Sin the Height. Cos the Radius. — Boom Geometry

The boom is a right triangle whether anyone on the crew admits it. The hypotenuse is the boom length you have out. The angle at the pin, from the horizontal, is on the indicator. Opposite that angle is height. Adjacent is radius. Sin and cos. That is the whole trick.

This is a straight main boom on a mobile crane. No luffing jib. No lattice fly until you treat that as a different triangle. If the boom is not a straight stick from pin to sheave, stop using this and get the range diagram for the configuration you actually have.

WHAT θ IS

θ is boom angle from the horizontal. That is how CSA Z150 and ASME B30.5 charts talk, and it is what most angle indicators read. Straight up is 90°. Laid over is toward 0°. Do not invent a second convention on the radio.

If someone gives you the angle from vertical, it is the same triangle with the names swapped: height becomes L × cos of that angle, radius becomes L × sin of that angle. Do not use both and average them.

The boom is doing sin() and cos() whether you are or not.

THE FORMULAS

L is the boom length in use — the telescoped length, not the stowed length. H is tip height above the boom foot pin. R is horizontal from the pin to the sheave, along the ground.

Load-chart radius is measured from the centre of rotation, not from the pin. The pin sits a distance D in front of the slew centre. That D is on the range diagram for that crane — not in your head.

Hook height above ground is another correction: pin height above the pads, minus the block hanging under the sheave, minus whatever parts of line you have reeved. Do not tell a rigger 'the boom is 28 metres' when the hook is 25. They are rigging to the hook.

WORKED BOOM — 30 m

Keep L, H and R in the same unit. Metres in, metres out. The sine does not care about tonnes. Capacity lives on the chart, at the radius you actually have.

1. 01

   70° SIN 70° = 0.940 COS 70° = 0.342

   H = 30 × 0.940 = 28.2 m above the pin. R = 30 × 0.342 = 10.3 m from the pin. Steep. Short radius. This is where most charts still have muscle.

2. 02

   60° SIN 60° = 0.866 COS 60° = 0.500

   H = 26.0 m. R = 15.0 m. You gave away 2.2 m of height and bought 4.7 m of radius. The load did not get heavier. The moment did.

3. 03

   45° SIN 45° = 0.707 COS 45° = 0.707

   H = 21.2 m. R = 21.2 m. Height and radius from the pin are the same number. If you needed 26 m of height, you no longer have it.

4. 04

   30° SIN 30° = 0.500 COS 30° = 0.866

   H = 15.0 m. R = 26.0 m. You are long, low, and usually out of the fat part of the chart. Booming down is not 'making it easier.'

Add D and those radii grow. A 1.5 m pin offset at 60° turns 15.0 m from the pin into about 16.5 m from the slew. Charts are picky about that metre. Guessing D is how you pick a radius you do not have.

THE TABLE YOU SHOULD KNOW COLD

Boom down: sin falls, cos rises. Height comes off, radius goes on. The LMI is watching radius. If you boom down to clear a pick and you have not re-checked the chart at the new radius, you are hoping. Hope is not a control.

USE IT

Phone calculator. Degree mode. sin() for height. cos() for radius. Check: sin(30) is 0.5. cos(60) is 0.5. If those are not true, you are in radians and every number you read out will sound official and be wrong.

WHAT THIS TRIANGLE DOES NOT DO

• It does not replace the load chart, the range diagram, or the LMI. Those already did this math for this machine.
• It does not include a jib, an offset, a stowed-jib deduction, or a sheave that does not sit on the boom centreline.
• It does not give hook height. Subtract the block. Subtract the reeving. Measure if you do not know.
• It does not make a long-radius pick legal because the height still 'looks like enough.'
• It does not fix an unknown boom length. If you do not know L, you do not have a triangle. You have a guess.

If you cannot explain the radius, you cannot use the radius.

Use this to see the shape of the lift before you argue with the chart. Then open the chart at the radius you actually have. The operator works to the chart, CSA Z150, ASME B30, the manufacturer, and the site. The triangle is so you are not surprised when booming down eats the capacity.

07 — CRANE OPERATIONS →

SIN() FOR SLING TENSION →

READ PROVEN →