Hydraulic and Pneumatic Cylinder Calculator (Force, Speed, Sizing, Buckling)
Enter the bore, rod, pressure and stroke of a linear cylinder, hydraulic or pneumatic. The calculator returns the piston, rod and annulus areas, the extend, retract and regeneration forces, the speeds, times and flows for a given pump flow, the swept volumes, and a rod buckling check against Euler. Size mode works backwards from a required force to the minimum bore and the next standard bore. Values are SI by default (mm, bar, L/min, kN) with an inch toggle that switches to psi, US gal/min, lbf and the NFPA bore series. The inputs are stored in the page URL so a case can be shared, exported as CSV or printed as a report.
Results
Areas
- Piston area
- 3117.245 mm²
- Rod area
- 1017.876 mm²
- Annulus area
- 2099.369 mm²
- Area ratio
- 1.485
Forces
- Extend force
- 56.11 kN
- Retract force
- 37.79 kN
- Regeneration force
- 18.32 kN
Speeds and flows
- Extend speed
- 213.9 mm/s
- Retract speed
- 317.6 mm/s
- Regeneration speed
- 655.0 mm/s
- Extend time
- 2.338 s
- Retract time
- 1.575 s
- Rod-side outflow while extending
- 26.94 L/min
- Cap-side outflow while retracting
- 59.39 L/min
Volumes and power
- Swept volume, cap side
- 1.559 L
- Swept volume, rod side
- 1.050 L
- Fluid power
- 13.33 kW
Buckling
Below target- Rod second moment
- 82,448 mm⁴
- Effective length
- 1000.000 mm
- Euler critical load
- 170.9 kN
- Buckling safety factor
- 3.05
- Target safety factor
- 3.5
Euler column with the extend force as the rod load.
Every value below uses the inputs above and updates as you edit them. The lines work in MPa, mm, mm² and N (1 MPa × 1 mm² = 1 N), so each product closes without hidden factors.
1. Areas
Pressure on the cap side acts on the full piston area. On the rod side the rod takes up part of the piston, so pressure acts on the annulus that is left.
Piston area: A_p = π × D² / 4 = π × 63.000² / 4 = 3117.245 mm²
Rod area: A_r = π × d² / 4 = π × 36.000² / 4 = 1017.876 mm²
Annulus area: A_a = A_p − A_r = 3117.245 mm² − 1017.876 mm² = 2099.369 mm²
Area ratio: φ = A_p / A_a = 3117.245 mm² / 2099.369 mm² = 1.485
2. Forces
Force is pressure times the area it acts on, times the efficiency. A back pressure on the return side pushes against the stroke on the other area. In regeneration the rod side feeds the cap side, so the two faces cancel except for the rod area.
Pressure: p = 200.0 bar (20.00 MPa)
Extend force: F_ext = p × A_p × η = 20.00 MPa × 3117.245 mm² × 0.90 = 56.11 kN
Retract force: F_ret = p × A_a × η = 20.00 MPa × 2099.369 mm² × 0.90 = 37.79 kN
Regeneration force: F_reg = p × A_r × η = 20.00 MPa × 1017.876 mm² × 0.90 = 18.32 kN
3. Speeds and flows
With an incompressible fluid the piston moves at the flow divided by the area being filled. The oil pushed out of the other side is that speed times the other area, so the cap side returns more than the pump delivers while retracting.
Flow: Q = 40.00 L/min = 666667 mm³/s
Extend speed: v_ext = Q / A_p = 666667 mm³/s / 3117.245 mm² = 213.9 mm/s
Retract speed: v_ret = Q / A_a = 666667 mm³/s / 2099.369 mm² = 317.6 mm/s
Regeneration speed: v_reg = Q / A_r = 666667 mm³/s / 1017.876 mm² = 655.0 mm/s
Extend time: t_ext = L / v_ext = 500.000 mm / 213.9 mm/s = 2.338 s
Retract time: t_ret = L / v_ret = 500.000 mm / 317.6 mm/s = 1.575 s
Rod-side outflow while extending: Q_out = Q × A_a / A_p = 40.00 L/min × 2099.369 mm² / 3117.245 mm² = 26.94 L/min
Cap-side outflow while retracting: Q_out = Q × A_p / A_a = 40.00 L/min × 3117.245 mm² / 2099.369 mm² = 59.39 L/min
Fluid power: P = p × Q / 600 = 200.0 bar × 40.00 L/min / 600 = 13.33 kW
4. Sizing
The area needed is the load, times the safety factor, divided by the pressure and the efficiency. The bore that gives that area is then rounded up to the next bore in the ISO 3320 series.
Switch the mode to Size to work out the minimum bore for a required force. The ticks show where the entered bore sits in the series.
5. Buckling
The extended rod is checked as an Euler column. The mounting sets the effective length factor K (1.0 here), and the critical load falls with the square of the effective length.
Rod second moment: I = π × d⁴ / 64 = π × 36.000⁴ / 64 = 82448 mm⁴
Effective length: L_eff = K × L = 1.0 × 1000.000 mm = 1000.000 mm
Euler critical load: F_cr = π² × E × I / L_eff² = π² × 210000 MPa × 82448 mm⁴ / 1000.000² mm² = 170.9 kN
Buckling safety factor: SF = F_cr / F_ext = 170.9 kN / 56.11 kN = 3.05 < 3.5, below target
Cross-section
Forces
Method
Areas and forces. The piston area is A_p = π D² / 4 and the rod area A_r = π d² / 4, where D is the bore and d the rod diameter. The annulus area on the rod side is A_a = A_p − A_r and the area ratio is A_p / A_a. The extend force is F = (p × A_p − p_b × A_a) × η and the retract force F = (p × A_a − p_b × A_p) × η, where p is the supply pressure, p_b an optional back pressure on the return side and η a single mechanical efficiency covering seal friction and other losses. In regeneration (hydraulic only) the rod-side port is connected to the cap side, so the net area is the rod area and F = p × A_r × η.
Speeds, flows and power. With a pump flow Q and an incompressible fluid, the extend speed is v = Q / A_p, the retract speed v = Q / A_a and the regeneration speed v = Q / A_r. Stroke times are t = L / v. While extending, the rod side returns Q × A_a / A_p; while retracting, the cap side returns Q × A_p / A_a, which is larger than the pump flow and sizes the return line and valve. Swept volumes are A_p × L and A_a × L. Fluid power is p × Q, shown as an indicative input power for both fluids.
Sizing. For a required force F with a load safety factor SF, the effective area needed is A_req = F × SF / (p × η). For extend this is the piston area; for retract the piston area must also cover the rod, A_p = A_req + A_r. The minimum bore is D = √(4 A_p / π), which is then snapped up to the next bore in the standard series: ISO 3320 for hydraulic cylinders, ISO 6432 and ISO 15552 for pneumatic cylinders, or NFPA T3.6.7 inch bores with the inch toggle. All other results are then given for the snapped bore, and the pressure needed to deliver the force at the bore entered in Analyse mode is reported alongside.
Buckling. The rod is treated as an Euler column (Budynas and Nisbett, Shigley’s Mechanical Engineering Design). Its second moment of area is I = π d⁴ / 64. The free length L is the distance between the mounting points at full extension, 2 × stroke unless entered, and the effective length is K × L, with K = 1.0 for a clevis at both ends, 2.0 for a fixed cap with a free rod end, 0.7 for a fixed cap with a guided rod end and 0.5 for both ends fixed. The critical load is F_cr = π² E I / (K L)² with E = 210 GPa for steel, and the safety factor is F_cr divided by the extend force (or by F × SF in Size mode). A factor of at least 3.5 is flagged as a pass.
Assumptions
- The rod behaves as a slender Euler column; the barrel and the rod-to-piston joint add no flexibility.
- There is no side load on the rod.
- One efficiency factor covers seal friction and mechanical losses for every stroke.
- The fluid is incompressible for speeds and flows; leakage is ignored.
- Pneumatic speeds are ideal and indicative only. Real speed depends on valve flow capacity, air compressibility and the load.
- No cushioning, pressure drop in lines and ports, or acceleration forces are included.
- Standard bores and rods are limited to the series listed in the standards table.
Standards and references
| Standard | Note |
|---|---|
| ISO 3320:2013, Fluid power systems and components, cylinder bores and piston rod diameters | Hydraulic bore series 25 to 500 mm and rod series 12 to 360 mm used for snapping. |
| ISO 6020-2:2015, Hydraulic fluid power, mounting dimensions for single rod cylinders, 16 MPa series | Typical bore and rod pairings for industrial hydraulic cylinders. |
| ISO 15552:2018 and ISO 6432:2015, Pneumatic fluid power cylinders | Pneumatic bore series 8 to 320 mm used for snapping. |
| NFPA T3.6.7 R3-2009, Square head industrial fluid power cylinders, mounting dimensions | Inch bores 1.5 to 14 in and rods 0.625 to 10 in used with the inch toggle. |
| Budynas, R. G. and Nisbett, J. K., Shigley’s Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 | Euler column formula and end-condition constants for the rod buckling check. |
| ISO 4413:2010 and ISO 4414:2010, Hydraulic and pneumatic fluid power, general rules and safety requirements | Requirement to design against buckling and overload; context for the safety factor. |
Worked example: 63 mm hydraulic cylinder
A hydraulic cylinder with a 63 mm bore and a 36 mm rod runs at 200 bar with an efficiency of 0.9, a pump flow of 40 L/min and a 500 mm stroke. It is mounted with a clevis at both ends, free length 1000 mm (2 × stroke), steel rod E = 210 GPa. This is the default case loaded in the calculator.
| Result | Value |
|---|---|
| Piston area | 3117.245 mm² |
| Rod area | 1017.876 mm² |
| Annulus area | 2099.369 mm² |
| Area ratio | 1.485 |
| Extend force | 56.11 kN |
| Retract force | 37.79 kN |
| Regeneration force | 18.32 kN |
| Extend speed | 213.9 mm/s |
| Retract speed | 317.6 mm/s |
| Regeneration speed | 655.0 mm/s |
| Extend time | 2.338 s |
| Retract time | 1.575 s |
| Rod-side outflow while extending | 26.94 L/min |
| Cap-side outflow while retracting | 59.39 L/min |
| Swept volume, cap side | 1.559 L |
| Swept volume, rod side | 1.050 L |
| Fluid power | 13.33 kW |
| Rod second moment | 82,448 mm⁴ |
| Euler critical load | 170.9 kN |
| Buckling safety factor | 3.05, below 3.5, so flagged |
Frequently asked questions
Why is the retract force smaller than the extend force?
On retract the pressure acts on the annulus, the piston area minus the rod area, so the same pressure gives less force. The default 63/36 mm cylinder gives 56.11 kN extending and 37.79 kN retracting at 200 bar. Retract speed is higher for the same reason: the same flow fills a smaller volume.
What is regeneration and when is it used?
In a regenerative circuit the oil leaving the rod side is fed back into the cap side while extending. The net area is then the rod area, so the cylinder extends much faster with much less force: 655 mm/s and 18.32 kN for the default case. It suits a fast approach stroke before a slow, full-force pressing stroke. Regeneration is shown for hydraulic cylinders only.
What does the efficiency factor cover?
It is one factor for the difference between the theoretical force p × A and the force at the rod end, mainly seal friction and mechanical losses. 0.9 is a common design value for hydraulic cylinders; pneumatic cylinders at low pressure are often lower. Use 1.0 for the theoretical force.
How does the bore snapping work in Size mode?
The calculator works out the minimum bore that delivers the required force at the stated pressure and efficiency, then picks the smallest bore in the standard series that is equal or larger. For 50 kN extending at 160 bar and η 0.9 the minimum bore is 66.49 mm and the next ISO 3320 bore is 80 mm. If the force is beyond the largest bore in the series, the results use the minimum bore and say so.
How accurate are pneumatic speeds?
Only as a first estimate. Air is compressible and real speed is set by the valve flow capacity, the supply line, the exhaust restriction and the load, so the calculated speed is an ideal value. Size pneumatic valves with the manufacturer’s flow data and treat the calculated speed as an upper bound.
What does a buckling safety factor of 3.5 mean?
The Euler critical load is at least 3.5 times the working compressive load on the rod. A margin of 3 to 4 is common practice for cylinder rods because real rods have misalignment, gland clearance and an initial bow that Euler theory ignores. Below the target the calculator flags the rod so it can be checked against the manufacturer’s buckling chart.
What free length should I use for buckling?
Use the distance between the two mounting points with the rod fully extended. For a clevis-mounted cylinder that is pin to pin. If it is left blank the calculator uses 2 × stroke, a rough allowance for the barrel and the extended rod.