The information required for the calculation is:

• Do - Outside diameter of the element (inches).
• Di - Inside diameter of the element (inches).
• L - Length of the element (inches).
• w - Weight of the element per inch of length (pounds per inch).
• E - Modulus of Elasticity (10,000,000 for Aluminum)

The basic formula for a round tube is:

"I" is the moment of inertia for the geometric shape. It is calculated as:

I = π * ( Do⁴ - Di⁴ ) / 64
   = 3.1416 * ( Do⁴ - Di⁴ ) / 64
   = 0.0491 * ( Do⁴ - Di⁴ )

As an example, I for a piece of tubing 1/2" OD and 3/8" ID is:

I = π * (0.5000⁴ - 0.3750⁴ ) / 64
   = 3.1416 * ( 0.0625 - 0.0198 ) / 64
   = 3.1416 * 0.0427 / 64
   = 0.0021

"w" is the weight per inch of the tubing. This may be available directly from the supplier. It can be calculated by calculating the weight of a solid aluminum bar for the outside diameter and then subtracting the weight of a solid aluminum bar for the inside diameter. Aluminum weighs approximately 0.098 pounds per cubic inch. The formula for calculating the weight of a 1 inch long round aluminum bar is:

Weight = π * 0.098 * D² / 4

As an example, a 1 inch length of a 1/2" round aluminum bar would weigh:

Weight = π * 0.098 * D² / 4
            = 3.1416 * 0.098 * 0.5000² / 4
            = 3.1416 * 0.098 * 0.0625 / 4
            = 0.0048 pounds

Likewise, a 1 inch length of a 3/8" round aluminum bar would weigh 0.0015 pounds.
for a 1 inch piece of aluminum tubing 1/2" OD and 3/8" ID:
w = 0.0048 - 0.0015 = 0.0033 pounds per inch.

Putting it all together, the estimated droop for a 48 inch long piece of 1/2" OD by 3/8" ID aluminum tubing is:

Droop = ( w * L⁴ ) / ( 8 * E * I )
           = ( 0.0033 * 48⁴ ) / ( 8 * 10,000,000 * 0.0021 )
           = ( 0.0033 * 5,308,416 ) / ( 8 * 10,000,000 * 0.0021 )
           = ( 17,518 ) / ( 168,000 )
           = 0.104 inches