Part 1 - introduction

Introduction

Hi.

My name is Adrian Pope and I am an engineer (for those who know me, it says it all really).

I took a general engineering degree at university, and was taught as much mechanical, electrical, thermodynamic and fluid mechanics in the first two years as structures: I specialised in the third year. Hence, when I left university, I knew a little about everything, with a bias towards structures, but that was all.

When I started working, I was practically useless for a few years unless heavily supervised. I thought that this was because I was taught how to think about things from first principles, rather than being taught in sufficient depth to be able to do my job. But now, I am not so sure. It is entirely possible that I was just plain ignorant, but I also recall that everything was taught as though it was a law of physics, whereas  most of it was actually special cases of a few fundamental rules: it was therefore very hard to tell the wood from the trees.  Luckily I worked for some very good engineers when I started, and they taught me what was important.

After thirty years of work, mainly in the bridges field, I am still a hard core engineer who gets to play with the difficult jobs rather than pushing pieces of paper around.

Aside: why do people strive so hard to get a degree in a difficult subject like engineering and then become paper wranglers, aka managers, before they become really good at the fun bits?

But as part of my job, I also get the fun job of technically training a number of graduates: since I coach (asking questions to guide them to give me the answer) rather than teaching (just telling them what to do) some of my grads seem to think that I take pleasure in tormenting them.

As if ….

The thing that I have found is that a lot of graduates leave university with the same level of preparedness as I did, i.e. not very much. They are intelligent people, but they have been shown how to apply a number of solutions, but taught every solution as though it were a fundamental principle of physics.

They have been taught how to analyse a reinforced concrete section using one method at serviceability, a different way at ultimate and other ways if there is an axial force. They will also have been taught different methods for elastic and plastic steel sections, with and without axial forces. Each of these will have been taught as a separate theory. When I recently ran a course for one hour, which described how all of the above used exactly the same single theory, with variations caused by the use of different limits and material stress strain diagrams, it took another two hours to convince them that I was not just teasing them.

The purpose of this document/web page/brain implant (delete as appropriate) is to try to cut through the fog that was created at university and to remind you that a lot of the work you do is really quite simple. Once you realise that most people are basically muddling through, and trying to keep their heads above water, you should lose that sense of panic that most engineers suffer, from time to time, and be able to learn how to apply our really quite simple set of rules to more and more complicated situations.

As my grads will tell you, I am well known for my rambling style of communication. This guide aims to be generally rigorous in its science, but is very tongue in cheek: this is intended to amuse, but if you find that sort of thing annoying, please remember how much you paid for this (nothing) and therefore realise that you got good value for your money.  If you like small breaks in what you are reading, then please read the “Asides”. Some of these are snarky comments about the sort of people who get in the way of me doing my job, whilst others are anecdotes that illustrate one of the points that I am making. If you want to plough through and risk falling asleep, then ignore them.

Please be warned: because I have broken everything down to a very low level, there are a lot of words. I suggest that you read a few sections at a time, rereading the slightly earlier bit each time to remind you where the line of argument is going. Or read it all in one go if you so desire, it matters not a bit to me.

The covering my rear end bit: never trust anything anyone ever says. The work in this blog goes down to first principles, so I leave it up to you to judge whether or not I have got anything wrong. It is intended to make you think and get out of the habit of blindly obeying orders, so please do not blindly follow it.

Cutting through the jargon

A lot of my lecturers seemed to assume that:

·        every student attended every lecture,

·        every student hung on every word that they spoke,

·        these attentive students immediately understood what every phrase meant (if the lecturer actually defined them, which I am not even sure about), and

·        these models of diligence were then able to apply those phrases to immediately build up increasingly complicated houses of cards, sorry theories.

Unfortunately, with one card missing from the house, understanding collapsed, and we just had to apply the resulting method without actually understanding where it came from. This is fine if you are doing the same thing all of the time and you know that the theorem is applicable, but if you do not know the basis then it is all too easy to apply the right theory in the wrong situation.

Aside: when I started coaching, one of my grads was designing a three span continuous highway bridge. He created 100+ live load cases, and produced beautiful diagrams showing the envelopes of moments in the spans and over the supports. Superimposed on this beautiful work of art (it was for his ICE submission of course) he plotted lines showing the capacities of the various reinforcement bar sizes and then produced detailed curtailment drawings. He showed these capacities for sagging and hogging all the way along and I was looking forward to seeing an equally beautiful diagram showing his sagging steel in the bottom of the slab in the spans and his hogging steel in the top of the slabs over the supports. I was amazed when I found that he had put all of the steel in the bottom of the slab, with none at all in the top. When I asked him why he put the steel there, he said that his lecturer had told them tension steel goes in the bottom. This is a classic example of a lecturer giving a rule that is correct for a lot of situations, i.e. for simply supported spans, but which will not work for all. If he had been taught that the steel goes on the tension side of the concrete then it would have been correct for all situations. I can only assume that this was due to a disconnect between one lecturer teaching section design (and hence using the simple example of a simply supported beam), with another teaching general analysis, which would have covered hogging and sagging.

It is very important to fully understand the basics before you launch into the guts of a problem, so please concentrate on the foundations. But to make things a bit easier, I will try to translate the “Estate Agent” English, often used by professionals to show how smart they are. This might then make it easier for you to understand what is going on.

Hopefully a lot of what you read will make you say “But that is obvious, why am I wasting my time reading this guff?”. If that is the case, you will have to judge for yourself whether to read on to the more complex bits. But even if it is familiar, I can promise you that a lot of people with whom you work will not understand the basics and you might find my explanations useful as a starting point to develop your own ways of breaking down complex ideas into simple concepts. One thing that I have found is that most engineers have visual minds, and can understand an idea better with a simple visual demonstration, hence the large number of diagrams

Plane sections remain plane

I regularly heard my lecturers saying “... and therefore, since plane sections remain plane ...”. I am absolutely sure (OK, I do tell lies sometimes) that someone explained what this meant during my first lecture, but I only worked out what it meant in my mid-twenties. Since I did not recall seeing it on the blackboard (for you youngsters, blackboards were painted timber that lecturers used to write on with bits of soft coloured stone called chalk), I was not originally sure if this meant plain (i.e. simple or unadorned) or plane (i.e. a flat surface): just to avoid confusion, the word used here is plane, which means flat.

The simplest way to understand what this means is to consider an artist’s rubber (or eraser for those brought up in the colonies). Figure 1 shows this rubber with red lines on the surface that represent a series of parallel planes cutting through the “Structure”. The blue lines are reference lines along which we will be interested in length.

Figure 1 – unbent rubber with lines drawn on the surface

Figure 2 shows exactly the same rubber, but now it has been bent. The form of the bend is not actually relevant, but is represents the behaviour of beam with equal but opposite bending moments applied to the ends.

Figure 2 - bent rubber

The red lines in Figure 1, on the unbent rubber, are straight and represent a plane section. The lines in Figure 2, on the bent rubber, are also straight, showing that the sections are still plane. This is the meaning of “Plane sections remain plane”.

SO WHAT ?

This actually leads on to the fundamental principle in section analysis. Since the original planes were parallel, the distance along the beam between corresponding points was the same anywhere on the section: a plot of the lengths of the blue lines (measured between a pair of red lines) against height forms a rectangle. On the bent rubber, a plot of the length of the blue lines, measured along the curve, against height (as shown in Figure 3) now forms a rhombus, i.e. pairs of triangles have been added to and subtracted from the original.

Figure 3 - length of blue lines before and after bending

Hopefully, none of you will be surprised by the following definition (and if you are then it may be time to become an accountant), then

strain = change in length / original length..

The width of the triangles stuck on the ends of the rectangle are the change in length at the various levels. Since the original length was constant over the full height of the section, the strain at any level is therefore proportional to the local width of the triangles.

Hence, all the “Plane sections remain plane” phrase means is that the strain diagram is a straight line.

Now why didn’t they just say that in the first place?

There is, however, one fundamental point that you need to bear in mind: if plane sections do not remain plane then it is likely that you have two structural sections sharing load rather than one large one. As an example the behaviour of two wooden planks, each with 150x25mm cross section, sitting on top of each other (OK, let us put rollers in between to make absolutely sure that they can slide over each other – pedants) is very different from those same pieces of wood if they are bonded together to form a single 150x50mm cross section. The first is two shallow sections sharing a load whilst the latter is one much deeper section. As you should be able to work out now (and certainly should be able to by the end of this chatty guide) the two planks sharing load will deflect four times as much as the deeper section.

Aside: before the advent of AS levels, which put too much strain on the timetable, I used to teach simplified bridge design to sixth formers for an organisation called Surrey SATRO (don’t ask). As part of this, I used to demonstrate the above principle using carefully prepared pieces of timber (50x5mm cross sections) in a pair and as a glued composite: I used that well known British Standard load aka a can of baked beans (and I could also show linear behaviour by measuring deflections with one two or three of these carefully calibrated weights). But then I got a bit showy, and tried the same demonstration with two of the 25mm thick planks mentioned above. I acted as the subtle applied load (well over a tenth of a tonne) to find the deflection when they share the load. I then launched into an involved anecdote (there’s a surprise) whilst I drove around 100 nails into the planks to join them together. Imagine my surprise when the measured deflection was around 3/4 of the pair of planks, rather than 1/4. My demonstration went so badly that I completely corpsed and had to be helped to a source of water to stop my laughter. When I looked closely, I could see that the oval nails, that I had used, had actually cut into the wood as the two pieces of wood tried (and pretty much succeeded) to slide over each other. If I do the demonstration now, I have gone back to the small sections and pre prepare the jointing using Resin W adhesive. Even geniuses have their off days ….

Fundamental: I am effectively restating from a couple of paragraphs before but if plane sections do not remain plane, i.e. a straight line on the side of the beam does not remain straight, then our theories do not apply. In the case of the planks with the nails, the two planks would have slid over each other and the straight line would have had a step in it, with the two planks acting as two separate structures sharing the load. This is why we have to check/design for longitudinal shear capacity to make sure that the various elements of a section will work together: we need to have enough weld/bolts/rivets/adhesive to join the parts together properly.

Strain compatibility

This goes together with the plane section remains plane bit. In order for our theories to work, the strain diagram has to be a straight line. This means, for example, that a reinforcing bar, embedded inside and bonded to some concrete, must have the same strain as that concrete. The strains are therefore the same and “Compatible”.

Hence, all that “Strain compatibility” means is that all of the various elements in a section, whether of the same material or not, follow the same straight strain diagram.

Subtleties: nearly all forms of structural section follow strain compatibility. Rebar is bonded to the concrete in reinforced concrete, bonded prestress is bonded (what a surprise !) to the surrounded concrete, and the concrete slab in a composite steel/concrete deck is kept honest by shear studs sticking into the concrete. The only exception to the rule of strain compatibility is unbounded prestress. In this form of construction, the prestressing strands often stretch through the inside of a voided concrete deck, like large piano wires,  and are therefore only affected by the average strain of the surrounding concrete between fixing points. This is just a variation on a theme rather than a fundamentally different problem.

Centre of Gravity

Unless you are designing see-saws, there ain’t no such animal.

Moment of inertia

Please refer to Centre of Gravity above.

Centroid

In structural engineering, we work with cross sections that have area, not mass. The centroid is area equivalent of the centre of gravity for weights. Rather than working out the point about which the moments (weights times perpendicular distances) balance, we work out the point where the first moment of area (areas times perpendicular distances) balance.

Second Moment of Area

This is the structural equivalent of the mechanical engineer’s moment of inertia. Again, it is worked out in the same way, but using areas instead of weights.

Aside: have you ever wondered why we use the term “I” to represent second moment of area ? Yes, you have guessed it, it stands for Inertia, as in Moment of … Whoever said that engineers were sensible ?

Neutral Axis

This is the line across a section where the axial stress is zero. A structural section may not necessarily have a neutral axis, especially where large axial forces are present.

Fundamental: PLEASE NOTE VERY CAREFULLY that although the centroid can be the same as the neutral axis, it usually is not. It is only on steel sections, working in the linear elastic range with ZERO axial force, that the two are the same. It is amazing how many experienced engineers mix up the terms and, although they know that the term is being misapplied, the grads that they are teaching can get very confused about such a simple pair of ideas.