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Hydraulic Cylinders Calculator — Force, Buckling & Cushioning (ISO 6020 / ISO 6022)

Governing standard: ISO 6020 / ISO 6022· ISO 6020-1/2 (16 MPa & 25 MPa series) · ISO 6022 (160 bar series) · Euler/Johnson rod buckling

How ISO 6020 works — the method explained

The MechanixCalc hydraulic cylinders calculator sizes and verifies hydraulic and pneumatic cylinders to ISO 6020 and ISO 6022. Enter the bore diameter, rod diameter, stroke, supply pressure and back pressure, and the tool returns the extension force, retraction force, flow rate, hydraulic power, piston speed, and the rod buckling safety factor — all in a single pass for both the advance and retract strokes.

It is built for fluid power and machine-design engineers who need fast, auditable cylinder sizing — whether selecting bore size to hit a target force, checking rod slenderness against an Euler or Johnson buckling limit, or verifying that an end-of-stroke cushion will arrest a moving mass within the allowable peak pressure. Both hydraulic (ISO 6020/6022) and pneumatic (ISO 6432/15552) cylinder modes are included.

What this calculator does

  • ISO 6020/6022 hydraulic and ISO 6432/15552 pneumatic cylinder force and flow calculation
  • Piston rod buckling safety factor — Euler (long rod) and Johnson (short rod) column theory with selectable end conditions
  • End-of-stroke cushioning and deceleration analysis — kinetic energy, peak cushion pressure and g-force
  • Flow rate, hydraulic power, and speed vs pressure charts for extend and retract strokes
  • Bore size selection chart for a target force across the supply pressure range
  • Speed and cycle-time analysis with differential area speed ratio
  • Branded PDF engineering report with full methodology shown

Method & formulas

Extension and retraction force (ISO 6020 / ISO 6022)

The net output force depends on which face of the piston the supply pressure acts on. On extension (advance), pressure acts on the full bore area. On retraction, pressure acts on the annular area between the bore and the rod — so the retraction force is always lower than the extension force for the same supply pressure. Both results are reduced by the back pressure on the opposing face and by mechanical efficiency.

Extension force
F_ext = P · A_bore − P_back · A_ann − F_friction

where F_ext = net extension force (N); P = supply pressure (Pa); A_bore = π/4 · D² (m²); P_back = back-pressure (Pa); A_ann = π/4 · (D² − d²) (m²); F_friction = seal friction (N)

Retraction force
F_ret = P · A_ann − P_back · A_bore − F_friction

where F_ret = net retraction force (N); A_ann = annular (rod-side) piston area (m²)

Piston rod buckling — Euler and Johnson column theory

A cylinder rod under compression is a column that can buckle if the slenderness ratio (effective length / radius of gyration) is high enough. The calculator selects automatically between the Euler formula (for slender rods, where buckling precedes yielding) and the Johnson parabolic formula (for short, stocky rods, where the two failure modes interact). The effective length Le depends on the mounting end conditions — fixed-free gives the most conservative Le = 2L; pin-pin gives Le = L.

Euler critical load (long rods)
F_cr = π² · E · I / Le²

where F_cr = critical buckling load (N); E = elastic modulus (Pa); I = π/64 · d⁴ (m⁴, solid round rod); Le = K · L = effective length (m); K = end-condition factor (1.0 pin-pin, 2.0 fixed-free)

Johnson critical load (short rods)
F_cr = A · Sy · [1 − (Sy · Le² / (4π² · E · r²))]

where A = rod cross-section area (m²); Sy = yield strength (Pa); r = √(I/A) = radius of gyration (m). Johnson governs when λ = Le/r < λ_c = π · √(2E/Sy).

Flow rate and hydraulic power

The volumetric flow rate needed to achieve a target piston speed (or the speed produced by a given pump flow) follows directly from the piston area and stroke. Hydraulic power is the product of supply pressure and flow rate. The differential area ratio between the bore face and the rod-side face is why retraction is faster than extension at equal pump flow — a key consideration when selecting a directional control valve.

Extend flow rate
Q_ext = A_bore · v_piston

where Q_ext = volumetric flow rate (m³/s); v_piston = required piston speed (m/s). Retraction: Q_ret = A_ann · v_piston — lower flow for the same speed due to the smaller annular area.

Worked example

Find the extension force, retraction force and rod buckling safety factor for a hydraulic cylinder with 80 mm bore, 45 mm rod and 15 MPa supply pressure; pin-pin rod mounting, L = 600 mm, steel rod (E = 206 GPa, Sy = 350 MPa). Neglect back pressure and friction for this illustration.

Given

  • Bore diameter D80 mm
  • Rod diameter d45 mm
  • Supply pressure P15 MPa
  • Rod length L600 mm (pin-pin, K = 1.0)
  • Elastic modulus E206 GPa
  • Yield strength Sy350 MPa

Result

  • Extension force F_ext≈ 75.4 kN
  • Retraction force F_ret≈ 51.5 kN
  • Rod buckling safety factor SF≈ 6.48 (Johnson formula — well above the min. 2.5)
  1. Compute the bore area: A_bore = π/4 × 80² = π/4 × 6400 = 5026.5 mm².
  2. Compute the rod area: A_rod = π/4 × 45² = π/4 × 2025 = 1590.4 mm².
  3. Compute the annular (rod-side) area: A_ann = A_bore − A_rod = 5026.5 − 1590.4 = 3436.1 mm².
  4. Extension force: F_ext = P × A_bore = 15 N/mm² × 5026.5 mm² = 75 398 N ≈ 75.4 kN.
  5. Retraction force: F_ret = P × A_ann = 15 × 3436.1 = 51 542 N ≈ 51.5 kN.
  6. Rod second moment of area: I = π/64 × 45⁴ = π/64 × 4 100 625 = 201 174 mm⁴.
  7. Effective length (pin-pin, K = 1.0): Le = 1.0 × 600 = 600 mm.
  8. Slenderness ratio: r = √(I/A_rod) = √(201 174 / 1590.4) = √126.5 = 11.25 mm; λ = Le/r = 600 / 11.25 = 53.3.
  9. Critical slenderness: λ_c = π × √(2 × 206 000 / 350) = π × √1177.1 = π × 34.3 = 107.8. Since λ = 53.3 < λ_c = 107.8, the Johnson formula governs.
  10. Johnson critical load: F_cr = A_rod × Sy × [1 − (Sy × Le²) / (4π² × E × r²)] = 1590.4 × 350 × [1 − (350 × 360 000) / (4 × 9.87 × 206 000 × 126.5)] = 556 640 × [1 − 126 000 000 / 1 028 578 080] = 556 640 × 0.8775 = 488 450 N ≈ 488.5 kN.
  11. Buckling safety factor: SF = F_cr / F_ext = 488 500 / 75 400 ≈ 6.48.

Illustrative only — verify with your actual back pressure, seal friction, dynamic loads and required safety factor for the application. The calculator applies all corrections automatically.

Frequently asked questions

Which standard does this hydraulic cylinder calculator use?

Force and area calculations follow ISO 6020-1 and ISO 6020-2 (hydraulic fluid power cylinders, 16 MPa and 25 MPa series) and ISO 6022 (160 bar series). Pneumatic mode follows ISO 6432 and ISO 15552. Rod buckling uses classical Euler (long rods) and Johnson parabolic (short rods) column theory. The governing equations are shown in the generated PDF report.

How does the tool decide between the Euler and Johnson buckling formula?

The calculator computes the slenderness ratio λ = Le/r (effective length divided by the radius of gyration of the rod cross-section) and compares it to the critical slenderness λ_c = π·√(2E/Sy). When λ > λ_c the rod is slender and Euler governs; when λ ≤ λ_c the rod is stocky and the Johnson parabolic correction is applied. The selected formula and intermediate values are shown in the results panel.

Why is the retraction force lower than the extension force at the same pressure?

On extension (advance), supply pressure acts on the full bore area. On retraction, it acts on the annular area between the bore and the rod — a smaller area by the rod cross-section. The retraction force deficit grows as the rod diameter increases relative to the bore, which is also why retraction is faster than extension at equal pump flow.

What does the end-of-stroke cushioning analysis cover?

The cushioning panel computes the kinetic energy of the moving mass at cushion entry, the required deceleration force, the resulting peak cushion pressure, and the deceleration g-force, then compares the peak pressure to the allowable limit. A velocity profile during cushioning is plotted. Note: the cushioning model is an engineering estimate (the cushion orifice is not standardised) — treat the results as sizing guidance and verify against the cylinder manufacturer's data.

Is the hydraulic cylinder calculator free?

You can run it during a free 30-minute preview with no sign-up required, and a free 14-day account trial unlocks every MechanixCalc tool with no credit card. The branded PDF engineering report and saved calculations are part of a paid plan.

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