Prelude - part II

Today we pick up where we left off last week on our whistlestop tour of beam bending.

At the end of the last week we established that:

1. If we apply a weight to a beam, it deflects downwards.
2. The deflection causes the top half of the beam to compress, and causes tension in the bottom half.
3. At the mid point of the beam there must be some amount of stress. 

Today, we are going to derive what happens all the way along the beam by paying particular attention to the very end points. We are in fact, going to begin by leaping right in with what happens at the ends of our particular beam.

If you recall the last post, I specifically said the end supports in our first example must allow the beam to rotate. There are both good practical and good mathematical reasons for this stipulation; the former we will cover another day, and the latter we are going to skip right over. For today we are going to start by accepting that the ends of the beam are allowed to rotate and that we call this a pinned support. Then we will consider what that in turn must mean for our bending stress at the ends of the beam. 

Pinned Supports

To explain why we have pinned supports and what they are I’m pretty excited to bring out the structural engineer’s most trusty, yet utterly unglamorous assistant to aid with the explanation: The ruler. Yes, the ruler. That 30cm shatterproof slice of secondary school nostalgia (but not those hinged ones that fold in half, sorry, but they were always rubbish) is every structural engineer’s best desk buddy. 

Behold, our beam as represented in compass-point graffitied, hi-res digital art!

I know. Thrilling right? No really, the humble ruler truly is a useful bit of kit when trying to understand bending, I promise.

If you don’t have a ruler to hand to play along, I want you to imagine you are holding a ruler as drawn above, otherwise, go root around in your pencil case then come straight back.

Ready now? 

Great - let’s go. Balance the ruler on your fingers as shown in the drawing. Now keep your left hand motionless but raise and lower your right hand while concentrating on what you feel with your left fingertip where it supports the ruler. You feel the ruler rocking on your finger? It’s rotating freely. No matter how quickly or how forcefully you move your right hand you won’t change that fact - the ruler will always spin about the finger of your left hand. It sounds trite, but this is the perfect way to think of a pinned support.

Now why do we want pinned supports? The simple fact is that bending stress can never occur at a pin. Without something to actively resist the ruler, any bending you apply disappears at the pin. Try it yourself with your ruler! The simplest way to demonstrate it is as below:

You create a pin support like before with your index finger of one hand, let a decent length of ruler project past your finger, and force the beam to bend with your other hand. You can see that between the support (your finger) and applied rotation (your fist) the ruler bends, but at and after the pin the ruler is completely unbent - perfectly straight in fact, that is unless you’ve been abusing your ruler beforehand and if that is the case, frankly I can offer you no help there. 

The lesson to learn from this exercise is that bending, as we have defined it, cannot exist at a pinned support. Now, if bending cannot exist, the stresses - our tensions and compressions from the previous lesson - cannot exist either. If you stop and think about this we can say that we have now defined along our beam three points at which we have some idea of the bending stresses.

1. We have the left support, where the bending stresses are zero.
2. We have the middle of the beam (where the load was applied) where the stress is ... something.
3. We have the right support, where the bending stresses are zero.

Let's put that on a drawing shall we?

Again, don't fret about the unfriendly diagram - just like last time it shows us what we already know. The pink shape I've sketched over the beam shows us what we have derived: there's no stress at either end, and some amount of stress in the middle.

I have made (and will gloss over) two assumptions .

1. I have shown a linear distribution of bending stress - that is the stress varies in a straight line from 'something' at the middle to nothing at either end giving us a distinct triangular shape.
2. I have assumed that the biggest stress occurs in the middle, where the load is.

I also know that I've left the actual amount of bending stress somewhat loosely defined in the middle there as 'some' stress, but frankly you've suffered enough if this is all new to you. Trust me when I say that there is a lot of unavoidable (but very pretty) maths to get to the true answer, which probably isn't even a topic for another day - it will take a lot to make that interesting enough to read for pleasure. 

But there it is - from nothing but a bit of common sense and a piece of plastic we've derived a way of showing bending stress distribution under a point load, and at the same time learned about pinned supports.

Next time I write on this topic, we can start getting into connections proper.

As always, comments and corrections are welcome either via email (martyn@martynpie.com) or via twitter (@martynpie).