# Shear and Moment Diagrams from First Principles > Shear and moment diagrams, built from equilibrium rather than memorised. We cut a simply supported beam at an arbitrary position, draw the free body of what is left, and read the internal shear and bending moment straight off two equilibrium equations, first for a single point load and then for a uniformly distributed load. Plotting those functions gives the diagrams. A slice of beam of length dx then turns the patterns we noticed into two derivatives, dV/dx equals minus w and dM/dx equals V, and into their integrals: the change in shear is the area under the load, the change in moment is the area under the shear. Every jump, slope and area rule follows from those, including the applied couple that jumps the moment diagram while leaving the shear untouched. The lecture closes by sketching both diagrams for an unseen beam and locating the maximum moment before computing it. - Canonical watch page: [Shear and Moment Diagrams from First Principles](https://academa.ai/lectures/deriving-shear-moment-diagrams) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Engineering - Published: 2026-08-28T19:09:19.623Z - Updated: 2026-08-28T19:09:19.623Z - Duration: PT1075S (17 minutes 55 seconds) - Chapters: 6 - Views: 0 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M14TY448ABTYCPADC1FDHE42/0/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14TY448ABTYCPADC1FDHE42/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M14TY448ABTYCPADC1FDHE42/0/dark/poster.jpg) ## Description Cut a beam, take the free body, and derive shear and moment diagrams from equilibrium, then from the two derivatives that govern them. ## Chapters - [00:00–04:9.645 · Cutting the Beam](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=0) - [04:9.645–06:34.051 · Drawing the Two Diagrams](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=249.64514583333334) - [06:34.051–09:25.493 · A Distributed Load](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=394.05110416666673) - [09:25.493–12:36.003 · Load, Shear and Moment](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=565.4929791666667) - [12:36.003–15:45 · An Overhang and a Couple](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=756.0034166666667) - [15:45–17:55 · Sketching a Beam You Have Not Seen](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=944.9995833333334) ## Transcript ### [00:00 · Cutting the Beam](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=0) Shear and moment diagrams are usually handed to you as a list of rules. Jump here by the size of the load. Slope there. Today we derive every one of them from a single idea: a piece of a beam in equilibrium is still in equilibrium after you cut it. Here is where we are going. Underneath a loaded beam you draw two pictures: the shear diagram, and the bending moment diagram. By the end you will sketch both of these for a beam you have not seen, and know where the biggest moment sits before computing anything. But not by pattern matching. Every value in them falls out of one free body diagram. So, the beam. Six metres between the supports, a pin at A, a roller at B, and one point load of twelve kilonewtons, four metres from the left. Before we cut anything we need the reactions. Take moments about A. The reaction at B acts six metres out, the load pulls down four metres out, and those two have to balance. That gives eight kilonewtons at B. Vertical equilibrium then leaves four kilonewtons at A. Both supports push upward. Now the question that matters. A beam does not fail at its supports. It fails somewhere inside, where the material is carrying an internal force and an internal moment. What are they, and how do they change as you walk along? Here is the one move the whole subject rests on. Cut the beam at a distance x from A, and throw away everything to the right. What is left is a piece with the reaction at A still pushing up on it. That piece cannot be in equilibrium on its own. Four kilonewtons up and nothing down. So the part we threw away must have been holding it, at the cut face, with a force and with a moment. Call them V and M. Those are the internal shear and the internal bending moment at position x. Draw them the standard way every time: V pointing down on the cut face, and M bending the beam into a smile. Then a negative answer simply means it acts the other way. Vertical equilibrium of the piece. Four up, V down, nothing else. So V is four kilonewtons. Moments about the cut. The reaction acts a distance x away, and M is the only other thing on the piece. So M is four x. Look at what x did. Nothing at all to the shear: four kilonewtons wherever you cut, as long as you stay left of the load. But the moment grows in proportion to x. Watch the piece get longer. That is what we mean by shear and moment as functions of position. One cut, two equilibrium equations, and we have both of them for the whole left hand stretch of this beam. But only that stretch. Move the cut past the load, and the free body is a different picture: the twelve kilonewton load now sits on the piece we kept. Vertical equilibrium now has three forces. Four up, twelve down, and V drawn downward as before. So V is four minus twelve, which is minus eight kilonewtons. And notice: the shear did not slide down to minus eight. It jumped, and it jumped by exactly twelve, the size of the load. Hold on to that. Moments about the cut again. The reaction still acts x away, and the load now acts on our piece as well, a lever arm of x minus four, turning it the other way. That tidies to forty eight minus eight x. Check it at the far end. Put x equal to six and the moment comes out zero, which is exactly what a roller has to give you. A roller cannot resist any moment at all. So here is the answer. Two expressions for the shear and two for the moment, one pair on each side of the load. Nothing there was remembered. A cut, a free body, and two equilibrium equations, done twice. Now let us draw them. ### [04:9.645 · Drawing the Two Diagrams](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=249.64514583333334) We have four expressions and a beam. Drawing the shear and moment diagrams is nothing more than plotting them, underneath the beam, on the same horizontal scale. So: position along the beam across the bottom, internal shear up the side. On the left stretch the shear is a constant four. The diagram starts from zero off the end of the beam, jumps straight up to four where the pin pushes in, and then runs flat all the way to the load. On the right stretch it is minus eight, so a horizontal line eight below the axis, running from the load across to B. And those two do not meet. At the load they differ by twelve, so the diagram falls vertically, right there, by twelve. The jump is the load. Then the roller pushes up eight and closes the diagram back to zero. That is a free check on your arithmetic. A shear diagram that does not return to zero at the far end means a reaction is wrong. Watch the shear as the cut walks along. Nothing happens to it until it reaches the load, and then in one step it is on the other side of the axis. Now the moment, on its own axis underneath the shear. On the left stretch it was four x, a straight line climbing out of the origin. It arrives at sixteen kilonewton metres under the load. On the right, forty eight minus eight x brings it back down, in another straight line, to zero at B. For a point load, then, the moment diagram is a triangle, and its peak sits directly under the load. Put the two side by side and two things stare at you. Where the shear is constant, the moment is straight. And where the shear is bigger, the moment climbs faster. Look at the numbers. On the left the shear is four and the moment gains four units per metre. On the right the shear is minus eight and the moment loses eight per metre. The slope of the moment diagram is the shear. And here is the one that earns its keep. The moment peaks exactly where the shear crosses the axis, which on this beam happens inside the vertical jump at the load. We have not proved any of that. We have noticed it, on one beam. So let us take a completely different kind of load and see whether it survives. ### [06:34.051 · A Distributed Load](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=394.05110416666673) Second beam. The same six metre span and the same supports, but instead of one point load it carries four kilonewtons on every metre of its length. A uniformly distributed load. The total is four times six, twenty four kilonewtons, and by symmetry each support takes half. Twelve up at each end. Cut it at x and keep the left piece, exactly as before. Here is the only new idea in this example. The load sitting on our piece is four kilonewtons per metre times x metres, and it acts through the middle of that stretch. Vertical equilibrium. Twelve up, four x down, V down. So the shear is twelve minus four x. It is not constant any more. It falls off at four per metre, which is exactly the load intensity. Moments about the cut. The reaction gives twelve x. The load on our piece gives four x, times its lever arm of x over two, turning the other way. So the moment is twelve x minus two x squared. A parabola. Watch the free body grow. The resultant of the load grows with it, and its arrow slides out to stay at the middle of the piece. Plot the shear. Twelve minus four x is a straight ramp, starting at plus twelve at the pin and finishing at minus twelve at the roller, crossing the axis at the middle of the span. And the moment is a parabola, zero at both supports. By symmetry its top is at three metres: twelve times three, minus two times nine, eighteen kilonewton metres. Both of the things we noticed on the first beam survived, and they got sharper. The shear was flat when there was no load; now it slopes, at exactly minus the load intensity. The shear is straight, and the moment has gone up one degree to a parabola. And the maximum moment is where the shear passes through zero. Not under the heaviest load, not at a support. Where the shear crosses the axis. There is a third thing hiding here, and it is the useful one. Shade the area under the shear diagram, from the left end out to some position x. Now watch the shaded area and the moment together. At three metres the shaded triangle is half of three times twelve, which is eighteen. And the moment there is eighteen. Keep going. Past the middle the shear is negative, so the new shading counts against you, and by the far end the positive and negative areas have cancelled exactly. The moment is back to zero. So the area under the shear, between two points, is the change in the moment between those two points. We have now seen it twice. Time to prove it once. ### [09:25.493 · Load, Shear and Moment](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=565.4929791666667) Both beams told us the same three things, so let us prove them once, for any beam and any load. Take a slice of beam of length d x, out of a beam carrying a distributed load w. On its left face the rest of the beam pushes with a shear V and a moment M. On its right face, a distance d x along, both have had a chance to change: V plus d V, and M plus d M. Vertical equilibrium of the slice. V up on the left, the load w d x pressing down on the top, and V plus d V down on the right. The two V's cancel. What is left is d V equals minus w d x. Divide through by d x, and the derivative of the shear with respect to position is minus the load intensity. Read that as a picture. Where there is no load, the shear is flat. Where the load is uniform, the shear is a straight ramp sloping downward. And where the load is heavy, the shear falls steeply. Now take moments about the right hand face. The two moments oppose each other and leave d M. The shear on the left face acts a distance d x away. And the load has a lever arm of half d x. That last term has d x squared in it. As the slice shrinks it dies away faster than everything else, so it goes. What survives is d M equals V d x. The derivative of the moment is the shear. The slope of the moment diagram, at any point at all, is the height of the shear diagram at that same point. That is the relation we kept noticing. Integrate the two of them and out comes the other half of the folklore. The change in shear between two sections is minus the area under the load diagram between them. And the change in moment between two sections is the area under the shear diagram between them. Exactly the shaded triangle we were watching a moment ago. Now every rule you were ever handed is a line in this table. No load: the shear is constant and the moment is straight. A uniform load: the shear is straight and the moment is a parabola. Each one is a degree higher than the last. A point force is an enormous load intensity over no length at all. Its area is finite, so the shear jumps by the size of the force, and the moment, whose slope is the shear, simply kinks. And a concentrated couple contributes nothing to vertical equilibrium, so the shear does not notice it at all. It sits in the moment equation instead, so the moment jumps by the size of the couple. That is the line people get wrong most often. And finally the sentence you actually use. The moment is stationary where its derivative vanishes, and its derivative is the shear. So the biggest moment sits where the shear diagram crosses the axis, or at a jump that carries it across. Two derivatives and two areas. Everything else about these diagrams is a consequence of them. ### [12:36.003 · An Overhang and a Couple](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=756.0034166666667) Third beam, and it has both of the things people find awkward. It runs eight metres, but the supports are at zero and at six, so the last two metres hang past the roller. The loads are twelve kilonewtons down at two metres, four kilonewtons down at the free tip, and a couple of eight kilonewton metres applied at five metres, twisting counterclockwise. Reactions first. Moments about A. The twelve acts at two, the four acts at eight, the reaction at B acts at six, and the couple goes straight into the sum as eight. It has no lever arm at all. A couple is the same about every point. That comes to forty eight, so the reaction at B is eight kilonewtons, and vertical equilibrium then leaves eight kilonewtons at A as well. Both supports push upward. Now build both diagrams without cutting anything, using only what we proved. Start at zero off the left end. At A the reaction pushes up eight, so the shear jumps to plus eight, and with no load between there and the twelve, it runs flat. At two metres the twelve kilonewton load drops it by twelve, from plus eight to minus four. Flat again after that. There is no distributed load anywhere on this beam. Here is where the couple acts, at five metres, and the shear passes straight through it without so much as a flinch. There is no vertical force in a couple, so there is nothing for the shear to notice. At B the reaction adds eight, taking it from minus four up to plus four. It runs flat along the overhang, and the four kilonewton load at the tip brings it home to zero. The diagram closes. Now the moment, and every piece of it is an area under that shear. From A out to two metres, a rectangle eight high and two wide: sixteen. So the moment climbs in a straight line to sixteen kilonewton metres. From two to five the shear is minus four, so the moment falls at four per metre. Three metres of that is minus twelve, taking it from sixteen down to four. And at five metres, the couple. The shear ignored it. The moment cannot. It drops vertically by eight, the size of the couple, from plus four to minus four. Below the axis now, and the shear is still minus four, so the moment goes on falling at four per metre for the last metre to the roller. Minus eight. Over the overhang the shear is plus four, so the moment climbs at four per metre for two metres and lands on zero exactly at the free tip. It has to. There is nothing beyond the tip to bend it. Check that by hand if you like. Cut the overhang anywhere and keep the right piece: one four kilonewton load, a lever arm of eight minus x, bending the beam the wrong way up. Negative, and shrinking to nothing at the tip. Two things to carry away from this beam. A couple jumps the moment and leaves the shear alone. And an overhang drives the moment below the axis, which means tension on the top of the beam, which is where the steel has to go. ### [15:45 · Sketching a Beam You Have Not Seen](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=944.9995833333334) Last beam, and you have not seen it. Eight metres, simply supported, with six kilonewtons per metre spread over the left half only and nothing on the right half. Here is the whole method, in five lines. Reactions, then jumps and slopes, then areas, then the maximum, then the check that it closes. One: reactions, and these you do have to compute. The load totals twenty four kilonewtons acting through the middle of the left half, which is much nearer A. So A takes eighteen of it, and B takes only six. Everything after this comes out of the two derivatives. Two: the shear. It starts at zero, jumps to plus eighteen at A, and then under the distributed load it slopes down at six per metre for four metres. That is a fall of twenty four, from plus eighteen to minus six. Past the load there is nothing pressing down, so the shear runs flat at minus six all the way to B, where the six kilonewton reaction closes it to zero. Three: the moment. Zero at a simple support. Its slope is the shear, which starts big and positive and is falling, so the moment leaves A steeply and bends over. A parabola, concave down. Four: it peaks where the shear crosses the axis. Eighteen divided by six is three, so the maximum sits three metres in. Not four, and certainly not at midspan. After that the shear is a constant minus six, so the moment comes down as a straight line, and it has to arrive at zero at B. Five: it does, and the sketch is finished. None of that needed the bending moment function. If you do want the number, it is the area under the shear out to three metres: half of three times eighteen, twenty seven kilonewton metres. So: cut, equilibrium, and two derivatives. The jumps, the slopes and the areas are not rules to remember. They are what those two derivatives look like once you draw them. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M14TY448ABTYCPADC1FDHE42/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [Cutting the Beam](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=0) Span: 00:00–04:9.645 (0s–249.64514583333334s). #### Objects - beam: a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)) - card: a Title that says "Engineering Statics — Shear and Moment Diagrams from First Principles" - cut: a VariableNumber (initial\_value=1.3, format\_spec='.1f') - cut\_line: a Line \[yellow\] drawn in beam (start=(\, 0.8), end=(\, -0.95), dashed=True) - fb\_arm: a Line \[gray\] labelled "x - 4" drawn in beam (start=(4.0, -4.28), end=(\, -4.28)) - fb\_line: a Line \[blue\] drawn in beam (start=(0.0, -3.1), end=(\, -3.1)) - fb\_load: a Vector \[red\] labelled "P" drawn in beam (start=(4.0, -2.34), end=(4.0, -3.04)) - fb\_m: a CurvedArrow \[yellow\] labelled "M" drawn in beam (start=((cut - 0.8), -3.62), end=((cut - 0.8), -2.58), bend=0.55) - fb\_ra: a Vector \[green\] labelled "R\_A" drawn in beam (start=(0.0, -4.05), end=(0.0, -3.16)) - fb\_v: a Vector \[yellow\] labelled "V" drawn in beam (start=(\, -3.1), end=(\, -3.98)) - fb\_x: a Line \[gray\] labelled "x" drawn in beam (start=(0.0, -4.62), end=(\, -4.62)) - head\_answer: a Heading that says "Shear and Moment as Functions of $x$" - head\_beam: a Heading that says "The Beam, and Its Reactions" - head\_cut: a Heading that says "Cut It, and Keep the Left Piece" - head\_goal: a Heading that says "Where We Are Going" - head\_right: a Heading that says "Move the Cut Past the Load" - here\_a: a Math \[gray\] that says "$4 thin upright("m")$" drawn in beam - here\_b: a Math \[gray\] that says "$2 thin upright("m")$" drawn in beam - lab\_left: a Tex \[text\] that says "For $0 \< x \< 4$:" - lab\_right: a Tex \[text\] that says "For $4 \< x \< 6$:" - left\_pair: a Derivation \[text\] that says "$V &= 4 \\ M &= 4 x$" - load: a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)) - load\_name: a Math \[red\] that says "$P = 12 thin upright("kN")$" drawn in beam - moment\_work: a Derivation \[text\] that says "$sum M\_(upright("cut")) &= 0 \\ M - 4 x &= 0 \\ M(x) &= 4 x$" - name\_a: a Math \[text\] that says "$A$" drawn in beam - name\_b: a Math \[text\] that says "$B$" drawn in beam - pin: a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35) - prev\_m: an Axes (x\_range=(0.0, 6.0), y\_range=(-2.0, 20.0), aspect=(7.0, 2.4)) - prev\_m\_left: a FunctionPlot \[magenta\] drawn in prev\_m (function=\, x\_range=(0.0, 4.0)) - prev\_m\_right: a FunctionPlot \[magenta\] drawn in prev\_m (function=\, x\_range=(4.0, 6.0)) - prev\_v: an Axes (x\_range=(0.0, 6.0), y\_range=(-12.0, 8.0), aspect=(7.0, 2.4)) - prev\_v\_jump: a Line \[yellow\] drawn in prev\_v (start=(4.0, 4.0), end=(4.0, -8.0)) - prev\_v\_left: a FunctionPlot \[yellow\] drawn in prev\_v (function=\, x\_range=(0.0, 4.0)) - prev\_v\_right: a FunctionPlot \[yellow\] drawn in prev\_v (function=\, x\_range=(4.0, 6.0)) - promise: a Text \[text\] that says "Every number in these two pictures will come out of one equilibrium statement written on a cut piece of beam." - r\_a: a Vector \[green\] drawn in beam (start=(0.0, -1.8), end=(0.0, -0.72)) - r\_a\_name: a Math \[green\] that says "$R\_A = 4$" drawn in beam - r\_b: a Vector \[green\] drawn in beam (start=(6.0, -1.8), end=(6.0, -0.72)) - r\_b\_name: a Math \[green\] that says "$R\_B = 8$" drawn in beam - reactions: a Derivation \[text\] that says "$sum M\_A &= 0 \\ 6 R\_B - 12 (4) &= 0 \\ R\_B &= 8 thin upright("kN") \\ R\_A &= 12 - 8 = 4 thin upright("kN")$" - right\_pair: a Derivation \[text\] that says "$V &= -8 \\ M &= 48 - 8 x$" - right\_work: a Derivation \[text\] that says "$V(x) &= 4 - 12 = -8 thin upright("kN") \\ M(x) &= 4 x - 12 (x - 4) \\ &= 48 - 8 x$" - roller: a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35) - shear\_work: a Derivation \[text\] that says "$sum F\_y &= 0 \\ 4 - V &= 0 \\ V(x) &= 4 thin upright("kN")$" - span: a Line \[blue\] drawn in beam (end=(6.0, 0.0)) #### Beats ##### [00:00](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=0) Narration: Shear and moment diagrams are usually handed to you as a list of rules. Jump here by the size of the load. Slope there. Today we derive every one of them from a single idea: a piece of a beam in equilibrium is still in equilibrium after you cut it. Board: Empty. Actions: - [00:00](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=0): card is shown on the screen, written out. - [00:1.5](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1.5): card: enter:write-left-to-right. - [00:16.358](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=16.358): card is hidden from the screen — left the board. ##### [00:17.558](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=17.558) Narration: Here is where we are going. Underneath a loaded beam you draw two pictures: the shear diagram, and the bending moment diagram. Board: Unchanged from the preceding beat in this scene. Actions: - [00:17.558](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=17.558): head\_goal is shown on the screen, written out. - [00:22.655](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=22.654999999999998): prev\_v is shown on the screen, written out. - [00:22.955](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=22.955): prev\_v\_left is shown on the screen, drawn. - [00:23.455](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=23.455): prev\_v\_right is shown on the screen, drawn. - [00:24.048](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=24.048): prev\_m is shown on the screen, written out. - [00:24.155](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=24.154999999999998): prev\_v\_jump is shown on the screen, drawn. - [00:24.348](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=24.348): prev\_m\_left is shown on the screen, drawn. - [00:24.848](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=24.848): prev\_m\_right is shown on the screen, drawn. ##### [00:26.483](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=26.482999999999997) Narration: By the end you will sketch both of these for a beam you have not seen, and know where the biggest moment sits before computing anything. But not by pattern matching. Every value in them falls out of one free body diagram. Board: prev\_v — an Axes (x\_range=(0.0, 6.0), y\_range=(-12.0, 8.0), aspect=(7.0, 2.4)); prev\_m — an Axes (x\_range=(0.0, 6.0), y\_range=(-2.0, 20.0), aspect=(7.0, 2.4)); head\_goal — a Heading that says "Where We Are Going"; prev\_v\_left — a FunctionPlot \[yellow\] drawn in prev\_v (function=\, x\_range=(0.0, 4.0)); prev\_v\_right — a FunctionPlot \[yellow\] drawn in prev\_v (function=\, x\_range=(4.0, 6.0)); prev\_v\_jump — a Line \[yellow\] drawn in prev\_v (start=(4.0, 4.0), end=(4.0, -8.0)); prev\_m\_left — a FunctionPlot \[magenta\] drawn in prev\_m (function=\, x\_range=(0.0, 4.0)); prev\_m\_right — a FunctionPlot \[magenta\] drawn in prev\_m (function=\, x\_range=(4.0, 6.0)) Actions: - [00:26.483](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=26.482999999999997): promise is shown on the screen, written out. - [00:39.625](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=39.625499999999995): head\_goal is hidden from the screen — left the board. - [00:39.625](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=39.625499999999995): prev\_m is hidden from the screen — left the board. - [00:39.625](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=39.625499999999995): prev\_m\_left is hidden from the screen — prev\_m left the board. - [00:39.625](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=39.625499999999995): prev\_m\_right is hidden from the screen — prev\_m left the board. - [00:39.625](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=39.625499999999995): prev\_v is hidden from the screen — left the board. - [00:39.625](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=39.625499999999995): prev\_v\_left is hidden from the screen — prev\_v left the board. - [00:39.625](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=39.625499999999995): prev\_v\_right is hidden from the screen — prev\_v left the board. - [00:39.625](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=39.625499999999995): prev\_v\_jump is hidden from the screen — prev\_v left the board. - [00:39.625](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=39.625499999999995): promise is hidden from the screen — left the board. ##### [00:40.825](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=40.8255) Narration: So, the beam. Six metres between the supports, a pin at A, a roller at B, and one point load of twelve kilonewtons, four metres from the left. Board: Empty. Actions: - [00:40.825](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=40.8255): head\_beam is shown on the screen, written out. - [00:40.825](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=40.8255): beam is shown on the screen, written out. - [00:41.882](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=41.882): span is shown on the screen, written out. - [00:44.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=44.958): pin is shown on the screen, written out. - [00:45.058](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=45.058): name\_a is shown on the screen, written out. - [00:46.085](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=46.085): roller is shown on the screen, written out. - [00:46.185](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=46.185): name\_b is shown on the screen, written out. - [00:47.443](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=47.443000000000005): load is shown on the screen, written out. - [00:47.643](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=47.64300000000001): load\_name is shown on the screen, written out. - [00:49.556](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=49.556000000000004): here\_a is shown on the screen, written out. - [00:49.756](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=49.75600000000001): here\_b is shown on the screen, written out. ##### [00:51.712](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=51.7115) Narration: Before we cut anything we need the reactions. Take moments about A. The reaction at B acts six metres out, the load pulls down four metres out, and those two have to balance. Board: beam — a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)); head\_beam — a Heading that says "The Beam, and Its Reactions"; span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35); name\_a — a Math \[text\] that says "$A$" drawn in beam; roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35); name\_b — a Math \[text\] that says "$B$" drawn in beam; load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12 thin upright("kN")$" drawn in beam; here\_a — a Math \[gray\] that says "$4 thin upright("m")$" drawn in beam; here\_b — a Math \[gray\] that says "$2 thin upright("m")$" drawn in beam Actions: - [00:55.067](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=55.067): beam moves to a new place on the board. - [00:55.067](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=55.067): reactions is shown on the screen, written out. - [01:2.125](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=62.125): reactions is shown on the screen, written out. ##### [01:3.538](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=63.538) Narration: That gives eight kilonewtons at B. Vertical equilibrium then leaves four kilonewtons at A. Both supports push upward. Board: Unchanged from the preceding beat in this scene. Actions: - [01:4.49](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=64.49): reactions is shown on the screen, written out. - [01:4.69](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=64.69): r\_b is shown on the screen, written out. - [01:5.09](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=65.09): r\_b\_name is shown on the screen, written out. - [01:7.857](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=67.857): reactions is shown on the screen, written out. - [01:8.057](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=68.057): r\_a is shown on the screen, written out. - [01:8.457](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=68.45700000000001): r\_a\_name is shown on the screen, written out. ##### [01:12.346](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=72.346) Narration: Now the question that matters. A beam does not fail at its supports. It fails somewhere inside, where the material is carrying an internal force and an internal moment. What are they, and how do they change as you walk along? Board: beam — a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)); head\_beam — a Heading that says "The Beam, and Its Reactions"; span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35); name\_a — a Math \[text\] that says "$A$" drawn in beam; roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35); name\_b — a Math \[text\] that says "$B$" drawn in beam; load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12 thin upright("kN")$" drawn in beam; here\_a — a Math \[gray\] that says "$4 thin upright("m")$" drawn in beam; here\_b — a Math \[gray\] that says "$2 thin upright("m")$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.8), end=(6.0, -0.72)); r\_b\_name — a Math \[green\] that says "$R\_B = 8$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.8), end=(0.0, -0.72)); r\_a\_name — a Math \[green\] that says "$R\_A = 4$" drawn in beam Actions: - [01:27.149](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=87.149): head\_beam is hidden from the screen — left the board. - [01:27.149](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=87.149): reactions is hidden from the screen — left the board. ##### [01:27.749](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=87.74900000000001) Narration: Here is the one move the whole subject rests on. Cut the beam at a distance x from A, and throw away everything to the right. What is left is a piece with the reaction at A still pushing up on it. Board: beam — a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)); span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35); name\_a — a Math \[text\] that says "$A$" drawn in beam; roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35); name\_b — a Math \[text\] that says "$B$" drawn in beam; load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12 thin upright("kN")$" drawn in beam; here\_a — a Math \[gray\] that says "$4 thin upright("m")$" drawn in beam; here\_b — a Math \[gray\] that says "$2 thin upright("m")$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.8), end=(6.0, -0.72)); r\_b\_name — a Math \[green\] that says "$R\_B = 8$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.8), end=(0.0, -0.72)); r\_a\_name — a Math \[green\] that says "$R\_A = 4$" drawn in beam Actions: - [01:27.749](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=87.74900000000001): head\_cut is shown on the screen, written out. - [01:30.988](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=90.98800000000001): cut\_line is shown on the screen, written out. - [01:36.178](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=96.17800000000001): fb\_line is shown on the screen, written out. - [01:36.378](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=96.37800000000001): fb\_x is shown on the screen, written out. - [01:38.454](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=98.45400000000002): fb\_ra is shown on the screen, written out. ##### [01:40.238](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=100.23800000000001) Narration: That piece cannot be in equilibrium on its own. Four kilonewtons up and nothing down. So the part we threw away must have been holding it, at the cut face, with a force and with a moment. Call them V and M. Board: beam — a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)); span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35); name\_a — a Math \[text\] that says "$A$" drawn in beam; roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35); name\_b — a Math \[text\] that says "$B$" drawn in beam; load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12 thin upright("kN")$" drawn in beam; here\_a — a Math \[gray\] that says "$4 thin upright("m")$" drawn in beam; here\_b — a Math \[gray\] that says "$2 thin upright("m")$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.8), end=(6.0, -0.72)); r\_b\_name — a Math \[green\] that says "$R\_B = 8$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.8), end=(0.0, -0.72)); r\_a\_name — a Math \[green\] that says "$R\_A = 4$" drawn in beam; head\_cut — a Heading that says "Cut It, and Keep the Left Piece"; cut\_line — a Line \[yellow\] drawn in beam (start=(\, 0.8), end=(\, -0.95), dashed=True); fb\_line — a Line \[blue\] drawn in beam (start=(0.0, -3.1), end=(\, -3.1)); fb\_x — a Line \[gray\] labelled "x" drawn in beam (start=(0.0, -4.62), end=(\, -4.62)); fb\_ra — a Vector \[green\] labelled "R\_A" drawn in beam (start=(0.0, -4.05), end=(0.0, -3.16)) Actions: - [01:50.222](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=110.22200000000002): fb\_v is shown on the screen, written out. - [01:51.105](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=111.10500000000002): fb\_m is shown on the screen, written out. ##### [01:54.375](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=114.37450000000001) Narration: Those are the internal shear and the internal bending moment at position x. Draw them the standard way every time: V pointing down on the cut face, and M bending the beam into a smile. Then a negative answer simply means it acts the other way. Board: beam — a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)); span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35); name\_a — a Math \[text\] that says "$A$" drawn in beam; roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35); name\_b — a Math \[text\] that says "$B$" drawn in beam; load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12 thin upright("kN")$" drawn in beam; here\_a — a Math \[gray\] that says "$4 thin upright("m")$" drawn in beam; here\_b — a Math \[gray\] that says "$2 thin upright("m")$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.8), end=(6.0, -0.72)); r\_b\_name — a Math \[green\] that says "$R\_B = 8$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.8), end=(0.0, -0.72)); r\_a\_name — a Math \[green\] that says "$R\_A = 4$" drawn in beam; head\_cut — a Heading that says "Cut It, and Keep the Left Piece"; cut\_line — a Line \[yellow\] drawn in beam (start=(\, 0.8), end=(\, -0.95), dashed=True); fb\_line — a Line \[blue\] drawn in beam (start=(0.0, -3.1), end=(\, -3.1)); fb\_x — a Line \[gray\] labelled "x" drawn in beam (start=(0.0, -4.62), end=(\, -4.62)); fb\_ra — a Vector \[green\] labelled "R\_A" drawn in beam (start=(0.0, -4.05), end=(0.0, -3.16)); fb\_v — a Vector \[yellow\] labelled "V" drawn in beam (start=(\, -3.1), end=(\, -3.98)); fb\_m — a CurvedArrow \[yellow\] labelled "M" drawn in beam (start=((cut - 0.8), -3.62), end=((cut - 0.8), -2.58), bend=0.55) Actions: - None. ##### [02:10.602](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=130.60150000000002) Narration: Vertical equilibrium of the piece. Four up, V down, nothing else. So V is four kilonewtons. Board: Unchanged from the preceding beat in this scene. Actions: - [02:10.95](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=130.95000000000005): shear\_work is shown on the screen, written out. - [02:13.632](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=133.63200000000003): shear\_work is shown on the screen, written out. - [02:15.315](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=135.31500000000003): shear\_work is shown on the screen, written out. ##### [02:19.817](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=139.81650000000002) Narration: Moments about the cut. The reaction acts a distance x away, and M is the only other thing on the piece. So M is four x. Board: Unchanged from the preceding beat in this scene. Actions: - [02:20.165](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=140.16500000000002): moment\_work is shown on the screen, written out. - [02:24.844](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=144.84400000000002): moment\_work is shown on the screen, written out. - [02:27.084](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=147.084): moment\_work is shown on the screen, written out. ##### [02:28.561](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=148.56050000000002) Narration: Look at what x did. Nothing at all to the shear: four kilonewtons wherever you cut, as long as you stay left of the load. But the moment grows in proportion to x. Watch the piece get longer. Board: Unchanged from the preceding beat in this scene. Actions: - [02:39.062](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=159.062): cut\_line is redrawn as the numbers it depends on change. - [02:39.062](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=159.062): fb\_line is redrawn as the numbers it depends on change. - [02:39.062](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=159.062): fb\_x is redrawn as the numbers it depends on change. - [02:39.062](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=159.062): fb\_v is redrawn as the numbers it depends on change. - [02:39.062](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=159.062): fb\_m is redrawn as the numbers it depends on change. - [02:39.062](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=159.062): fb\_arm is redrawn as the numbers it depends on change. - [02:39.062](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=159.062): cut ticks to 3.6. ##### [02:41.566](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=161.56600000000003) Narration: That is what we mean by shear and moment as functions of position. One cut, two equilibrium equations, and we have both of them for the whole left hand stretch of this beam. Board: Unchanged from the preceding beat in this scene. Actions: - [02:51.365](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=171.36500000000004): head\_cut is hidden from the screen — left the board. - [02:51.365](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=171.36500000000004): moment\_work is hidden from the screen — left the board. - [02:51.365](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=171.36500000000004): shear\_work is hidden from the screen — left the board. ##### [02:52.565](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=172.56500000000003) Narration: But only that stretch. Move the cut past the load, and the free body is a different picture: the twelve kilonewton load now sits on the piece we kept. Board: beam — a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)); span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35); name\_a — a Math \[text\] that says "$A$" drawn in beam; roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35); name\_b — a Math \[text\] that says "$B$" drawn in beam; load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12 thin upright("kN")$" drawn in beam; here\_a — a Math \[gray\] that says "$4 thin upright("m")$" drawn in beam; here\_b — a Math \[gray\] that says "$2 thin upright("m")$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.8), end=(6.0, -0.72)); r\_b\_name — a Math \[green\] that says "$R\_B = 8$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.8), end=(0.0, -0.72)); r\_a\_name — a Math \[green\] that says "$R\_A = 4$" drawn in beam; cut\_line — a Line \[yellow\] drawn in beam (start=(\, 0.8), end=(\, -0.95), dashed=True); fb\_line — a Line \[blue\] drawn in beam (start=(0.0, -3.1), end=(\, -3.1)); fb\_x — a Line \[gray\] labelled "x" drawn in beam (start=(0.0, -4.62), end=(\, -4.62)); fb\_ra — a Vector \[green\] labelled "R\_A" drawn in beam (start=(0.0, -4.05), end=(0.0, -3.16)); fb\_v — a Vector \[yellow\] labelled "V" drawn in beam (start=(\, -3.1), end=(\, -3.98)); fb\_m — a CurvedArrow \[yellow\] labelled "M" drawn in beam (start=((cut - 0.8), -3.62), end=((cut - 0.8), -2.58), bend=0.55) Actions: - [02:52.565](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=172.56500000000003): head\_right is shown on the screen, written out. - [02:54.817](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=174.81700000000004): cut\_line is redrawn as the numbers it depends on change. - [02:54.817](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=174.81700000000004): fb\_line is redrawn as the numbers it depends on change. - [02:54.817](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=174.81700000000004): fb\_x is redrawn as the numbers it depends on change. - [02:54.817](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=174.81700000000004): fb\_v is redrawn as the numbers it depends on change. - [02:54.817](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=174.81700000000004): fb\_m is redrawn as the numbers it depends on change. - [02:54.817](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=174.81700000000004): fb\_arm is redrawn as the numbers it depends on change. - [02:54.817](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=174.81700000000004): cut ticks to 5.0. - [03:1.446](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=181.44600000000003): fb\_load is shown on the screen, written out. - [03:1.746](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=181.74600000000004): fb\_arm is shown on the screen, written out. ##### [03:2.801](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=182.80100000000002) Narration: Vertical equilibrium now has three forces. Four up, twelve down, and V drawn downward as before. So V is four minus twelve, which is minus eight kilonewtons. Board: beam — a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)); span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35); name\_a — a Math \[text\] that says "$A$" drawn in beam; roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35); name\_b — a Math \[text\] that says "$B$" drawn in beam; load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12 thin upright("kN")$" drawn in beam; here\_a — a Math \[gray\] that says "$4 thin upright("m")$" drawn in beam; here\_b — a Math \[gray\] that says "$2 thin upright("m")$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.8), end=(6.0, -0.72)); r\_b\_name — a Math \[green\] that says "$R\_B = 8$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.8), end=(0.0, -0.72)); r\_a\_name — a Math \[green\] that says "$R\_A = 4$" drawn in beam; cut\_line — a Line \[yellow\] drawn in beam (start=(\, 0.8), end=(\, -0.95), dashed=True); fb\_line — a Line \[blue\] drawn in beam (start=(0.0, -3.1), end=(\, -3.1)); fb\_x — a Line \[gray\] labelled "x" drawn in beam (start=(0.0, -4.62), end=(\, -4.62)); fb\_ra — a Vector \[green\] labelled "R\_A" drawn in beam (start=(0.0, -4.05), end=(0.0, -3.16)); fb\_v — a Vector \[yellow\] labelled "V" drawn in beam (start=(\, -3.1), end=(\, -3.98)); fb\_m — a CurvedArrow \[yellow\] labelled "M" drawn in beam (start=((cut - 0.8), -3.62), end=((cut - 0.8), -2.58), bend=0.55); head\_right — a Heading that says "Move the Cut Past the Load"; fb\_load — a Vector \[red\] labelled "P" drawn in beam (start=(4.0, -2.34), end=(4.0, -3.04)); fb\_arm — a Line \[gray\] labelled "x - 4" drawn in beam (start=(4.0, -4.28), end=(\, -4.28)) Actions: - [03:10.777](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=190.77700000000002): right\_work is shown on the screen, written out. ##### [03:15.94](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=195.94000000000003) Narration: And notice: the shear did not slide down to minus eight. It jumped, and it jumped by exactly twelve, the size of the load. Hold on to that. Board: Unchanged from the preceding beat in this scene. Actions: - [03:20.909](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=200.90900000000005): right\_work (the "-8" part) is emphasized. - [03:25.518](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=205.51800000000006): right\_work (the "-8" part) is no longer emphasized. ##### [03:27.442](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=207.44200000000004) Narration: Moments about the cut again. The reaction still acts x away, and the load now acts on our piece as well, a lever arm of x minus four, turning it the other way. That tidies to forty eight minus eight x. Board: Unchanged from the preceding beat in this scene. Actions: - [03:32.969](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=212.96900000000002): right\_work is shown on the screen, written out. - [03:39.133](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=219.13300000000004): right\_work is shown on the screen, written out. ##### [03:42.015](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=222.01450000000003) Narration: Check it at the far end. Put x equal to six and the moment comes out zero, which is exactly what a roller has to give you. A roller cannot resist any moment at all. Board: Unchanged from the preceding beat in this scene. Actions: - [03:46.247](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=226.247): right\_work is indicated — a transient flash. - [03:52.017](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=232.01700000000002): head\_right is hidden from the screen — left the board. - [03:52.017](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=232.01700000000002): right\_work is hidden from the screen — left the board. ##### [03:52.617](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=232.61700000000002) Narration: So here is the answer. Two expressions for the shear and two for the moment, one pair on each side of the load. Board: beam — a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)); span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35); name\_a — a Math \[text\] that says "$A$" drawn in beam; roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35); name\_b — a Math \[text\] that says "$B$" drawn in beam; load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12 thin upright("kN")$" drawn in beam; here\_a — a Math \[gray\] that says "$4 thin upright("m")$" drawn in beam; here\_b — a Math \[gray\] that says "$2 thin upright("m")$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.8), end=(6.0, -0.72)); r\_b\_name — a Math \[green\] that says "$R\_B = 8$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.8), end=(0.0, -0.72)); r\_a\_name — a Math \[green\] that says "$R\_A = 4$" drawn in beam; cut\_line — a Line \[yellow\] drawn in beam (start=(\, 0.8), end=(\, -0.95), dashed=True); fb\_line — a Line \[blue\] drawn in beam (start=(0.0, -3.1), end=(\, -3.1)); fb\_x — a Line \[gray\] labelled "x" drawn in beam (start=(0.0, -4.62), end=(\, -4.62)); fb\_ra — a Vector \[green\] labelled "R\_A" drawn in beam (start=(0.0, -4.05), end=(0.0, -3.16)); fb\_v — a Vector \[yellow\] labelled "V" drawn in beam (start=(\, -3.1), end=(\, -3.98)); fb\_m — a CurvedArrow \[yellow\] labelled "M" drawn in beam (start=((cut - 0.8), -3.62), end=((cut - 0.8), -2.58), bend=0.55); fb\_load — a Vector \[red\] labelled "P" drawn in beam (start=(4.0, -2.34), end=(4.0, -3.04)); fb\_arm — a Line \[gray\] labelled "x - 4" drawn in beam (start=(4.0, -4.28), end=(\, -4.28)) Actions: - [03:52.617](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=232.61700000000002): head\_answer is shown on the screen, written out. - [03:54.788](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=234.788): lab\_left is shown on the screen, written out. - [03:55.088](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=235.08800000000002): left\_pair is shown on the screen, written out. - [03:55.588](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=235.58800000000002): left\_pair is shown on the screen, written out. - [03:58.41](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=238.41000000000003): lab\_right is shown on the screen, written out. - [03:58.71](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=238.71000000000004): right\_pair is shown on the screen, written out. - [03:59.21](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=239.21000000000004): right\_pair is shown on the screen, written out. ##### [04:0.288](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=240.288) Narration: Nothing there was remembered. A cut, a free body, and two equilibrium equations, done twice. Now let us draw them. Board: beam — a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)); span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35); name\_a — a Math \[text\] that says "$A$" drawn in beam; roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35); name\_b — a Math \[text\] that says "$B$" drawn in beam; load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12 thin upright("kN")$" drawn in beam; here\_a — a Math \[gray\] that says "$4 thin upright("m")$" drawn in beam; here\_b — a Math \[gray\] that says "$2 thin upright("m")$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.8), end=(6.0, -0.72)); r\_b\_name — a Math \[green\] that says "$R\_B = 8$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.8), end=(0.0, -0.72)); r\_a\_name — a Math \[green\] that says "$R\_A = 4$" drawn in beam; cut\_line — a Line \[yellow\] drawn in beam (start=(\, 0.8), end=(\, -0.95), dashed=True); fb\_line — a Line \[blue\] drawn in beam (start=(0.0, -3.1), end=(\, -3.1)); fb\_x — a Line \[gray\] labelled "x" drawn in beam (start=(0.0, -4.62), end=(\, -4.62)); fb\_ra — a Vector \[green\] labelled "R\_A" drawn in beam (start=(0.0, -4.05), end=(0.0, -3.16)); fb\_v — a Vector \[yellow\] labelled "V" drawn in beam (start=(\, -3.1), end=(\, -3.98)); fb\_m — a CurvedArrow \[yellow\] labelled "M" drawn in beam (start=((cut - 0.8), -3.62), end=((cut - 0.8), -2.58), bend=0.55); fb\_load — a Vector \[red\] labelled "P" drawn in beam (start=(4.0, -2.34), end=(4.0, -3.04)); fb\_arm — a Line \[gray\] labelled "x - 4" drawn in beam (start=(4.0, -4.28), end=(\, -4.28)); lab\_left — a Tex \[text\] that says "For $0 \< x \< 4$:"; lab\_right — a Tex \[text\] that says "For $4 \< x \< 6$:"; head\_answer — a Heading that says "Shear and Moment as Functions of $x$" Actions: - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): beam is hidden from the screen — left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): span is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): pin is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): name\_a is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): roller is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): name\_b is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): load is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): load\_name is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): here\_a is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): here\_b is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): r\_b is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): r\_b\_name is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): r\_a is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): r\_a\_name is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): cut\_line is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): fb\_line is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): fb\_x is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): fb\_ra is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): fb\_v is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): fb\_m is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): fb\_load is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): fb\_arm is hidden from the screen — beam left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): head\_answer is hidden from the screen — left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): lab\_left is hidden from the screen — left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): lab\_right is hidden from the screen — left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): left\_pair is hidden from the screen — left the board. - [04:8.603](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=248.6034791666667): right\_pair is hidden from the screen — left the board. ### Scene 2: [Drawing the Two Diagrams](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=249.64514583333334) Span: 04:9.645–06:34.051 (249.64514583333334s–394.05110416666673s). #### Objects - beam: a Figure (x\_range=(-0.9, 7.0), y\_range=(-1.7, 2.5), aspect=(7.9, 4.2)) - head\_moment: a Heading that says "Plotting the Moment" - head\_shear: a Heading that says "Plotting the Shear" - lab\_left: a Tex \[text\] that says "For $0 \< x \< 4$:" - lab\_right: a Tex \[text\] that says "For $4 \< x \< 6$:" - left\_pair: a Derivation \[text\] that says "$V &= 4 \\ M &= 4 x$" - load: a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)) - load\_name: a Math \[red\] that says "$P = 12$" drawn in beam - m\_axes: an Axes (x\_range=(0.0, 6.0), y\_range=(-2.0, 20.0), aspect=(7.9, 2.8)) - m\_dot: a PlotPoint \[green\] drawn in m\_axes (target='m\_left', x=\) - m\_left: a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(0.0, 4.0)) - m\_peak: a Point \[green\] labelled "16" drawn in m\_axes (location=(4.0, 16.0)) - m\_right: a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(4.0, 6.0)) - marker: a Line \[green\] drawn in beam (start=(\, 0.7), end=(\, -0.7), dashed=True) - marker\_x: a VariableNumber (initial\_value=0.6) - note\_slope: a Math \[text\] that says "$upright("slope of ") M = V$" - pin: a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.3, -0.55), (0.3, -0.55)), fill\_opacity=0.35) - probe: a VariableNumber (initial\_value=0.6) - probe\_m: a VariableNumber (initial\_value=3.7) - r\_a: a Vector \[green\] drawn in beam (start=(0.0, -1.5), end=(0.0, -0.65)) - r\_a\_name: a Math \[green\] that says "$4$" drawn in beam - r\_b: a Vector \[green\] drawn in beam (start=(6.0, -1.5), end=(6.0, -0.65)) - r\_b\_name: a Math \[green\] that says "$8$" drawn in beam - right\_pair: a Derivation \[text\] that says "$V &= -8 \\ M &= 48 - 8 x$" - roller: a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.7, -0.55), (6.3, -0.55)), fill\_opacity=0.35) - span: a Line \[blue\] drawn in beam (end=(6.0, 0.0)) - v\_axes: an Axes (x\_range=(0.0, 6.0), y\_range=(-12.0, 8.0), aspect=(7.9, 2.8)) - v\_close: a Line \[yellow\] drawn in v\_axes (start=(6.0, -8.0), end=(6.0, 0.0)) - v\_dot: a PlotPoint \[green\] drawn in v\_axes (target='v\_left', x=\) - v\_drop: a Line \[yellow\] drawn in v\_axes (start=(4.0, 4.0), end=(4.0, -8.0)) - v\_left: a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)) - v\_right: a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 6.0)) - v\_start: a Line \[yellow\] drawn in v\_axes (end=(0.0, 4.0)) #### Beats ##### [04:9.645](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=249.64514583333334) Narration: We have four expressions and a beam. Drawing the shear and moment diagrams is nothing more than plotting them, underneath the beam, on the same horizontal scale. Board: Empty. Actions: - [04:9.645](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=249.64514583333334): head\_shear is shown on the screen, written out. - [04:9.645](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=249.64514583333334): beam is shown on the screen, written out. - [04:10.481](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=250.48114583333336): lab\_left is shown on the screen, written out. - [04:10.681](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=250.68114583333335): left\_pair is shown on the screen, written out. - [04:11.081](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=251.08114583333335): left\_pair is shown on the screen, written out. - [04:11.681](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=251.68114583333335): lab\_right is shown on the screen, written out. - [04:11.7](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=251.70014583333335): span is shown on the screen, written out. - [04:11.8](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=251.80014583333335): pin is shown on the screen, written out. - [04:12](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=252.00014583333333): roller is shown on the screen, written out. - [04:12.3](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=252.30014583333335): load is shown on the screen, written out. - [04:12.481](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=252.48114583333336): right\_pair is shown on the screen, written out. - [04:12.7](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=252.70014583333335): load\_name is shown on the screen, written out. - [04:13.2](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=253.20014583333335): r\_a is shown on the screen, written out. - [04:13.481](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=253.48114583333336): right\_pair is shown on the screen, written out. - [04:13.8](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=253.80014583333335): r\_a\_name is shown on the screen, written out. - [04:14.5](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=254.50014583333333): r\_b is shown on the screen, written out. - [04:15.3](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=255.30014583333335): r\_b\_name is shown on the screen, written out. ##### [04:20.148](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=260.14814583333333) Narration: So: position along the beam across the bottom, internal shear up the side. Board: lab\_left — a Tex \[text\] that says "For $0 \< x \< 4$:"; lab\_right — a Tex \[text\] that says "For $4 \< x \< 6$:"; beam — a Figure (x\_range=(-0.9, 7.0), y\_range=(-1.7, 2.5), aspect=(7.9, 4.2)); head\_shear — a Heading that says "Plotting the Shear"; span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.3, -0.55), (0.3, -0.55)), fill\_opacity=0.35); roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.7, -0.55), (6.3, -0.55)), fill\_opacity=0.35); load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.5), end=(0.0, -0.65)); r\_a\_name — a Math \[green\] that says "$4$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.5), end=(6.0, -0.65)); r\_b\_name — a Math \[green\] that says "$8$" drawn in beam Actions: - [04:21.1](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=261.10014583333333): v\_axes is shown on the screen, written out. ##### [04:26.089](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=266.08914583333336) Narration: On the left stretch the shear is a constant four. The diagram starts from zero off the end of the beam, jumps straight up to four where the pin pushes in, and then runs flat all the way to the load. Board: lab\_left — a Tex \[text\] that says "For $0 \< x \< 4$:"; lab\_right — a Tex \[text\] that says "For $4 \< x \< 6$:"; beam — a Figure (x\_range=(-0.9, 7.0), y\_range=(-1.7, 2.5), aspect=(7.9, 4.2)); v\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-12.0, 8.0), aspect=(7.9, 2.8)); head\_shear — a Heading that says "Plotting the Shear"; span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.3, -0.55), (0.3, -0.55)), fill\_opacity=0.35); roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.7, -0.55), (6.3, -0.55)), fill\_opacity=0.35); load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.5), end=(0.0, -0.65)); r\_a\_name — a Math \[green\] that says "$4$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.5), end=(6.0, -0.65)); r\_b\_name — a Math \[green\] that says "$8$" drawn in beam Actions: - [04:32.811](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=272.81114583333334): v\_start is shown on the screen, drawn. - [04:36.341](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=276.34114583333337): v\_left is shown on the screen, drawn. ##### [04:38.682](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=278.6821458333333) Narration: On the right stretch it is minus eight, so a horizontal line eight below the axis, running from the load across to B. Board: lab\_left — a Tex \[text\] that says "For $0 \< x \< 4$:"; lab\_right — a Tex \[text\] that says "For $4 \< x \< 6$:"; beam — a Figure (x\_range=(-0.9, 7.0), y\_range=(-1.7, 2.5), aspect=(7.9, 4.2)); v\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-12.0, 8.0), aspect=(7.9, 2.8)); head\_shear — a Heading that says "Plotting the Shear"; span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.3, -0.55), (0.3, -0.55)), fill\_opacity=0.35); roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.7, -0.55), (6.3, -0.55)), fill\_opacity=0.35); load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.5), end=(0.0, -0.65)); r\_a\_name — a Math \[green\] that says "$4$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.5), end=(6.0, -0.65)); r\_b\_name — a Math \[green\] that says "$8$" drawn in beam; v\_start — a Line \[yellow\] drawn in v\_axes (end=(0.0, 4.0)); v\_left — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)) Actions: - [04:41.55](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=281.5501458333333): v\_right is shown on the screen, drawn. ##### [04:46.335](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=286.33514583333334) Narration: And those two do not meet. At the load they differ by twelve, so the diagram falls vertically, right there, by twelve. The jump is the load. Board: lab\_left — a Tex \[text\] that says "For $0 \< x \< 4$:"; lab\_right — a Tex \[text\] that says "For $4 \< x \< 6$:"; beam — a Figure (x\_range=(-0.9, 7.0), y\_range=(-1.7, 2.5), aspect=(7.9, 4.2)); v\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-12.0, 8.0), aspect=(7.9, 2.8)); head\_shear — a Heading that says "Plotting the Shear"; span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.3, -0.55), (0.3, -0.55)), fill\_opacity=0.35); roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.7, -0.55), (6.3, -0.55)), fill\_opacity=0.35); load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.5), end=(0.0, -0.65)); r\_a\_name — a Math \[green\] that says "$4$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.5), end=(6.0, -0.65)); r\_b\_name — a Math \[green\] that says "$8$" drawn in beam; v\_start — a Line \[yellow\] drawn in v\_axes (end=(0.0, 4.0)); v\_left — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)); v\_right — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 6.0)) Actions: - [04:51.542](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=291.54214583333334): v\_drop is shown on the screen, drawn. ##### [04:56.415](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=296.4151458333333) Narration: Then the roller pushes up eight and closes the diagram back to zero. That is a free check on your arithmetic. A shear diagram that does not return to zero at the far end means a reaction is wrong. Board: lab\_left — a Tex \[text\] that says "For $0 \< x \< 4$:"; lab\_right — a Tex \[text\] that says "For $4 \< x \< 6$:"; beam — a Figure (x\_range=(-0.9, 7.0), y\_range=(-1.7, 2.5), aspect=(7.9, 4.2)); v\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-12.0, 8.0), aspect=(7.9, 2.8)); head\_shear — a Heading that says "Plotting the Shear"; span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.3, -0.55), (0.3, -0.55)), fill\_opacity=0.35); roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.7, -0.55), (6.3, -0.55)), fill\_opacity=0.35); load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.5), end=(0.0, -0.65)); r\_a\_name — a Math \[green\] that says "$4$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.5), end=(6.0, -0.65)); r\_b\_name — a Math \[green\] that says "$8$" drawn in beam; v\_start — a Line \[yellow\] drawn in v\_axes (end=(0.0, 4.0)); v\_left — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)); v\_right — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 6.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(4.0, 4.0), end=(4.0, -8.0)) Actions: - [04:58.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=298.5401458333333): v\_close is shown on the screen, drawn. ##### [05:9.74](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=309.7396458333333) Narration: Watch the shear as the cut walks along. Nothing happens to it until it reaches the load, and then in one step it is on the other side of the axis. Board: lab\_left — a Tex \[text\] that says "For $0 \< x \< 4$:"; lab\_right — a Tex \[text\] that says "For $4 \< x \< 6$:"; beam — a Figure (x\_range=(-0.9, 7.0), y\_range=(-1.7, 2.5), aspect=(7.9, 4.2)); v\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-12.0, 8.0), aspect=(7.9, 2.8)); head\_shear — a Heading that says "Plotting the Shear"; span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.3, -0.55), (0.3, -0.55)), fill\_opacity=0.35); roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.7, -0.55), (6.3, -0.55)), fill\_opacity=0.35); load — a Vector \[red\] drawn in beam (start=(4.0, 1.7), end=(4.0, 0.08)); load\_name — a Math \[red\] that says "$P = 12$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.5), end=(0.0, -0.65)); r\_a\_name — a Math \[green\] that says "$4$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.5), end=(6.0, -0.65)); r\_b\_name — a Math \[green\] that says "$8$" drawn in beam; v\_start — a Line \[yellow\] drawn in v\_axes (end=(0.0, 4.0)); v\_left — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)); v\_right — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 6.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(4.0, 4.0), end=(4.0, -8.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(6.0, -8.0), end=(6.0, 0.0)) Actions: - [05:9.74](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=309.7396458333333): marker is shown on the screen, written out. - [05:9.74](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=309.7396458333333): v\_dot is shown on the screen, written out. - [05:11.388](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=311.38814583333334): marker is redrawn as the numbers it depends on change. - [05:11.388](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=311.38814583333334): v\_dot is redrawn as the numbers it depends on change. - [05:11.388](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=311.38814583333334): probe ticks to 3.7. - [05:11.388](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=311.38814583333334): marker\_x ticks to 3.7. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): lab\_left moves to a new place on the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): lab\_right moves to a new place on the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): left\_pair moves to a new place on the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): right\_pair moves to a new place on the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): v\_axes moves to a new place on the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): beam is hidden from the screen — left the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): span is hidden from the screen — beam left the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): pin is hidden from the screen — beam left the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): roller is hidden from the screen — beam left the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): load is hidden from the screen — beam left the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): load\_name is hidden from the screen — beam left the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): r\_a is hidden from the screen — beam left the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): r\_a\_name is hidden from the screen — beam left the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): r\_b is hidden from the screen — beam left the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): r\_b\_name is hidden from the screen — beam left the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): marker is hidden from the screen — beam left the board. - [05:18.54](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=318.53964583333334): head\_shear is hidden from the screen — left the board. ##### [05:19.14](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=319.13964583333336) Narration: Now the moment, on its own axis underneath the shear. On the left stretch it was four x, a straight line climbing out of the origin. Board: lab\_left — a Tex \[text\] that says "For $0 \< x \< 4$:"; lab\_right — a Tex \[text\] that says "For $4 \< x \< 6$:"; v\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-12.0, 8.0), aspect=(7.9, 2.8)); v\_start — a Line \[yellow\] drawn in v\_axes (end=(0.0, 4.0)); v\_left — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)); v\_right — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 6.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(4.0, 4.0), end=(4.0, -8.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(6.0, -8.0), end=(6.0, 0.0)); v\_dot — a PlotPoint \[green\] drawn in v\_axes (target='v\_left', x=\) Actions: - [05:19.14](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=319.13964583333336): head\_moment is shown on the screen, written out. - [05:19.14](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=319.13964583333336): m\_axes is shown on the screen, written out. - [05:25.92](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=325.9201458333333): m\_left is shown on the screen, drawn. ##### [05:29.016](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=329.0161458333333) Narration: It arrives at sixteen kilonewton metres under the load. On the right, forty eight minus eight x brings it back down, in another straight line, to zero at B. Board: lab\_left — a Tex \[text\] that says "For $0 \< x \< 4$:"; lab\_right — a Tex \[text\] that says "For $4 \< x \< 6$:"; v\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-12.0, 8.0), aspect=(7.9, 2.8)); v\_start — a Line \[yellow\] drawn in v\_axes (end=(0.0, 4.0)); v\_left — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)); v\_right — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 6.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(4.0, 4.0), end=(4.0, -8.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(6.0, -8.0), end=(6.0, 0.0)); v\_dot — a PlotPoint \[green\] drawn in v\_axes (target='v\_left', x=\); m\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-2.0, 20.0), aspect=(7.9, 2.8)); head\_moment — a Heading that says "Plotting the Moment"; m\_left — a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(0.0, 4.0)) Actions: - [05:30.177](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=330.1771458333334): m\_peak is shown on the screen, written out. - [05:37.294](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=337.29414583333335): m\_right is shown on the screen, drawn. ##### [05:40.425](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=340.42464583333333) Narration: For a point load, then, the moment diagram is a triangle, and its peak sits directly under the load. Board: lab\_left — a Tex \[text\] that says "For $0 \< x \< 4$:"; lab\_right — a Tex \[text\] that says "For $4 \< x \< 6$:"; v\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-12.0, 8.0), aspect=(7.9, 2.8)); v\_start — a Line \[yellow\] drawn in v\_axes (end=(0.0, 4.0)); v\_left — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)); v\_right — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 6.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(4.0, 4.0), end=(4.0, -8.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(6.0, -8.0), end=(6.0, 0.0)); v\_dot — a PlotPoint \[green\] drawn in v\_axes (target='v\_left', x=\); m\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-2.0, 20.0), aspect=(7.9, 2.8)); head\_moment — a Heading that says "Plotting the Moment"; m\_left — a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(0.0, 4.0)); m\_peak — a Point \[green\] labelled "16" drawn in m\_axes (location=(4.0, 16.0)); m\_right — a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(4.0, 6.0)) Actions: - [05:44.825](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=344.82514583333335): m\_peak is indicated — a transient flash. ##### [05:47.678](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=347.6776458333334) Narration: Put the two side by side and two things stare at you. Where the shear is constant, the moment is straight. And where the shear is bigger, the moment climbs faster. Board: Unchanged from the preceding beat in this scene. Actions: - [05:52.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=352.48414583333334): v\_left is emphasized. - [05:54.016](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=354.0161458333333): m\_left is emphasized. - [05:55.804](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=355.80414583333334): v\_left is no longer emphasized. - [05:55.804](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=355.80414583333334): m\_left is no longer emphasized. ##### [05:58.831](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=358.8311458333334) Narration: Look at the numbers. On the left the shear is four and the moment gains four units per metre. On the right the shear is minus eight and the moment loses eight per metre. The slope of the moment diagram is the shear. Board: Unchanged from the preceding beat in this scene. Actions: - [06:10.243](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=370.24314583333336): note\_slope is shown on the screen, written out. - [06:11.938](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=371.93814583333335): A box is drawn around note\_slope. ##### [06:13.305](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=373.30514583333337) Narration: And here is the one that earns its keep. The moment peaks exactly where the shear crosses the axis, which on this beam happens inside the vertical jump at the load. Board: lab\_left — a Tex \[text\] that says "For $0 \< x \< 4$:"; lab\_right — a Tex \[text\] that says "For $4 \< x \< 6$:"; v\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-12.0, 8.0), aspect=(7.9, 2.8)); v\_start — a Line \[yellow\] drawn in v\_axes (end=(0.0, 4.0)); v\_left — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)); v\_right — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 6.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(4.0, 4.0), end=(4.0, -8.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(6.0, -8.0), end=(6.0, 0.0)); v\_dot — a PlotPoint \[green\] drawn in v\_axes (target='v\_left', x=\); note\_slope — a Math \[text\] that says "$upright("slope of ") M = V$"; m\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-2.0, 20.0), aspect=(7.9, 2.8)); head\_moment — a Heading that says "Plotting the Moment"; m\_left — a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(0.0, 4.0)); m\_peak — a Point \[green\] labelled "16" drawn in m\_axes (location=(4.0, 16.0)); m\_right — a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(4.0, 6.0)) Actions: - [06:13.305](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=373.30514583333337): m\_dot is shown on the screen, written out. - [06:18.112](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=378.11214583333333): v\_dot is redrawn as the numbers it depends on change. - [06:18.112](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=378.11214583333333): m\_dot is redrawn as the numbers it depends on change. - [06:18.112](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=378.11214583333333): probe ticks to 0.5. - [06:18.112](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=378.11214583333333): probe\_m ticks to 0.5. - [06:21.722](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=381.72214583333334): v\_drop is indicated — a transient flash. ##### [06:23.495](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=383.49514583333337) Narration: We have not proved any of that. We have noticed it, on one beam. So let us take a completely different kind of load and see whether it survives. Board: lab\_left — a Tex \[text\] that says "For $0 \< x \< 4$:"; lab\_right — a Tex \[text\] that says "For $4 \< x \< 6$:"; v\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-12.0, 8.0), aspect=(7.9, 2.8)); v\_start — a Line \[yellow\] drawn in v\_axes (end=(0.0, 4.0)); v\_left — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)); v\_right — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 6.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(4.0, 4.0), end=(4.0, -8.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(6.0, -8.0), end=(6.0, 0.0)); v\_dot — a PlotPoint \[green\] drawn in v\_axes (target='v\_left', x=\); note\_slope — a Math \[text\] that says "$upright("slope of ") M = V$"; m\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-2.0, 20.0), aspect=(7.9, 2.8)); head\_moment — a Heading that says "Plotting the Moment"; m\_left — a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(0.0, 4.0)); m\_peak — a Point \[green\] labelled "16" drawn in m\_axes (location=(4.0, 16.0)); m\_right — a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(4.0, 6.0)); m\_dot — a PlotPoint \[green\] drawn in m\_axes (target='m\_left', x=\) Actions: - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): head\_moment is hidden from the screen — left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): lab\_left is hidden from the screen — left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): lab\_right is hidden from the screen — left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): left\_pair is hidden from the screen — left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): m\_axes is hidden from the screen — left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): m\_left is hidden from the screen — m\_axes left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): m\_peak is hidden from the screen — m\_axes left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): m\_right is hidden from the screen — m\_axes left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): m\_dot is hidden from the screen — m\_axes left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): note\_slope is hidden from the screen — left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): right\_pair is hidden from the screen — left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): v\_axes is hidden from the screen — left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): v\_start is hidden from the screen — v\_axes left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): v\_left is hidden from the screen — v\_axes left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): v\_right is hidden from the screen — v\_axes left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): v\_drop is hidden from the screen — v\_axes left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): v\_close is hidden from the screen — v\_axes left the board. - [06:33.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=393.00943750000005): v\_dot is hidden from the screen — v\_axes left the board. ### Scene 3: [A Distributed Load](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=394.05110416666673) Span: 06:34.051–09:25.493 (394.05110416666673s–565.4929791666667s). #### Objects - area\_claim: a Text \[text\] that says "The shaded area under the shear diagram, out to any position, is the bending moment there." - beam: a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)) - cut: a VariableNumber (initial\_value=2.2, format\_spec='.1f') - cut\_line: a Line \[yellow\] drawn in beam (start=(\, 0.8), end=(\, -0.95), dashed=True) - fb\_half: a Line \[gray\] labelled "frac(x, 2)" drawn in beam (start=(0.0, -4.3), end=((cut / 2.0), -4.3)) - fb\_line: a Line \[blue\] drawn in beam (start=(0.0, -3.1), end=(\, -3.1)) - fb\_m: a CurvedArrow \[yellow\] labelled "M" drawn in beam (start=((cut - 0.8), -3.62), end=((cut - 0.8), -2.58), bend=0.55) - fb\_ra: a Vector \[green\] labelled "12" drawn in beam (start=(0.0, -4.05), end=(0.0, -3.16)) - fb\_res: a Vector \[red\] labelled "w x" drawn in beam (start=((cut / 2.0), -2.15), end=((cut / 2.0), -3.04)) - fb\_v: a Vector \[yellow\] labelled "V" drawn in beam (start=(\, -3.1), end=(\, -3.98)) - fb\_x: a Line \[gray\] labelled "x" drawn in beam (start=(0.0, -4.72), end=(\, -4.72)) - fun\_m: a Math \[text\] that says "$M(x) = 12 x - 2 x^2$" - fun\_v: a Math \[text\] that says "$V(x) = 12 - 4 x$" - head\_beam: a Heading that says "Four Kilonewtons on Every Metre" - head\_plot: a Heading that says "A Ramp and a Parabola" - m\_axes: an Axes (x\_range=(0.0, 6.0), y\_range=(-2.0, 22.0), aspect=(7.9, 2.8)) - m\_peak: a Point \[green\] labelled "18" drawn in m\_axes (location=(3.0, 18.0)) - m\_plot: a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(0.0, 6.0)) - m\_rider: a PlotPoint \[green\] drawn in m\_axes (target='m\_plot', x=\) - moment\_work: a Derivation \[text\] that says "$sum M\_(upright("cut")) &= 0 \\ M - 12 x + (4 x) frac(x, 2) &= 0 \\ M(x) &= 12 x - 2 x^2$" - pin: a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35) - r\_a: a Vector \[green\] drawn in beam (start=(0.0, -1.8), end=(0.0, -0.72)) - r\_a\_name: a Math \[green\] that says "$12$" drawn in beam - r\_b: a Vector \[green\] drawn in beam (start=(6.0, -1.8), end=(6.0, -0.72)) - r\_b\_name: a Math \[green\] that says "$12$" drawn in beam - roller: a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35) - shear\_work: a Derivation \[text\] that says "$sum F\_y &= 0 \\ 12 - 4 x - V &= 0 \\ V(x) &= 12 - 4 x$" - span: a Line \[blue\] drawn in beam (end=(6.0, 0.0)) - sweep: a VariableNumber (initial\_value=0.5) - udl\_0: a Vector \[red\] drawn in beam (start=(0.0, 1.4), end=(0.0, 0.08)) - udl\_1: a Vector \[red\] drawn in beam (start=(1.0, 1.4), end=(1.0, 0.08)) - udl\_2: a Vector \[red\] drawn in beam (start=(2.0, 1.4), end=(2.0, 0.08)) - udl\_3: a Vector \[red\] drawn in beam (start=(3.0, 1.4), end=(3.0, 0.08)) - udl\_4: a Vector \[red\] drawn in beam (start=(4.0, 1.4), end=(4.0, 0.08)) - udl\_5: a Vector \[red\] drawn in beam (start=(5.0, 1.4), end=(5.0, 0.08)) - udl\_6: a Vector \[red\] drawn in beam (start=(6.0, 1.4), end=(6.0, 0.08)) - udl\_name: a Math \[red\] that says "$w = 4 thin upright("kN/m")$" drawn in beam - udl\_top: a Line \[red\] drawn in beam (start=(0.0, 1.45), end=(6.0, 1.45)) - v\_area: an AreaUnder \[green\] drawn in v\_axes (x\_range=(0.0, \), target='v\_plot') - v\_axes: an Axes (x\_range=(0.0, 6.0), y\_range=(-15.0, 15.0), aspect=(7.9, 2.8)) - v\_close: a Line \[yellow\] drawn in v\_axes (start=(6.0, -12.0), end=(6.0, 0.0)) - v\_plot: a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 6.0)) - v\_start: a Line \[yellow\] drawn in v\_axes (end=(0.0, 12.0)) - v\_zero: a Point \[green\] labelled "x = 3" drawn in v\_axes (location=(3.0, 0.0)) #### Beats ##### [06:34.051](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=394.05110416666673) Narration: Second beam. The same six metre span and the same supports, but instead of one point load it carries four kilonewtons on every metre of its length. A uniformly distributed load. Board: Empty. Actions: - [06:34.051](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=394.05110416666673): head\_beam is shown on the screen, written out. - [06:34.051](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=394.05110416666673): beam is shown on the screen, written out. - [06:36.361](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=396.36110416666673): span is shown on the screen, written out. - [06:36.461](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=396.46110416666676): pin is shown on the screen, written out. - [06:36.661](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=396.66110416666675): roller is shown on the screen, written out. - [06:39.647](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=399.64710416666674): udl\_top is shown on the screen, written out. - [06:39.747](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=399.74710416666676): udl\_0 is shown on the screen, written out. - [06:39.947](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=399.94710416666675): udl\_1 is shown on the screen, written out. - [06:40.247](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=400.24710416666676): udl\_2 is shown on the screen, written out. - [06:40.647](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=400.64710416666674): udl\_3 is shown on the screen, written out. - [06:41.147](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=401.14710416666674): udl\_4 is shown on the screen, written out. - [06:41.747](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=401.74710416666676): udl\_5 is shown on the screen, written out. - [06:42.447](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=402.44710416666675): udl\_6 is shown on the screen, written out. - [06:43.269](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=403.26910416666675): udl\_name is shown on the screen, written out. ##### [06:45.774](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=405.77360416666676) Narration: The total is four times six, twenty four kilonewtons, and by symmetry each support takes half. Twelve up at each end. Board: beam — a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)); head\_beam — a Heading that says "Four Kilonewtons on Every Metre"; span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35); roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35); udl\_top — a Line \[red\] drawn in beam (start=(0.0, 1.45), end=(6.0, 1.45)); udl\_0 — a Vector \[red\] drawn in beam (start=(0.0, 1.4), end=(0.0, 0.08)); udl\_1 — a Vector \[red\] drawn in beam (start=(1.0, 1.4), end=(1.0, 0.08)); udl\_2 — a Vector \[red\] drawn in beam (start=(2.0, 1.4), end=(2.0, 0.08)); udl\_3 — a Vector \[red\] drawn in beam (start=(3.0, 1.4), end=(3.0, 0.08)); udl\_4 — a Vector \[red\] drawn in beam (start=(4.0, 1.4), end=(4.0, 0.08)); udl\_5 — a Vector \[red\] drawn in beam (start=(5.0, 1.4), end=(5.0, 0.08)); udl\_6 — a Vector \[red\] drawn in beam (start=(6.0, 1.4), end=(6.0, 0.08)); udl\_name — a Math \[red\] that says "$w = 4 thin upright("kN/m")$" drawn in beam Actions: - [06:52.948](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=412.9481041666667): r\_a is shown on the screen, written out. - [06:53.148](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=413.1481041666667): r\_a\_name is shown on the screen, written out. - [06:53.548](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=413.54810416666675): r\_b is shown on the screen, written out. - [06:54.148](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=414.1481041666667): r\_b\_name is shown on the screen, written out. ##### [06:55.127](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=415.12710416666675) Narration: Cut it at x and keep the left piece, exactly as before. Here is the only new idea in this example. The load sitting on our piece is four kilonewtons per metre times x metres, and it acts through the middle of that stretch. Board: beam — a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)); head\_beam — a Heading that says "Four Kilonewtons on Every Metre"; span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35); roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35); udl\_top — a Line \[red\] drawn in beam (start=(0.0, 1.45), end=(6.0, 1.45)); udl\_0 — a Vector \[red\] drawn in beam (start=(0.0, 1.4), end=(0.0, 0.08)); udl\_1 — a Vector \[red\] drawn in beam (start=(1.0, 1.4), end=(1.0, 0.08)); udl\_2 — a Vector \[red\] drawn in beam (start=(2.0, 1.4), end=(2.0, 0.08)); udl\_3 — a Vector \[red\] drawn in beam (start=(3.0, 1.4), end=(3.0, 0.08)); udl\_4 — a Vector \[red\] drawn in beam (start=(4.0, 1.4), end=(4.0, 0.08)); udl\_5 — a Vector \[red\] drawn in beam (start=(5.0, 1.4), end=(5.0, 0.08)); udl\_6 — a Vector \[red\] drawn in beam (start=(6.0, 1.4), end=(6.0, 0.08)); udl\_name — a Math \[red\] that says "$w = 4 thin upright("kN/m")$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.8), end=(0.0, -0.72)); r\_a\_name — a Math \[green\] that says "$12$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.8), end=(6.0, -0.72)); r\_b\_name — a Math \[green\] that says "$12$" drawn in beam Actions: - [06:55.533](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=415.5331041666667): cut\_line is shown on the screen, written out. - [06:57.46](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=417.4601041666667): fb\_line is shown on the screen, written out. - [06:57.66](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=417.6601041666667): fb\_x is shown on the screen, written out. - [06:58.06](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=418.06010416666675): fb\_ra is shown on the screen, written out. - [07:6.586](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=426.58610416666676): fb\_res is shown on the screen, written out. - [07:8.92](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=428.92010416666676): fb\_half is shown on the screen, written out. - [07:9.22](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=429.2201041666667): fb\_v is shown on the screen, written out. - [07:9.72](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=429.7201041666667): fb\_m is shown on the screen, written out. ##### [07:11.006](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=431.0061041666667) Narration: Vertical equilibrium. Twelve up, four x down, V down. So the shear is twelve minus four x. It is not constant any more. It falls off at four per metre, which is exactly the load intensity. Board: beam — a Figure (x\_range=(-1.0, 7.4), y\_range=(-5.0, 2.6), aspect=(8.4, 7.6)); head\_beam — a Heading that says "Four Kilonewtons on Every Metre"; span — a Line \[blue\] drawn in beam (end=(6.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.32, -0.62), (0.32, -0.62)), fill\_opacity=0.35); roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.68, -0.62), (6.32, -0.62)), fill\_opacity=0.35); udl\_top — a Line \[red\] drawn in beam (start=(0.0, 1.45), end=(6.0, 1.45)); udl\_0 — a Vector \[red\] drawn in beam (start=(0.0, 1.4), end=(0.0, 0.08)); udl\_1 — a Vector \[red\] drawn in beam (start=(1.0, 1.4), end=(1.0, 0.08)); udl\_2 — a Vector \[red\] drawn in beam (start=(2.0, 1.4), end=(2.0, 0.08)); udl\_3 — a Vector \[red\] drawn in beam (start=(3.0, 1.4), end=(3.0, 0.08)); udl\_4 — a Vector \[red\] drawn in beam (start=(4.0, 1.4), end=(4.0, 0.08)); udl\_5 — a Vector \[red\] drawn in beam (start=(5.0, 1.4), end=(5.0, 0.08)); udl\_6 — a Vector \[red\] drawn in beam (start=(6.0, 1.4), end=(6.0, 0.08)); udl\_name — a Math \[red\] that says "$w = 4 thin upright("kN/m")$" drawn in beam; r\_a — a Vector \[green\] drawn in beam (start=(0.0, -1.8), end=(0.0, -0.72)); r\_a\_name — a Math \[green\] that says "$12$" drawn in beam; r\_b — a Vector \[green\] drawn in beam (start=(6.0, -1.8), end=(6.0, -0.72)); r\_b\_name — a Math \[green\] that says "$12$" drawn in beam; cut\_line — a Line \[yellow\] drawn in beam (start=(\, 0.8), end=(\, -0.95), dashed=True); fb\_line — a Line \[blue\] drawn in beam (start=(0.0, -3.1), end=(\, -3.1)); fb\_x — a Line \[gray\] labelled "x" drawn in beam (start=(0.0, -4.72), end=(\, -4.72)); fb\_ra — a Vector \[green\] labelled "12" drawn in beam (start=(0.0, -4.05), end=(0.0, -3.16)); fb\_res — a Vector \[red\] labelled "w x" drawn in beam (start=((cut / 2.0), -2.15), end=((cut / 2.0), -3.04)); fb\_half — a Line \[gray\] labelled "frac(x, 2)" drawn in beam (start=(0.0, -4.3), end=((cut / 2.0), -4.3)); fb\_v — a Vector \[yellow\] labelled "V" drawn in beam (start=(\, -3.1), end=(\, -3.98)); fb\_m — a CurvedArrow \[yellow\] labelled "M" drawn in beam (start=((cut - 0.8), -3.62), end=((cut - 0.8), -2.58), bend=0.55) Actions: - [07:11.354](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=431.35410416666673): beam moves to a new place on the board. - [07:11.354](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=431.35410416666673): shear\_work is shown on the screen, written out. - [07:15.58](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=435.58010416666673): shear\_work is shown on the screen, written out. - [07:17.542](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=437.5421041666667): shear\_work is shown on the screen, written out. ##### [07:26.688](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=446.68760416666674) Narration: Moments about the cut. The reaction gives twelve x. The load on our piece gives four x, times its lever arm of x over two, turning the other way. So the moment is twelve x minus two x squared. A parabola. Board: Unchanged from the preceding beat in this scene. Actions: - [07:26.85](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=446.8501041666667): moment\_work is shown on the screen, written out. - [07:33.155](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=453.1551041666667): moment\_work is shown on the screen, written out. - [07:40.411](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=460.41110416666675): moment\_work is shown on the screen, written out. ##### [07:41.887](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=461.88710416666675) Narration: Watch the free body grow. The resultant of the load grows with it, and its arrow slides out to stay at the middle of the piece. Board: Unchanged from the preceding beat in this scene. Actions: - [07:42.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=462.0091041666667): cut\_line is redrawn as the numbers it depends on change. - [07:42.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=462.0091041666667): fb\_line is redrawn as the numbers it depends on change. - [07:42.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=462.0091041666667): fb\_x is redrawn as the numbers it depends on change. - [07:42.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=462.0091041666667): fb\_res is redrawn as the numbers it depends on change. - [07:42.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=462.0091041666667): fb\_half is redrawn as the numbers it depends on change. - [07:42.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=462.0091041666667): fb\_v is redrawn as the numbers it depends on change. - [07:42.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=462.0091041666667): fb\_m is redrawn as the numbers it depends on change. - [07:42.009](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=462.0091041666667): cut ticks to 4.6. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): beam is hidden from the screen — left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): span is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): pin is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): roller is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): udl\_top is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): udl\_0 is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): udl\_1 is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): udl\_2 is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): udl\_3 is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): udl\_4 is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): udl\_5 is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): udl\_6 is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): udl\_name is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): r\_a is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): r\_a\_name is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): r\_b is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): r\_b\_name is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): cut\_line is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): fb\_line is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): fb\_x is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): fb\_ra is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): fb\_res is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): fb\_half is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): fb\_v is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): fb\_m is hidden from the screen — beam left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): head\_beam is hidden from the screen — left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): moment\_work is hidden from the screen — left the board. - [07:48.592](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=468.59210416666673): shear\_work is hidden from the screen — left the board. ##### [07:49.792](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=469.7921041666667) Narration: Plot the shear. Twelve minus four x is a straight ramp, starting at plus twelve at the pin and finishing at minus twelve at the roller, crossing the axis at the middle of the span. Board: Empty. Actions: - [07:49.792](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=469.7921041666667): head\_plot is shown on the screen, written out. - [07:49.792](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=469.7921041666667): v\_axes is shown on the screen, written out. - [07:50.094](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=470.09410416666674): fun\_v is shown on the screen, written out. - [07:53.577](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=473.57710416666674): v\_plot is shown on the screen, drawn. - [07:54.25](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=474.25010416666674): v\_start is shown on the screen, drawn. - [07:58.069](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=478.06910416666676): v\_close is shown on the screen, drawn. - [07:58.65](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=478.6501041666667): v\_zero is shown on the screen, written out. ##### [08:1.636](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=481.6356041666667) Narration: And the moment is a parabola, zero at both supports. By symmetry its top is at three metres: twelve times three, minus two times nine, eighteen kilonewton metres. Board: fun\_v — a Math \[text\] that says "$V(x) = 12 - 4 x$"; v\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-15.0, 15.0), aspect=(7.9, 2.8)); head\_plot — a Heading that says "A Ramp and a Parabola"; v\_start — a Line \[yellow\] drawn in v\_axes (end=(0.0, 12.0)); v\_plot — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 6.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(6.0, -12.0), end=(6.0, 0.0)); v\_zero — a Point \[green\] labelled "x = 3" drawn in v\_axes (location=(3.0, 0.0)) Actions: - [08:1.636](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=481.6356041666667): m\_axes is shown on the screen, written out. - [08:2.164](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=482.16410416666673): fun\_m is shown on the screen, written out. - [08:2.721](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=482.72110416666675): m\_plot is shown on the screen, drawn. - [08:11.348](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=491.34810416666676): m\_peak is shown on the screen, written out. ##### [08:14.014](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=494.0141041666667) Narration: Both of the things we noticed on the first beam survived, and they got sharper. The shear was flat when there was no load; now it slopes, at exactly minus the load intensity. The shear is straight, and the moment has gone up one degree to a parabola. Board: fun\_v — a Math \[text\] that says "$V(x) = 12 - 4 x$"; fun\_m — a Math \[text\] that says "$M(x) = 12 x - 2 x^2$"; v\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-15.0, 15.0), aspect=(7.9, 2.8)); m\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-2.0, 22.0), aspect=(7.9, 2.8)); head\_plot — a Heading that says "A Ramp and a Parabola"; v\_start — a Line \[yellow\] drawn in v\_axes (end=(0.0, 12.0)); v\_plot — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 6.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(6.0, -12.0), end=(6.0, 0.0)); v\_zero — a Point \[green\] labelled "x = 3" drawn in v\_axes (location=(3.0, 0.0)); m\_plot — a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(0.0, 6.0)); m\_peak — a Point \[green\] labelled "18" drawn in m\_axes (location=(3.0, 18.0)) Actions: - None. ##### [08:29.754](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=509.7536041666667) Narration: And the maximum moment is where the shear passes through zero. Not under the heaviest load, not at a support. Where the shear crosses the axis. Board: Unchanged from the preceding beat in this scene. Actions: - [08:31.971](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=511.97110416666675): v\_zero is indicated — a transient flash. - [08:37.416](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=517.4161041666667): m\_peak is indicated — a transient flash. ##### [08:39.793](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=519.7926041666667) Narration: There is a third thing hiding here, and it is the useful one. Shade the area under the shear diagram, from the left end out to some position x. Board: Unchanged from the preceding beat in this scene. Actions: - [08:44.901](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=524.9011041666668): v\_area is shown on the screen, faded in. - [08:45.301](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=525.3011041666667): m\_rider is shown on the screen, written out. - [08:48.593](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=528.5931041666668): area\_claim is shown on the screen, written out. ##### [08:50.482](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=530.4816041666668) Narration: Now watch the shaded area and the moment together. At three metres the shaded triangle is half of three times twelve, which is eighteen. And the moment there is eighteen. Board: fun\_v — a Math \[text\] that says "$V(x) = 12 - 4 x$"; fun\_m — a Math \[text\] that says "$M(x) = 12 x - 2 x^2$"; area\_claim — a Text \[text\] that says "The shaded area under the shear diagram, out to any position, is the bending moment there."; v\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-15.0, 15.0), aspect=(7.9, 2.8)); m\_axes — an Axes (x\_range=(0.0, 6.0), y\_range=(-2.0, 22.0), aspect=(7.9, 2.8)); head\_plot — a Heading that says "A Ramp and a Parabola"; v\_start — a Line \[yellow\] drawn in v\_axes (end=(0.0, 12.0)); v\_plot — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 6.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(6.0, -12.0), end=(6.0, 0.0)); v\_zero — a Point \[green\] labelled "x = 3" drawn in v\_axes (location=(3.0, 0.0)); m\_plot — a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(0.0, 6.0)); m\_peak — a Point \[green\] labelled "18" drawn in m\_axes (location=(3.0, 18.0)); v\_area — an AreaUnder \[green\] drawn in v\_axes (x\_range=(0.0, \), target='v\_plot'); m\_rider — a PlotPoint \[green\] drawn in m\_axes (target='m\_plot', x=\) Actions: - [08:51.132](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=531.1321041666667): v\_area is redrawn as the numbers it depends on change. - [08:51.132](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=531.1321041666667): m\_rider is redrawn as the numbers it depends on change. - [08:51.132](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=531.1321041666667): sweep ticks to 3.0. ##### [09:2.065](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=542.0646041666668) Narration: Keep going. Past the middle the shear is negative, so the new shading counts against you, and by the far end the positive and negative areas have cancelled exactly. The moment is back to zero. Board: Unchanged from the preceding beat in this scene. Actions: - [09:2.413](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=542.4131041666667): v\_area is redrawn as the numbers it depends on change. - [09:2.413](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=542.4131041666667): m\_rider is redrawn as the numbers it depends on change. - [09:2.413](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=542.4131041666667): sweep ticks to 5.9. ##### [09:15.32](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=555.3201041666667) Narration: So the area under the shear, between two points, is the change in the moment between those two points. We have now seen it twice. Time to prove it once. Board: Unchanged from the preceding beat in this scene. Actions: - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): area\_claim is hidden from the screen — left the board. - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): fun\_m is hidden from the screen — left the board. - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): fun\_v is hidden from the screen — left the board. - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): head\_plot is hidden from the screen — left the board. - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): m\_axes is hidden from the screen — left the board. - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): m\_plot is hidden from the screen — m\_axes left the board. - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): m\_peak is hidden from the screen — m\_axes left the board. - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): m\_rider is hidden from the screen — m\_axes left the board. - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): v\_axes is hidden from the screen — left the board. - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): v\_start is hidden from the screen — v\_axes left the board. - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): v\_plot is hidden from the screen — v\_axes left the board. - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): v\_close is hidden from the screen — v\_axes left the board. - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): v\_zero is hidden from the screen — v\_axes left the board. - [09:24.451](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=564.4513125000001): v\_area is hidden from the screen — v\_axes left the board. ### Scene 4: [Load, Shear and Moment](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=565.4929791666667) Span: 09:25.493–12:36.003 (565.4929791666667s–756.0034166666667s). #### Objects - consequences: a Table \[text\] that says "On a stretch Shear Moment no load constant straight uniform $w$ straight parabola point force $P$ jumps by $P$ kink couple $M\_0$ unchanged jumps by $M\_0$" (rows=(('On a stretch', 'Shear', 'Moment'), ('no load', 'constant', '…, header=True) - dx\_dim: a Line \[gray\] labelled "dif x" drawn in elem (start=(1.0, -2.0), end=(3.0, -2.0)) - elem: a Figure (x\_range=(-1.4, 4.8), y\_range=(-2.7, 3.2), aspect=(6.2, 5.9)) - force\_work: a Derivation \[text\] that says "$sum F\_y &= 0 \\ V - w thin dif x - (V + dif V) &= 0 \\ dif V &= -w thin dif x \\ frac(dif V, dif x) &= -w$" - head\_int: a Heading that says "The Same Two Statements, Integrated" - head\_max: a Heading that says "Where the Maximum Moment Is" - head\_slice: a Heading that says "A Slice of Beam, of Length $d x$" - integrals: a Derivation \[text\] that says "$V\_2 - V\_1 &= - integral\_(x\_1)^(x\_2) w thin dif x \\ M\_2 - M\_1 &= integral\_(x\_1)^(x\_2) V thin dif x$" - load\_a: a Vector \[red\] drawn in elem (start=(1.0, 1.95), end=(1.0, 0.88)) - load\_b: a Vector \[red\] drawn in elem (start=(2.0, 1.95), end=(2.0, 0.88)) - load\_c: a Vector \[red\] drawn in elem (start=(3.0, 1.95), end=(3.0, 0.88)) - load\_name: a Math \[red\] that says "$w$" drawn in elem - load\_top: a Line \[red\] drawn in elem (start=(1.0, 2.0), end=(3.0, 2.0)) - m\_in: a CurvedArrow \[yellow\] labelled "M" drawn in elem (start=(0.3, 1.1), end=(0.3, -0.4)) - m\_out: a CurvedArrow \[yellow\] labelled "M + dif M" drawn in elem (start=(3.7, -0.4), end=(3.7, 1.1)) - moment\_work: a Derivation \[text\] that says "$sum M &= 0 \\ dif M - V thin dif x + frac(w (dif x)^2, 2) &= 0 \\ dif M &= V thin dif x \\ frac(dif M, dif x) &= V$" - note\_max: a Text \[text\] that says "The bending moment is stationary wherever the shear crosses the axis. Check the ends and every jump in the shear as well." - rule\_max: a Math \[text\] that says "$frac(dif M, dif x) = V = 0$" - slab: a Polygon \[blue\] drawn in elem (vertices=((1.0, 0.0), (3.0, 0.0), (3.0, 0.8), (1.0, 0.8)), fill\_opacity=0.25) - v\_in: a Vector \[yellow\] labelled "V" drawn in elem (start=(1.0, -1.2), end=(1.0, -0.05)) - v\_out: a Vector \[yellow\] labelled "V + dif V" drawn in elem (start=(3.0, -0.05), end=(3.0, -1.2)) #### Beats ##### [09:25.493](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=565.4929791666667) Narration: Both beams told us the same three things, so let us prove them once, for any beam and any load. Take a slice of beam of length d x, out of a beam carrying a distributed load w. Board: Empty. Actions: - [09:25.493](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=565.4929791666667): head\_slice is shown on the screen, written out. - [09:25.493](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=565.4929791666667): elem is shown on the screen, written out. - [09:30.834](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=570.8339791666667): load\_top is shown on the screen, written out. - [09:30.934](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=570.9339791666667): load\_a is shown on the screen, written out. - [09:31.134](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=571.1339791666667): load\_b is shown on the screen, written out. - [09:31.434](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=571.4339791666667): load\_c is shown on the screen, written out. - [09:31.834](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=571.8339791666667): load\_name is shown on the screen, written out. - [09:32.215](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=572.2149791666667): slab is shown on the screen, written out. - [09:32.415](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=572.4149791666667): dx\_dim is shown on the screen, written out. ##### [09:38.736](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=578.7359791666668) Narration: On its left face the rest of the beam pushes with a shear V and a moment M. On its right face, a distance d x along, both have had a chance to change: V plus d V, and M plus d M. Board: elem — a Figure (x\_range=(-1.4, 4.8), y\_range=(-2.7, 3.2), aspect=(6.2, 5.9)); head\_slice — a Heading that says "A Slice of Beam, of Length $d x$"; slab — a Polygon \[blue\] drawn in elem (vertices=((1.0, 0.0), (3.0, 0.0), (3.0, 0.8), (1.0, 0.8)), fill\_opacity=0.25); dx\_dim — a Line \[gray\] labelled "dif x" drawn in elem (start=(1.0, -2.0), end=(3.0, -2.0)); load\_top — a Line \[red\] drawn in elem (start=(1.0, 2.0), end=(3.0, 2.0)); load\_a — a Vector \[red\] drawn in elem (start=(1.0, 1.95), end=(1.0, 0.88)); load\_b — a Vector \[red\] drawn in elem (start=(2.0, 1.95), end=(2.0, 0.88)); load\_c — a Vector \[red\] drawn in elem (start=(3.0, 1.95), end=(3.0, 0.88)); load\_name — a Math \[red\] that says "$w$" drawn in elem Actions: - [09:41.535](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=581.5349791666667): v\_in is shown on the screen, written out. - [09:42.963](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=582.9629791666667): m\_in is shown on the screen, written out. - [09:48.268](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=588.2679791666667): v\_out is shown on the screen, written out. - [09:48.668](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=588.6679791666667): m\_out is shown on the screen, written out. ##### [09:52.891](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=592.8914791666667) Narration: Vertical equilibrium of the slice. V up on the left, the load w d x pressing down on the top, and V plus d V down on the right. The two V's cancel. Board: elem — a Figure (x\_range=(-1.4, 4.8), y\_range=(-2.7, 3.2), aspect=(6.2, 5.9)); head\_slice — a Heading that says "A Slice of Beam, of Length $d x$"; slab — a Polygon \[blue\] drawn in elem (vertices=((1.0, 0.0), (3.0, 0.0), (3.0, 0.8), (1.0, 0.8)), fill\_opacity=0.25); dx\_dim — a Line \[gray\] labelled "dif x" drawn in elem (start=(1.0, -2.0), end=(3.0, -2.0)); load\_top — a Line \[red\] drawn in elem (start=(1.0, 2.0), end=(3.0, 2.0)); load\_a — a Vector \[red\] drawn in elem (start=(1.0, 1.95), end=(1.0, 0.88)); load\_b — a Vector \[red\] drawn in elem (start=(2.0, 1.95), end=(2.0, 0.88)); load\_c — a Vector \[red\] drawn in elem (start=(3.0, 1.95), end=(3.0, 0.88)); load\_name — a Math \[red\] that says "$w$" drawn in elem; v\_in — a Vector \[yellow\] labelled "V" drawn in elem (start=(1.0, -1.2), end=(1.0, -0.05)); m\_in — a CurvedArrow \[yellow\] labelled "M" drawn in elem (start=(0.3, 1.1), end=(0.3, -0.4)); v\_out — a Vector \[yellow\] labelled "V + dif V" drawn in elem (start=(3.0, -0.05), end=(3.0, -1.2)); m\_out — a CurvedArrow \[yellow\] labelled "M + dif M" drawn in elem (start=(3.7, -0.4), end=(3.7, 1.1)) Actions: - [09:53.048](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=593.0479791666667): elem moves to a new place on the board. - [09:53.048](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=593.0479791666667): force\_work is shown on the screen, written out. - [09:58.574](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=598.5739791666667): force\_work is shown on the screen, written out. - [10:3.206](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=603.2059791666667): force\_work (the "V" part) is slashed through — it cancels. - [10:3.206](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=603.2059791666667): force\_work (the "V#2" part) is slashed through — it cancels. ##### [10:4.758](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=604.7584791666667) Narration: What is left is d V equals minus w d x. Divide through by d x, and the derivative of the shear with respect to position is minus the load intensity. Board: Unchanged from the preceding beat in this scene. Actions: - [10:5.443](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=605.4429791666668): force\_work is shown on the screen, written out. - [10:11.597](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=611.5969791666668): force\_work is shown on the screen, written out. ##### [10:16.701](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=616.7014791666667) Narration: Read that as a picture. Where there is no load, the shear is flat. Where the load is uniform, the shear is a straight ramp sloping downward. And where the load is heavy, the shear falls steeply. Board: Unchanged from the preceding beat in this scene. Actions: - [10:17.723](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=617.7229791666667): force\_work (the "-w" part) is emphasized. - [10:29.02](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=629.0199791666666): force\_work (the "-w" part) is no longer emphasized. ##### [10:30.758](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=630.7579791666667) Narration: Now take moments about the right hand face. The two moments oppose each other and leave d M. The shear on the left face acts a distance d x away. And the load has a lever arm of half d x. Board: Unchanged from the preceding beat in this scene. Actions: - [10:31.698](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=631.6979791666668): moment\_work is shown on the screen, written out. - [10:41.218](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=641.2179791666667): moment\_work is shown on the screen, written out. ##### [10:43.698](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=643.6984791666667) Narration: That last term has d x squared in it. As the slice shrinks it dies away faster than everything else, so it goes. What survives is d M equals V d x. Board: Unchanged from the preceding beat in this scene. Actions: - [10:50.282](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=650.2819791666667): moment\_work (the "frac(w (dif x)^2, 2)" part) is slashed through — it cancels. - [10:51.675](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=651.6749791666667): moment\_work is shown on the screen, written out. ##### [10:55.479](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=655.4794791666667) Narration: The derivative of the moment is the shear. The slope of the moment diagram, at any point at all, is the height of the shear diagram at that same point. That is the relation we kept noticing. Board: Unchanged from the preceding beat in this scene. Actions: - [10:55.99](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=655.9899791666667): moment\_work is shown on the screen, written out. - [11:2.085](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=662.0849791666667): A box is drawn around moment\_work. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): elem is hidden from the screen — left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): slab is hidden from the screen — elem left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): dx\_dim is hidden from the screen — elem left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): load\_top is hidden from the screen — elem left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): load\_a is hidden from the screen — elem left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): load\_b is hidden from the screen — elem left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): load\_c is hidden from the screen — elem left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): load\_name is hidden from the screen — elem left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): v\_in is hidden from the screen — elem left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): m\_in is hidden from the screen — elem left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): v\_out is hidden from the screen — elem left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): m\_out is hidden from the screen — elem left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): force\_work is hidden from the screen — left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): head\_slice is hidden from the screen — left the board. - [11:7.484](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=667.4839791666667): moment\_work is hidden from the screen — left the board. ##### [11:8.684](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=668.6839791666666) Narration: Integrate the two of them and out comes the other half of the folklore. The change in shear between two sections is minus the area under the load diagram between them. Board: Empty. Actions: - [11:8.684](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=668.6839791666666): head\_int is shown on the screen, written out. - [11:13.224](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=673.2239791666667): integrals is shown on the screen, written out. ##### [11:19.21](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=679.2104791666667) Narration: And the change in moment between two sections is the area under the shear diagram between them. Exactly the shaded triangle we were watching a moment ago. Board: head\_int — a Heading that says "The Same Two Statements, Integrated" Actions: - [11:19.721](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=679.7209791666667): integrals is shown on the screen, written out. ##### [11:29.11](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=689.1104791666667) Narration: Now every rule you were ever handed is a line in this table. No load: the shear is constant and the moment is straight. A uniform load: the shear is straight and the moment is a parabola. Each one is a degree higher than the last. Board: Unchanged from the preceding beat in this scene. Actions: - [11:32.28](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=692.2799791666667): consequences is shown on the screen, written out. - [11:33.093](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=693.0929791666667): consequences is shown on the screen, written out. - [11:37.133](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=697.1329791666667): consequences is shown on the screen, written out. ##### [11:44.734](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=704.7339791666667) Narration: A point force is an enormous load intensity over no length at all. Its area is finite, so the shear jumps by the size of the force, and the moment, whose slope is the shear, simply kinks. Board: Unchanged from the preceding beat in this scene. Actions: - [11:51.514](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=711.5139791666668): consequences is shown on the screen, written out. ##### [11:57.315](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=717.3149791666667) Narration: And a concentrated couple contributes nothing to vertical equilibrium, so the shear does not notice it at all. It sits in the moment equation instead, so the moment jumps by the size of the couple. That is the line people get wrong most often. Board: Unchanged from the preceding beat in this scene. Actions: - [12:6.893](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=726.8929791666667): consequences is shown on the screen, written out. - [12:7.393](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=727.3929791666667): consequences (the "row=5" part) is emphasized. - [12:9.424](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=729.4239791666666): consequences (the "row=5" part) is no longer emphasized. - [12:11.572](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=731.5719791666667): consequences is hidden from the screen — left the board. - [12:11.572](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=731.5719791666667): head\_int is hidden from the screen — left the board. - [12:11.572](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=731.5719791666667): integrals is hidden from the screen — left the board. ##### [12:12.172](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=732.1719791666667) Narration: And finally the sentence you actually use. The moment is stationary where its derivative vanishes, and its derivative is the shear. So the biggest moment sits where the shear diagram crosses the axis, or at a jump that carries it across. Board: Empty. Actions: - [12:12.172](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=732.1719791666667): head\_max is shown on the screen, written out. - [12:16.34](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=736.3399791666667): rule\_max is shown on the screen, written out. - [12:23.759](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=743.7589791666667): note\_max is shown on the screen, written out. - [12:25.721](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=745.7209791666667): A box is drawn around rule\_max. ##### [12:28.306](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=748.3064791666667) Narration: Two derivatives and two areas. Everything else about these diagrams is a consequence of them. Board: rule\_max — a Math \[text\] that says "$frac(dif M, dif x) = V = 0$"; note\_max — a Text \[text\] that says "The bending moment is stationary wherever the shear crosses the axis. Check the ends and every jump in the shear as well."; head\_max — a Heading that says "Where the Maximum Moment Is" Actions: - [12:34.962](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=754.96175): head\_max is hidden from the screen — left the board. - [12:34.962](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=754.96175): note\_max is hidden from the screen — left the board. - [12:34.962](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=754.96175): rule\_max is hidden from the screen — left the board. ### Scene 5: [An Overhang and a Couple](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=756.0034166666667) Span: 12:36.003–15:45 (756.0034166666667s–944.9995833333334s). #### Objects - beam: a Figure (x\_range=(-1.0, 9.4), y\_range=(-2.6, 2.7), aspect=(10.4, 5.3)) - couple: a CurvedArrow \[red\] drawn in beam (start=(5.55, 0.15), end=(4.45, 0.15), bend=0.9) - couple\_mark: a Line \[gray\] labelled "M\_0" drawn in v\_axes (start=(5.0, -6.5), end=(5.0, 6.5), dashed=True) - couple\_name: a Math \[red\] that says "$M\_0 = 8$" drawn in beam - fact\_1: a Text \[text\] that says "A couple leaves the shear untouched and jumps the moment." - fact\_2: a Text \[text\] that says "Over the overhang the moment is negative, so the beam is in tension along its top face." - head\_beam: a Heading that says "An Overhang, and a Couple at Five Metres" - head\_build: a Heading that says "Built From Jumps and Areas" - load\_1: a Vector \[red\] drawn in beam (start=(2.0, 1.7), end=(2.0, 0.08)) - load\_1\_name: a Math \[red\] that says "$12$" drawn in beam - load\_2: a Vector \[red\] drawn in beam (start=(8.0, 1.7), end=(8.0, 0.08)) - load\_2\_name: a Math \[red\] that says "$4$" drawn in beam - m\_axes: an Axes (x\_range=(0.0, 8.0), y\_range=(-11.0, 20.0), aspect=(9.0, 2.7)) - m\_jump: a Line \[magenta\] drawn in m\_axes (start=(5.0, 4.0), end=(5.0, -4.0)) - m\_s1: a Line \[magenta\] drawn in m\_axes (end=(2.0, 16.0)) - m\_s2: a Line \[magenta\] drawn in m\_axes (start=(2.0, 16.0), end=(5.0, 4.0)) - m\_s3: a Line \[magenta\] drawn in m\_axes (start=(5.0, -4.0), end=(6.0, -8.0)) - m\_s4: a Line \[magenta\] drawn in m\_axes (start=(6.0, -8.0), end=(8.0, 0.0)) - name\_a: a Math \[text\] that says "$A$" drawn in beam - name\_b: a Math \[text\] that says "$B$" drawn in beam - pin: a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.34, -0.66), (0.34, -0.66)), fill\_opacity=0.35) - r\_a: a Vector \[green\] drawn in beam (start=(0.0, -1.9), end=(0.0, -0.76)) - r\_a\_name: a Math \[green\] that says "$8$" drawn in beam - r\_b: a Vector \[green\] drawn in beam (start=(6.0, -1.9), end=(6.0, -0.76)) - r\_b\_name: a Math \[green\] that says "$8$" drawn in beam - reactions: a Derivation \[text\] that says "$sum M\_A &= 0 \\ 6 R\_B &= 12 (2) + 4 (8) - 8 \\ &= 48 \\ R\_B &= 8 thin upright("kN") \\ R\_A &= 16 - 8 = 8 thin upright("kN")$" - roller: a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.66, -0.66), (6.34, -0.66)), fill\_opacity=0.35) - span: a Line \[blue\] drawn in beam (end=(8.0, 0.0)) - v\_axes: an Axes (x\_range=(0.0, 8.0), y\_range=(-8.0, 11.0), aspect=(9.0, 2.7)) - v\_close: a Line \[yellow\] drawn in v\_axes (start=(8.0, 4.0), end=(8.0, 0.0)) - v\_drop: a Line \[yellow\] drawn in v\_axes (start=(2.0, 8.0), end=(2.0, -4.0)) - v\_s1: a Line \[yellow\] drawn in v\_axes (start=(0.0, 8.0), end=(2.0, 8.0)) - v\_s2: a Line \[yellow\] drawn in v\_axes (start=(2.0, -4.0), end=(6.0, -4.0)) - v\_s3: a Line \[yellow\] drawn in v\_axes (start=(6.0, 4.0), end=(8.0, 4.0)) - v\_up\_a: a Line \[yellow\] drawn in v\_axes (end=(0.0, 8.0)) - v\_up\_b: a Line \[yellow\] drawn in v\_axes (start=(6.0, -4.0), end=(6.0, 4.0)) #### Beats ##### [12:36.003](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=756.0034166666667) Narration: Third beam, and it has both of the things people find awkward. It runs eight metres, but the supports are at zero and at six, so the last two metres hang past the roller. Board: Empty. Actions: - [12:36.003](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=756.0034166666667): head\_beam is shown on the screen, written out. - [12:36.003](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=756.0034166666667): beam is shown on the screen, written out. - [12:40.09](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=760.0904166666667): span is shown on the screen, written out. - [12:41.448](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=761.4484166666667): pin is shown on the screen, written out. - [12:41.548](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=761.5484166666666): name\_a is shown on the screen, written out. - [12:41.748](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=761.7484166666667): roller is shown on the screen, written out. - [12:42.048](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=762.0484166666666): name\_b is shown on the screen, written out. ##### [12:46.832](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=766.8319166666666) Narration: The loads are twelve kilonewtons down at two metres, four kilonewtons down at the free tip, and a couple of eight kilonewton metres applied at five metres, twisting counterclockwise. Board: beam — a Figure (x\_range=(-1.0, 9.4), y\_range=(-2.6, 2.7), aspect=(10.4, 5.3)); head\_beam — a Heading that says "An Overhang, and a Couple at Five Metres"; span — a Line \[blue\] drawn in beam (end=(8.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.34, -0.66), (0.34, -0.66)), fill\_opacity=0.35); name\_a — a Math \[text\] that says "$A$" drawn in beam; roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.66, -0.66), (6.34, -0.66)), fill\_opacity=0.35); name\_b — a Math \[text\] that says "$B$" drawn in beam Actions: - [12:47.737](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=767.7374166666667): load\_1 is shown on the screen, written out. - [12:47.937](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=767.9374166666666): load\_1\_name is shown on the screen, written out. - [12:50.036](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=770.0364166666667): load\_2 is shown on the screen, written out. - [12:50.236](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=770.2364166666666): load\_2\_name is shown on the screen, written out. - [12:52.381](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=772.3814166666667): couple is shown on the screen, written out. - [12:52.681](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=772.6814166666667): couple\_name is shown on the screen, written out. ##### [12:57.3](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=777.3004166666667) Narration: Reactions first. Moments about A. The twelve acts at two, the four acts at eight, the reaction at B acts at six, and the couple goes straight into the sum as eight. It has no lever arm at all. A couple is the same about every point. Board: beam — a Figure (x\_range=(-1.0, 9.4), y\_range=(-2.6, 2.7), aspect=(10.4, 5.3)); head\_beam — a Heading that says "An Overhang, and a Couple at Five Metres"; span — a Line \[blue\] drawn in beam (end=(8.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.34, -0.66), (0.34, -0.66)), fill\_opacity=0.35); name\_a — a Math \[text\] that says "$A$" drawn in beam; roller — a Polygon \[gray\] drawn in beam (vertices=((6.0, 0.0), (5.66, -0.66), (6.34, -0.66)), fill\_opacity=0.35); name\_b — a Math \[text\] that says "$B$" drawn in beam; load\_1 — a Vector \[red\] drawn in beam (start=(2.0, 1.7), end=(2.0, 0.08)); load\_1\_name — a Math \[red\] that says "$12$" drawn in beam; load\_2 — a Vector \[red\] drawn in beam (start=(8.0, 1.7), end=(8.0, 0.08)); load\_2\_name — a Math \[red\] that says "$4$" drawn in beam; couple — a CurvedArrow \[red\] drawn in beam (start=(5.55, 0.15), end=(4.45, 0.15), bend=0.9); couple\_name — a Math \[red\] that says "$M\_0 = 8$" drawn in beam Actions: - [12:59.529](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=779.5294166666666): beam moves to a new place on the board. - [12:59.529](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=779.5294166666666): reactions is shown on the screen, written out. - [13:6.495](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=786.4954166666666): reactions is shown on the screen, written out. - [13:9.246](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=789.2464166666666): reactions (the "- 8" part) is emphasized. - [13:11.824](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=791.8244166666667): reactions (the "- 8" part) is no longer emphasized. ##### [13:13.504](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=793.5039166666667) Narration: That comes to forty eight, so the reaction at B is eight kilonewtons, and vertical equilibrium then leaves eight kilonewtons at A as well. Both supports push upward. Board: Unchanged from the preceding beat in this scene. Actions: - [13:14.432](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=794.4324166666667): reactions is shown on the screen, written out. - [13:15.593](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=795.5934166666667): reactions is shown on the screen, written out. - [13:15.893](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=795.8934166666667): r\_b is shown on the screen, written out. - [13:16.493](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=796.4934166666667): r\_b\_name is shown on the screen, written out. - [13:18.031](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=798.0314166666667): reactions is shown on the screen, written out. - [13:18.331](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=798.3314166666667): r\_a is shown on the screen, written out. - [13:18.931](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=798.9314166666667): r\_a\_name is shown on the screen, written out. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): beam is hidden from the screen — left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): span is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): pin is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): name\_a is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): roller is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): name\_b is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): load\_1 is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): load\_1\_name is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): load\_2 is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): load\_2\_name is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): couple is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): couple\_name is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): r\_b is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): r\_b\_name is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): r\_a is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): r\_a\_name is hidden from the screen — beam left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): head\_beam is hidden from the screen — left the board. - [13:23.976](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=803.9764166666666): reactions is hidden from the screen — left the board. ##### [13:24.576](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=804.5764166666667) Narration: Now build both diagrams without cutting anything, using only what we proved. Start at zero off the left end. At A the reaction pushes up eight, so the shear jumps to plus eight, and with no load between there and the twelve, it runs flat. Board: Empty. Actions: - [13:24.576](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=804.5764166666667): head\_build is shown on the screen, written out. - [13:24.576](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=804.5764166666667): v\_axes is shown on the screen, written out. - [13:35.617](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=815.6174166666667): v\_up\_a is shown on the screen, drawn. - [13:39.553](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=819.5534166666666): v\_s1 is shown on the screen, drawn. ##### [13:40.942](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=820.9424166666666) Narration: At two metres the twelve kilonewton load drops it by twelve, from plus eight to minus four. Flat again after that. There is no distributed load anywhere on this beam. Board: v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-8.0, 11.0), aspect=(9.0, 2.7)); head\_build — a Heading that says "Built From Jumps and Areas"; v\_up\_a — a Line \[yellow\] drawn in v\_axes (end=(0.0, 8.0)); v\_s1 — a Line \[yellow\] drawn in v\_axes (start=(0.0, 8.0), end=(2.0, 8.0)) Actions: - [13:43.473](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=823.4734166666667): v\_drop is shown on the screen, drawn. - [13:47.246](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=827.2464166666666): v\_s2 is shown on the screen, drawn. ##### [13:52.49](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=832.4904166666666) Narration: Here is where the couple acts, at five metres, and the shear passes straight through it without so much as a flinch. There is no vertical force in a couple, so there is nothing for the shear to notice. Board: v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-8.0, 11.0), aspect=(9.0, 2.7)); head\_build — a Heading that says "Built From Jumps and Areas"; v\_up\_a — a Line \[yellow\] drawn in v\_axes (end=(0.0, 8.0)); v\_s1 — a Line \[yellow\] drawn in v\_axes (start=(0.0, 8.0), end=(2.0, 8.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(2.0, 8.0), end=(2.0, -4.0)); v\_s2 — a Line \[yellow\] drawn in v\_axes (start=(2.0, -4.0), end=(6.0, -4.0)) Actions: - [13:54.522](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=834.5224166666667): couple\_mark is shown on the screen, drawn. - [14:1.848](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=841.8484166666667): v\_s2 is indicated — a transient flash. ##### [14:4.166](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=844.1664166666667) Narration: At B the reaction adds eight, taking it from minus four up to plus four. It runs flat along the overhang, and the four kilonewton load at the tip brings it home to zero. The diagram closes. Board: v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-8.0, 11.0), aspect=(9.0, 2.7)); head\_build — a Heading that says "Built From Jumps and Areas"; v\_up\_a — a Line \[yellow\] drawn in v\_axes (end=(0.0, 8.0)); v\_s1 — a Line \[yellow\] drawn in v\_axes (start=(0.0, 8.0), end=(2.0, 8.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(2.0, 8.0), end=(2.0, -4.0)); v\_s2 — a Line \[yellow\] drawn in v\_axes (start=(2.0, -4.0), end=(6.0, -4.0)); couple\_mark — a Line \[gray\] labelled "M\_0" drawn in v\_axes (start=(5.0, -6.5), end=(5.0, 6.5), dashed=True) Actions: - [14:5.571](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=845.5714166666667): v\_up\_b is shown on the screen, drawn. - [14:10.343](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=850.3434166666667): v\_s3 is shown on the screen, drawn. - [14:14.487](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=854.4874166666667): v\_close is shown on the screen, drawn. ##### [14:18.339](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=858.3389166666667) Narration: Now the moment, and every piece of it is an area under that shear. From A out to two metres, a rectangle eight high and two wide: sixteen. So the moment climbs in a straight line to sixteen kilonewton metres. Board: v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-8.0, 11.0), aspect=(9.0, 2.7)); head\_build — a Heading that says "Built From Jumps and Areas"; v\_up\_a — a Line \[yellow\] drawn in v\_axes (end=(0.0, 8.0)); v\_s1 — a Line \[yellow\] drawn in v\_axes (start=(0.0, 8.0), end=(2.0, 8.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(2.0, 8.0), end=(2.0, -4.0)); v\_s2 — a Line \[yellow\] drawn in v\_axes (start=(2.0, -4.0), end=(6.0, -4.0)); couple\_mark — a Line \[gray\] labelled "M\_0" drawn in v\_axes (start=(5.0, -6.5), end=(5.0, 6.5), dashed=True); v\_up\_b — a Line \[yellow\] drawn in v\_axes (start=(6.0, -4.0), end=(6.0, 4.0)); v\_s3 — a Line \[yellow\] drawn in v\_axes (start=(6.0, 4.0), end=(8.0, 4.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(8.0, 4.0), end=(8.0, 0.0)) Actions: - [14:18.339](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=858.3389166666667): v\_axes moves to a new place on the board. - [14:18.339](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=858.3389166666667): m\_axes is shown on the screen, written out. - [14:29.843](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=869.8434166666667): m\_s1 is shown on the screen, drawn. ##### [14:33.81](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=873.8104166666667) Narration: From two to five the shear is minus four, so the moment falls at four per metre. Three metres of that is minus twelve, taking it from sixteen down to four. Board: v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-8.0, 11.0), aspect=(9.0, 2.7)); m\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-11.0, 20.0), aspect=(9.0, 2.7)); head\_build — a Heading that says "Built From Jumps and Areas"; v\_up\_a — a Line \[yellow\] drawn in v\_axes (end=(0.0, 8.0)); v\_s1 — a Line \[yellow\] drawn in v\_axes (start=(0.0, 8.0), end=(2.0, 8.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(2.0, 8.0), end=(2.0, -4.0)); v\_s2 — a Line \[yellow\] drawn in v\_axes (start=(2.0, -4.0), end=(6.0, -4.0)); couple\_mark — a Line \[gray\] labelled "M\_0" drawn in v\_axes (start=(5.0, -6.5), end=(5.0, 6.5), dashed=True); v\_up\_b — a Line \[yellow\] drawn in v\_axes (start=(6.0, -4.0), end=(6.0, 4.0)); v\_s3 — a Line \[yellow\] drawn in v\_axes (start=(6.0, 4.0), end=(8.0, 4.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(8.0, 4.0), end=(8.0, 0.0)); m\_s1 — a Line \[magenta\] drawn in m\_axes (end=(2.0, 16.0)) Actions: - [14:37.455](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=877.4554166666667): m\_s2 is shown on the screen, drawn. ##### [14:44.244](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=884.2439166666667) Narration: And at five metres, the couple. The shear ignored it. The moment cannot. It drops vertically by eight, the size of the couple, from plus four to minus four. Board: v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-8.0, 11.0), aspect=(9.0, 2.7)); m\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-11.0, 20.0), aspect=(9.0, 2.7)); head\_build — a Heading that says "Built From Jumps and Areas"; v\_up\_a — a Line \[yellow\] drawn in v\_axes (end=(0.0, 8.0)); v\_s1 — a Line \[yellow\] drawn in v\_axes (start=(0.0, 8.0), end=(2.0, 8.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(2.0, 8.0), end=(2.0, -4.0)); v\_s2 — a Line \[yellow\] drawn in v\_axes (start=(2.0, -4.0), end=(6.0, -4.0)); couple\_mark — a Line \[gray\] labelled "M\_0" drawn in v\_axes (start=(5.0, -6.5), end=(5.0, 6.5), dashed=True); v\_up\_b — a Line \[yellow\] drawn in v\_axes (start=(6.0, -4.0), end=(6.0, 4.0)); v\_s3 — a Line \[yellow\] drawn in v\_axes (start=(6.0, 4.0), end=(8.0, 4.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(8.0, 4.0), end=(8.0, 0.0)); m\_s1 — a Line \[magenta\] drawn in m\_axes (end=(2.0, 16.0)); m\_s2 — a Line \[magenta\] drawn in m\_axes (start=(2.0, 16.0), end=(5.0, 4.0)) Actions: - [14:50.292](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=890.2924166666667): m\_jump is shown on the screen, drawn. - [14:52.069](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=892.0694166666667): fact\_1 is shown on the screen, written out. ##### [14:55.757](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=895.7574166666667) Narration: Below the axis now, and the shear is still minus four, so the moment goes on falling at four per metre for the last metre to the roller. Minus eight. Board: fact\_1 — a Text \[text\] that says "A couple leaves the shear untouched and jumps the moment."; v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-8.0, 11.0), aspect=(9.0, 2.7)); m\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-11.0, 20.0), aspect=(9.0, 2.7)); head\_build — a Heading that says "Built From Jumps and Areas"; v\_up\_a — a Line \[yellow\] drawn in v\_axes (end=(0.0, 8.0)); v\_s1 — a Line \[yellow\] drawn in v\_axes (start=(0.0, 8.0), end=(2.0, 8.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(2.0, 8.0), end=(2.0, -4.0)); v\_s2 — a Line \[yellow\] drawn in v\_axes (start=(2.0, -4.0), end=(6.0, -4.0)); couple\_mark — a Line \[gray\] labelled "M\_0" drawn in v\_axes (start=(5.0, -6.5), end=(5.0, 6.5), dashed=True); v\_up\_b — a Line \[yellow\] drawn in v\_axes (start=(6.0, -4.0), end=(6.0, 4.0)); v\_s3 — a Line \[yellow\] drawn in v\_axes (start=(6.0, 4.0), end=(8.0, 4.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(8.0, 4.0), end=(8.0, 0.0)); m\_s1 — a Line \[magenta\] drawn in m\_axes (end=(2.0, 16.0)); m\_s2 — a Line \[magenta\] drawn in m\_axes (start=(2.0, 16.0), end=(5.0, 4.0)); m\_jump — a Line \[magenta\] drawn in m\_axes (start=(5.0, 4.0), end=(5.0, -4.0)) Actions: - [15:0.575](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=900.5754166666667): m\_s3 is shown on the screen, drawn. ##### [15:6.028](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=906.0284166666667) Narration: Over the overhang the shear is plus four, so the moment climbs at four per metre for two metres and lands on zero exactly at the free tip. It has to. There is nothing beyond the tip to bend it. Board: fact\_1 — a Text \[text\] that says "A couple leaves the shear untouched and jumps the moment."; v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-8.0, 11.0), aspect=(9.0, 2.7)); m\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-11.0, 20.0), aspect=(9.0, 2.7)); head\_build — a Heading that says "Built From Jumps and Areas"; v\_up\_a — a Line \[yellow\] drawn in v\_axes (end=(0.0, 8.0)); v\_s1 — a Line \[yellow\] drawn in v\_axes (start=(0.0, 8.0), end=(2.0, 8.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(2.0, 8.0), end=(2.0, -4.0)); v\_s2 — a Line \[yellow\] drawn in v\_axes (start=(2.0, -4.0), end=(6.0, -4.0)); couple\_mark — a Line \[gray\] labelled "M\_0" drawn in v\_axes (start=(5.0, -6.5), end=(5.0, 6.5), dashed=True); v\_up\_b — a Line \[yellow\] drawn in v\_axes (start=(6.0, -4.0), end=(6.0, 4.0)); v\_s3 — a Line \[yellow\] drawn in v\_axes (start=(6.0, 4.0), end=(8.0, 4.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(8.0, 4.0), end=(8.0, 0.0)); m\_s1 — a Line \[magenta\] drawn in m\_axes (end=(2.0, 16.0)); m\_s2 — a Line \[magenta\] drawn in m\_axes (start=(2.0, 16.0), end=(5.0, 4.0)); m\_jump — a Line \[magenta\] drawn in m\_axes (start=(5.0, 4.0), end=(5.0, -4.0)); m\_s3 — a Line \[magenta\] drawn in m\_axes (start=(5.0, -4.0), end=(6.0, -8.0)) Actions: - [15:9.243](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=909.2434166666667): m\_s4 is shown on the screen, drawn. ##### [15:17.599](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=917.5989166666667) Narration: Check that by hand if you like. Cut the overhang anywhere and keep the right piece: one four kilonewton load, a lever arm of eight minus x, bending the beam the wrong way up. Negative, and shrinking to nothing at the tip. Board: fact\_1 — a Text \[text\] that says "A couple leaves the shear untouched and jumps the moment."; v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-8.0, 11.0), aspect=(9.0, 2.7)); m\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-11.0, 20.0), aspect=(9.0, 2.7)); head\_build — a Heading that says "Built From Jumps and Areas"; v\_up\_a — a Line \[yellow\] drawn in v\_axes (end=(0.0, 8.0)); v\_s1 — a Line \[yellow\] drawn in v\_axes (start=(0.0, 8.0), end=(2.0, 8.0)); v\_drop — a Line \[yellow\] drawn in v\_axes (start=(2.0, 8.0), end=(2.0, -4.0)); v\_s2 — a Line \[yellow\] drawn in v\_axes (start=(2.0, -4.0), end=(6.0, -4.0)); couple\_mark — a Line \[gray\] labelled "M\_0" drawn in v\_axes (start=(5.0, -6.5), end=(5.0, 6.5), dashed=True); v\_up\_b — a Line \[yellow\] drawn in v\_axes (start=(6.0, -4.0), end=(6.0, 4.0)); v\_s3 — a Line \[yellow\] drawn in v\_axes (start=(6.0, 4.0), end=(8.0, 4.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(8.0, 4.0), end=(8.0, 0.0)); m\_s1 — a Line \[magenta\] drawn in m\_axes (end=(2.0, 16.0)); m\_s2 — a Line \[magenta\] drawn in m\_axes (start=(2.0, 16.0), end=(5.0, 4.0)); m\_jump — a Line \[magenta\] drawn in m\_axes (start=(5.0, 4.0), end=(5.0, -4.0)); m\_s3 — a Line \[magenta\] drawn in m\_axes (start=(5.0, -4.0), end=(6.0, -8.0)); m\_s4 — a Line \[magenta\] drawn in m\_axes (start=(6.0, -8.0), end=(8.0, 0.0)) Actions: - None. ##### [15:31.435](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=931.4349166666667) Narration: Two things to carry away from this beam. A couple jumps the moment and leaves the shear alone. And an overhang drives the moment below the axis, which means tension on the top of the beam, which is where the steel has to go. Board: Unchanged from the preceding beat in this scene. Actions: - [15:37.275](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=937.2754166666667): fact\_2 is shown on the screen, written out. - [15:40.398](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=940.3984166666667): fact\_2 (the "tension" part) is emphasized. - [15:42.267](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=942.2674166666667): fact\_2 (the "tension" part) is no longer emphasized. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): fact\_1 is hidden from the screen — left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): fact\_2 is hidden from the screen — left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): head\_build is hidden from the screen — left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): m\_axes is hidden from the screen — left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): m\_s1 is hidden from the screen — m\_axes left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): m\_s2 is hidden from the screen — m\_axes left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): m\_jump is hidden from the screen — m\_axes left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): m\_s3 is hidden from the screen — m\_axes left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): m\_s4 is hidden from the screen — m\_axes left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): v\_axes is hidden from the screen — left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): v\_up\_a is hidden from the screen — v\_axes left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): v\_s1 is hidden from the screen — v\_axes left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): v\_drop is hidden from the screen — v\_axes left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): v\_s2 is hidden from the screen — v\_axes left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): couple\_mark is hidden from the screen — v\_axes left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): v\_up\_b is hidden from the screen — v\_axes left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): v\_s3 is hidden from the screen — v\_axes left the board. - [15:43.958](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=943.9579166666667): v\_close is hidden from the screen — v\_axes left the board. ### Scene 6: [Sketching a Beam You Have Not Seen](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=944.9995833333334) Span: 15:45–17:54.754 (944.9995833333334s–1074.7540625000001s). #### Objects - beam: a Figure (x\_range=(-1.0, 9.4), y\_range=(-2.2, 2.7), aspect=(10.4, 4.9)) - head: a Heading that says "Half Loaded, and Sketched by Eye" - m\_axes: an Axes (x\_range=(0.0, 8.0), y\_range=(-2.0, 30.0), aspect=(9.0, 2.7)) - m\_curve: a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(0.0, 4.0)) - m\_line: a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(4.0, 8.0)) - m\_peak: a Point \[green\] labelled "27" drawn in m\_axes (location=(3.0, 27.0)) - pin: a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.34, -0.66), (0.34, -0.66)), fill\_opacity=0.35) - r\_a: a Vector \[green\] drawn in beam (start=(0.0, -1.7), end=(0.0, -0.76)) - r\_a\_name: a Math \[green\] that says "$18$" drawn in beam - r\_b: a Vector \[green\] drawn in beam (start=(8.0, -1.7), end=(8.0, -0.76)) - r\_b\_name: a Math \[green\] that says "$6$" drawn in beam - roller: a Polygon \[gray\] drawn in beam (vertices=((8.0, 0.0), (7.66, -0.66), (8.34, -0.66)), fill\_opacity=0.35) - span: a Line \[blue\] drawn in beam (end=(8.0, 0.0)) - steps: a Block \[text\] that says "Reactions, from equilibrium. Shear: start at zero, jump by each force, slope $-w$ between them. Moment: start at zero, add the area under the shear, jump at each couple. Maximum moment where the shear crosses the axis. Both diagrams must c…" - udl\_0: a Vector \[red\] drawn in beam (start=(0.0, 1.4), end=(0.0, 0.08)) - udl\_1: a Vector \[red\] drawn in beam (start=(1.0, 1.4), end=(1.0, 0.08)) - udl\_2: a Vector \[red\] drawn in beam (start=(2.0, 1.4), end=(2.0, 0.08)) - udl\_3: a Vector \[red\] drawn in beam (start=(3.0, 1.4), end=(3.0, 0.08)) - udl\_4: a Vector \[red\] drawn in beam (start=(4.0, 1.4), end=(4.0, 0.08)) - udl\_name: a Math \[red\] that says "$w = 6 thin upright("kN/m")$" drawn in beam - udl\_top: a Line \[red\] drawn in beam (start=(0.0, 1.45), end=(4.0, 1.45)) - v\_axes: an Axes (x\_range=(0.0, 8.0), y\_range=(-9.0, 22.0), aspect=(9.0, 2.7)) - v\_close: a Line \[yellow\] drawn in v\_axes (start=(8.0, -6.0), end=(8.0, 0.0)) - v\_flat: a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 8.0)) - v\_ramp: a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)) - v\_up: a Line \[yellow\] drawn in v\_axes (end=(0.0, 18.0)) - v\_zero: a Point \[green\] labelled "x = 3" drawn in v\_axes (location=(3.0, 0.0)) #### Beats ##### [15:45](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=944.9995833333334) Narration: Last beam, and you have not seen it. Eight metres, simply supported, with six kilonewtons per metre spread over the left half only and nothing on the right half. Board: Empty. Actions: - [15:45](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=944.9995833333334): head is shown on the screen, written out. - [15:45](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=944.9995833333334): beam is shown on the screen, written out. - [15:48.158](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=948.1575833333334): span is shown on the screen, written out. - [15:48.258](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=948.2575833333334): pin is shown on the screen, written out. - [15:48.458](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=948.4575833333333): roller is shown on the screen, written out. - [15:50.062](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=950.0615833333334): udl\_top is shown on the screen, written out. - [15:50.162](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=950.1615833333334): udl\_0 is shown on the screen, written out. - [15:50.362](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=950.3615833333333): udl\_1 is shown on the screen, written out. - [15:50.662](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=950.6615833333334): udl\_2 is shown on the screen, written out. - [15:51.062](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=951.0615833333334): udl\_3 is shown on the screen, written out. - [15:51.562](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=951.5615833333334): udl\_4 is shown on the screen, written out. - [15:52.314](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=952.3135833333333): udl\_name is shown on the screen, written out. ##### [15:55.607](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=955.6070833333333) Narration: Here is the whole method, in five lines. Reactions, then jumps and slopes, then areas, then the maximum, then the check that it closes. Board: beam — a Figure (x\_range=(-1.0, 9.4), y\_range=(-2.2, 2.7), aspect=(10.4, 4.9)); head — a Heading that says "Half Loaded, and Sketched by Eye"; span — a Line \[blue\] drawn in beam (end=(8.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.34, -0.66), (0.34, -0.66)), fill\_opacity=0.35); roller — a Polygon \[gray\] drawn in beam (vertices=((8.0, 0.0), (7.66, -0.66), (8.34, -0.66)), fill\_opacity=0.35); udl\_top — a Line \[red\] drawn in beam (start=(0.0, 1.45), end=(4.0, 1.45)); udl\_0 — a Vector \[red\] drawn in beam (start=(0.0, 1.4), end=(0.0, 0.08)); udl\_1 — a Vector \[red\] drawn in beam (start=(1.0, 1.4), end=(1.0, 0.08)); udl\_2 — a Vector \[red\] drawn in beam (start=(2.0, 1.4), end=(2.0, 0.08)); udl\_3 — a Vector \[red\] drawn in beam (start=(3.0, 1.4), end=(3.0, 0.08)); udl\_4 — a Vector \[red\] drawn in beam (start=(4.0, 1.4), end=(4.0, 0.08)); udl\_name — a Math \[red\] that says "$w = 6 thin upright("kN/m")$" drawn in beam Actions: - [15:57.523](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=957.5225833333334): beam moves to a new place on the board. - [15:57.523](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=957.5225833333334): steps is shown on the screen, written out. ##### [16:6.877](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=966.8765833333333) Narration: One: reactions, and these you do have to compute. The load totals twenty four kilonewtons acting through the middle of the left half, which is much nearer A. So A takes eighteen of it, and B takes only six. Everything after this comes out of the two derivatives. Board: steps — a Block \[text\] that says "Reactions, from equilibrium. Shear: start at zero, jump by each force, slope $-w$ between them. Moment: start at zero, add the area under the shear, jump at each couple. Maximum moment where the shear crosses the axis. Both diagrams must c…"; beam — a Figure (x\_range=(-1.0, 9.4), y\_range=(-2.2, 2.7), aspect=(10.4, 4.9)); head — a Heading that says "Half Loaded, and Sketched by Eye"; span — a Line \[blue\] drawn in beam (end=(8.0, 0.0)); pin — a Polygon \[gray\] drawn in beam (vertices=((0.0, 0.0), (-0.34, -0.66), (0.34, -0.66)), fill\_opacity=0.35); roller — a Polygon \[gray\] drawn in beam (vertices=((8.0, 0.0), (7.66, -0.66), (8.34, -0.66)), fill\_opacity=0.35); udl\_top — a Line \[red\] drawn in beam (start=(0.0, 1.45), end=(4.0, 1.45)); udl\_0 — a Vector \[red\] drawn in beam (start=(0.0, 1.4), end=(0.0, 0.08)); udl\_1 — a Vector \[red\] drawn in beam (start=(1.0, 1.4), end=(1.0, 0.08)); udl\_2 — a Vector \[red\] drawn in beam (start=(2.0, 1.4), end=(2.0, 0.08)); udl\_3 — a Vector \[red\] drawn in beam (start=(3.0, 1.4), end=(3.0, 0.08)); udl\_4 — a Vector \[red\] drawn in beam (start=(4.0, 1.4), end=(4.0, 0.08)); udl\_name — a Math \[red\] that says "$w = 6 thin upright("kN/m")$" drawn in beam Actions: - [16:7.922](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=967.9215833333334): steps (the "Reactions" part) is emphasized. - [16:17.872](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=977.8715833333333): r\_a is shown on the screen, written out. - [16:18.072](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=978.0715833333334): r\_a\_name is shown on the screen, written out. - [16:19.811](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=979.8105833333334): r\_b is shown on the screen, written out. - [16:20.011](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=980.0105833333333): r\_b\_name is shown on the screen, written out. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): beam is hidden from the screen — left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): span is hidden from the screen — beam left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): pin is hidden from the screen — beam left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): roller is hidden from the screen — beam left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): udl\_top is hidden from the screen — beam left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): udl\_0 is hidden from the screen — beam left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): udl\_1 is hidden from the screen — beam left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): udl\_2 is hidden from the screen — beam left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): udl\_3 is hidden from the screen — beam left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): udl\_4 is hidden from the screen — beam left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): udl\_name is hidden from the screen — beam left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): r\_a is hidden from the screen — beam left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): r\_a\_name is hidden from the screen — beam left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): r\_b is hidden from the screen — beam left the board. - [16:23.944](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=983.9435833333333): r\_b\_name is hidden from the screen — beam left the board. ##### [16:24.544](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=984.5435833333333) Narration: Two: the shear. It starts at zero, jumps to plus eighteen at A, and then under the distributed load it slopes down at six per metre for four metres. That is a fall of twenty four, from plus eighteen to minus six. Board: steps — a Block \[text\] that says "Reactions, from equilibrium. Shear: start at zero, jump by each force, slope $-w$ between them. Moment: start at zero, add the area under the shear, jump at each couple. Maximum moment where the shear crosses the axis. Both diagrams must c…"; head — a Heading that says "Half Loaded, and Sketched by Eye" Actions: - [16:24.544](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=984.5435833333333): v\_axes is shown on the screen, written out. - [16:25.937](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=985.9365833333334): steps (the "Reactions" part) is no longer emphasized. - [16:25.937](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=985.9365833333334): steps (the "Shear" part) is emphasized. - [16:28.723](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=988.7225833333333): v\_up is shown on the screen, drawn. - [16:32.346](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=992.3455833333334): v\_ramp is shown on the screen, drawn. ##### [16:40.213](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1000.2130833333333) Narration: Past the load there is nothing pressing down, so the shear runs flat at minus six all the way to B, where the six kilonewton reaction closes it to zero. Board: steps — a Block \[text\] that says "Reactions, from equilibrium. Shear: start at zero, jump by each force, slope $-w$ between them. Moment: start at zero, add the area under the shear, jump at each couple. Maximum moment where the shear crosses the axis. Both diagrams must c…"; head — a Heading that says "Half Loaded, and Sketched by Eye"; v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-9.0, 22.0), aspect=(9.0, 2.7)); v\_up — a Line \[yellow\] drawn in v\_axes (end=(0.0, 18.0)); v\_ramp — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)) Actions: - [16:43.789](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1003.7885833333333): v\_flat is shown on the screen, drawn. - [16:47.795](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1007.7945833333333): v\_close is shown on the screen, drawn. ##### [16:49.869](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1009.8685833333334) Narration: Three: the moment. Zero at a simple support. Its slope is the shear, which starts big and positive and is falling, so the moment leaves A steeply and bends over. A parabola, concave down. Board: steps — a Block \[text\] that says "Reactions, from equilibrium. Shear: start at zero, jump by each force, slope $-w$ between them. Moment: start at zero, add the area under the shear, jump at each couple. Maximum moment where the shear crosses the axis. Both diagrams must c…"; head — a Heading that says "Half Loaded, and Sketched by Eye"; v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-9.0, 22.0), aspect=(9.0, 2.7)); v\_up — a Line \[yellow\] drawn in v\_axes (end=(0.0, 18.0)); v\_ramp — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)); v\_flat — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 8.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(8.0, -6.0), end=(8.0, 0.0)) Actions: - [16:49.869](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1009.8685833333334): m\_axes is shown on the screen, written out. - [16:51.274](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1011.2735833333334): steps (the "Moment" part) is emphasized. - [16:51.274](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1011.2735833333334): steps (the "Shear" part) is no longer emphasized. - [17:0.991](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1020.9905833333333): m\_curve is shown on the screen, drawn. ##### [17:5.33](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1025.3295833333334) Narration: Four: it peaks where the shear crosses the axis. Eighteen divided by six is three, so the maximum sits three metres in. Not four, and certainly not at midspan. Board: steps — a Block \[text\] that says "Reactions, from equilibrium. Shear: start at zero, jump by each force, slope $-w$ between them. Moment: start at zero, add the area under the shear, jump at each couple. Maximum moment where the shear crosses the axis. Both diagrams must c…"; head — a Heading that says "Half Loaded, and Sketched by Eye"; v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-9.0, 22.0), aspect=(9.0, 2.7)); m\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-2.0, 30.0), aspect=(9.0, 2.7)); v\_up — a Line \[yellow\] drawn in v\_axes (end=(0.0, 18.0)); v\_ramp — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)); v\_flat — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 8.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(8.0, -6.0), end=(8.0, 0.0)); m\_curve — a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(0.0, 4.0)) Actions: - [17:6.77](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1026.7695833333335): steps (the "Maximum moment" part) is emphasized. - [17:6.77](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1026.7695833333335): steps (the "Moment" part) is no longer emphasized. - [17:7.78](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1027.7795833333334): v\_zero is shown on the screen, written out. - [17:13.028](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1033.0275833333333): m\_peak is shown on the screen, written out. ##### [17:18.167](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1038.1670833333333) Narration: After that the shear is a constant minus six, so the moment comes down as a straight line, and it has to arrive at zero at B. Five: it does, and the sketch is finished. Board: steps — a Block \[text\] that says "Reactions, from equilibrium. Shear: start at zero, jump by each force, slope $-w$ between them. Moment: start at zero, add the area under the shear, jump at each couple. Maximum moment where the shear crosses the axis. Both diagrams must c…"; head — a Heading that says "Half Loaded, and Sketched by Eye"; v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-9.0, 22.0), aspect=(9.0, 2.7)); m\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-2.0, 30.0), aspect=(9.0, 2.7)); v\_up — a Line \[yellow\] drawn in v\_axes (end=(0.0, 18.0)); v\_ramp — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)); v\_flat — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 8.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(8.0, -6.0), end=(8.0, 0.0)); m\_curve — a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(0.0, 4.0)); v\_zero — a Point \[green\] labelled "x = 3" drawn in v\_axes (location=(3.0, 0.0)); m\_peak — a Point \[green\] labelled "27" drawn in m\_axes (location=(3.0, 27.0)) Actions: - [17:22.486](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1042.4855833333334): m\_line is shown on the screen, drawn. - [17:26.898](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1046.8975833333334): steps (the "Maximum moment" part) is no longer emphasized. - [17:26.898](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1046.8975833333334): steps (the "close at the far end" part) is emphasized. - [17:28.175](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1048.1745833333334): steps (the "close at the far end" part) is no longer emphasized. ##### [17:29.61](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1049.6100833333335) Narration: None of that needed the bending moment function. If you do want the number, it is the area under the shear out to three metres: half of three times eighteen, twenty seven kilonewton metres. Board: steps — a Block \[text\] that says "Reactions, from equilibrium. Shear: start at zero, jump by each force, slope $-w$ between them. Moment: start at zero, add the area under the shear, jump at each couple. Maximum moment where the shear crosses the axis. Both diagrams must c…"; head — a Heading that says "Half Loaded, and Sketched by Eye"; v\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-9.0, 22.0), aspect=(9.0, 2.7)); m\_axes — an Axes (x\_range=(0.0, 8.0), y\_range=(-2.0, 30.0), aspect=(9.0, 2.7)); v\_up — a Line \[yellow\] drawn in v\_axes (end=(0.0, 18.0)); v\_ramp — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(0.0, 4.0)); v\_flat — a FunctionPlot \[yellow\] drawn in v\_axes (function=\, x\_range=(4.0, 8.0)); v\_close — a Line \[yellow\] drawn in v\_axes (start=(8.0, -6.0), end=(8.0, 0.0)); m\_curve — a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(0.0, 4.0)); v\_zero — a Point \[green\] labelled "x = 3" drawn in v\_axes (location=(3.0, 0.0)); m\_peak — a Point \[green\] labelled "27" drawn in m\_axes (location=(3.0, 27.0)); m\_line — a FunctionPlot \[magenta\] drawn in m\_axes (function=\, x\_range=(4.0, 8.0)) Actions: - [17:39.155](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1059.1545833333334): m\_peak is indicated — a transient flash. ##### [17:41.891](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1061.8905833333333) Narration: So: cut, equilibrium, and two derivatives. The jumps, the slopes and the areas are not rules to remember. They are what those two derivatives look like once you draw them. Board: Unchanged from the preceding beat in this scene. Actions: - [17:53.712](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1073.7123958333334): head is hidden from the screen — left the board. - [17:53.712](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1073.7123958333334): m\_axes is hidden from the screen — left the board. - [17:53.712](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1073.7123958333334): m\_curve is hidden from the screen — m\_axes left the board. - [17:53.712](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1073.7123958333334): m\_peak is hidden from the screen — m\_axes left the board. - [17:53.712](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1073.7123958333334): m\_line is hidden from the screen — m\_axes left the board. - [17:53.712](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1073.7123958333334): steps is hidden from the screen — left the board. - [17:53.712](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1073.7123958333334): v\_axes is hidden from the screen — left the board. - [17:53.712](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1073.7123958333334): v\_up is hidden from the screen — v\_axes left the board. - [17:53.712](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1073.7123958333334): v\_ramp is hidden from the screen — v\_axes left the board. - [17:53.712](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1073.7123958333334): v\_flat is hidden from the screen — v\_axes left the board. - [17:53.712](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1073.7123958333334): v\_close is hidden from the screen — v\_axes left the board. - [17:53.712](https://academa.ai/lectures/deriving-shear-moment-diagrams?t=1073.7123958333334): v\_zero is hidden from the screen — v\_axes left the board.