NonDimensionalization

This is just a very long way of saying that Truss Solver does not know about units. Thus you must enter all numerical quantities into Truss Solver in a consistant way. The process of converting values with units to values without units suitable for entry into Truss Solver is called NonDimensionalization.

The process of nondimensionalization starts with the choice of scales for the problem. The subject of dimensional analysis considers how to determine the required set of scales for any given problem. Basic scales are length, time, mass, electron charge, etc. For static truss problems, time and mass do not explicity appear, thus we can use force (mass*length/time²) as a basic scale. Thus, to nondimensionalize your truss problem you must first pick:

• A length scale: Ls
• A force scale: Fs

Other than choosing these scales so that you enter consistant values into Truss Solver, they should also be chosen with consideration for numerical accuracy. Computers only have limited ranges of numbers they can represent in a given precision floating point number. Typical single precision numbers (REAL in FORTRAN, float in C/C++) can only have exponents in the range +36 to -36, or something close to that. In other words there is a limit on the largest and smallest numbers that the program can handle. Since there are a range of calculations done in any program, it is usually best to pick your input values near the center of this range which is the value 1.0.

This is done by picking length and force scales that are typical values for the problem under consideration. If you are considering atoms you use nanometers, if you are considering the solar system you use light-minutes (or AU -- 93 million miles, the radius of Earth's orbit). You probably have been doing this without thinking already. Similarly you want to pick a force scale that reflects the typical size of the forces involved in the problem. You need not go over board with this, pick round numbers, 1 mm, or 1 m, for length and 1 N or 1 kN for force. Whatever seems reasonable.

Once you have picked a length scale Ls ( 1 m for example) and a force scale Fs (1 kN for example) then nondimensional quantities are calculated by dividing by combinations of the scales that remove all the units. Nondimensional quantities are typically denoted with a * on the symbol.

Thus, we calculate a nondimensional length by

L* = L / Ls.

A nondimensional force is found as

F* = F / Fs.

Area has units of length squared, thus

A* = A / Ls².

Elastic Modulus has units of force per area (stress), thus

E* = E Ls² / Fs.

Enter the nondimensional quantities (*'ed values) into Truss Solver.

Numbers that are calculated by Truss Solver are then also nondimensional with the same scales. Use the formula's above, only this time you know the nondimensional values, to calculate the dimensional result values.

Note that dimensional displacements have units of length, thus, given u* from a Truss Solver Results File as the X-direction displacement of a node, then

u = Ls u*

is how you calculate the dimensional displacement.