Hydraulic Cylinder Push & Pull Force Formula (F = P·A·η)
Calculate extension thrust force, retraction pull force, effective annulus area, and speed ratios for single-rod double-acting hydraulic cylinders.
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Primary Mathematical Expression
Nomenclature & Variables
| Symbol | Variable Name | Metric Unit | Imperial Unit | Description |
|---|---|---|---|---|
| F | Cylinder output force | kN | lbf | The mechanical thrust (push) or tension (pull) force generated by fluid pressure. |
| P | Operating pressure | bar | psi | The working fluid pressure supplied by the hydraulic pump. |
| d_{bore} | Cylinder bore diameter | mm | in | The inside diameter of the cylinder tube. |
| d_{rod} | Piston rod diameter | mm | in | The diameter of the piston rod. |
| A | Effective pressurized area | mm² | in² | The cross-sectional area acting against hydraulic fluid. |
| \eta | Mechanical efficiency | - | - | Seal friction and mechanical transmission efficiency (typically 0.90 - 0.95). |
Step-by-Step Derivation
- 1
Hydraulic force originates from Pascal's Principle: fluid pressure applied to an enclosed surface exerts a uniform perpendicular force per unit area: F = P * A.
- 2
For cylinder extension (Push Stroke), the entire bore surface area is exposed to pressurized fluid: A_{bore} = \pi * d_{bore}² / 4.
- 3
The ideal push force is F_{push, ideal} = P * A_{bore}. Accounting for piston seal friction: F_{push} = P * (\pi * d_{bore}² / 4) * \eta.
- 4
For cylinder retraction (Pull Stroke), the piston rod reduces the available pressurized area to an annular ring: A_{annulus} = A_{bore} - A_{rod} = \pi * (d_{bore}² - d_{rod}²) / 4.
- 5
The actual pull force is F_{pull} = P * [\pi * (d_{bore}² - d_{rod}²) / 4] * \eta.
- 6
Therefore, pull force is always strictly less than push force by the ratio (1 - d_{rod}² / d_{bore}²).
Worked Example Calculation
Calculate the push force and pull force of a hydraulic cylinder with a bore diameter of 100 mm and a rod diameter of 50 mm operating at 200 bar with 95% mechanical efficiency.
- •Calculate Piston Area: A_{bore} = \pi \times 100² / 4 = 7,853.98 mm².
- •Calculate Annulus Area: A_{annulus} = \pi \times (100² - 50²) / 4 = 5,890.49 mm².
- •Convert pressure: 200 bar = 20 MPa = 20 N/mm².
- •Compute Push Force: F_{push} = (20 N/mm² \times 7,853.98 mm² \times 0.95) / 1,000 = 149.23 kN.
- •Compute Pull Force: F_{pull} = (20 N/mm² \times 5,890.49 mm² \times 0.95) / 1,000 = 111.92 kN.
Engineering Assumptions
- •Hydraulic fluid is virtually incompressible.
- •Pressure is distributed uniformly across the entire active piston and annulus areas.
- •Mechanical seal drag accounts for approximately 5% frictional loss (\eta = 0.95).
Design Limitations
- •Does not account for dynamic pressure surges or back-pressure on the discharge port.
- •Rod column buckling is estimated using Euler's ideal column formula as a first-order screening check. Detailed standards-certified structural buckling analysis for production hydraulic cylinders (accounting for stop tubes, guide clearances, cylinder/rod step-changes, and mounting misalignment) requires ISO/TS 13725.
Academic References & Standards
Fluid power cylinders - Mounting dimensions for single rod cylinders, 160 bar compact series
International standard for industrial single-rod hydraulic cylinders.
Fluid power systems - Cylinder dimensions and column strength
National Fluid Power Association standard.
Hydraulic fluid power — Method for evaluating the buckling load of an oil-hydraulic cylinder
Standard method for evaluating compressive buckling loads in hydraulic cylinders.