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## About this book

This book highlights the mechanics of the elastic elements made of steel alloys with focus on the metal springs for automotive industry. The industry and scientific organizations study intensively the foundations of design of spring elements and permanently improve the mechanical properties of spring materials. The development responsibilities of spring manufacturing company involve the optimal application of the existing material types. Thus, the task entails in the target-oriented evaluation of the mechanical properties and the subsequent design of the springs, which makes full use of the attainable material characteristics. The book stands as a valuable reference for professionals in practice as well as an advanced learning resource for students of structural and automotive engineering

## Table of Contents

### Chapter 1. Principles of Spring Design

Abstract
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. This chapter is the entry section of first part, which studies the springs from the viewpoint of design.
Vladimir Kobelev

### Chapter 2. Stress Distributions Over Cross-Section of Wires

Abstract
The stress distribution over the cross-section wire of helical springs is examined 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 StVenant’s 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. This Chapter is the next section of first part, which studies the springs from viewpoint of design.
Vladimir Kobelev

### Chapter 3. “Equivalent Columns” for Helical Springs

Abstract
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. This chapter is the third section of first part, which studies the springs from the design viewpoint.
Vladimir Kobelev

### Chapter 4. Disk Springs

Abstract
In the current chapter, we examine the disk spring using the models of thin and moderately thick isotropic shells. The equations developed here are based on common assumptions and are simple enough to be applied to the applied analysis. The analysis of thin-walled disk springs could be performed using basic spreadsheet tools, removing the need to perform an onerous finite element analysis. Further, the theory of linear and progressive disk wave springs is presented. This chapter is the fourth section of first part, which studies the design and modelling of springs.
Vladimir Kobelev

### Chapter 5. Thin-Walled Rods with Semi-opened Profiles

Abstract
The semi-opened cross tubular sections are manufactured from the standard tubes by their flattening. For the reason that the significant variation of mechanical parameters along the length is possible, the flattening could be easily adjusted for optimal design. From the design viewpoint, the semi-opened cross tubular sections fill the gap between classical closed and open one-wall sections. Due to the high variability of the section’s geometry, the simple analytical model is essential for primary design purposes and estimation of numerous opposing static effects. The effective torsion stiffness $${r}_{t}$$, effective bending stiffness $${r}_{z}$$ and effective bending spring rate $${r}_{c}$$ of the twist-beam in terms of section properties of the twist-beam with the semi-opened cross-section was expressed with the analytical formulas. This chapter is the final section of first part, which studies the elastic elements from viewpoint of modeling and design.
Vladimir Kobelev

### Chapter 6. Coiling of Helical Springs

Abstract
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 (SAE AE-22, Sect. 8.3.33). 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 unloading is linear elastic. This chapter is the primary section of second part, which studies mechanical problems arising during the manufacturing of helical springs.
Vladimir Kobelev

### Chapter 7. Presetting and Residual Stresses in Springs

Abstract
In this chapter, the method for calculation of residual stress and enduring deformation of helical springs is developed. The method is based on the deformational formulation of plasticity theory and common kinematic hypotheses (SAE AE-22, Sect. 8.3.33). Two principal types of the helical springs—the compression springs and the torsion springs are studied. For the first type (axial compression or tension springs) the spring wire is twisted. The basic approach neglects the pitch and curvature of the coil, substituting the helical wire by the straight cylindrical bar. The helical spring is in the state of screw dislocation, accordingly, the wire is twisted. The elastic–plastic torsion of the straight bar with the circular cross-section is examined. For the second type (torsion helical springs) the helical wire is in the state edge dislocation, so the wire is in state of flexure. The elastic–plastic bending of the straight bar with the rectangular and circular cross-section is studied using the Bernoulli’s hypothesis. The material for both types of springs is nonlinearly-hardening elastic–plastic with the elastic unloading. The hyperbolic, Ramberg–Osgood, Ludwik-Hollomon, Swift-Voce, Johnson-Cook laws for the material are surveyed. For both problems the elastic–plastic active deformation and elastic spring-back allow the closed-form solutions. Further, this Chapter examines the calculation of remaining deformation and residual stress for helical springs after long-lasting presetting process. The article extends the model for the immediate presetting process accounting the creep deformation of the spring. The method is based on plasticity theory for the instant flow overexposed by the relaxation over the long-term presetting. In this article the following method is used. The plastic deformation of the helical spring with the circular cross-section occurs instantly. If the shortening of the spring in the tool holder persists, the relaxation of stresses occurs and the force of the spring reduces. As the consequence, after the elastic unloading of the long-time presetting, the residual stresses spring reduce gradually with the squeezing time as well. The final length of the springs considerably shortens with the increasing preset duration. The advantage of the discovered closed form solutions is the calculation without the necessity of complex finite-element simulation of spring length loss and residual stresses after presetting process. The analytical expressions are proposed and the exact calibration applied for evaluation of factors for presetting processes. This Chapter is the closing section of second part, which studies manufacturing processes of the helical springs.
Vladimir Kobelev

### Chapter 8. Creep and Relaxation of Springs

Abstract
In this Chapter, we investigate the time-depending behavior of spring elements under constant and oscillating load. The common creep laws are implemented for the description of material. The models are established for the relaxation of stresses under constant stroke and for creep under persistent load. For basic spring elements, closed-form solutions for the models of creep and relaxation are found. The introduction of the “scaled” time variable simplifies thÝe presentation. The explanation of the experimental procedure for the experimental acquisition of creep models is presented. This Chapter forms the first section of the third part, which examines the lifecycle of springs.
Vladimir Kobelev

### Chapter 9. Fatigue of Spring Materials

Abstract
The present chapter explains the customary ways for accounting of the stress amplitude for fatigue life of springs. The fatigue of spring materials for fully reversed, uniaxial loading is reviewed. Two different estimation methods for fatigue life are briefly discussed. The first method implements the stress-life and strain-life procedures. The second method describes the fatigue crack growths (FCG) per cycle. Two analogous unifications of FCG functions are proposed. The expressions for spring length over the number of cycles are derived. This chapter is the second section of final part, which investigates the lifecycle of the elastic elements.
Vladimir Kobelev

### Chapter 10. Factors Affecting the Fatigue Life of Springs

Abstract
In this chapter, the effects of mean stress, multiaxial loading, and residual stresses on fatigue life of springs are reviewed. We study the fatigue life of the homogeneously stressed material subjected to the cyclic load with a non-zero mean stress. Traditional methods for the estimation of fatigue life are based on Goodman and Haigh diagrams. The formal analytical descriptions, namely stress-life and strain-life approaches, turn out to be more appropriate for the numerical methods. The reported above method of fatigue analysis, which is describes the crack growths per cycle, is extended. The extensions enable the accounting of the mean stress of load cycle. The closed form expressions for the crack length over the number of cycles are derived. The complement effects on the fatigue life of springs, which eventually significantly influence the fatigue life of springs, are concisely pronounced. This chapter is the second section of the third part, which investigates the lifecycle of the elastic elements.
Vladimir Kobelev

### Chapter 11. Failure Analysis Based on Weakest Link Concept

Abstract
In this Chapter we begin to study the statistical effects in failure analysis of springs. It presents the procedure for dimensional analysis of failures of the helical springs. The dimensional analysis spring element is examined with the weakest-link concept. This concept is applicable to the extremely brittle material, for which the failure of one link causes the complete destruction of the complete structural element. The variation of stress over the surface of the wire is accounted for the analytical calculation of Weibull failure probability. Chapter delivers analytical formulas for failure probability of helical springs. The derived solution clarifies the effects of spring index and wire diameter on the fatigue life. The methodology is valid for different types of springs or basic structural elements.
Vladimir Kobelev

### Chapter 12. Statistical Effects on Fatigue of Spring Materials

Abstract
In the present Chapter, we continue to study the stochastic influences on fatigue life of springs. At first, we evaluate the probabilistic effects prevailing at low amplitudes of cyclic stresses. The question is the calculation of failure probability as the function of stress amplitude. The answer to this question results from the examination of the experimental fatigue life data. The experimental data demonstrate different behaviors in the regions of low and high amplitudes of stress. To describe this phenomenon, we introduce the randomization to the crack propagation, which accompanies by the accidental deviation and branching of crack. The randomization of crack propagation escalates with the reducing stress amplitude. The high inhomogeneity of the polycrystalline structure on the micro level cause hypothetically the random propagation. This hypothesis leads to the mathematical model for the randomly propagating crack. The differential equation with stochastic coefficients describes the randomly travelling crack. This equation is analogous to the equation of the enforced Brownian motion. The examples of the solutions for the Brownian stochastic differential equation are presented. This Chapter is the final section of the last part of the book, which studies the lifecycle of springs.
Vladimir Kobelev

### Backmatter

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