Hydraulic Cylinder Sizing: Formulas, Calculations, and Selection Steps
Hydraulic cylinder sizing determines whether a cylinder can do the job. Undersize the cylinder, and it cannot move the load. Oversize it, and the machine wastes energy, space, and money. The sizing process connects the application’s force and speed requirements to the cylinder’s bore, rod diameter, and stroke.
This guide covers the formulas used in hydraulic cylinder sizing, the calculation steps for force, speed, and rod strength, and the factors that affect the final selection. It is written for equipment engineers, machine designers, and buyers who need to specify a cylinder for a specific application. For the complete selection process, see this guide to hydraulic cylinder selection.
The Core Formula: Force, Pressure, and Area
Hydraulic cylinder sizing starts with the relationship between force, pressure, and piston area.
Force = Pressure × Area
The piston area depends on the bore diameter:
Piston area = π × (Bore ÷ 2)²
For a cylinder with a 100 mm bore, the piston area is π × (50 mm)² = 7,854 mm². At a working pressure of 200 bar (20 MPa), the extension force is:
Force = 20 MPa × 7,854 mm² = 157,080 N ≈ 16 tonnes
To find the bore required for a known force and pressure, rearrange the formula:
Bore = 2 × √(Force ÷ (π × Pressure))
This is the starting point for hydraulic cylinder sizing. The bore determines the extension force at a given pressure. If the available pressure is limited, a larger bore is needed to produce the same force. If the bore is fixed, higher pressure produces more force.
Calculating Extension and Retraction Force
A double-acting cylinder produces different forces in extension and retraction because the rod reduces the effective area on the rod side.
Extension force = Pressure × Full piston area
Retraction force = Pressure × (Piston area − Rod area)
The rod area is:
Rod area = π × (Rod diameter ÷ 2)²
For a cylinder with a 100 mm bore and a 50 mm rod, the full piston area is 7,854 mm² and the rod area is 1,963 mm². The annular area is 7,854 − 1,963 = 5,891 mm². At 200 bar, the retraction force is:
Retraction force = 20 MPa × 5,891 mm² = 117,820 N ≈ 12 tonnes
The retraction force is about 75 percent of the extension force. A larger rod reduces the retraction force further; a smaller rod brings the two forces closer together. Engineers account for this difference when the cylinder must produce a specific force in both directions.
Calculating Cylinder Speed
The cylinder’s speed depends on the flow rate and the effective area.
Extension speed = Flow rate ÷ Full piston area
Retraction speed = Flow rate ÷ Annular area
For a cylinder with a 7,854 mm² piston area and a pump flow of 30 L/min (500,000 mm³/s):
Extension speed = 500,000 ÷ 7,854 = 64 mm/s
For the same cylinder with a 5,891 mm² annular area:
Retraction speed = 500,000 ÷ 5,891 = 85 mm/s
The retraction speed is higher because the annular area is smaller. This is a characteristic of double-acting cylinders: retraction is faster than extension for the same flow rate.
If the application requires a specific speed, the flow rate must match the area. A larger bore slows the cylinder for a given flow; a smaller bore speeds it up. The pump, the valve, and the hose size all affect the actual flow rate the cylinder receives.
Rod Diameter and Buckling Strength
The rod diameter affects both the retraction force and the rod’s resistance to buckling. A long-stroke cylinder operating in compression can buckle if the rod is too thin.
The buckling strength depends on the rod diameter, the stroke length, the mounting configuration, and the load. The calculation uses the Euler buckling formula or the Rankine formula, depending on the slenderness ratio.
Euler buckling load = π² × E × I ÷ (K × L)²
Where E is the modulus of elasticity, I is the moment of inertia of the rod cross-section, K is the effective length factor (which depends on the mounting), and L is the stroke length.
For a rod with a circular cross-section, the moment of inertia is:
I = π × (Rod diameter)⁴ ÷ 64
A rod that is too thin for the stroke and the load will buckle before it reaches the cylinder’s rated force. The rod diameter must be increased until the buckling load exceeds the maximum compressive load with a safety factor.
The mounting configuration affects the effective length. A cylinder with both ends pivoted (clevis mounting at both ends) has a higher effective length factor than one with one end fixed. This means a pivoted cylinder needs a larger rod for the same stroke and load.
Stroke Length and Installation Dimensions
The stroke length determines how far the cylinder can move the load. It must match the machine’s required travel plus a margin for adjustment.
Stroke = Required travel + Margin
The margin accounts for manufacturing tolerances, thermal expansion, and the need to avoid bottoming out the piston against the end cap. A typical margin is 5 to 10 mm.
The stroke also affects the cylinder’s retracted and extended lengths. The retracted length is the distance from the cap end to the rod end when the cylinder is fully retracted. The extended length is the retracted length plus the stroke.
These dimensions must fit the machine’s installation space. The mounting configuration, the port location, and the clearance around the cylinder all constrain the dimensions. A cylinder that meets the force and speed requirements but does not fit the installation space cannot be used.
Accounting for Pressure Losses and Safety Margin
The calculations above use the theoretical pressure and flow. In a real system, pressure losses in the hoses, valves, and fittings reduce the pressure available at the cylinder. The sizing calculation should use the actual pressure at the cylinder, not the pump’s rated pressure.
A safety margin should also be applied to the force calculation. The margin accounts for:
Friction in the cylinder and the machine
Load variation during operation
Shock loads and impact
Wear over the cylinder’s service life
A typical safety margin is 1.2 to 1.5 times the calculated force. The margin ensures the cylinder can handle the load even when conditions are less favorable than the design point.
The system pressure should also be checked against the cylinder’s rated pressure. A cylinder rated for 250 bar should not be operated at 300 bar, even if the force calculation requires it. In that case, a larger bore at the lower pressure is the safer choice.
A Step-by-Step Sizing Example
A practical example shows how the formulas work together.
Application: A hydraulic press needs to produce 50 tonnes of force. The available system pressure is 200 bar. The stroke is 400 mm. The cylinder mounts with a clevis at both ends.
Step 1: Calculate the required piston area.
Area = Force ÷ Pressure = (50,000 kg × 9.81 m/s²) ÷ 20 MPa = 490,500 N ÷ 20 MPa = 24,525 mm²
Step 2: Calculate the required bore.
Bore = 2 × √(24,525 ÷ π) = 2 × √7,806 = 2 × 88.4 = 176.7 mm
The nearest standard bore is 180 mm. Using 180 mm gives a piston area of 25,447 mm², which produces about 51.9 tonnes at 200 bar—enough for the application with a small margin.
Step 3: Select the rod diameter.
For a 400 mm stroke with clevis mounting at both ends, the effective length is 400 mm (K = 1 for pivoted-pivoted). The buckling load must exceed 490,500 N with a safety factor. A 90 mm rod has a moment of inertia of π × 90⁴ ÷ 64 = 3,220,000 mm⁴. Using E = 210,000 MPa and K = 1:
Buckling load = π² × 210,000 × 3,220,000 ÷ 400² = 41,800,000 N
The buckling load far exceeds the applied load, so the 90 mm rod is adequate. In practice, the rod diameter is often selected based on the retraction force requirement and the standard rod-to-bore ratio rather than buckling alone.
Step 4: Check the installation dimensions.
The 180 mm bore cylinder with a 400 mm stroke has a retracted length that depends on the mounting style. The extended length is the retracted length plus 400 mm. Both must fit the machine’s available space.
Step 5: Confirm the speed.
If the press must extend in 5 seconds, the required flow is:
Flow = Area × Stroke ÷ Time = 25,447 mm² × 400 mm ÷ 5 s = 2,035,760 mm³/s = 122 L/min
The pump and valve must deliver 122 L/min at 200 bar to meet the cycle time.
This example shows how the sizing calculation connects the force requirement to the bore, the stroke to the speed, and the mounting to the rod diameter. Each step builds on the previous one, and a change in one parameter affects the others.
What to Confirm Before Finalizing the Size
The sizing calculation produces a preliminary cylinder size. Before finalizing, confirm:
Actual system pressure at the cylinder, not the pump rating
Pressure losses in the hoses and valves
Safety margin for friction, shock, and wear
Rod buckling for long-stroke compression applications
Installation dimensions including retracted length, extended length, and mounting
Port size and location for the hydraulic connections
Cushioning requirements at the end of the stroke
Seal compatibility with the hydraulic fluid and the operating temperature
Hydraulic cylinder sizing is a calculation process, but it is also a design process. The formulas give the minimum size, and the application requirements—space, mounting, speed, safety—determine the final choice. For a deeper look at the components that make up the cylinder, see this guide to hydraulic cylinder parts. For the full selection process that includes the sizing calculation, see this guide to hydraulic cylinder selection.