Durability of Springs

The calculation formulas for linear helical springs with an inconstant wire diameter and with a variable mean diameter of spring are presented. Based on these formulas the optimization of spring for given spring rate and strength of the wire is performed. The design principles for optimal leaf springs are briefly presented.

The stress distribution over the cross-section wire of helical springs is studied in this chapter. For simplification the pitch of the helical spring is neglected and the traditional representation of one coil as an incomplete torus is used. This model generalizes the Saint-Venant torsion problem of an elliptical straight rod accounting the curvature rod. The closed form solution for the torsion problem of an incomplete torus is discussed.

In this chapter the helical spring is substituted by a flexible rod that is located along the axis of helix. This rod possesses the same mechanical features, as the spring itself. Its bending, torsion and compression stiffness are equal to the corresponding stiffness of the helical spring. This rod is known as an “equivalent column” of the helical spring. The “equivalent column” equations are considerable easier to handle than the original equations of the helical spring. The integral spring properties, as an axial and transversal stiffness, buckling loads, fundamental frequencies could be directly determined using the “equivalent column” equations. In contrast, the local properties, like stresses in the wire or contact forces, could be evaluated only with the more complicated equations of the helical elastic rod.

In this chapter the stability and transversal vibrations of the spring are studied from the unified point of view, which is based on the “equivalent column” concept. Buckling refers to the loss of stability up to the sudden and violent failure of straight bars or beams under the action of pressure forces, whose line of action is the column axis. This concept is applied for the stability of helical springs.

An alternative approach method is based on the dynamic criterion for the spring stability. The equations for transverse (lateral) vibrations of the compressed coil springs were derived. This solution expresses the fundamental natural frequency of the transverse vibrations of the column as the function of the axial force, as well as the variable length of the spring.

In this chapter the method for calculation of residual stress and plastic bending and torsion moments for combined bending-torsion load is developed. The Bernoulli’s hypothesis is assumed for the deformation of the bar. The analysis was provided using deformational theory of plasticity with a nonlinear stress strain law describing active plastic deformation. The curvature and twist of the bar during the plastic loading increase proportionally, such that the ratio curvature to twist remains constant. The complete solutions based on this approximate material law provide closed analytical solution. The spring-back from the plastic state is assumed to be linear elastic.

In the current chapter the disk spring using the variation methods and equations of thin and moderately thick isotropic and anisotropic shells are investigated in closed analytical form. The equations developed here are based on common assumptions and are simple enough to be applied to the analysis. The analysis of isotropic and anisotropic thin-walled disk springs could be performed using basic spreadsheet tools, removing the need to perform an onerous finite element analysis.

The theory of linear and progressive disk wave springs is presented.

In this chapter the time-depending behavior of spring elements under steady load is studied. The common creep laws are implemented for the description of material. For basic spring elements the Norton-Bailey, Garofalo and Naumenko-Altenbach-Gorash constitutive models were studied. Analytical models are developed for the relaxation of stresses and creep under constant load. Closed-form solutions of the analytical models of creep and relaxation are found. The explanation of the experimental procedure for the experimental acquisition of creep models is given.

The aim of this chapter is to examine further the detailed behavior of simple structures with fractional creep laws. The relaxation of stresses for common and fractional Norton-Bailey constitutive models was studied for basic elements in torsion and bending. The unified formula for several regions of creep law is studied. The new expression is based on the experimental data and merges the primary, secondary and tertiary regions of creep curve in a single time-dependent formula.

In the present and the next chapter, an approach is developed to account the stress gradient effect on fatigue life of springs. The applied method of the analytical description is based on two steps. The first step provides the description of fatigue life of the homogeneously stressed material subjected to the cyclic load. This problem is studied in this chapter. Common methods for the estimation of fatigue life, based on Goodman and Haigh diagrams, stress-life and strain-life approaches, are briefly summarized. More attention is paid to different method of fatigue analysis, which is describes the crack growths per cycle. The expressions for spring length over the number of cycles are derived. The second step uses the weak-link concept for the non-homogeneously loaded structural elements. The estimation of the fatigue life utilizes the closed-form solutions for fatigue crack propagation from this chapter. The weak-link is applied for the evaluation of fatigue life of helical spring in Chap. 9.

This chapter introduces the methodology for calculation of fatigue life of structural elements with particular application to helical springs. The methodology could be extended also for different types of springs or basic structural elements.

The spring element is investigated using the weak-link concept in this chapter. The variation of stress over the surface of the wire is accounted for the analytical calculation of failure probability. This chapter provides closed form analytical formulas for failure probability of helical compression springs. The derived solution explains the influence of spring index and wire diameter on the fatigue life.

A thin-walled beam is an extraordinary type of the torsion spring. This type of torsion spring serves as the cross-connection member in several industrial and automotive structures. The thin-walled beam is an elongated elastic structural element whose distinctive geometric dimensions are all of different orders of magnitude. Thin-walled beams can be classified by their geometric features. Two classes of thin-walled beam cross-sections are notable, namely thin-walled beams with the open cross-section and thin-walled beams with the closed cross-section (Vlasov 1961; Timoshenko 1945; Flügge and Marguerre 1950). In the Chap. 10 an intermediate class of thin-walled beam cross-sections is studied. The cross-section of the beam is closed, but the shape of cross-section is elongated and curved. The walls, which form the cross-section, are nearly equidistant. The Saint-Venant free torsion behavior of the beam is similar to the behavior of closed cross-section beams. However, the total warping function of the semi-opened cross-section is similar to the warping function of open cross-section beam.

Generally speaking, particular difference between thin-walled rods is in the presence of warping of cross-section and the corresponding force factor, the bi-moment. It is shown, that the thin-walled beams with closed profiles behave differently, depending on the form of the cross-section. There exist two types of the thin-walled beams with closed profiles. The first type of thin-walled beams possesses closed profiles with nearly equal diameters in all directions, so that there is no distinguished direction. The torsion of such closed beams is predominantly of Saint-Venant type. The effect of bi-moment on the twist behavior of the thin-walled beams of the first type is in most cases negligible in comparison to the Saint-Venant torsion. The second type of thin-walled beams embraces closed profiles formed by equidistant walls, elongated profiles and star-like profiles. The contribution of Saint-Venant torsion stiffness of beams with this type of profile is of the same order of magnitude, as the contribution of sectorial stiffness and the effect of bi-moment should be accounted.

In Chap. 11 analytical methods design and analysis of a rear twist beam axle in the concept design phase are reported. The twist-beam axle is a specific type of semi-solid suspensions. The twist-beam axles are commonly designed with the purpose of improving the vehicle dynamics performance, improving load carrying capacity, reducing weight and cost. The principal element of common twist-beam axle is the thin-walled twist beam. This beam behaves elastically, delivering certain roll rate for the vehicle, but must be sufficiently stiff for bending and guarantee the chamber and lateral stiffness of the vehicle. Insufficient camber and lateral stiffness cause oversteering of the vehicle, unacceptable from viewpoint of vehicle dynamics.