Engineers are supposed to do calculations. That’s what we do. We calculate everything from strength and mass flow, temperature rise and heat transfer, fuel consumption and top speed as well as simple things such as the total weight of a newly designed product. However, what do these calculations mean?

## Do we really have to calculate all of these things?

Ancient engineers over the ages did a great job designing complex machines and constructions: war chariots and composite bows, aqueducts and towers. They created marvellous designs even before the introduction of the Arabic numerals, before the invention of algebra and functions.

## If calculating is so important, how could they be so successful without it?

The truth is that calculations are not essential for design, as long as the product performs! The engineer is tested by the finished product, not by the way they got there. Calculations are though a good prediction tool and thus serve to save time and money. However, if all else fails we can always do what our fathers did: build and test, rebuild and retest.

## What do calculations actually mean?

Linguistically * calculation* comes from

*which is the Latin word for small stones used for reckoning. It started from simple counting but what do we really do when we count? We symbolize a quantity of real objects by a mathematical abstract term (the number), or as the early Romans did, represent the real objects by another group of much smaller objects, the pebbles. In a nutshell, this is the whole idea of calculating: representing phenomena in the real world by something else, something that is easier to grasp.*

**calculus**## Complex situations and basic calculations

When we say that a stone will take 5 seconds to fall from a tower 125 metres tall we actually mean is that it will be the same time that a car running at 100 km/h will travel 138 metres. We use the numeric results as an exchange, much like we do with money when we sell a cow to buy 2.5 lambs.

Numbers are easy to use when we need to exchange features of the same units which serve as our “coins”. However, we can still exchange features which have different units, by somewhat more complex exchange rates: exchange of more than two features at a time. If we cannot exchange cows with lambs (well, who wants half a lamb?) we can always add some apples. If we cannot exchange time with distance we can still do it if we add speed to the basket. Calculations are always about exchange, the benefit of calculating is the possibility to exercise these exchanges without the need to do them in reality. Just like playing make believe.

## What situations will encourage us to play these games?

We might want to prove that a product we made is safe and performs. It might be a good idea to smash an imaginary car into an imaginary brick wall but the market might not buy it. There will always be questions about how well the imaginary scenarios mimic the reality and eventually the car maker will have to do some real tests too. However, running these imaginary scenarios during the development of the product can lead to a better design solution – a win win for all parties.

Calculations use our knowledge of the physical phenomena, allowing us to mathematically model situations in a way that is easier to understand. However, our physical knowledge is only formulated for simple cases or for basic laws, defined in differential forms. Accurately modelling a complex system is always a big task and even when we use FEA programs it still can lead to mathematical models that are costly to make. Indeed the mathematical modelling may be too complex or expansive for even large powerful computers to solve.

## Finding a balance between mathematical modelling and real life

We can often simplify the modelling of a system, but if we overdo it the results of the calculation will be meaningless and irrelevant to the real life problem. Finding a balance between simplifying the mathematical model and making it easy to understand is the key. We now need to ask ourselves, how accurate should the model and the calculation be? Should we make an accurate complex model and use high accuracy calculation, or would we be better off making a rough model and using only approximate calculations?

The answer lies in the following rule:

**The lazy man’s law: ***Use the crudest model and simplest calculations that are just accurate enough for the purpose.*

Good reading!!

Calculations are a means to try to avoid failure – failure to perform as needed, failure to perform at all, structural failure, catastrophic failure with resultant death or serious injury, etc. Build, break and redesign can often be more expensive than trying to model, design and predict, and for complicated systems or processes, this is always true. Many times throughout my career I have been amazed by the insight provided by my calculations, and how often my seat-of-the pants impression was wrong or inadequate. Calculations separate engineers from people that just make things.

toler-fes, I must disagree.

No doubt that the ability to make incresingly better calculations is helping us to make better and more complex designs, and make them cheaper and faster. Calculations are great tools in the toolbox of the modern engineer, though I must wonder if it is right to call finite elements or other computer modelling tools “calculations” per se, or just “prediction tools” based on “mathematical modeling”.

I resent the seperation you make between “engineers” and “people who just make things”. I am proud to be a chain in the line of ages, that included people like Leonardo da Vinci, who was not an engineer by your definition. indeed, “calculations” help us in building our “insight”, which we later implement as another tool in our “seat-of-the pants”.

I bring here three quotes from Wikipedia:

Vasa (or Wasa) is a Swedish warship built between 1626 and 1628. The ship foundered after sailing about 1,300 m (1,400 yd) into her maiden voyage on 10 August 1628.

Vasa was dangerously unstable and top-heavy with too much weight in the upper structure of the hull. Despite this lack of stability she was ordered to sea and foundered only a few minutes after encountering a wind stronger than a breeze.

The 1940 Tacoma Narrows Bridge, the first Tacoma Narrows Bridge, was a suspension bridge in the U.S. state of Washington that spanned the Tacoma Narrows strait of Puget Sound between Tacoma and the Kitsap Peninsula. It opened to traffic on July 1, 1940, and dramatically collapsed into Puget Sound on November 7 of the same year.

The Space Shuttle Challenger disaster occurred on January 28, 1986, when the NASA Space Shuttle orbiter Challenger (OV-099) (mission STS-51-L) broke apart 73 seconds into its flight, leading to the deaths of its seven crew members, which included five NASA astronauts and two Payload Specialists. The spacecraft disintegrated over the Atlantic Ocean, off the coast of Cape Canaveral, Florida, at 11:39 EST (16:39 UTC). Disintegration of the vehicle began after an O-ring seal in its right solid rocket booster (SRB) failed at liftoff.

All quotes tell us about famous and expensive engineering failures. All made by the top engineers of their time. Why did’t they do the “simple” calculations that would have prevented it?

In the case of the Vasa, they did not have the tools for that. even today, you need a complex computer modeling to modelize the stability of a ship in the water. Including the effect of the wind make it even more difficult, especially if you wat to do it “acurately”. The sinking of the Vasa started the rule that no major ship would be built without prior modeling and putting the model to the water. In those days real physical modelling was a lot cheaper and faster (and possible) than mathematical modeling.

So was in the case of the Tacoma Narrows Bridge, centuries later. A problem too difficult to calculate was ignored. Wind tunnel experiments were not yet thought of as desirable and neede for construction projects. I suppose that a “builder of things” with good “seat-of-the pants” feeling from building kids swings might have helped… maybe not…

As for the Challenger and its O-ring, I wonder what calculation would have prevented it. You need to gut feeling to suspect that there might be a problem, and once you have the suspicion, you would have find your way to test it, this way or another.

So yes, calculations are great tools, but they are only one more (awesome) tool in our toolbox.

Actually, the challenger o-ring was calculated to fail and the engineer knew it. He brought it to the attention of his supervisor who chose to ignore the calculation. The rest is history.

Also, as a machine designer engineer, I frequently use my experience to skip calculations (depending on what the part is). I never skip calculations on safety items or critical items.

Kyle Goltsch

I like your remark! It is very true and rooted in real life. I shall make use of it to further clarify several points.

You said: “I frequently use my experience to skip calculations (depending on what the part is)”.

Indeed, this is exactly the type of knowledge (you may say “experience” that defines the “seasoned” engineer from the “half baked” one. You KNOW certain things without calculating them even before you start the design, ehile the less experienced has to “finish” the design of the part/system so that he will be able to calculate it, and then “redesign” or “reshape” it to prevent failure or to cut weight. This knowledge, not only makes your work much faster, but it also makes your design a lot better, as a good part of the optimization is done by you intuitively, and optimizes the complete system,while the optimization of the not so experienced guy starts with a system more remote from the “best” system, and is bound to fall into a “local” optimum, which will probably be somewhat inferior to yours.

You said: “I never skip calculations on safety items or critical items”.

I shall add to it: and since even you (and me, and every human being) is prone to err, if the real critical items we shall add a real testing to our calculations. Sometimes we are even bound to do so by the law. Some of the most common errors are our failure to identify the most dangerous MODE OF FALILURE, which is what we try to clculate.

You said: “Actually, the challenger o-ring was calculated to fail and the engineer knew it”.

Yes, I know. It was written everywhere. But these are the words of the journalists, trying to convey the information to the public, and probably having only half understanding of our profession anyway. I always wondered what they were “calculating” there? Was ha comparing the temperature of the o-ring to the air temperature and to the catalog properties of the rubber? Was he analizing the speed of the heating up of the rubber to the rate of its burning? Did he analize the thermal shrinkage of the rubber versus the size of the groove? Some of these calculations are very basic, and others are very complex. The crude ones are not very reliable, and the complex are probably too complex, unless you try to make some assumptions about them. I suspect that what really was there is that this engineer had a bad gutt feeling, suspicion, based on his experience, and then he might have tried to base it on some rough calculations, mostly in order to convince his superiors. However, his superiors, being engineers themselves, knew that his “calculations” or suspicions are “ststistical” in nature, and what they meant is some reduction of a sefety factor, which they decided to live with it, you may say “gamble”. We all gamble all the time, but we always hope to win and have faith in our chances.

I know that I shall get fire for this last phrase, so I shall clarify: all technical calculations are, to some degree! The statistical nature of our calculations has many sources: tolerances, dispersion of tested strength data, some crudeness of our formulas or the size of our elements in the final element analysis. To compenstae for it we use the “safety factor”, sometimes called the “ignorance factor”. Indeed, this is something for another article.

Great article.

Calculations are important, though some not as important as others. By important, I mean that it has a more far reaching impact on the final or actual product. Some products designed and built without accurate calculations will not fail, and if they do, are not life threatening or has little impact to the user.

However, my opinion is that it is important that an engineer be based and rooted in calculations based design.

We would never have achieved so much without calculating our design models. Could we have landed on Mars or build a mega structure like the Burj al Khalifa aka the tallest building in the world, just by modelling and some crude calculations?

Thanks.

However, I don’t see calculations as something very different in nature than modeling. The difference between testing a model aurplane in a wind tunnel and testing a computer 3D model in a simulation program seems to me no more than changing one tool with another. Both rely on theories and assumptions of silmilarity and suffer from certain inaccuracies. The decision to use one tool and not the other lies more on economic basis – which tool gives quicker and cheaper results, and this, is ever changing, determined by the certain point in time we are, as the relative cost of the technologies canges. Indeed, in this point in time we are now, it seems that the computer model has the upper hand. Is this situation here to stay forever? Who can tell?

There is one type of calculation though, which seems to have an edge over computer modeling, and this is the analytical tool. Analysis base on the old analytical formulas, wher, sometimes, one can reach far reaching conclusions, which can be very useful in the early stages of the design work. Unfortunately this kind of computing is extremely difficult, unless you make it extremely simplified. i may write about few interesting such analyses in the future. all other today’s calculations seem to fall in the category of the mathematical modelling, which is much the same as the physical modelling, with few, not so great, differences.

Great article!