1. FATIGUE 2014 MCG, Melbourne, Australia, 2-7 March 2014
School of Mechanical Engineering

2. An Investigation into Fatigue Crack Growth through a Weld Toe Residual Stress Field
Alan K. HELLIER1,a *, Gangadhara B. PRUSTY1,b, Michael ANDARY1,
Ania M. PARADOWSKA1,2,c and David G. CARR3,d
1School of Mechanical and Manufacturing Engineering, The University of New South Wales,
Sydney, NSW 2052, Australia
2Bragg Institute, Australian Nuclear Science and Technology Organisation,
Locked Bag 2001, Kirrawee DC, NSW 2232, Australia
3Institute of Materials Engineering, Australian Nuclear Science and Technology Organisation,
*Corresponding author

3. Introduction
Welds are a common method of joining metallic components in buildings, bridges and offshore structures (with 350 grade structural steel often being used in these applications).
However, welds are susceptible to fatigue crack initiation and slow but accelerating growth arising due to fluctuating service loads, often eventually resulting in fracture unless detected and remedied.
An existing Brennan-Dover-Karé-Hellier (BDKH) parametric equation is available for T-butt weld toe stress intensity factor (SIF) geometric Y-factor at the deepest point of a semi-elliptical surface crack, subject to tension (membrane) loading.
The equation is wide ranging and accurate, having been published back in 1999 in the International Journal of Fatigue.
To the best of our knowledge this equation has never been used except by us, probably because it is very complicated.

4. Structural Integrity Parametric Equations
Parametric equations are derived from a large number of 2-D FE analyses or a smaller number of 3-D FE analyses (or possibly from a number of physical tests on scale-models in the laboratory).
Structural integrity parametric equations may give approximations to stress intensity factors (SIFs) for cracks, stress concentration factors (SCFs) for notches, or stress distributions.
Parametric equation predictions are virtually instantaneous.
Even inaccurate parametric equations are useful provided the distribution of percentage errors from original data is quantified.
For example, if the greatest underestimate is known to be 50% then a safety factor of 2 may be applied to all predicted values in order to always be conservative.

5. Welded T-butt Plate Joints
These are found in a wide variety of structural components.
In addition, other more complex geometrical joints are often simplified for stress analysis purposes by approximating them as 2-D T-butt plate models.
These include skewed T-joints, cruciform welded joints, tubular welded joints, and pipe–plate joints.
Accurate predictions of SIFs, SCFs and through-thickness stress distributions aid efficient structural design and in-service structural integrity assessment management.
They can be used together or for SIF weight function calculations using linear elastic fracture mechanics (LEFM) modelling.
This approach can give accurate assessments of fatigue crack propagation rates with relative ease.

6. Geometry of T-butt Weldments Studied

7. Two-dimensional Finite Element Analyses
80 2-D plane stress FE analyses of T-butt welds were conducted.
Plane stress was used to ensure the resulting applied stresses are the highest possible (to be conservative).
By contrast, when measuring the resistance to crack growth, plane strain gives the lowest possible stress (again to be conservative).
Weldment angle (), weld toe radius (), weld attachment width (L), and plate thickness (T) were varied for tension (membrane) loading.
The geometry validity limits for all the equations developed are:-
Weld angle: 30°<<60°
Weld toe radius: 0.01< /T<0.066
Attachment width: 0.3<L/T<4.0
The weld attachment width is twice the attachment plate thickness.

8. Matrix of Geometrical Parameters Analysed

9. Finite Element Mesh (Dimensions in mm)

10. Finite Element Fillet Radius Detail

11. Derivation of HBC Full SCF Parametric Equation
A simple Hellier-Brennan-Carr (HBC) SCF parametric equation was used as the starting point in the derivation of a more accurate, full expression for weld toe surface SCF.
Terms were fitted to the error in the SCF prediction, starting with a linear expression in , ( /T) and (L/T), moving successively to a quadratic, cubic, and finally up to a full quartic (including all cross terms).
Initial values of the coefficients were obtained by linear regression.
All coefficients were then iteratively optimised to give the best percentage error distribution, before going on to fit higher order terms to the residual errors.
Terms which, after exhaustive experimentation, gave no significant improvement in the percentage error distribution were discarded.
The remaining optimisation procedure was as for the simple SCF.

12. HBC Full SCF Parametric Equation
SCFmw0 = – 0.302 ( /T) (L/T)
 ( /T)2 – 0.633(L/T)2
– 0.614 (L/T)3
– 0.018(L/T)4 – 35.5( /T)
– 0.153(L/T) ( /T)(L/T)
+ 30.62( /T) – 0.2192(L/T)
– 64.3( /T) (L/T)2
– 54.5( /T)2(L/T) ( /T)(L/T)2
+ 0.68( /T)–0.299(L/T) (1)
The histogram of percentage errors is narrow and sharply peaked, with the values all lying between –3.8% and +3.1%.
The residual standard deviation is

13. Full SCF Scatter Diagram Showing Errors

14. Full SCF Histogram of Percentage Errors

15. HBC Stress Distribution Parametric Equation
(2)

16. Stress Distribution Equation Scatter Diagram Showing Errors

17. Stress Distribution Equation Histogram of Residual Errors
Membrane Stress
Bend Stress

18. BDKH SIF Y-factor Parametric Equations
Existing Brennan-Dover-Karé-Hellier (BDKH) parametric equations are available for T-butt weld toe stress intensity factor (SIF) geometric Y-factor at the deepest point of a semi-elliptical surface crack, subject to tension (membrane) and pure bending loadings.
These are based on the same 80 2-D FE analyses as the HBC equation, where a is the crack depth and c is the semi-crack width.
The following additional crack geometry validity limits apply:-
Range of crack aspect ratios: 0<a/c<1.0
Range of crack depths: 0.01<a/T<1.0
These equations were commissioned by the Marine Technology Support Unit of the UK Health and Safety Executive back in 1995.
A comprehensive validity study was subsequently conducted on both equations involving the generation of 160 tables and 81 plots.

19. Weld Showing Semi-elliptical Crack Geometry

20. Fatigue Crack Propagation Life Programs
Three simple fatigue crack propagation life programs have been written in fully internally-documented ‘virtually perfect’ FORTRAN.
FATIGUE1 uses the BDKH tension SIF Y-factor parametric equation with the Paris Law to predict the (Stage 2) fatigue propagation life of a T-butt weld containing an initial semi-elliptical surface flaw.
FATIGUE2 uses the BDKH tension SIF Y-factor parametric equation in conjunction with the Forman Equation to predict the semi-elliptical flaw fatigue life in the presence of a mean applied stress (i.e. with fixed applied stress ratio R = SigMin/SigMax ≥ 0).
FATIGUE3 uses both the BDKH tension SIF Y-factor parametric equation and the HBC tension stress distribution parametric equation in conjunction with the Forman Equation to predict the semi-elliptical flaw fatigue life in the presence of a known residual stress distribution from the literature (for 350 grade structural steel).

21. Paris Law and Forman Equation

22. Assumed T-butt and Crack Geometry for Runs
Weld angle,  = 45°
Weld toe radius,  = 0.4 mm
Main plate thickness, T = 10 mm
 /T = 0.04
Welded attachment width, L = 28 mm
L/T = 2.8 (chosen to be the same as for Monahan’s SCF and stress distribution equations)
Initial semi-elliptical crack depth, ai = 2 mm
Initial semi-elliptical crack width, ci = 30 mm
Initial full crack width, 2ci = 60 mm
Semi-elliptical crack aspect ratio, a/c = 0.067
No. of crack increments is taken as 10,000 to ensure convergence.

23. FATIGUE1 Properties and Loading Conditions
Properties for 350 grade structural steel will be used:-
Fatigue crack growth threshold, ΔKth = 3.2 MPa√m (only considered to ensure that the applied stress range is enough to exceed this)
Fatigue crack growth coefficient in Paris Law, C = 8.57 x 10-12
(in SI units)
Fatigue crack growth exponent in Paris Law, m = 3 (dimensionless)
Applied stresses for FATIGUE1:-
Minimum applied membrane stress, SigMin = 0 MPa
Maximum applied membrane stress, SigMax = 50 MPa
R = SigMin/SigMax = 0

24. Fatigue Crack Growth Curve from FATIGUE1

25. FATIGUE2 Properties and Loading Conditions
Properties for 350 grade structural steel will be used:-
Fatigue crack growth threshold, ΔKth = 3.2 MPa√m
Fatigue crack growth coefficient in Forman Equation, C = 3.7 x 10-9 (in SI units)
Value of C gives the same fatigue life with R = 0 as the Paris Law
Fatigue crack growth exponent in Forman Equation, m = 3
Fracture toughness Kc = 440 MPa√m (got from JIc = 114 kN/m)
Applied stresses for FATIGUE 2:-
Minimum applied membrane stress, SigMin = 10, 20 and 30 MPa
Maximum applied membrane stress, SigMax = 60, 70 and 80 MPa
R = SigMin/SigMax = 0.167, and 0.375

26. Fatigue Crack Growth Curves from FATIGUE2

27. Residual Stress Distribution from the Literature

28. FATIGUE3 Properties and Loading Conditions
Properties for 350 grade structural steel will be used:-
Fatigue crack growth threshold, ΔKth = 3.2 MPa√m
Fatigue crack growth coefficient in Forman Equation, C = 3.7 x 10-9 (in SI units)
Value of C gives the same fatigue life with R = 0 as the Paris Law
Fatigue crack growth exponent in Forman Equation, m = 3
Fracture toughness Kc = 440 MPa√m
Applied stresses for FATIGUE 3:-
Minimum applied membrane stress, SigMin = 0 MPa
Maximum applied membrane stress, SigMax = 50 MPa
R = SigMin/SigMax = 0 but actual R will vary with residual stress.

29. Fatigue Crack Growth Curves from FATIGUE3

30. Conclusions
A BDKH parametric equation is available for T-butt weld toe stress intensity factor (SIF) geometric Y-factor at the deepest point of a semi-elliptical surface crack, subject to tension (membrane) loading.
HBC parametric equations for T-butt weld toe SCF and through-thickness stress distribution in the anticipated plane of Mode I crack growth have recently been derived, also subject to tension (membrane) loading.
These are available as a function of weldment angle, weld toe radius, weld attachment width, and plate thickness.
They are considered to be accurate and wide-ranging.
They have also all now been coded in fully internally-documented ‘virtually perfect’ FORTRAN and integrated into 3 programs: FATIGUE1, FATIGUE2 and FATIGUE3.

31. Conclusions (continued)
FATIGUE1 uses the BDKH tension SIF Y-factor parametric equation with the Paris Law to predict the (Stage 2) fatigue propagation life of a T-butt weld containing an initial semi-elliptical surface flaw.
FATIGUE2 uses the BDKH tension SIF Y-factor parametric equation in conjunction with the Forman Equation to predict the semi-elliptical flaw fatigue life in the presence of a mean applied stress (i.e. with fixed applied stress ratio R = SigMin/SigMax ≥ 0).
FATIGUE3 uses both the BDKH tension SIF Y-factor parametric equation and the HBC tension stress distribution parametric equation in conjunction with the Forman Equation to predict the semi-elliptical flaw fatigue life in the presence of a known residual stress distribution from the literature (for 350 grade structural steel).
Graphs of crack length a versus number of fatigue cycles N were generated in a matter of seconds for 10,000 crack increments.

32. Conclusions (continued)
Although only one weld geometry and crack geometry have been considered, some tentative conclusions may be drawn.
The fatigue crack growth curve from FATIGUE1 is exponential and asymptotic when the fatigue life is reached. The predicted fatigue propagation life is reasonable.
The fatigue crack growth curves from FATIGUE2 show that increasing the R-ratio leads to shorter fatigue lives as expected.
The fatigue crack growth curves from FATIGUE3 show that a typical residual stress distribution does shorten the fatigue propagation life, but not by much.
However, the fatigue initiation life is expected to be considerably reduced due to the high tensile stress at the weld toe surface.
It has been assumed that crack closure occurs when R is negative.

33. Further Work
A number of T-butt welded joints with typical geometry will be fabricated from 350 grade structural steel.
One as-received specimen will have its through-thickness residual stress distribution determined firstly by neutron diffraction at ANSTO Bragg Institute, and secondly by the contour method at ANSTO Materials. These will be compared.
Others will be thermally stress-relieved and shot-peened.
All will be notched and tested on an INSTRON machine subject to either tension (membrane) loading or pure bending loading.
We need to find ~$A60,000 to purchase a state-of-the-art crack microgauge to monitor crack growth.
The fatigue crack growth curves from the mechanical testing will be compared with those predicted by our software.