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Reply #68: A simple example for structural scaling. [View All]

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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Sat Oct-08-05 06:52 AM
Response to Reply #54
68. A simple example for structural scaling.
In order to show how complicated it can be to build a scale model that accurately simulates the behavior of the original, I have written a quick example showing how to scale a simple column.

In engineering, the amount of "stress" in a structural member is related to the force exerted on that member. The general formula is that the stress equals the force divided by the cross-sectional area of the structural member. For the imperial system, stress is usually in units of pounds per square inch (psi), while force is in pounds-force (lbf) and cross-sectional area is in square inches.

If we decide to model the stress in a simple column - say, a cylinder - by using a 1/4 scale model, we need to make sure the stress in the model is the same as it would be in the full-size version.

Note: Uppercase variables represent the original while lowercase are used for the 1/4 scale model

The formula for the cross-sectional area of a cylinder is easy - area equals pi times the square of the radius.

a = pi*r^2

Since we are using a 1/4 scale model, the model radius (r) is one-fourth the radius of the original (R).

r = R/4

This means that

a = pi*(R/4)^2 = (pi*R^2)/16 (1/4 scale model)

while

A = pi*R^2 (original)

Since

P = F/A

and we want P(model) = P(original)

f/a = F/A

or

f/((pi*R^2)/16) = F/(pi*R^2)

If we do a little algebra, we can get

f = F/16

What does this mean? It means when you use a 1/4 scale model, in order to correctly model the stress of the original (assuming similar material properties) you need to cut the force applied to the model by a factor of 16 - not 4 - even though the model is 1/4 scale.

It gets even more complicated when considering other behavior to be modelled - choices have to be made because the scaling of various factors might not be by the same amount, or even in the same direction!


I hope this helps. :)
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