🚀 Spring Deflection Calculator & Simulator
- Select Material Properties: Choose a spring material such as carbon steel, stainless steel, or phosphor bronze to specify the shear modulus (G).
- Specify Geometric Variables: Adjust the sliders for wire diameter (d), mean coil diameter (D), and active coils (n) to match your component catalog specifications.
- Set Applied Load: Adjust the external force (F) applied to the spring to exert a compressive load.
- Observe Real-time Compressive Deformation: Watch the spring in the 2D simulator compress in real time according to elastic formulas, and check the visual feedback as the helical coil turns into a magenta glow in proportion to the shear stress load.
- Review Solid Height and Failure Risk: Check if the spring has reached its solid height (fully closed), and verify through the safety factor that it is operating safely below the shear yield strength.
📚 View Detailed Mechanical Engineering Explanation & Spring Design Standards (KS/ISO) ▼
1. Derivation of the Stiffness Formula for Helical Coil Springs
A coil spring is a typical elastic mechanical element that stores energy and provides damping by winding a wire into a helix to utilize torsional deflection. When a compressive load F is applied to the spring, a uniform torsional moment of T = F · D / 2 acts on the cross-section of the wire.
The spring rate (stiffness, k) of a coil spring, derived by integrating the torsional strain energy, is defined as follows:
k = (G · d4) / (8 · D3 · n) [N/mm]
Where the geometric parameters are as follows:
- G: Shear Modulus of the Material (GPa) – Approximately 79.3 GPa for steel materials and around 69 GPa for stainless steel.
- d: Wire Diameter (mm) – Has a dominant effect on spring stiffness, as it is proportional to the fourth power.
- D: Mean Coil Diameter (mm) – Calculated by subtracting the wire diameter (d) from the outer coil diameter (Do).
- n: Active Coils – The number of turns that actually participate in deflection.
2. Shear Stress Analysis Incorporating the Wahl Correction Factor
Inside the spring wire, in addition to pure torsional stress, direct shear stress caused by wire bending and stress concentration due to curvature occur strongly on the inner fiber. To correct for this, mechanical engineer A. M. Wahl proposed the following Wahl Correction Factor (KW):
KW = (4C – 1)/(4C – 4) + 0.615/C (where spring index C = D / d)
The maximum shear stress τ in the spring wire incorporating this factor is calculated as follows, and this stress must remain below the shear yield limit of the spring material to prevent permanent set:
τ = KW × (8 · F · D) / (π · d3) [MPa]
3. Intrinsic Limits of Springs and Solid Height
When a spring is excessively loaded, the coil wires come into contact with each other. The height at this limit state is called the solid height (hs). Under solid conditions, no further elastic deflection is possible, and the system behaves as a rigid solid block, transmitting massive shock forces directly to the seating surfaces and the frame. Therefore, standard design specifications (such as KS B 2400) recommend limiting the maximum operating deflection to no more than 80% of the solid deflection to ensure a safety margin.