The most used springs are the following:

  • Torsion bar
  • Cylindrical propeller spring
  • Flat spiral spring
  • Crossbow
  • Disc springs

They are built in suitable high strength steels. Only stainless steel, bronze and brass springs are used in special cases. Springs all special are those made of rubber. Others rely on the compression of a gas. In the suspensions, damping devices always act in parallel with the springs. The springs and springs in rubber give rise to a not inconsiderable dissipation of energy in the deformation.

The force-displacement curve F (f) is significant, as in the helical spring, or torque-rotation C (q), as in the torsion bar. This curve, or characteristic, can be linear or not. In the first case k = Flf (or k = C / q) is the stiffness (l / k flexibility) having force / length, or torque / angle dimensions. More generally, k = dF / df, (or k = dC / dB). A practically linear diagram with the stiffness independent of the arrow is common for the metal springs. The diagram with increasing stiffness with the arrow is typical of the rubber springs. The diagram with decreasing stiffness with the arrow is typical of disc springs.

The areas subtended by the curves of these graphs represent the elastic potential energy accumulated in the spring upon application of the force F. Fig.9.3 shows examples of springs loaded indirectly. In Fig.9.3a, where the propeller spring is loaded by means of a lever of length l, the relationship between stiffness k at the end of the lever and the stiffness K m of the helical spring, located at distance a from the fulcrum, is, small trips:

k = Km (a / l) ^ 2

In the case of Fig.9.3b of the torsion bar loaded by means of an arm, the relationship between the stiffness k at the end of the arm itself, of length b, and the stiffness K t of the torsion bar is, for small displacements:

k = K t / b ^ 2

In the systems of Fig.9.3, for non-small displacements, the relationships between loads and displacements at the end of the lever are no longer linear, even with linear characteristic springs. For 2 springs in series and in parallel the resulting stiffness is respectively:

k = K1 * K2 / (K1 + K2) or 1 / k = 1 / K1 + 1 / K2 K = K1 + K2

The sizing of the springs is generally carried out for subsequent attempts, starting from the design data. It is advisable to provide springs with minimum weight and minimum overall dimensions. This is the reason why materials with high yield strength and high resistance to fatigue are used.


The springs are made of steels with a high yield strength. The elastic modulus and the tangential elasticity modulus are assumed equal respectively to: E = 206000 N / mm ^ 2 G = 81400 N / mm ^ 2. C and alloyed steels are used. Among the former, for example, C72 and C98 carbon steels. The addition of silicon, up to 2%, increases yield strength and impact resistance. Vanadium (0.1 - 0.2%) and Tungsten (> 1.2%) improve mechanical characteristics and resistance to heat. The materials for high-strength springs are those of Silicon-Vanadium and Chromium-Silicon-Vanadium. Fatigue strength depends not only on the size, composition and condition of the material, but also on the surface finish of the springs. For propeller springs, particularly for small wire diameters, the harmonic wire is normally used, made of drawn carbon steel, with very high resistance. The UNI 3823 provides wires of four classes (A, B, C, D), with increasing resistance from A to D. The first two are recommended when the stresses are predominantly static and the latter for dynamic applications. Shows the minimum breaking strength of the harmonic steel of classes C and D according to the diameter of the wire.Shows the fatigue strength of the harmonic steel. The curves, reported for some wire diameters, provide for the case of oscillating tension the maximum tangential stress as a function of the minimum one. These tensions are already affected by an adequate safety factor.


The design and verification criteria for the most common springs are shown. In the calculation of springs we distinguish between static use with few repetitions of the load, or of deformation, and dynamic use with consequent fatigue stresses.

Torsion bar
The torsion bar achieves an elastic connection between elements with relative rotation motion. The use of this spring is sometimes made difficult by its considerable length. The torsion bars have the shapes shown in Fig.9.6. The diameter of the heads must be at least 1.4 times the diameter of the bar with a full circular section. The useful length to consider in the calculations is a little larger than the cylindrical section, to take into account the elasticity of the extreme parts.

The formulas to be used for the project and the verification are for the bar with a circular section full of useful length 1 and diameter d:

t = 16 * C / (p) * d ^ 3 t = C * l / G * J0

J = (p) * d ^ 4/32 moment of inertia of the section

t = maximum cutting stress

C = applied torque

G = tangential elasticity modulus of the material

The stiffness is given by: K = C / q = p * G * d ^ 4/32 * l

The permissible static stress, when a limit switch accurately determines the maximum deflection angle q, can be set equal to about 0.9 times the shear yield stress equal, in turn, to 0.6 - 0.7 times the stress stress yielding normal stress of the material used to build the bar. In this way, working voltages of: 600 - 800 N / mm ^ 2 are recommended for the most used materials. In the case of alternating deformation between two values q min and q max and the corresponding stresses must be calculated and the resistance test carried out using the Smith diagram, or an equivalent diagram related to the fatigue strength, for the spring material. The resistance to fatigue depends as it is also known from the surface condition of the pieces. Hollow bars are rarely used, even if they are lighter with the same performance


They are widely used for the low construction cost, for the ease of insertion in different spaces (the side encumbrance and the length can in fact vary inversely to give rise to equivalent springs) and because they do not require a strict coaxiality between the fixed and mobile. The propeller can be left or right. The torque (Fig.9.7) which stresses the thread is F * R, where F is the load and R is the radius of the helix or of the loop. The nominal stress, as it neglects some effects, is: t n = 16 * F * R / p * d ^ 3

The arrow (indicating with iu the number of useful turns, equal to the total number of turns less extreme supported sections that do not participate in the deformation) is:

f = 64 * iu * (F * R) ^ 3 / G * d ^ 4

and the stiffness:

K = F / f = G * d ^ 4/64 * iu * R ^ 3

Once the maximum load Fmax, the stiffness k and the stress t, have been assigned, it must first be fixed or the diameter of the loop, or the diameter of the wire, to proceed with the sizing. The nominal stress must be corrected with a factor l> 1 that takes into account the overstresses related to the curved beam effect and the shear stress. Said factor is plotted in Fig.9.8 as a function of the winding ratio c = 2R / d. The actual stress therefore, to be compared with the permissible value for the material used, is: t = l * t n

The compression helix springs are presented as shown in Fig.9.7 with the coils close together and ground to create flat resting surfaces on an arc of about 270º. Fig.9.9 shows two variants for the terminal turns that require to suitably conform the support areas. The compression springs must be centered, inside or outside, at both ends. When the fixed and mobile bearing surfaces do not remain sufficiently perpendicular to the axis of the loop, orientable supports must be provided. To prevent lateral inflection in the compression springs, the free length L * (discharge spring length) and the maximum arrow are subject to the following limitations dictated by experience:

L * / 2 * R <= 0:55

For lower free length values there is no danger of lateral inflection. For higher values, an external guide or a sufficiently long guide must be provided.

The pitch p of the propeller is normally held, with a discharged spring, in the following values:

p <= (0.3-0.5) * R

The ratio of winding factor c = 2R / d between the average diameter of the loop and the diameter of the wire is practically less than <4. The recommended sum of the minimum distances between the turns is (with a number of useful turns):

S = x * d * iu

This verification must obviously be performed in the maximum arrow situation:

Large compression helix springs are sometimes made with rectangular section bars, replacing the wire, with the advantage of accumulating more energy for the same space requirement. However, they require processing cycles for hot deformation that are a little complicated. The tension springs (Fig.9.1 1) are almost always with the coils in contact and preloaded during construction. The minimum load at which the spring is made must be slightly higher than the preload. In this way the shortest possible spring is obtained.

The ends are shaped in various ways. Not infrequently they constitute the weak zone of the spring. The break occurs by flexing the transition from the loop to the pull eyelet, in particular if the connecting radius is tight or if the sign of the bending tool remains. We recommend, for static use, when there are no uncertainties about working conditions, tensions:

t amm = 0.5 * s R for compression springs

t amm = 0.45 * s R for extension springs

When the compression or tension spring works with an oscillating deformation between fmin
and fmax, ie with the load between Fmin = k * fmin and Fmax = k * fmax the actual stresses must be calculated:

t min = l * (16 * Fmin * R / p * d ^ 3) t max = l * (16 * Fmax * R / p * d ^ 3)

and voltage fluctuations:

D * t lim = t max-t min.