Section Index Owatonna Steele County Amateur Radio 08-Oct-2014

## Horizontal Element Droop

When designing antennas with horizontal elements (Yagis), one consideration is how much the horizontal elements will droop (sag) under their own weight. The information presented here is intended to provide an estimate. The calculations assume the horizontal element is single piece of aluminum tubing.

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:

Droop = ( w * L4 ) / ( 8 * E * I )

"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

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