Fundamentals of Springs Mechanics

Foreword
Preface
Introduction
	Aims and Methods of the Book
	Structure of the Book
	Target Audience of the Book
Contents
Symbols
Part I Design of Mechanical Springs
1 Principles of Spring Design
	1.1 Compression, Extension, and Torque of Helical Springs
		1.1.1 Forces and Moments in Coil Springs
		1.1.2 Elastic Energy of the Coil Spring
		1.1.3 Compression and Twist Spring Rates
		1.1.4 Change in Diameter Due to Simultaneous Compression and Torque Application
	1.2 Design Formulas for Compression-Extension Springs
		1.2.1 Stiffness and Stored Energy of Cylindrical Helical Springs
		1.2.2 Stresses in Spring Wire
		1.2.3 Fatigue Life and Damage Accumulation Criteria
	1.3 Helical Springs of Minimal Mass
		1.3.1 Restricted Optimization Problem
		1.3.2 Optimization of Helical Springs for Maximal Stress
		1.3.3 Design for Fatigue Life
		1.3.4 Spring Quality Parameter for Helical Springs
	1.4 Semi-elliptic Longitudinal and Transverse Leaf Springs of Minimal Mass
		1.4.1 Rectangular Cross-Section
		1.4.2 Circular Cross-Section
	1.5 Multi-material Design of Springs
	1.6 Conclusions
	References
2 Stress Distributions Over Wire Cross-Section
	2.1 Warping Function
	2.2 Prandtl Stress Function
	2.3 Shear Stresses on Surface of Elliptic and Circular Wires
	2.4 Shear Stresses on Surface of Ovate Wire
	2.5 Quasi-elliptical Cross-Section
	2.6 Hollow Ovate Wire
	2.7 Helical Spring Deformation Dislocation Character
		2.7.1 Screw and Edge Dislocations
		2.7.2 Torsion of Helical Spring
	2.8 Conclusions
	References
3 “Equivalent Columns” for Helical Spring
	3.1 Static Stability Criteria of Helical Springs
	3.2 Static “Equivalent Column” Equations
	3.3 Dynamic “Equivalent Column” Equations
	3.4 Natural Frequency of Transverse Vibrations
	3.5 Stability Conditions and Buckling of Spring
	3.6 Buckling of Twisted, Compressed, and Tensioned Helical Spring
		3.6.1 Instability of Twisted Helical Spring
		3.6.2 Instability of Helical Springs Under Torque and Axial Force
		3.6.3 Instability of Tension Spring
	3.7 Spatial Models for Dynamic Behavior of Helical Springs
	References
4 Disk Springs
	4.1 Thick Shell Model for Disk Springs
		4.1.1 Mechanical Models of Elastic Disk Springs
		4.1.2 Geometry of Disk Spring in Undeformed State
		4.1.3 Mass of Disk Spring with Variable Material Thickness
		4.1.4 Load-Caused Alteration of Strain and Curvature
		4.1.5 Disk Springs of Moderate Material Thickness
	4.2 Disk Springs of Moderate Thickness
		4.2.1 Deformation of Thick Conical Shell
		4.2.2 Variation Method for Thick Shell Models of Disk Springs
		4.2.3 Comparison of Calculation Techniques
	4.3 Statics of Thin Disk Springs
		4.3.1 Forces and Moments in Disk Springs
		4.3.2 Strain Energy of Thin Disk Springs
		4.3.3 Almen and Laszlo Method for Thin Disk Springs
		4.3.4 Stresses in Disk Springs
	4.4 Disk Wave Springs
		4.4.1 Application Fields of Disk Wave Springs
		4.4.2 Design Formulas for Linear Disk Wave Springs
		4.4.3 Design Formulas for Non-linear Disk Wave Springs
	References
5 Radially Constrained Disk Springs
	5.1 Shell Model for Conical Disk Springs
		5.1.1 Introduction
		5.1.2 Models of Elastic Disk Springs
		5.1.3 Geometry of Disk Spring in Undeformed State
		5.1.4 Variations of Strain and Curvature Due to Axial Contraction of Conical Spring
		5.1.5 Spring Travel and Heights of Disk Springs
	5.2 Statics of Disk Springs Using Equations of Axisymmetric Elasticity
		5.2.1 Deformation of Conical Shell
		5.2.2 Variation Method for Shell Models of Disk Springs
	5.3 Forces and Stresses in Disk Springs
	5.4 Deformation of Conical Spring with Both Radially Constrained Edges
	5.5 Comparison of Spring Constants for Differently Confined Disk Springs
	5.6 Finite Element Simulation of the Disk Springs with the Free Gliding and Radially Constrained Edges
	5.7 Conclusions
	References
6 Disk Springs with Variable Thickness
	6.1 Models of Elastic Disk Springs
	6.2 Geometry of Disk Spring in Undeformed State
	6.3 Load-Caused Variations of Strain and Curvature
	6.4 Statics of Disk Springs Using Equations of Axisymmetric Elasticity
	6.5 Linearly Variable Thickness of Disk Spring
	6.6 Quadratically Variable Thickness of Disk Spring
	6.7 Variable Thickness Along the Parallels
	6.8 Radial Forces on the Radially Constrained Disk Springs
	6.9 Verification of the Analytical Results with the Finite-Element Simulation
	6.10 Stresses in Disk Springs
	References
7 Thin-Walled Rods with Semi-open Profiles
	7.1 Thin-Walled Rods with Semi-open Profiles
		7.1.1 Open, Closed and Semi-open Wall Sections
		7.1.2 Baseline of Semi-open Cross-Section
		7.1.3 Main Hypotheses of Thin-Walled Open-Profile Bars
	7.2 Deformation Behavior of Cross-Sections
		7.2.1 Deformation of Rods with Opened and Closed Profiles
		7.2.2 Deformation of Rods with Semi-open Profiles
	7.3 Statics of Semi-open Profile Bars
		7.3.1 Normal Stresses in Semi-open Profile Bars
		7.3.2 Torque and Bi-moment
		7.3.3 Tangential Stresses in bar Cross-Sections
		7.3.4 Average Tangential Stress and Equilibrium Conditions
		7.3.5 Strain Energy of Semi-open Rod
	7.4 Applications of Thin-Walled Rods with Semi-open Cross-Sections
		7.4.1 Semi-solid Axis with Twist Beam
		7.4.2 Mechanical Models of Twist-Beam Axle
	7.5 Elastic Behavior of Twist-Beam Axles Under Load
		7.5.1 Loads and Displacements of Twist-Beam Axles
		7.5.2 Roll Stiffness of Twist-Beam Axle
		7.5.3 Lateral Stiffness of Twist-Beam Axle
		7.5.4 Camber Stiffness of Twist-Beam Axle
	7.6 Deformation of Semi-open Beam Under Terminal Load
		7.6.1 Bending of Semi-open Beam Due to End Moments
		7.6.2 Torsion Stiffness of Beam with Constant Section Due to Terminal Torques
		7.6.3 Stresses in the Beam with Constant Section Due to Terminal Torques
		7.6.4 Equivalent Tensile Stress Due to Simultaneous Bending and Torsion
		7.6.5 Stiffness Properties of Semi-open Profiles for Automotive Applications
		7.6.6 Semi-open Beams with Variable Cross-Sections
	References
Part II Manufacturing of Springs
8 Coiling of Helical Springs
	8.1 Elastic–Plastic Bending and Torsion of Wire
	8.2 Modified Ramberg–Osgood’s Law
	8.3 Plastic Deformation of Wire During Coiling
	8.4 Behavior of Wire in Manufacturing Process
	8.5 Elastic Spring-Back and Appearance of Residual Stresses
	8.6 Post-coiling Shape of Helical Spring
	8.7 Conclusions
	References
9 Presetting and Residual Stresses in Springs
	9.1 Elastic–Plastic Deformation During the Presetting Process of Helical Springs
	9.2 Implicit Formulations for the Stress–Strain Curves of Plastic Materials
	9.3 Analysis of Active Plastic Torsion and Spring-Back of Circular Wire for Presetting Assessment of Helical Compression Springs
		9.3.1 Plastic Deformation of Wire During Active Plastic Twisting of a Solid Rod
		9.3.2 Torque Moment During the Active Plastic Presetting in the Wire Cross Section for Hyperbolic Law
		9.3.3 Torque Moment During the Active Plastic Presetting in the Wire Cross Section for Ramberg–Osgood Law
		9.3.4 Elastic Spring-Back and Residual Stresses Appearing
	9.4 Evaluation of Helical Torsion Spring Presetting by Analysis of Active Plastic Bending of Rectangular Wire
		9.4.1 Plastic Deformation of Wire During Active Plastic Bending Process
		9.4.2 Bending Moment During the Active Plastic Presetting in the Wire Cross Section for Hyperbolic Law
		9.4.3 Bending Moment During the Active Plastic Presetting in the Wire Cross Section for Ramberg–Osgood Law
		9.4.4 Elastic Recovery and Evaluation of Residual Stresses
	9.5 Explicit Formulations for Plastic Stress–Strain Curves
		9.5.1 Relationships of Ramberg–Osgood and Johnson–Cook Formulas
		9.5.2 Torsion of the Rod with Circular Cross-Section
		9.5.3 Bending of the Rod with Circular Cross-Section
		9.5.4 Bending of the Rod with Rectangular Cross-Section
	9.6 Time-Delayed Presetting
		9.6.1 Instantaneous Ideal Elastic-Ideal Plastic Flow
		9.6.2 Equations of Creep During Time-Delayed Presetting
		9.6.3 Creep Deformation After Instant Plastic Flow
		9.6.4 Elastic Spring-Back and Occurrence of Residual Stresses
		9.6.5 Creep Deformation After Instant Plastic Flow for Garofalo Law
	9.7 Conclusions
	9.8 Summary of Principal Results
	References
Part III Service Life and Durability of Springs
10 Creep and Relaxation of Springs
	10.1 Operational Damage of Spring Elements
	10.2 Common Creep Constitutive Equations
		10.2.1 Constitutive Equations for Creep of Spring Elements
		10.2.2 Time-Dependent Constitutive Equations
		10.2.3 Experimental Acquisition of Creep Laws
		10.2.4 Time-Invariant Constitutive Equations
	10.3 Scalar Constitutive Equations for Uniaxial Stresses
		10.3.1 Norton-Bailey Law
		10.3.2 Garofalo Creep Law
		10.3.3 Exponential Law
	10.4 Creep and Relaxation of Twisted Rods
		10.4.1 Constitutive Equations for Relaxation in Torsion
		10.4.2 Torque Relaxation for Norton-Bailey Law
		10.4.3 Torque Relaxation for Garofalo Law
		10.4.4 Torque Relaxation for Exponential Law
	10.5 Creep and Relaxation of Helical Coiled Springs
		10.5.1 Phenomena of Relaxation and Creep
		10.5.2 Relaxation of Helical Springs
		10.5.3 Creep of Helical Compression Springs
	10.6 Creep and Relaxation of Beams in State of Pure Bending
		10.6.1 Constitutive Equations for Relaxation in Bending
		10.6.2 Relaxation of Bending Moment for Norton-Bailey Law
		10.6.3 Relaxation of Bending Moment for Garofalo Law
		10.6.4 Relaxation of Bending Moment for Exponential Law
		10.6.5 Creep in State of Bending
	10.7 Creep and Relaxation of Disk Springs
		10.7.1 Creep of Disk Springs
		10.7.2 Relaxation of Disk Springs
	10.8 Cyclic Creep and Fatigue-Creep Interaction
	10.9 Temperature Influence on Creep
	10.10 Conclusions
	References
11 Fatigue of Spring Materials
	11.1 Phenomenon of Fatigue
		11.1.1 Fatigue Influence Factors
		11.1.2 Stages of Fatigue Fracture
	11.2 Crack Initiation Approach for Uniaxial Stress State
		11.2.1 Stress-Life Approach for Symmetric Cycle
		11.2.2 Strain-Life Approach for Symmetric Cycle
	11.3 Crack Propagation Approach
		11.3.1 Crack Growths Functions of Paris-Erdogan Type
		11.3.2 Fatigue Crack Growths Functions for Crack Under Cyclic Loading
	11.4 Fatigue Crack Growths
		11.4.1 Unification of Paris Law
		11.4.2 Unification of Paris Law Type I
		11.4.3 Unification of Fatigue Law Type II
	11.5 Conclusions
	References
12 Factors Affecting the Fatigue Life of Springs
	12.1 Fatigue Life Estimation Based on Empirical Damage Models
		12.1.1 Influence of Stress Ratio, Amplitude and Mean Stress
		12.1.2 Evaluation of Fatigue Life with Goodman Diagrams
		12.1.3 Evaluation of Fatigue Life with Haigh Diagrams
		12.1.4 Consistence of Fatigue Life Diagrams and Creep Diagrams
		12.1.5 Accumulation of Damage and Sequence Effects
	12.2 Stress Ratio Influence in Uni-axial Fatigue
		12.2.1 Stress-Life Approach with Variable Stress Ratio
		12.2.2 Equivalent Stress Amplitude
		12.2.3 Strain-Life Approach with Variable Stress Ratio
	12.3 Stress Ratio Influence on Fatigue Crack Growth
		12.3.1 Influence of Stress Ratio on Fatigue Threshold
		12.3.2 Models for Estimation of Stress Ratio Influence
		12.3.3 Stress Ratio Influence in NASGRO Model
	12.4 Influence of Defects on Fatigue Resistance
		12.4.1 Influence of Area of Defects for Uniaxial Stress
		12.4.2 Influence of Area of Defects for Shear Stress
		12.4.3 Influence of Corrosion-Induced Defects
	12.5 Influence of Multiaxial Stresses
		12.5.1 Stress-Life Approach in Multiaxial Fatigue
		12.5.2 Strain-Life Approach in Multiaxial Fatigue
	12.6 Influence of Environment and Manufacturing
		12.6.1 Influence of Wire Diameter
		12.6.2 Influence of Shot Peening
	References
13 Failure Analysis Based on Weakest Link Concepts
	13.1 Evaluation of Failure Probability of Springs
	13.2 Weakest Link Concepts for Homogeneously Loaded Elements
		13.2.1 Failure Probability of Elements with Distributed Defects
		13.2.2 Application of Weakest Link for Fatigue
	13.3 Analysis of Springs with Weakest Link Method
		13.3.1 Failure Probability of Helical Springs
		13.3.2 Effect of Spring Index on Immediate Failure
		13.3.3 Effect of Spring Index on Fatigue Life
	13.4 Conclusions
	13.5 Summary of Principal Results
	References
14 Statistical Effects on Fatigue of Spring Materials
	14.1 Fatigue Analysis at Very High Number of Cycles
		14.1.1 Fatigue Strength and Failure Mechanisms
		14.1.2 Experimental Investigation for Low-Stress Springs’ Fatigue
		14.1.3 Modelling Hypothesis for Low-Stress Springs’ Fatigue
	14.2 Statistical Assessment of Springs Fatigue Test
		14.2.1 Fatigue Scattering of Spring Steels
		14.2.2 Uniformity of Stress on Wire Surface and Averaged Fatigue Stress
		14.2.3 Experimental Acquisition for Sensitivity to Stress Ratio and Spring Indices
		14.2.4 Median S–N Curves
		14.2.5 Valuation of S–N Lines for Prearranged Failure Probabilities
		14.2.6 S–N Curves of Spring Steels in VHCF Range
	14.3 Scale-Dependent Propagation of Straight Crack
		14.3.1 S–N Curves of Spring Steels in Transition Range
		14.3.2 Piecewise Linear Paris-Irvin Equation
	14.4 Scale-Dependent Propagation of Probabilistic Crack
		14.4.1 Simulation of Crack Propagation with Random Deviation
		14.4.2 Closed-Form Solution of the Stochastic Differential Equation
	14.5 Conclusions
	References
Appendix A Models of Spring Materials
A.1 Plastic Deformation
Creep Deformation
Appendix B Some Special Functions
Appendix C Statistical Assessment for Scattering of Fatigue Data
Appendix D Influence of Mean Stress on Fatigue Life
D.1 Fatigue Strength Diagrams
D.2 Mean Stress Sensitivity According to Schütz
D.3 Mean Stress Sensitivity According to Smith–Watson–Topper
D.4 Mean Stress Sensitivity According to Walker
D.5 Mean Stress Sensitivity According to Bergmann
D.6 Recalculation Between Bergmann and Walker Criteria
References
Index