Learning topic
Design of Helical Springs
Calculate helical spring stiffness, axial deflection, and shear stress from wire diameter, coil diameter, active coils, shear modulus, and the Wahl factor.
A cylindrical helical spring converts an axial force primarily into torsion of the wire in each coil. This allows the spring to store elastic energy and produce relatively large axial displacements within compact dimensions.
Main geometric parameters
Design parameters include wire diameter $d$, mean coil diameter $D$, number of active coils $n$, and the spring index $C=D/d$. End coils may serve a structural function and are not always counted as active coils.
Torsion of the spring wire
An axial force $F$ produces a principal torque in the wire of approximately $T=FD/2$. Therefore, the basic spring model is based on the torsion theory of a circular bar.
For a solid circular section of diameter $d$:
$$J_p=\frac{\pi d^4}{32},\qquad W_p=\frac{J_p}{d/2}=\frac{\pi d^3}{16}.$$
For a hollow circular section with outer diameter $D$ and inner diameter $d$:
$$J_p=\frac{\pi(D^4-d^4)}{32},\qquad W_p=\frac{J_p}{D/2}=\frac{\pi(D^4-d^4)}{16D}.$$
- $J_p$ — polar second moment of area;
- $W_p$ — polar section modulus;
- $D$, $d$ — section diameters.
The calculator below is for a solid circular section.
Deflection and stiffness
For a close-coiled cylindrical tension or compression spring in the basic wire-torsion model:
$$\delta=\frac{8FD^3n}{Gd^4},\qquad k=\frac{F}{\delta}=\frac{Gd^4}{8D^3n}.$$
- $\delta$ — axial deflection;
- $F$ — axial force;
- $D$ — mean coil diameter;
- $d$ — wire diameter;
- $n$ — number of active coils;
- $G$ — shear modulus;
- $k$ — spring stiffness.
This is a basic educational model. More accurate stress calculations account for direct shear and wire curvature using appropriate correction factors.
The formula shows the strong influence of wire diameter: stiffness is proportional to $d^4$. Increasing the mean coil diameter $D$ or the number of active coils $n$ reduces stiffness.
Shear stresses
The simplest torsion model gives a nominal shear stress due to torque. In a real helical spring, direct shear and wire curvature also affect the maximum stress. More accurate calculations therefore use correction factors related to the spring index, such as the Wahl factor in common engineering models.
Strength and stiffness
Spring design requires at least two checks: maximum shear stress must remain within the adopted allowable limit, and axial deformation must provide the required force–displacement characteristic.
Example of geometric influence
If $d$ is increased by a factor of 1.2 while $D$, $n$, and $G$ remain unchanged, the basic model predicts a stiffness increase by $1.2^4\approx2.07$. Thus a relatively small increase in wire diameter can more than double spring stiffness.
Limits of the simple model
The basic formulas are most appropriate for close-coiled springs with small helix angle and elastic material behavior. Practical design may also require consideration of end coils, coil contact in compression, fatigue, buckling of long springs, manufacturing effects, and applicable standards.
About this topic
Close-coiled helical springs are crucial machine components operating primarily under wire torsion driven by axial loading. This page provides engineering theory for spring coil design: calculating torsional torque, curvature correction factors, maximum shear stresses, total spring deflection, and determining active coil counts for required stiffness.