Have you ever performed an analysis on a seemingly robust structure only to discover stresses that exceed the allowable limit by orders of magnitude? As experienced analysts, this is likely a problem you rarely encounter; however, when first delving into CAE, many engineers run into confusion when interchanging between unit systems. So, why does this happen?
It is first important to understand that physics problems, which are ultimately what we are solving when performing FEA, can be defined in a number of ways. For example, it is intuitive that a one-pound ball falling one foot in one second is the same thing as a one-pound ball falling 12 inches in one second. But is it as obvious that a 0.45-kilogram ball falling 0.30 meters in one second describes the same problem?
For most of us, yes, this is pretty obvious – we all know how to convert between pounds/kilograms and feet/meters. But what happens to force? Or stress? Or any other derived unit? This is where working with disparate, yet equivalent, unit systems can become tricky (particularly if it is a unit system that you aren’t naturally familiar with). But don’t fret – if you can understand three simple rules, you’ll never be flummoxed by unit systems again!
Unit System Fundamentals
When discussing unit systems, there are three fundamental rules that must be understood:
1) All unit systems are defined by a set of Base Units.
2) All unit systems include Derived Units.
3) All unit systems must be Consistent.
Base Units
When we talk about Base Units, what we are referring to are seven mutually independent quantities that can be used to describe the fundamental properties of a physical system: Length, Mass, Time, Temperature, Current, Luminous Intensity, and Quantity of Substance. Any desired unit can be selected to describe any of the aforementioned measures (for example, meters, feet, or inches could be used to define length and slugs, grams, or tonnes could be used to describe mass); however, it must be noted that whichever units are selected as the Base Units will dictate the units that all Derived Units must use (which we will discuss in more detail shortly). For this reason, it is strongly suggested that analysts use a standardized unit system, such as SI or Imperial.
Derived Units
Now that we understand the concept of Base Units, let us discuss their successors: Derived Units. As the name implies, Derived Units are derived from the base units. For example, we all understand that density is a normalized measure of the amount of mass within a given volume: this is a Derived Unit that is calculated simply by dividing our units of mass by our units of volume (Density = Mass / Volume). Referencing the SI and Imperial examples shown in the table above, this would mean that the units for density would become kg/m3 and slug/ft3, respectively (remember, the unit of length must be cubed to represent volume, which is also a Derived Unit). This same concept can be extended for a litany of other measurements and, as we saw with the density example, Derived Units can be used to further derive more Derived Units.
Stress, for instance, is derived from force and area (Stress = Force / Area); force is derived from mass and acceleration (Force = Mass * Acceleration); acceleration is derived from length and time (Acceleration = Length / Time^2, both Base Units); and, lastly, area is derived from length (Area = Length^2, a Base Unit). When we work our way backwards like this, it becomes very apparent that all of our familiar engineering units – density, force, stress, energy, power, etc. – all stem from the same set of Base Units. This means that every value we calculate when performing FEA (or even hand calculations, for that matter) is inherently related to the others (via the Base Units), which gives rise to our final (and most important) rule: unit systems must be Consistent!
Consistent Unit Systems
The notion of a Consistent Unit System can be explained quite simply: the inputs and outputs of your calculations will always speak the same language. This means that if you define a problem using standard SI units, the calculated answer will also be with respect to those same SI units. Alternatively, if you define your problem using mixed units, your solution will also be presented with respect to the mixed units. This is perhaps best illustrated by a simple example:
Imagine a simulation in which length is measured in mm (SI) and mass is measured in slugs (Imperial). What are our Derived Units for stress? Since we know that stress is derived from force and area, let us first start by defining the appropriate units for force (which is derived from acceleration and mass).
So, what does this actually mean? Well, if you’re interested in reviewing stress results, it means that they will be presented using strange units: Slug / mm.s^2. And, technically, there is absolutely nothing wrong with this – this is a real answer with real meaning. But does it make sense to you? Can you conceptualize what this stress value means with respect to the material limits? Since engineering data is most commonly presented with respect to standard SI or Imperial units (and not a mix of the two), the answer is almost certainly “no” – although you may be predicting the correct answer, this value of stress is not relatable to most people and is therefore not of much value to anybody reviewing the results.
To avoid this confusion (along with unintentional errors), it is highly advisable that analysts avoid mixing units and instead utilize one of the standardized unit systems shown below.
Standardized Unit Systems
Now that we’ve covered the basic operating principles and can recognize the relationship between Base Units and Derived Units, and we understand the importance of Consistent Units, let’s explore some of the advantages, disadvantages, and differences between the two most commonly used unit systems.
International System of Units (SI)
Otherwise known as the metric system, the International System of Units (SI) is the most commonly used unit system globally. First introduced in France in the late 1700’s, the metric system was originally developed based on consistent measures of length and weight that could be derived from nature. More specifically, the metric unit of length (meter) was originally based on the dimensions of Earth (one ten-millionth of the distance from the Equator to the North Pole) and the unit of mass (kilogram) was based on the mass of one volumetric liter of water. However, with advancements in science and technology, the definitions have become much more precise over the years: today, for instance, the definition of a meter is the length of the path travelled by light in a vacuum in 3.34 x 10^-9 seconds. But other than precision, which could certainly be achieved for other unit systems, why would some favor SI units over the Imperial System?
When we think about the metric system, one of the first things that comes to mind is the ease of scalability: all units are relatable by a base factor of 10; and the consistent nomenclature used makes understanding scale rather intuitive. The most relatable example of ease of scale can be highlighted by comparing the units of length between the metric and imperial systems.
1 Yard = 3 Feet = 36 Inches - Variable scale factors between units of length
1 Meter = 10 Decimeters = 100 Centimeters = 1000 Millimeters - Constant scale factor between units of length
Additionally, the consistent nomenclature used makes it easy to understand scale regardless of the type of measurement being referenced: whether we are discussing mass, length, or volume, the prefix “milli” always means 10^-3 x Base Units. For example, one milligram is one-thousandth of one gram, one millimeter is one-thousandth of one meter, and one milliliter is one-thousandth of one liter. Pretty easy to understand, right? Let’s compare that with a similar example using the U.S. Customary System of Measurements (which is another name for the Imperial System): one pound is one-fourteenth of one stone, one inch is one-twelfth of one foot, and one quart is one-fourth of one gallon. When you consider all of the other possible intermediary measures available within the Imperial System (ounces, pints, tons, yards, etc.), it’s easy to understand how the lack of consistency between naming conventions and scale factors can make complex unit conversions quite challenging (or, at least very tedious!).
Although the metric system provides a very convenient and scalable means for understanding units, there is one primary disadvantage: it is not well-suited for working with fractions. For example, one-sixth of one meter is roughly 167.66 millimeters, which is a bit clunky. While this may be somewhat confusing when first learning fractions, it certainly isn’t an impediment for experienced engineers.
More challenging, perhaps (at least for my American comrades), is familiarizing oneself with a new unit system. When you are used to thinking about temperature in terms of degrees Fahrenheit, it is not immediately intuitive what the equivalent value is in degrees Celsius; when you’re used to driving 70 miles per hour on the expressway, it’s strange to see signs posted in kilometers. Why? Simply because these numbers are not relatable to us – we don’t have a baseline or benchmark understanding of what these units “feel like” because most of us spent the better parts of our lives mostly oblivious to the metric system. But, as a full-fledged convert, I can assure you that it doesn’t take long to gain an appreciation for the metric system – in fact, at this point, I have a hard time thinking of things in Imperial Units and often find myself converting to metric!
U.S. Customary System of Measurements (Imperial System)
The current unit system used (almost exclusively) by the United States of America traces its origins back to 1824, when the Imperial System was officially adopted in Great Britain. However, the true origins of many of these measurements far predate the official adoption. For example, a length of one yard was originally based on the dimensions of a rod or stick, while one inch was equal to the length of three barleycorns. Another interesting historical measure is that of the “rod”, which was defined as the length of the left feet of 16 men lined up heel to toe as they emerged from church. While certainly interesting, it’s clear that these early measurements were not very precise (hence the need for standardization in 1824). Between 1824 and 1963, several differences existed between the US and British systems of measurement; however, in 1959 most major English-speaking nations agreed that the inch, yard, and pound would be defined as a function of their metric equivalents (e.g., the official definition of one pound is 0.4536 kilograms). Great Britain officially converted to the metric system only a few years later, in 1965; today, the US is the only major economy in the world still using a version of the Imperial System (whereas nearly all other industrialized nations use the metric system). This raises an interesting question – why is the US still using what seems to many to be an outdated and overly complex unit system?
The simple answer is this: that’s how we’ve always done things and the legacy costs of replacing the Imperial System outweigh the benefits of doing so. For example, many of the scientific and engineering standards used in the US are based on the Imperial System; road signs, weight limits, government documents (and corporate standards, for that matter) would all need to change if the US were to change unit systems. But would it be worth it to have a consistent global standard with reduced complexity? Other than the “it would be expensive to change” argument, there aren’t a lot of advantages to our current system of measures – as we discussed earlier, the scalability and nomenclature are inconsistent and the only real benefit of the Imperial System (compared to the metric system) is that it is well suited for fractions (which, again, doesn’t seem to be a huge advantage to anybody that understands a decimal system and owns a calculator).
Contrarily, there are a couple of disadvantages one may encounter when working with the U.S. Customary Units of Measurements. The most notable for us as engineers is the relationship between mass and force. Many of us hear the word “pounds” and immediately (and usually erroneously) think of mass. A heavier person is obviously more massive than a skinny person – so “pounds” (in this context) must be a measure of mass, right? Absolutely not! We must remember that the concept of weight is actually a measure of force: more specifically, it is the force resulting from the mass of your body being accelerated by Earth’s gravity. So, when we tell somebody we weigh 200 pounds, what we are really telling them is a measure of force, not a measure of mass. This idea can be extrapolated to explain why a person weighs less on the moon than they do on Earth – even though we have exactly the same mass on Earth and on the moon, the gravitational force on the moon is substantially less than that on Earth (and, therefore, the force exerted by our bodies – our weight – is reduced on the moon).
So, that brings us to an interesting point – if pounds are a measure of force, what units are used to describe mass in the Imperial System? The answer is easy: Slugs (or Blobs)!
And now onto the final part of the post, the part you've all been waiting for...
But what in the Blob is a Slug, anyway?
Well, to answer this question, we need to first think about the how force is calculated. But, before we do that, let’s add another layer of confusion to the mix: pounds can also be a measure of mass (in addition to a measure of force). So, let’s make a clear distinction between the two right now: the term pound-force (lbf) will always refer to a measure of force and the term pound-mass (lbm) will refer to a measure of mass. With that clear, let’s now look at the definition of a pound-force: A pound-force is the amount of force required to accelerate a slug at a rate of 1 ft/s^2.
But we still don’t know what a Slug is or where it comes from? Well, the answer here is simple – this is a product of Earth’s gravity. Another way to write the equation for a pound-force is as so:
Since we know the base units must be provided in terms of Slugs (and not lbm, as shown in the equation above), we can deduce that 1 Slug = 32.174049 lbm.
And, for the record, a blob is a slug multiplied by 12, since now we're choosing inches rather than feet for our base length unit, hence 1 Blob = 386.087056 lbm.
Final Thoughts
As we discussed in this article, a strong comprehension of unit systems is incredibly important for obtaining accurate results – understanding Base Units, Derived Units, and maintaining Consistency between them is paramount. With that said, it should also be obvious now that unit systems are nothing more than mathematical languages: identical information can be presented in two completely different forms, yet the underlying meaning is exactly the same (much like a bilingual speaker saying “hello” in two different languages). With that said, though, it is my opinion that some languages are easier to learn than others. And when it comes to unit systems, the simplicity and consistency of the metric system is far superior to the Imperial System in this analyst’s opinion (apologies to all of our friends in the aerospace industry)!
Hopefully by now you have a strong grasp on unit systems and how to work interoperably between them. But, if not, you can always reach out to the expert team at Fidelis and we’ll help get you on the right track (that being metric, of course 😊).