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So this is a distributed force being applied to a beam. The problem is to draw the shear and moment diagrams.
First thing to do is solve for the reactions, so let's walk through that process.
Draw axes for reference.
Use w, for the weight, as the y-axis.
These unknown constants need to be solved for, which can be done using boundary conditions.
So when x = 0, the graph tells us the weight is equal to 1000 N/m
Apply this reasoning to the other boundary.
Next is finding the resultant of this force, which is equal to the area under the curve.
This creates a perfect setup for an integral
So now that the resultant of the load has been found, we can replace the force with it ONLY to find the reactions.
But there's a problem. Where's the resultant being applied?
Center of masses will tell us exactly where it's at.
This is how a resultant is thought of, as the center of mass of the distributed force.
So we know that dm is = to density multiplied by dV
But that's a constant in both integrals so it'll drop out of the equation.
We also know that this is a 2D problem so the thickness will be constant as well
Now the bottom integral solves for the area, but that was already solved for.
Note the boundaries.
We know what w is equal to.
If we were to carry the units in the equations, we would see that the force units drop out.
Now we can finally find those reactions at the supports.
That concludes the reactions part of the problem.
Before beginning the diagrams, remember that we ALWAYS use the distributed force for it, not its resultant.
Always put units and a number that your referencing in your diagrams.
These lines are in sync with the member, which means a force happening 1 meter into the shear diagram is happening 1 meter into the member.
It'd be good practice to line these up under your member.
We'll begin with the shear since the moment diagram is derived from the shear.
First, you begin at one side of the member and make a cut, leaving a tiny piece.
Drop everything on the other side of the cut from the picture
Imagine this piece to be infinitely small, similar to a differential area.
If this were a Free Body Diagram, there would need to be an equal and opposite force to balance it.
Well, there is, it's called the internal stress or the shear force.
If there was some distance between these forces, there would also need to be a moment to keep it still, similar to a cantilever beam.
But there can't be a length within this piece, since it's infinitely small there's virtually no distance, therefore, no moment.
These internal stresses that are currently drawn represent the positive direction if you make this type of cut.
These stresses represent positive direction if you make a cut on the other side.
It can be seen here that this side requires a negative shear force.
So every shear diagram is a curve that is defined by an equation.
This relationship allows us to find the equation.
This integral defines shear.
This integral doesn't have bounds so remember there's an unknown constant
but it can easily be solved for.
We know when x = 0, the shear is equal to 931.
Simple, solve for C.
In order to draw this curve, we'll need to plot some points. Five equally divided points should create an accurate curve.
Use the shear equation.
So remember the other side of the member was supposed to have a negative shear, and it did.
The reaction at this point counters the shear, bringing it back to 0 which means it's in equilibrium.
Connecting the points creates this curve which is almost straight.
This suggests that this is an extremely large parabola, and if you look at our equation, it has large numbers as well.
So this does make sense.
The diagrams only require labeling equations and points of interest such as maximums, minimums, and critical points.
We need to define where this critical point is happening on the member.
Use the equation again, just where shear = 0. However,
this can't be solved with traditional methods, a graphing calculator or some computation device will be needed.
Using a graphing calculator to pinpoint the X value gives us 0.9487m
The bending moment is also defined with an equation.
We know that C here is equal to 0 because this didn't start with a moment so there's nothing to balance out.
Calculus tells us this critical point represents a maximum.
Instead of plotting points or using a calculator, we can graph this logically.
We see this slice has a great amount of area under it, this suggests the moment diagram will initially have a large slope.
Further down the shear diagram, the slices acquire less area at a gradual pace.
Therefore, the slope is gradually decreasing as well.
A slice further down shows very little area telling us the slope is very small, it looks like it's leveling out.
It reaches a maximum that can easily be found by plugging in the x value, since we know where it's happening.
That wraps up the problem, remember to always place units, reference your diagrams to a number, and label points of interest.