CHST formulas: the math to know, with worked examples

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The formulas are public engineering and arithmetic, written in our own words. Always confirm a limit or table value against the current rule or your own code copy before you rely on it.

Incident rates (TRIR and DART)

An incidence rate says how many cases happen per 100 full-time workers in a year. A full-time worker is counted as 2,000 hours, so the base is 200,000 hours.

Rate = (number of cases x 200,000) / hours worked

Use all recordable cases for the total recordable incident rate (TRIR). For the DART rate, count only cases with days away, restricted work or job transfer. The recording rules are in 1904.7, and the recordkeeping guide covers them.

Worked example. A contractor logs 7 recordable cases in 410,000 hours. Three of them involved days away, restricted work or transfer.

1. TRIR = 7 x 200,000 / 410,000 = 3.41
2. DART = 3 x 200,000 / 410,000 = 1.46

Check it with the OSHA incident rate calculator.

Fall clearance

Fall clearance asks whether a worker will stop before hitting the level below. Add up every distance the body travels, then compare the total with the room available.

Required clearance = free fall + deceleration distance + D-ring to feet + D-ring shift + safety factor

The personal fall arrest rules are in 1926.502(d). The free fall limit is in (d)(16)(iii) at 6 feet, and the maximum deceleration distance is in (d)(16)(iv) at 3.5 feet. D-ring shift (stretch of the harness and slip of the strap) and the safety factor are planning values. Use the numbers from the manufacturer's instructions.

Worked example. A 6 ft lanyard is tied off at D-ring height. Deceleration is 3.5 ft, the D-ring sits 5 ft above the worker's feet, harness shift is 1 ft and the planning safety factor is 2 ft.

1. Required clearance = 6 + 3.5 + 5 + 1 + 2 = 17.5 ft
2. The anchor is 20 ft above the level below, so the margin is 20 - 17.5 = 2.5 ft

The system clears, but a 2.5 ft margin is thin. A shorter lanyard would add room. Try your own numbers in the fall clearance calculator.

Ladder setup: the 4 to 1 rule

A non-self-supporting ladder rests at a slant: the base sits one quarter of the working length away from the wall. Working length is the distance along the ladder from the foot to the support point. This is in 1926.1053(b)(5)(i). The same rule says the side rails extend at least 3 ft above the landing, in 1926.1053(b)(1).

• Base distance = working length / 4
• Angle from level = arccos(1/4), about 75.5 degrees
• Working length for a landing height h = 4h / sqrt(15), about 1.033 x h

Worked example. A landing is 12 ft above the lower level.

1. Working length = 4 x 12 / sqrt(15) = 48 / 3.873 = 12.39 ft
2. Base distance = 12.39 / 4 = 3.10 ft
3. Add the 3 ft extension above the landing: the ladder must be at least 15.4 ft long, before any overlap on an extension ladder

Try other heights in the ladder angle calculator.

Trench sloping

Sloping lays the trench wall back so it does not collapse. The maximum allowable slope depends on the soil type and is written horizontal : vertical. The five foot protection point and the stable rock exception are in 1926.652(a)(1), and the slope table is in Appendix B to Subpart P.

Soil type	Maximum slope (H:V)
Stable rock	vertical
Type A	3/4 : 1
Type B	1 : 1
Type C	1 1/2 : 1

Run = depth x H and top width = bottom width + 2 x run. Appendix B covers excavations up to 20 ft deep. Deeper cuts need a design by a registered professional engineer.

Worked example. A trench is 10 ft deep with a 4 ft bottom width, sloped on both sides.

1. Type B: run = 10 x 1 = 10 ft, top width = 4 + 2 x 10 = 24 ft
2. Type C: run = 10 x 1.5 = 15 ft, top width = 4 + 2 x 15 = 34 ft

Moving from Type B to Type C adds ten feet of top width here, which is why soil classification comes first. Run both in the trench sloping calculator, and read the excavations guide for the rest of Subpart P.

Noise dose and time-weighted average

Noise exposure follows an exchange rate of 5 dB: each 5 dBA increase halves the time allowed. The general industry rule is 1910.95, with Table G-16a and the computation in its Appendix A. The construction rule, 1926.52, uses its own table (Table D-2) with the same 90 dBA, 8 hour starting point, and the health hazards guide covers the difference.

• Reference duration: T = 8 / 2^((L - 90) / 5) hours, where L is the sound level in dBA
• Dose: D = 100 x (C1/T1 + C2/T2 + ...), where C is the hours spent at each level
• 8-hour TWA from the dose: TWA = 16.61 x log10(D / 100) + 90

A dose of 100 percent equals 90 dBA over 8 hours. The 85 dBA action level in 1910.95(c) is a dose of 50 percent: T at 85 dBA is 16 hours, so 8 hours gives 8 / 16 = 50 percent.

Worked example. A worker spends 4 hours at 92 dBA and 2 hours at 95 dBA. The rest of the shift is below 80 dBA and adds nothing.

1. T at 92 dBA = 8 / 2^0.4 = 6.06 hours, so C/T = 4 / 6.06 = 0.660
2. T at 95 dBA = 8 / 2^1 = 4 hours, so C/T = 2 / 4 = 0.500
3. Sum = 1.160, so dose = 116 percent
4. TWA = 16.61 x log10(1.160) + 90 = 91.1 dBA

The dose is over 100 percent, so the exposure is above the limit and controls come first: engineering, then administrative, then hearing protection.

Sling loads and sling angle

When a sling leaves the hook at an angle, each leg carries more than its share of the weight. The angle that matters is measured from the horizontal.

Tension per leg = load / (number of legs x sin of the angle from horizontal)

Flatter legs mean higher tension: at 30 degrees from horizontal, each leg of a two-leg sling carries the full load. Rigging rules are in Subpart H, with slings at 1926.251. Compare the result with the rated capacity at that angle, which is on the sling's tag.

Worked example. A two-leg bridle lifts a 6,000 lb load with the legs at 60 degrees from horizontal.

1. Tension per leg = 6,000 / (2 x sin 60) = 6,000 / 1.732 = 3,464 lb
2. At 30 degrees: 6,000 / (2 x 0.5) = 6,000 lb per leg

The same load puts 73 percent more tension in each leg at the flatter angle. This example assumes the load is centered and the legs share it equally.

Basic unit conversions

Keep a few exact factors in your head so a metric or imperial answer does not slip you up.

• 1 ft = 0.3048 m (exact), so 1 m = 3.281 ft
• 1 in = 25.4 mm (exact)
• 1 lb = 0.45359 kg, so 1 kg = 2.2046 lb
• 1 psi = 6.895 kPa

Worked example. A 6 ft free fall limit is 6 x 0.3048 = 1.83 m. A 4,000 lb load is 4,000 x 0.45359 = 1,814 kg.

Common mix-ups

• TRIR counts all recordables; DART counts only the severe subset. Both use the 200,000 hour base.
• Both the ladder angle and the sling angle are measured from level. A flatter sling means more tension, while the ladder is fixed at 4 to 1.
• Noise time halves every 5 dBA, not every 3. That is the OSHA exchange rate, and other agencies use different ones.
• Round at the end. Rounding between steps is how a 91.1 becomes a 91.

Practice it

Take the free 10-question check. It spans the four CHST domains, needs no signup and explains every answer, so you see fast whether the math is a weak spot. The course goes deeper with full practice and explanations built to the blueprint. The CHST study guide shows where calculation practice fits in a six-week plan.

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Sources

Show the list

• 29 CFR 1904, Recording and Reporting Occupational Injuries and Illnesses and 1926.502, Fall protection systems criteria, OSHA.
• 1926.1053, Ladders and Subpart P, Excavations, OSHA.
• 1910.95, Occupational noise exposure and 1926.52, OSHA.