Determination of notch factors for welded cruciform joints based on numerical analysis and metamodeling

  • Markus Oswald
  • Christina Mayr
  • Klemens RotherEmail author
Research Paper


The effective notch stress approach to estimate fatigue strength of welded components requires knowledge of stress concentration factors of an idealized weld geometry using notch radii. This paper covers the estimation of those stress concentration factors for welded cruciform joints with double-filled welds, and K- or double-Y-butt welds with partial or full penetration. Thin, medium, and thick-walled joints are covered in the resulting estimations as well as different notch radii and weld angles. In comparison with different existing estimations, new methods using metamodeling (a) by the response surface method based on polynomial regression using coupling terms and (b) based on artificial neural networks are presented. Both methods show similar and superior quality. Much lower errors compared to existing estimation methods are obtained. The methods were trained by a large data base of reference results obtained by finite element analysis for 5973 design alternatives (samples) in total. Besides higher quality of prognosis, the new metamodels enhance the range of allowable parameters of the cruciform joints compared to existing ones. The resulting methods provide sound means of obtaining stress concentration factors fast and of sufficient quality which could also be embedded in more complex applications as programmed solutions.


Elastic analysis Finite element analysis Mathematical models Sampling Notches Cruciform joints 


Symbols, abbreviations

ANN (−)

artificial neural network

α (°)

flank angle

b (mm)

total model width

bi (−)

bias vectors for artificial neural networks

ck (−)

scalar multiplication parameter for the PRC method

d (mm)

total model depth

E (MPa)

modulus of elasticity

errrel (%)

relative error

F (N)


fk (−)

value of geometric multiplication parameter for the PRC method

g (−)

input vector for the ANN method

h (mm)

total model height

Kf (−)

fatigue notch factor

Kt (−)

stress concentration factor

Kt, EST (−)

stress concentration factor, estimated

Kt, FEM (−)

stress concentration factor, calculated by FEM

kt (−)

stress concentration output vector of the ANN method

Kt, AKS (−)

stress concentration factor of the Anthes et al. method

Kt, ANN (−)

stress concentration factor of the ANN method

Kt, PRC (−)

stress concentration factor of the PRC method

Kt, RAD (−)

stress concentration factor of Radaj’s method

Kt, RAI (−)

stress concentration factor of Rainer’s method

Kt, YL (−)

stress concentration factor of Yung and Lawrence’s method

Kw (−)

ratio of notch stress to structural stress

Kw, min (−)

minimum ratio of notch stress to structural stress

l1 (mm)

leg length

M (N/mm)


ν (−)

Poisson ratio

Φi (−)

artificial neural network layer potential

PRC (−)

polynomial regression with coupling terms

r (mm)

Notch radius

Sb (MPa)

nominal bending stress

St (MPa)

nominal tension stress

σe (MPa)

Notch stress

σw (MPa)

structural stress

ti (mm)

sheet thickness

w (mm)

length of root face

Wi (−)

weight matrices of artificial neural networks

xi,gain (−)

gain input vector for artificial neural networks

xi,offset (−)

offset input vector for artificial neural networks

y (−)

ratio of leg length to sheet thickness

yo,gain (−)

gain output vector of artificial neural networks

yo,offset (−)

offset output vector of artificial neural networks

z (−)

ratio of length of root face to first sheet thickness

Indices, superscripts

b, bend


t, tens


f. p.

full penetration


PRC method index

p. p.

partial penetration

r, root






The financial support is greatly acknowledged.

Funding information

The IGF project 19450 N of FOSTA (Forschungsvereinigung Stahlanwendung e.V.), Düsseldorf, is funded by the Federal Ministry of Economic Affairs and Energy via the AiF within the framework of the program for the promotion of the Industrielle Gemeinschaftsforschung (IGF) based on a resolution of the German Bundestag.


  1. 1.
    Hobbacher AF (2016) Recommendations for fatigue design of welded joints and components, 2nd edn. Springer Verlag, BerlinCrossRefGoogle Scholar
  2. 2.
    Radaj D, Sonsino CM, Fricke W (2006) Fatigue assessment of welded joints by local approaches, 2nd edn. Woodhead Publishing in materials. Woodhead Publishing Limited, CambridgeCrossRefGoogle Scholar
  3. 3.
    Deutscher Verband für Schweißen und verwandte Verfahren e.V. (2017) DVS 0905 - Industrielle Anwendung des Kerbspannungskonzepter für den Ermüdungsfestigkeitsnachweis von Schweißverbindungen(0905)Google Scholar
  4. 4.
    Rother K, Rudolph J (2011) Fatigue assessment of welded structures: practical aspects for stress analysis and fatigue assessment. Fatigue & Fracture of Engineering Materials & Structures 34:177–204. CrossRefGoogle Scholar
  5. 5.
    Köttgen VB, Olivier R, Seeger T (1992) Fatigue analysis of welded connections based on local stresses: IIW document XIII, pp 1408–1491Google Scholar
  6. 6.
    Radaj D (1990) Design and analysis of fatigue resistant welded structures. Woodhead Publishing Series in Welding and Other Joining Technologies, CambridgeCrossRefGoogle Scholar
  7. 7.
    Neuber H (1968) Über die Berücksichtigung der Spannungskonzentration bei Festigkeitsberechnungen. Konstruktion 20(7):245–251Google Scholar
  8. 8.
    Lawrence FV, Ho NJ, Mazumdar PK (1981) Predicting the fatigue resistance of welds. Annual Review of Material Science 11:401–425CrossRefGoogle Scholar
  9. 9.
    Yung JL, Lawrence FV (1985) Analytical and graphical aids for the fatigue design of weldments. Fatigue & Fracture of Engineering Materials & Structures 8(3):223–241CrossRefGoogle Scholar
  10. 10.
    Rainer G (1979) Errechnen von Spannungen in Schweißverbindungen mit der Methode der Finiten Elemente. In: Frankfurt/MainGoogle Scholar
  11. 11.
    Rainer G (1983) Parameterstudien mit finiten Elementen, Berechnung der Bauteilfestigkeit von Schweißverbindungen unter äußeren Beanspruchungen. Konstruktion 37(2):45–52Google Scholar
  12. 12.
    Haibach E (2006) Betriebsfestigkeit: Verfahren und Daten zur Bauteilberechnung, 3rd edn. Springer Verlag, BerlinGoogle Scholar
  13. 13.
    Radaj D (1986) Zur vereinfachten Darstellung der mehrparametrigen Formzahlabhängigkeit. Konstruktion 38(5):193–197Google Scholar
  14. 14.
    Radaj D, Zhang S (1990) Mehrparametrige Strukturoptimierung hinsichtlich Spannungserhöhungen. Konstruktion 42:289–292Google Scholar
  15. 15.
    Radaj D, Zhang S (1991) Multiparameter design optimisation in respect of stress concentrations. In: Springer-Verlag (ed) engineering optimisation in design processes. Springer, Berlin, pp 181–189CrossRefGoogle Scholar
  16. 16.
    Anthes RJ, Köttgen VB, Seeger T (1993) Kerbformzahlen von Stumpfstößen und Doppel-T-Stößen. Schweissen und Schneiden 45(12):685–688Google Scholar
  17. 17.
    Anthes RJ, Köttgen VB, Seeger T (1994) Einfluß der Nahtgeometrie auf die Dauerfestigkeit von Stumpf- und Doppel-T-Stößen. Schweissen und Schneiden 46(9):433–436Google Scholar
  18. 18.
    Ushirokawa O, Nakayama E (1983) Stress concentration factor at welded joints 23(4)Google Scholar
  19. 19.
    Tsuji I (1990) Estimation of stress concentration factor at weld toe of non-load-carrying fillet welded joints. West Japan Society of Naval Architects 80:241–251Google Scholar
  20. 20.
    Wächter M (2016) Zur Ermittlung von zyklischen Werkstoffkennwerten und Schädigungsparameterwöhlerlinien. Dissertation, TU ClausthalGoogle Scholar
  21. 21.
    Most T, Will J (2011) Sensitivity analysis using the metamodel of optimal prognosis. Weimar Optimization and Stochastic Days 2011, WeimarGoogle Scholar
  22. 22.
    Most T, Will J (2010) Recent advances in metamodel of optimal prognosis. Weimar Optimization and Stochastic Days 2010, WeimarGoogle Scholar
  23. 23.
    Hagan MT, Demuth HB, Beale MH et al (2014) Neural network design, 2nd edn. PWS Publishing Co, BostonGoogle Scholar
  24. 24.
    Rother K, Fricke W (2016) Effective notch stress approach for welds having low stress concentration. Int J Press Vessel Pip 147:12–20. CrossRefGoogle Scholar
  25. 25.
    ANSYS Homepage (2019) ANSYS, Inc.,, accessed on 04/18/2019
  26. 26.
    Dynardo optiSLang Homepage (2019), Dynardo GmbH, Weimar,, accessed on 04/18/2019
  27. 27.
    ANSYS Help 17.2 (2016) Ansys, Inc., help/ans_elem/Hlp_E_PLANE183.html, Accessed on 04/18/2019Google Scholar
  28. 28.
    SCF-predictor for cruciform joints and circular shafts,, accessed on 04/18/2019

Copyright information

© International Institute of Welding 2019

Authors and Affiliations

  1. 1.Dept. Mechanical, Automotive and Aerospace EngMunich University of Applied SciencesMunichGermany

Personalised recommendations