been a few years many aspects of the film come across differently nowadays from celebrity cameos to deleted scenes there's a lot about the movie that you may not have noticed the first time around at the beginning of deadpol 2 Wade Wilson is on top of the world he's reunited with his soulmate Vanessa and the two plan to start a family but they're tragically torn apart when she's killed by a straight bullet meant for her boyfriend Vanessa's death shapes the tone of the movie and wait character Arc as he sees visions of her that he interprets as encouragement to help Russell however Deadpool 2 has been criticized for making Vanessa a victim of quote fridging if you're not familiar with the term it's basically a plot device where a female character is put in Peril because the plot requires it did you ever see Commando it's like that think Alyssa Milano with Anie as her dad saving the day at the end director David Leech doesn't believe that Vanessa was fridged because the character emotionally underlines the whole movie but in truth Vanessa was initially never supposed to die at all leech admitted on the audio commentary we had versions where they broke up and she left him instead of dying according to Ryan Reynolds the whole tone of the movie would have changed if vanessa had lived as they might have actually had a child in the end leech felt to death was best for Deadpool's narrative as he wouldn't have had quite the same motivation if vanessa had survived in the original Deadpool fighting Francis probably would have gone a lot smoother if there were more than three mutants present to take him out before the big battle though Wade make sure the audience know knows exactly why we shouldn't expect to see Mega popular mutants like Wolverine or Cyclops this time around it's a big house it's funny that I only ever see two of you it's almost like the studio couldn't afford another X-Men before the climax of Deadpool 2 however this has proven not to be the case six of the most iconic members of the X-Men are clearly present in the x- Mansion it's just that they just don't want anything to do with Wade as annoying as Deadpool can be to deal with juggernaut's attack on an orphanage probably warranted some assistance from the more popular members of the superhero team so why didn't they show up once again the answer is more about behind the scenes limitations than plot Hol none of the X-Men were actually able to appear on the Deadpool 2 set because they were busy shooting dark Phoenix at the time as with many elements in today's superh hero movies The Cameo was added via green screen which means that Ryan Reynolds never actually got to hang out with James makoy and Evan Peters drawing production peters's recall to Cinema Blend we had a lot of different versions for when they opened that door and turn to us but the one they used was pretty chill whether he's stranded on mars or buying a zoo it's always a pleasure to see Matt Damon on the big screen even though he has yet to play a big shot superhero like his buddy Ben Affleck Damon has taken the opportunity to appear in multiple Marvel movies in 2017 stor Ragnarok Damon portrayed an Asgardian actor in the role of Loki and a year later he put on some Prosthetics to play a hillbilly who doesn't believe in the hygienic value of toilet paper toilet paper is plen fying appetizer but then huggy's Natural Care whip wips details of Damon's Cameo were kept under wraps and while his scene partner Alan tck is named in the credits Damon isn't he's credited as Dicky Greenleaf which is a reference to The Talented Mr Ripley and the identity that his character steals in that film his Cameo wasn't publicized or talked about in interviews until after the movie was released and plenty of viewers missed that it was him the first time around most of the casting crew members were kept in the dark as Reynolds and leech did everything they could to keep it Hush Hush Reynolds who co-wrote deadpol 2 pitched Damon the idea for the toilet paper scene over dinner director leech later admitted to kider we didn't even tell the crew his name wasn't on the cool sheet it was a fake name nobody really knew what that scene was about they were like why are we shooting these two red necks we just didn't tell anybody once cable arrives in the present Deadpool assembles the ex force in order to stop him from harming Russell the invisible hero vanisher joins the team but quickly meets his And Timely demise after strong winds blow him into electrical cables and zap him to a crisp it's only as he's electrocuted that vanisher becomes visible and once he is he's revealed to be played by Brad Pit you guys look amazing vanisher I have no doubt you look amazing too if you get the chance to put Brad Pit in your movie you should probably put him in your movie even if it's for Less Than 3 seconds of screen time fit had originally been in the running to play cable but back doubt due to a scheduling conflict even though Josh Brolin ended up playing the futuristic Warrior David Leech told told collider that pit still really wanted to be involved in the franchise he recalled he kind of left the door open when we left that meeting like if you guys ever need anything give me a call I'd love to be involved and Ryan just said oh I have an idea so he started to think about what it could be and the vanisher thing was perfect the only thing that pit requested as compensation for his appearance was a single cup of coffee well Matt Damon and Brad Pit's cameos were two of the most notable in Deadpool 2 it's probably Stan Le's that was the most anticipated late in his life the Marvel comic co-creator made a number of cameos in the MCU and other Marvel based films but his Deadpool to appearance wasn't what viewers were expecting unlike in the first movie where Lee played a strip club DJ his appearance in the sequel was purely two-dimensional Deadpool 2 celebrates Lee with a giant mural which can be seen when Domino is parachuting into the city David Leech explained to kollider that the colorful graffiti was included in the movie because scheduling issues precluded them from shooting Lee In the Flesh he elaborated we're always aware that you want to get a Stanley appearance in there but it was more like what would work and what was the time in place and our schedule was tight and things got away from us so by the time we were done with the movie he were thinking where could we plug him in during re-shoot leech denied at the time that Lee's absence was due to his rumored ill health sadly Lee's Health did begin to decline and he passed away just a few months after the release of Deadpool 2 his last ever Cameo was an Avengers endgame when it was announced that Domino would be a featured character in Deadpool 2 some fans were left one wondering how exactly the film would portray her power of perfect luck it seems that Ryan Reynolds noticed this because Deadpool's rant during Domino's attempt at saving Russell makes that exact point in a pretty ironic way in fact Deadpool is so insistent that luck isn't a superpower that he mocks the person who came up with the idea in the first place his insult doesn't make much sense unless you're familiar with the source material though what C out glass pipe suck a freak show comic book artist came up with that little Chestnut probably a guy who can't draw feet so who actually is a guy who can't draw feet that would be Rob leld comic book veteran and co-creator of the characters Deadpool Domino and cable leld is Infamous for avoiding having to draw feat as much as possible in his work even though many other artists in the medium have followed his lead over the years Ryan Reynolds reached out to clear the joke with life out who despite the implication that he smokes crack was thrilled to be included in the movie he told inverse there's no greatest shout out in the world Deadpool 2 was full of surprises when it first came out one of them being the presence of the super villain Juggernaut this wasn't the underwhelming Juggernaut from X-Men the Last Stand either this was a fully CGI Behemoth capable of ripping Deadpool in half like a Twinkie but who could play such a beast of a man just like Mike Myers and Austin Powers Ryan Reynolds plays both the anti-hero and the villain but he wasn't actually supposed to do it and he isn't named as the voice of Juggernaut in the credits after Deadpool 2 was released Reynolds revealed to Empire that he' just been standing in until they could find the more suitable voice actor which never ended up happening as he put it I just did it as a kind of temp it was this sort of Brooklyn Brawler voice that we modulated and post and cranked up and gave it all this base and reverberation we didn't settle on that just because it was quality we settled on that because we just didn't have any more budget left for other actors to jump in David Leech ended up doing motion capture for juggernaut's face which was brought to life thanks to the work of talented VI effect artist you know it has always been a dream of mine to see my face reflected in your helmet as you charge at me with murderous intent I don't mean right now in nearly every single time travel movie tampering with the past has disastrous consequences in the present Deadpool 2 is one of the rare exceptions armed with Cable's time travel device Wade makes some pretty major changes to the timeline with seemingly no ill effects at least not until Deadpool and Wolverine that is he saves Vanessa by killing her attacker with a cream cheese spreader takes out the mute version of Deadpool from X-Men Origins Wolverine and puts a bullet in Ryan Reynold's head before he can start production on Green Lantern it's a pretty happy ending overall and it creates a clear path for Marena backin to appear in Deadpool 3 as a mutant superhero copycat however Deadpool 2 almost had a completely different darker ending co-writer R Reese said the team was initially happy with their decision to kill Vanessa off without resurrecting her Reese told CBR we originally did not save her it was interesting we thought we might have to save her with the time machine because the audience would be mad at us but they were accepting of her being dead bringing Vanessa back was ultimately a last minute decision as they felt it was the best fit for the story and the future of the franchise when they called back hering up to let her know she was very receptive to the news one of Wade's time traveling Adventures that didn't make it into the theatrical release of Deadpool 2 involved him going back in time to kill Adolf Hitler as a baby in the deleted post credit scene he's ready to unleash hell on the newborn infant but he starts to struggle with the decision he eventually decides to let cable do the dirty work for him I think we both no I don't have what it takes to do this the scene was eventually released as part of the super duper cut in August 2018 Ryan Reynolds told The Hollywood Reporter that it caused a debate behind the scenes saying I always loved it because it created the same debate in our edit room as it does in the streets of the world it was debated back and forth and over and over again we were all a little nervous about it wondering is it too much in the end none of us could come to a consensus on how to handle it so our final decision was to just not handle it to just not put l in the movie
Shear force and bending moment diagrams are powerful, graphical methods that every mechanical and civil engineer should know how to use to analyse a beam under loading In this video. I ll explain exactly how to master these diagrams and we will see how they can be used to understand how a beam is loaded. I want to start by explaining what shear forces and bending moments actually are When a beam is loaded, internal forces develop within it to maintain equilibrium. These internal forces have two components: We have shear forces oriented in the vertical direction And we also have normal forces which are oriented along the axis of the beam. If the beam is sagging, the top of the beam will get shorter, and so the normal forces in the top of the beam will be compressive. The bottom of the beam will get longer, and so the normal forces in the bottom of the beam will be tensile. Each of the tensile normal forces has a corresponding compressive force, which is equal in magnitude, but opposite in direction. As such, these forces don t produce a net normal force, but they do produce a moment. This means that we can conveniently represent the internal forces acting on the beam cross section using just two resultants one shear force, which is a resultant of the vertical internal forces and one bending moment, which is a resultant of the normal internal forces. This is a very common way of representing the internal forces within a beam Drawing the shear force and bending moment diagrams is just figuring out what these internal forces are at each location along the beam. These resultant shear forces and bending moments will depend on the loads. Acting on the beam and the way in which the beam is supported, Beams can be loaded in a number of ways. The most common being concentrated forces, distributed forces and concentrated moments. Beams can also be supported in a number of different ways. They can have pinned supports roller supports or be fully fixed, which each restrain the beam in different ways Pinned supports, prevent vertical and horizontal displacements, but allow rotation Roller supports, prevent vertical displacement, but allow horizontal displacement and rotation Fixed supports, prevent all displacements and rotation. If a certain degree of freedom is restrained at a support, we will have a corresponding reaction force or reaction moment at that location. For example, rotations are permitted for a pinned support, so there is no reaction moment, but displacements in the vertical and horizontal directions are prevented. So we will have horizontal and vertical reaction forces. So how do you determine the shear forces and bending moments within a beam? There are three main steps we need to follow. First, we draw a free body diagram of the beam. This shows all of the applied and reaction loads acting on the beam. The next step is to calculate the magnitude of the reaction, forces and reaction moments at all of the beam supports. We do this using the concept of equilibrium To maintain equilibrium. All of the forces in the vertical and horizontal directions should cancel each other out. Similarly, all of the moments acting at every point along the beam should cancel each other out. This gives us a set of simple equations. We can solve to calculate the reaction, forces and moments If we can calculate all of the reaction loads using the three equilibrium equations. The beam is said to be statically determinate For some beam configurations like this one shown here. We won t be able to calculate all of the reaction loads because we have too many unknowns and not enough equilibrium. Equations, In this case the beam is said to be statically indeterminate. This beam has 4 reaction forces, but we only have 3 equilibrium equations To solve this beam, we would need to use slightly more complicated methods and consider boundary conditions In this video. I will only cover statically determinate cases, where we can use the equilibrium equations to calculate all of the reaction loads Once we have calculated all of the reaction loads. The third and final step is to figure out the internal shear forces and bending moments at every location. Along the beam To do this, we will use the concept of equilibrium again. If we cut our beam at any location, the internal forces and moments need to cancel out the external forces and moments so that equilibrium is maintained. This allows us to easily calculate the shear force and bending moment at each location along the beam. All we need to do is start from one side of the beam and move the location of the cut along the beam, calculating the shear forces and bending moments. As we go. Now is a good time to define the sign convention we will be using Applied forces. Will be positive if they are acting in the downwards direction For shear forces and bending moments? The positive sign convention will be as shown here If the beam is on the left side of our cut shear forces pointing downwards will be positive If the beam is on the right side of our cut shear forces, pointing upwards will be positive. Positive bending moments will be those that put the lower section of the beam into tension. Another way to think about it is that bending moments which cause sagging of the beam are positive and those that cause hogging of the beam are negative. Let’s take a look at an example of a beam with pinned and roller supports loaded by two concentrated forces. First, we draw the free body diagram. We can then use the equilibrium equations to determine the unknown reaction forces at Point A and Point B. The sum of the forces in the vertical direction is equal to zero, so R, A plus R B, is equal to 15 plus 6. Because H, A is the only horizontal force, it must be equal to zero. We also know that the sum of the moments about any point along our beam must be zero. Let’s consider the moments about Point B That gives us this equation, which we can solve to determine that R A is equal to 12 By substituting R A into the previous equation. We can deduce that R B is equal to 9. Now that all of the external loads acting on the beam are defined, we can draw the shear force and bending moment diagrams. We will start from the left hand side of the beam. Let’s draw the free body diagram for a location immediately to the right of the 12 kN reaction force To maintain equilibrium. The shear force must be equal to the reaction force. We can draw this on our shear force diagram. The shear force will be constant until we reach the next applied force. The bending moment must be equal to the 12 kN reaction force, multiplied by the distance X to the reaction force. This gives us the equation for a straight line, which we can draw on our bending moment diagram. We then repeat the process by moving the location of our cut further to the right. This time we place the cut immediately after the 15 kN force, and we draw the free body diagram again to determine the shear force and the bending moment. We repeat this process until we have covered the full length of the beam We end up with the complete shear force and bending moment diagrams for the beam. That example was a fairly simple one For cases with more complex loading, drawing the shear force and bending moment diagrams can be more difficult. There are relationships between the applied loads, shear forces and bending moments, which will help us better understand what our diagrams should look like. Let’s consider a beam loaded by an arbitrary distributed force. We can zoom in to look at an infinitesimally small segment of the beam with a width equal to D X and draw the free body diagram Over such a short section of the beam. The distributed force can be assumed to be uniform and we can replace it with an equivalent concentrated force By applying the equilibrium equations to this free body diagram. It is possible to demonstrate that the following relationships exist between the applied distributed force, the shear force graph and the bending moment graph. The quantity D V over D X is the slope of the shear force curve and at a given point along the beam, it is equal to minus the distributed force. Similarly, D M over D X is the slope of the bending moment curve and at a given point it is equal to the shear force. If we integrate the first equation, we can show that the change in shear force between two points is equal to the area. Under the loading diagram between those two points, And if we integrate the second equation, we can show that the change in bending moment between two points is equal to the area under the shear force curve. This is really useful information we can use to help construct or sense check our shear force and bending moment diagrams. Let’s take a look at an example. This beam has an applied, distributed, force and a concentrated force. Let s quickly draw the shear force and bending moment diagrams By using the free body diagram method. We can show that the bending moment curve for the section of the beam under the distributed force is defined by the quadratic equation: 4 X, 2. 34. X. 68. If we differentiate this equation, we get 8 X 34, which, based on the D M over D X equation above we now know, is the equation for the shear force curve. In this section of the beam. If we differentiate again, we get 8, which is the equation for the distributed force. This is a great way to sense, check your shear force and bending moment diagrams. Another way of checking your diagrams is using the area equations I mentioned earlier. The area under the shear force curve highlighted here is equal to 34 times 2, which is 68. This is equal to the change in bending moment over this section of the beam. We can also calculate the area under the shear force diagram for the beam section under the distributed force. The total area of this section is equal to 72 3 minus 12 3, which is 60. This is equal to the change in bending moment of 60 kNm over this section of the beam Where concentrated forces are applied. There is a sudden jump in the shear force diagram and where concentrated moments are applied. There is a sudden jump in the bending moment diagram. These equations will not be applicable across discontinuities in the diagrams. One final observation we can make based on these equations is that when the shear force is equal to zero, the bending moment curve will be at a local minimum or maximum. Let s look at one. Last example: Here we have a cantilever with an applied concentrated moment of 120 kNm and a distributed force of 6 kN m. Again we start by drawing the free body diagram, Because the support is fully fixed. We have vertical and horizontal reaction forces, R, A and H A and a reaction moment M A Let’s. Look at our first equilibrium equation. The sum of forces in the vertical direction is equal to zero. In this case, the only forces acting in the vertical direction are the reaction force, R, A and the distributed force. So R A is equal to 6 times 3, which is 18 H. A is the only force in the horizontal direction, so it must be equal to zero. Next, we can take the sum of the moments acting at point, A In calculating the moment caused by a uniformly distributed force. You can remember that it is equal to a concentrated force located in the middle point of the load. This gives us M A equals 21 To calculate our shear forces and bending moments. We will start on the left side of the beam and move towards the right. This is our first free body diagram The shear force calculation is easy, as we only need to consider the reaction force of 18 kN. The bending moment needs to take into account the reaction moment and the reaction force At X equals zero. The bending moment is equal to the reaction moment of 21 kNm As we move to the right. We also need to consider the moment caused by the 18 kN reaction force. This gives us the equation for a straight line. We can then move our cut to the right of the concentrated moment. The moment won’t affect the shear force, which will remain constant at 18 kN until we reach the distributed force, But it does cause the bending moment to suddenly drop by 120 kNm After the drop. The bending moment is again defined by a straight line. Things get a little more tricky when we reach the distributed force, We can replace the uniformly distributed force by an equivalent concentrated force with a magnitude of 6 multiplied by the length X over which the force is applied. This force is located at a distance of X. 2 from our cut, We can then calculate the shear force and bending moment equations using the normal approach. The bending moment in this section of the beam is defined by a quadratic equation. No loads are acting on the small one metre section to the right of the distributed force, so shear forces and bending moments in that section will be equal to zero. Although we can’t calculate displacements from these diagrams, we can use the bending moment information to predict the deformed shape of the beam Where the bending moment is positive, the beam will be sagging and where it is negative, it will be hogging Where the bending moment Is zero? The beam will be straight That will give us a deformed shape. That looks something like this: That’s it for this quick, look at shear forces and bending moments in beams. I hope you learned something new and if you enjoyed the video, please don t forget to subscribe.
