### Consistent Units in FEA by Kotur Raghavan

**General**

This article can be treated as a
refresher of basics related to units and dimensions. It is an obvious need that
one should work with consistent units in order to get meaningful results. This
is true in any problem associated with the physical world and structural
analysis is no exception. Over most of the world metric system is in use. The
United States, however, is still sticking to the British system of units. It is
necessary that engineers are conversant with both the metric and the British
system of units.

Fig. 1

**Description**

1.
In any structural analysis we deal with four
basic entities. They are LENGTH, FORCE (or MASS), TIME and TEMPERATURE. All
other material properties are derived from these basic ones.

2.
In British (or imperial) system INCH is the unit
used for length. In SI (abbreviation of *Système international) *system meter (m) is the standard unit. Due to the design office and shop floor preference, millimetre (mm) has also become an accepted unit of length.

3.
In gravitational system of units “force” is the
basic entity and “mass” is the derived entity. In the absolute system of units,
“mass” is basic and “force” is derived.

4.
In British as well as SI systems, “time” is
always specified is seconds.

5.
Temperature is either degree F or degree C or
the corresponding absolute values.

6.
In structural analysis, we use a number of other
entities or properties like Young’s modulus, coefficient of thermal expansion,
acceleration, pressure. They are all derived ones.

7.
It is absolutely essential that values assigned
to the various quantities used in analysis are consistent with one another.

8.
Traditionally, the British or imperial system is
gravitational one. Force (or weight) is the basic unit. On the surface of earth
one cubic foot of water weighs about 62.4 Lbs and this is the reference data.

9.
Thus we can define ** one pound force** as the “force” exerted by
1/62.4 or 0.01026 cubic foot of water on the surface of earth. As this is the
definition of weight, we can say that weight density of water is equal 62.4
pounds per cubic foot. It is equivalent to 62.4/1728 or 0.036111 pound per
cubic inch.

10.
Mass, on the other hand, is the amount of matter
contained in a body. According to Newton’s second law, force is a product of
mass and acceleration. As the acceleration due to gravity on the earth’s
surface is 386.4 inches per square second, mass density of water is equal to
0.03611/386.4 or 0.00009345 Lb-mass per cubic inch.

11.
As we know the specific gravity of steel is
about 7.8, mass density of steel is 0.00009345*7.8 or 0.000723 units. Analysts
who have worked for American clients will be familiar with this number.

12.
Absolute or SI units define mass as the basic
entity. **Kilogram** is the standard
unit for mass.

13.
By definition, one kilogram is the amount of
matter contained in 1000 cubic centimetre (one litre) of water.

14.
In SI units ** “newton”** is the derived unit for force. One newton
(abbreviation: N) is the force generated by one kilogram of mass accelerated at
one meter per square second.

15.
Thus the force exerted by one litre of water on
the surface of earth is about 9.8 N. This is the Kilogram we us in our daily
life in the context one’s body weight, weights of groceries, vegetables etc. A
person weighing “75 Kg” actually exerts a force of about 735 newtons in the vertically
downward direction on the structure supporting him.

16.
Of all the material properties we use, one that
is absolutely essential in any structural analysis task is the modulus of
elasticity or Young’s modulus. Its units are force per unit area. In British
system the unit is pounds per square inch (PSI) and in SI units the unit is
newtons per square millimetre or megapascals (MPa).

17.
You may notice that the units used for Young’s
modulus dictates the units to be used for all other entities. For example
moments and torque are specified as pound-inch or newton-mm. Accelerations are
to be expressed as inch per square second or millimetres per square second.
Stresses have the same dimensions as the modulus of elasticity.

18.
Here is an important point while working with
newtons and millimetres. By definition, one newton is the force exerted by one
kilogram of mass accelerated at 1 meter per second. As the length (and hence
acceleration) is expressed in millimetres mass density needs to be specified in
KKg per cubic millimetre.

19.
For example, steel has density of about 7800 Kg
per cubic meter. This is equal to 7.8e-06 kilogram per cubic millimetre. For
consistency, this has to be divided by 1000 and input as 7.8E-09 KKg per cubic millimetre.

20.
Using the same logic, it follows that the
appropriate unit for discrete masses is KKg.

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This is an important write-up

ReplyDeleteI think the most unambiguous feature of SI units is giving different names to Mass (Kg) and Force (N). The good old confusion of kgm and Kgf ,and both being referred to as Kg, is once for all gotten rid off.

Point No. 18 can be emphasized by the fact that the drawings from which we make FE model inputs are in mm and that's where we depart from SI

Depending upon task at hand We can formulate our own units as say, Newtons per kilometre square while modelling say earth or newtons per nanometre square for modelling tiny objects

PBG, You are right. Millimetres are popular because of the established shop practices. The machinist or the foreman gets a better feel of the dimensions when expressed in mm rather than in metres. The design office practice has got to meet the shop floor preferences.

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