Enter torque in Newton-meters and the effective radius in inches to calculate pressure in PSI. This calculator uses the circular piston model (T = P x pi x r^3), where the same dimension r serves as both the piston radius and the moment arm.
| Torque to Pressure | Pressure to Torque |
|---|---|
| 5 ft·lbf = 19.10 psi | 25 psi = 6.54 ft·lbf |
| 10 ft·lbf = 38.20 psi | 50 psi = 13.09 ft·lbf |
| 15 ft·lbf = 57.30 psi | 75 psi = 19.63 ft·lbf |
| 20 ft·lbf = 76.39 psi | 100 psi = 26.18 ft·lbf |
| 25 ft·lbf = 95.49 psi | 150 psi = 39.27 ft·lbf |
| 30 ft·lbf = 114.59 psi | 200 psi = 52.36 ft·lbf |
| 40 ft·lbf = 152.79 psi | 250 psi = 65.45 ft·lbf |
| 50 ft·lbf = 190.99 psi | 300 psi = 78.54 ft·lbf |
| 75 ft·lbf = 286.48 psi | 500 psi = 130.90 ft·lbf |
| 100 ft·lbf = 381.97 psi | 1000 psi = 261.80 ft·lbf |
| Formula: T = P x pi x r³ and P = T / (pi x r³). Radius fixed at 1 inch. | |
| Torque to Pressure | Pressure to Torque |
|---|---|
| 10 ft·lbf = 4.77 psi | 10 psi = 20.94 ft·lbf |
| 20 ft·lbf = 9.55 psi | 25 psi = 52.36 ft·lbf |
| 30 ft·lbf = 14.32 psi | 50 psi = 104.72 ft·lbf |
| 40 ft·lbf = 19.10 psi | 75 psi = 157.08 ft·lbf |
| 50 ft·lbf = 23.87 psi | 100 psi = 209.44 ft·lbf |
| 75 ft·lbf = 35.81 psi | 150 psi = 314.16 ft·lbf |
| 100 ft·lbf = 47.75 psi | 200 psi = 418.88 ft·lbf |
| 150 ft·lbf = 71.62 psi | 300 psi = 628.32 ft·lbf |
| 200 ft·lbf = 95.49 psi | 500 psi = 1047.20 ft·lbf |
| 300 ft·lbf = 143.24 psi | 1000 psi = 2094.40 ft·lbf |
| Formula: T = P x pi x r³ and P = T / (pi x r³). Radius fixed at 2 inches. | |
Nm to Psi Conversion Formula
The formula to convert torque (Nm) to pressure (PSI) using the circular piston model is:
PSI = \frac{T_{Nm} \times 8.85075}{\pi \times r^3}
Variables:
- PSI – pressure in pounds per square inch
- T – torque in Newton-meters (N·m)
- 8.85075 – conversion factor from N·m to in·lbf (1 N·m = 8.85074579 in·lbf)
- r – radius in inches; represents both the piston radius and the moment arm
Radius has a cubic effect on pressure: doubling r reduces the required pressure by a factor of 8. The table below shows PSI per 1 N·m of torque at common radii:
| Radius (in) | PSI per 1 N·m | PSI for 100 N·m | PSI for 500 N·m |
|---|---|---|---|
| 0.5 | 22.53 | 2,253 | 11,267 |
| 1.0 | 2.82 | 282 | 1,408 |
| 1.5 | 0.836 | 83.6 | 418 |
| 2.0 | 0.352 | 35.2 | 176 |
| 3.0 | 0.104 | 10.4 | 52.2 |
Why This Conversion Requires a Radius
Torque (N·m) and pressure (PSI) measure fundamentally different physical quantities: torque is rotational force times distance, while pressure is force per unit area. No single conversion factor exists between them. A geometric parameter is always required.
The formula above uses the circular piston model, where a pressurized circular piston of radius r applies a tangential force at a moment arm also equal to r. This gives T = P x (pi x r^2) x r = P x pi x r^3. This model applies to rotary vane actuators and equivalent piston-crank geometries where the piston and crank dimensions are equal.
For a hydraulic motor with a known displacement, use a different relationship: T(N·m) = P(bar) x D(cc/rev) / (20 x pi). The table below shows theoretical output torque for common motor displacements at typical operating pressures (assuming 100% mechanical efficiency):
| Displacement (cc/rev) | 1,000 PSI (69 bar) | 2,000 PSI (138 bar) | 3,000 PSI (207 bar) |
|---|---|---|---|
| 50 | 55 N·m | 110 N·m | 165 N·m |
| 100 | 110 N·m | 220 N·m | 329 N·m |
| 250 | 275 N·m | 549 N·m | 824 N·m |
| 500 | 549 N·m | 1,098 N·m | 1,647 N·m |
| 1000 | 1,098 N·m | 2,196 N·m | 3,294 N·m |
| Formula: T(N·m) = P(bar) x D(cc/rev) / (20 x pi). Actual torque is lower due to mechanical efficiency, typically 0.90 to 0.97 for gear motors and 0.85 to 0.95 for piston motors. | |||
How to Calculate Pressure from Torque (Nm)
- Determine the torque in Newton-meters (N·m).
- Determine the effective radius in inches. For a rotary vane actuator, this is the vane radius. For a crank-piston, this is the crank radius (which must equal the piston radius for this formula to apply).
- Apply the formula: PSI = (Torque x 8.85075) / (pi x Radius^3). Note the exponent is 3, not 2 — radius appears cubed because it accounts for both the piston area (r^2) and the moment arm (r).
- The result is the hydraulic pressure in pounds per square inch required to produce that torque at that radius.
- Verify your result with the calculator above. For a different system geometry (hydraulic motor with known displacement), use the displacement formula instead.
Example Problem:
A rotary vane actuator with a 2-inch vane radius must produce 50 N·m of torque. What hydraulic pressure is required?
PSI = (50 x 8.85075) / (pi x 2^3) = 442.54 / 25.13 = 17.60 PSI
