Shear and Moment

Question

Solve for the Shear forces and Moment of each point and Draw its shear and moment diagram.

A simply supported beam with a fixed support(A) at left end and roller support(F) at right end. Point A has no force, Point B has 10 kN, C to D has uniformly distributed load of 20 kN/m, E with 15 kN and F with no force applied.

Distance from point-point:

A-B(1m), B-C(1m), C-D (3m), D-E (2m), and E-F (1m).

Answer 1

Hello Sam!

This problem is quite lengthy, so I won't work it all out here, but I will explain how to approach it, and why I think it is tricky. If I am misinterpreting some aspect of this beam, particularly the constraint present at point A, please let me know. In particular, are you sure that the constraint at point A is a "fixed constraint" (like being welded to the wall), and not a "pin joint constraint"? This has big implications for the techniques we'd need in order to solve this one.

Now, the general methodology that we can use when trying to make shear & moment diagrams is making imaginary "cuts" at an arbitrary position "x" along the beam in each distinct section of the beam, redrawing either the left-side-of-the-cut or right-side-of-the-cut free-body diagram (which will have the internal shear and internal moment terms on the cut surface), and then using standard statics equations to solve for the internal shear and moment in each segment of the beam.

BUT before we can start all that process with the "cuts" and redrawing free-body diagrams and such... we almost always need to solve for our unknown reaction forces and moments first. If we don't know a least some of the reactions at A and F, then we almost certainly can't solve for shear/moment diagrams.

For 2D problems like this one, the standard way that we go about solving for the unknown reactions is by using the "net forces and torques must equal zero" equations.

These equations are:

1. The sum of all forces in the x-direction must equal zero.
2. The sum of all forces in the y-direction must equal zero.
3. The sum of all torques about the z-axis must equal zero.

Generally speaking, through using these three equations we can solve for up to three unknown parameters in our body.

Now, inspect our FBD for this problem... How many unknown reactions do we have?

Point A is a "Fixed" constraint, which implies that we have:

1. A horizontal A_x reaction force
2. A vertical A_y reaction force
3. A reaction moment M_a

Point F is on a roller bearing, which implies that we have:

1. Only a vertical reaction force F_y

So we start this problem with four unknown reactions, but we only have three statics equations at our disposal. If I am understanding the constraints at A and F correctly, this looks like a statically indeterminate beam, meaning that we need to be creative and find one more equation in order to solve for all our reactions BEFORE we even start thinking about shear/moment diagrams.

In this problem, the extra equation that I think enables its solution is that we know that, at point F, the vertical deflection of this beam must equal zero (since the beam can't physically move downwards at point F). If we do lots of lengthy and complicated beam deflection equations (preferably referencing presolved deflection formulas from your textbook), then we should be able to solve for the force required at point F to impose that condition of zero net beam deflection at point F.

AFTER all that, then you could solve for the rest of the reactions, then proceed with the shear/moment diagram analysis technique I described above.

Unless you have spent a lot of time talking about beam deflection equations / analysis, then its more likely the case that some information about the beam is being lost in translation. Sorry to not be of more help and please let me know if anything in my writeup doesn't match with the details in the actual problem!

Please let me know if there are any parts that I can help more with!

--Zach