# Stress-Based Uniaxial Fatigue Analysis Using Methods Described in FKM-Guideline

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## Abstract

The process of prevention of failure from structural fatigue is a process that should take place during the early development and design phases of a structure. In the ground vehicle industry, for example, the durability specifications of a new product are directly interweaved with the desired performance characteristics, materials selection, manufacturing methods, and safety characteristics of the vehicle. In the field of fatigue and durability analysis of materials, three main techniques have emerged: nominal stress-based analysis, local strain-based analysis, and fracture mechanics analysis. Each of these methods has their own strengths and domain of applicability—for example, if an initial crack or flaw size is known to exist in a structure, a fracture mechanics approach can give a meaningful estimate of the number of cycles it takes to propagate the initial flaw to failure. The development of the local strain-based fatigue analysis approach has been used to great success in the automotive industry, particularly for the analysis of measured strain time histories gathered during proving ground testing or customer usage. However, the strain life approach is dependent on specific material properties data and the ability to measure (or calculate) a local strain history. Historically, the stress-based fatigue analysis approach was developed first—and is sometimes considered an “old” approach—but the stress-based fatigue analysis methods have been continued to be developed. The major strengths of this approach include the ability to give both quantitative and qualitative estimates of fatigue life with minimal estimates on stress levels and material properties, thus making the stress-based approach very relevant in the early design phase of structures where uncertainties regarding material selection, manufacturing processes, and final design specifications may cause numerous design iterations. This article explains the FKM-Guideline approach to stress-based uniaxial fatigue analysis. The Forschungskuratorium Maschinenbau (FKM) was developed in 1994 in Germany and has since continued to be updated. The guideline was developed for the use of the mechanical engineering community involved in the design of machine components, welded joints, and related areas. It is our desire to make the failure prevention and design community aware of these guidelines through a thorough explanation of the method and the application of the method to detailed examples.

## Keywords

Structural fatigue Durability analysis Failure prevention Stress-based fatigue analysis Surface finish effect## List of symbols

*A*Fatigue parameter

*a*_{d}Constant in the size correction formula

*a*_{R}Roughness constant

*a*_{P}Peterson’s material constant

*a*_{N}Neuber’s material constant

*a*_{SS}Siebel and Stieler material parameter

*a*_{G}Material constant in the

*K*_{t}/*K*_{f}ratio*a*_{M}Material parameter in determining the mean stress sensitivity factor

*B*Width of a plate

*b*Slope (height-to-base ratio) of an

*S*–*N*curve in the HCF regime*b*_{M}Material parameter in determining the mean stress sensitivity factor

*b*_{W}Width of a rectangular section

*b*_{nw}Net width of a plate

*b*_{G}Material constant in the

*K*_{t}/*K*_{f}ratio*C*_{R}Reliability correction factor

*C*_{D}Size correction factor

*C*_{u,T}Temperature correction factor for ultimate strength

*C*_{σ}Stress correction factor for normal stress

*C*_{τ}Stress correction factor for shear stress

*C*_{b,L}Load correction factor for bending

*C*_{t,L}Load correction factor for torsion

*C*_{E,T}Temperature correction factor for the endurance limit

*C*_{σ,E}Endurance limit factor for normal stress

*C*_{S}Surface treatment factor

*C*_{σ,R}Roughness correction factor for normal stress

*C*_{τ,R}Roughness correction factor for shear stress

- COV
_{S} Coefficient of variations

*D*Diameter of a shaft

*D*_{PM}Critical damage value in the linear damage rule

*d*Net diameter of a notched shaft

*d*_{eff}Effective diameter of a cross section

*d*_{eff,min}Minimum effective diameter of a cross section

*G*Stress gradient along a local

*x*-axis- \( \bar{G} \)
Relative stress gradient

- \( \bar{G}_{\sigma } (r) \)
Relative normal stress gradient for plate or shaft based on notch radius

- \( \bar{G}_{\sigma } (d) \)
Relative normal stress gradient for plate or shaft based on component net diameter or width at notch

- \( \bar{G}_{\tau } (r) \)
Relative shear stress gradient for plate or shaft based on notch radius

- \( \bar{G}_{\tau } (d) \)
Relative shear stress gradient for plate or shaft based on component net diameter or width at notch

- HB
Brinell hardness

- 2
*h*_{T} Height of a rectangular section

*K*_{ax,f}Fatigue notch factor for a shaft under axial loading

*K*_{ax,t}Elastic stress concentration factor for a shaft under axial loading

*K*_{b,f}Fatigue notch factor for a shaft under bending

*K*_{b,t}Elastic stress concentration factor for a shaft under bending

*K*_{f}Fatigue notch factor or the fatigue strength reduction factor

*K*_{i,f}Fatigue notch factor for a superimposed notch

*K*_{s,f}Fatigue notch factor for a shaft under shear

*K*_{s,t}Elastic stress concentration factor for a shaft under shear

*K*_{t}Elastic stress concentration factor

*K*_{t,f}Fatigue notch factor for a shaft under torsion

*K*_{t,t}Elastic stress concentration factor for a shaft under torsion

*K*_{x,f}Fatigue notch factor for a plate under normal stress in

*x*-axis*K*_{y,f}Fatigue notch factor for a plate under normal stress in

*y*-axis- \( K_{{\tau_{xy} , {\text{f}}}} \)
Fatigue notch factor for a plate under shear

*K*_{x,t}Elastic stress concentration factor for a plate under normal stress in

*x*-axis*K*_{y,t}Elastic stress concentration factor for a plate under normal stress in

*y*-axis- \( K_{{\tau_{xy} , {\text{t}}}} \)
Elastic stress concentration factor for a plate under shear stress

*k*Slope factor (negative base-to-height ratio) of an

*S*–*N*curve in the HCF regime*M*_{i}Initial yielding moment

*M*_{o}Fully plastic yielding moment

*M*_{σ}Mean stress sensitivity factor in normal stress

*N*Number of cycles to a specific crack initiation length

- 2
*N* Number of reversals to a specific crack initiation length

*N*_{E}Endurance cycle limit

*N*_{f,i}Number of cycles to failure at the specific stress event

*n*_{K}*K*_{t}/*K*_{f}ratio or the supporting factor*n*_{K,σ}(*r*)*K*_{t}/*K*_{f}ratio for a shaft under normal stress based on the notch radius*n*_{K,σ}(*d*)*K*_{t}/*K*_{f}ratio for a shaft under normal stress based on component net diameter or width at notch*n*_{K,σ,x}(*r*)*K*_{t}/*K*_{f}ratio for a plate under normal stress in*x*-axis based on notch radius*n*_{K,σ,y}(*r*)*K*_{t}/*K*_{f}ratio for a plate under normal stress in*y*-axis based on notch radius*n*_{K,τ}(*r*)*K*_{t}/*K*_{f}ratio for a plate or shaft under shear stress based on notch radius*n*_{K,τ}(*d*)*K*_{t}/*K*_{f}ratio for a shaft under shear stress based on component net diameter or width at notch*n*_{i}Number of stress cycles

*O*Surface area of the section of a component

*q*Notch sensitivity factor

*R*_{r}Reliability value

*R*Stress ratio = ratio of minimum stress to maximum stress

*R*_{Z}Average roughness value of the surface based on German DIN system

*r*Notch root radius

*r*_{max}Larger of the superimposed notch radii

*S*Nominal stress

*S*_{C}Nominal stress of a notched component

*S*_{a}Stress amplitude

*S*_{m}Mean stress

*S*_{max}Maximum stress

*S*_{min}Minimum stress

*S*_{σ,a}Normal stress amplitude in a stress cycle

*S*_{σ,m}Mean normal stress in a stress cycle

*S*_{σ,max}Maximum normal stresses in a stress cycle

*S*_{σ,min}Minimum normal stresses in a stress cycle

*S*_{σ,ar}Equivalent fully reversed normal stress amplitude

*S*_{σ,E}Endurance limit for normal stress at 10

^{6}cycles*S*_{τ,E}Endurance limit for shear stress at 10

^{6}cycles*S*_{E}Endurance limit at 10

^{6}cycles*S*_{N,E}Nominal endurance limit of a notched component

*S*_{S,E,Smooth}Nominal endurance limit of a smooth component at 10

^{6}cycles*S*_{S,E,Notched}Nominal endurance limit of a notched component at 10

^{6}cycles*S*_{S,σ,E}Endurance limit of a smooth, polish component under fully reversed normal stress

*S*_{σ,FL}Fatigue limit in normal stress at 10

^{8}cycles*S*_{S,τ,E}Endurance limit of a smooth, polish component under fully reversed shear stress

*S*_{S,τ,u}Ultimate strength of a notched, shell-shaped component for shear stress

- \( S_{{{\text{S,}}\tau_{xy} , {\text{E}}}} \)
Endurance limit of a notched, shell-shaped component under fully reversed shear stress

*S*_{τ,FL}Fatigue limit in shear at 10

^{8}cycles*S*_{S,ax,E}Endurance limit of a notched, rod-shaped component under fully reversed axial loading

*S*_{S,ax,u}Ultimate strength of a notched, rod-shaped component in axial loading

*S*_{S,b,E}Endurance limit of a notched, rod-shaped component under fully reversed bending loading

*S*_{S,b,u}Ultimate strength of a notched, rod-shaped component in bending

*S*_{S,s,E}Endurance limit of a notched, rod-shaped component under fully reversed shear loading

*S*_{S,s,u}Ultimate strength of a notched, rod-shaped component in shear

*S*_{S,t,E}Endurance limit of a notched, rod-shaped component under fully reversed torsion loading

*S*_{S,t,u}Ultimate strength of a notched, rod-shaped component in torsion

*S*_{S,x,E}Endurance limit of a notched, shell-shaped component under fully reversed normal stress in

*x*-axis*S*_{S,x,u}Ultimate strength of a notched, shell-shaped component for normal stress in

*x*-axis*S*_{S,y,E}Endurance limit of a notched, shell-shaped component under fully reversed normal stress in

*y*-axis*S*_{S,y,u}Ultimate strength of a notched, shell-shaped component for normal stress in

*y*-axis*S*_{t,u}Ultimate tensile strength with R97.5

*S*_{t,u,min}Minimum ultimate tensile strength

*S*_{t,u,std}Mean ultimate tensile strength of a standard material test specimen

*S*_{t,y}Tensile yield strength with R97.5

*S*_{t,y,max}Maximum tensile yield strength

- \( S^{\prime}_{\text{f}} \)
Fatigue strength coefficient

- \( {\text{S}^{\prime}}_{{\sigma , {\text{f}}}} \)
Fatigue strength coefficient in normal stress

*T*Temperature in degrees Celsius

*t*_{c}Coating layer thickness in μm

*V*Volume of the section of a component

- σ
^{e} Fictitious or pseudo-stress

- σ
^{e}(*x*) Pseudo-stress distribution along

*x*- \( \sigma_{\text{E}}^{\text{e}} \)
Pseudo-endurance limit

- \( {{\sigma}}_{ \max }^{\text{e}} \)
Maximum pseudo-stress at

*x*= 0- \( \Upphi (z) \)
Standard normal density function

- \( \varphi = 1/(4\sqrt {t/r} + 2) \)
Parameter to calculate relative stress gradient

*γ*_{W}Mean stress fitting parameter in Walker’s mean stress formula

## Notes

### Acknowledgments

The following people should be recognized for their technical support in writing this report: Robert Burger, Peter Bauerle, and Richard Howell of Chrysler Group LLC.

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