The stress intensity factors of semi-elliptical cracks in a tubular welded T-joint under axial loading

Engineering

Printed

Fracture

Mechnics

Vol. 30, No. 1, pp. 25-35,

0013-7944188 $3.00+ .oo Pergamon Press pk.

1988

in Great Britain.

THE STRESS INTENSITY FACTORS OF SEMI-ELLIPTICAL CRACKS IN A TUBULAR WELDED T-JOINT UNDER AXIAL LOADING X. HUANG and J. W. HANCOCK Department of Mechanical Engineering, University of Glasgow, Scotland. Abstract-The

stress intensity factors Kt, Kn and Krrr of semi-elliptical cracks located near the chord-brace intersection of a tubular welded T-joint have been determined under axial loading conditions. A finite element analysis technique has been used, based on the line-spring concept of Rice and Levy to represent the crack, while the structure has been represented by shell elements. The stiffness of the joint is maintained until the crack penetrates the chord wall when the stiffness decays. The study provides a basis for a fracture mechanics analysis of the fatigue life of tubular welded joints.

INTRODUCTION

structures fabricated from tubular sections are subject to fatigue loading due to the effects of the marine environment. The most critical regions in such structures are generally recognised to be the tubular welded joints. Because of their technological importance, the associated stress concentrations have been studied by a range of techniques, including finite element analysis[l, 21, photoelasticity[3] and by strain-gauging acrylic[4] or steel models[5]. In these studies it is recognized that the stress concentration arises from two sources. Firstly, the forces and moments acting on any small element of a tube wall are largely determined by the overall geometry of the joint; that is the dimensions of the tubes, the angles of intersection, and the mode of loading. From the forces and moments a corresponding equilibrium set of stresses can be deduced by regarding the tubes as thin elastic shells. The maximum values of these stresses and their stress concentration factors are given in parametric formulae proposed by Kuang[2], and Wordsworth and Smedley[4]. Although the formulae depend on the overall geometry joint, they do not involve the geometry of local details such as the weld profile which provide a second source of stress concentration. The total stress concentration is thus determined both by the overall geometry, the loading of the joint and the detailed geometry of the weld. The overall geometry of the joint determines the forces and moments transmitted through the wall thickness and the corresponding stresses are then further amplified on a very local scale by the details of the weld geometry. In parallel with stress analysis there have been several major experimental investigations[6,7] of the fatigue behaviour of welded joints. These have measured the number of cycles to initate the first small cracks at sites of maximum stress concentration near the weld toe or in the weld. This is generally recognised to be a small fraction of the total life of the joint and depends on the details of the weld preparation. The life of the joint is however determined by the growth of the cracks through the wall thickness, which is determined not by the very localized stresses which caused the initiation of the first crack but largely by the forces and moments transmitted across the wall thickness. The long range stress distribution associated with these forces and moments are usually characterised by ‘hot spot’ stress concentration factors[2,4]. Although these stress concentration factors are widely used in the S-N approach to the design of tubular joints because of the simplicity of the method, the periodic inspection of critical joints in offshore structures frequently reveals the presence of cracks which compromise the integrity of the structure. In these circumstances maintenance and safety requirements demand that predictions of the crack growth be made, and this can only be achieved by fracture mechanics. The successful application of linear elastic fracture mechanics to tubular joints depends on the determination of the appropriate stress intensity factors. These have been determined experimentally in a series of papers by Dover and co-workers[8,9, lo]. Their method has been based on the measurement of the fatigue crack growth rates of semi-elliptical cracks in welded OFFSHORE

25

26

X. HUANG and J. W. HANCOCK

joints. A comparison with the calibration is known, has then inferred. The purpose of the semi-elliptical cracks by a finite

rates measured in small standard specimens for which the K enabled the stress intensity of the semi-elliptical crack to be present work is to calculate the stress intensity factors for element method.

COMPUTATIONAL

METHOD

Several methods for determining K for surface crack problems exist, notably threedimensional finite element analysis[l 11, the use of boundary integral equations[ 121, and generalised weight function methods[l3]. However in the present work the elastic line-spring technique devised by Rice and Levy[l4] has been adopted in conjunction with a shell analysis of the joint. The essential features of the line-spring method of Rice and Levy[l4] can be illustrated by considering a plate subject to a far-field loading consisting of a membrane force such as F, and a moment such as M, as illustrated in Fig. 1. These produce local forces F(x) and moments M(x) which act across each section (x) of the ligament left by the semi-elliptical crack. These local forces and moments vary along the length x of the untracked ligament and produce additional displacements 6(x) and rotations O(x) of the plate mid-surface which are attributable to the crack. The relationship between the local forces and moments, F(x), M(x), and the local displacements 6(x) and rotations O(x) is then of the form

\k 4 2

a= a(x)

(a)

\

X

(b)

Fig. 1. The line-spring model of Rice and Levy[l4].

SIFs of semi-elliptical

cracks in T-joint

27

the [S] is the local stiffness matrix. From the point of view of the plate the surface crack can then be modelled as a series of generalised line-springs acting across a discontinuity in a two-dimensional plate or shell. The local spring stiffnesses S vary with x in accord with the depth of the crack at that point and are determined by reference to the additional displacements and rotations produced by a crack in a simple edge-cracked plane strain specimen subject to tension and bending. Finite element solution of the structural problem now produces the displacements and rotations 6, 8 and the corresponding local forces and moments. The local mode I stress intensity factors are then obtained by reference to the K calibration of a single edge-cracked specimen subject to the forces and moments F(x), M(x). This method has then been further generalised in the same spirit by Desvaux[lS] to include mode II and mode III components. In general the elastic line-spring concept has been used to analyse the stress intensity factors of semi-elliptical cracks in simple geometries such as flat plates or pipes. However in the case of offshore structures cracks are located near the stress concentrations associated with geometrical discontinuities such as tube-tube intersections. In order to check the applicability of the line-spring technique to this type of problem some preliminary calculations have been performed on simple T-butt joints made from flat plates. In this simple system, the stress intensity factors have been determined by two different methods. Further details are given by Huang[l6], but here it is only necessary to consider one simple T-butt joint which was idealised as shown in Fig. 2. In the first case shown in Fig. 2(a) the joint was represented by 14 elastic 8-noded shell elements which do not allow the details of the weld geometry to be modelled while the crack was represented by an elastic line spring located at a position representative of the weld toe. In the second case the joint was represented by approximately 150 elastic 8-noded plane strain isoparametric elements. In contrast these elements allow the details of the local weld geometry to be modelled specifically, and three idealised weld profiles have been considered. The first case models a smoothly radiused weld profile, the second case a straight 45” weld profile, and finally the weld profile was removed completely. The meshes corresponding to these models are shown in Fig. 2(b) with the crack located in similar position at the weld toe. In each case the crack tip was modelled by a focused mesh with the mid-side nodes located at the quarter point positions. This procedure allows the crack tip element to adopt the correct form of displacement function for the elastic singularity as discussed by Barsoum[l7] and Henshell and Shaw[l8]. With these meshes the stress intensity factors have been determined with the aid of the ABAQUS finite

where

(a)

lb)

Fig. 2. The idealisations of the T-butt joint using shell element in Fig. 2(a), and isoparametric continuum elements in Fig. 2(b)[16]. In Fig. 2(a) the crack is located in the horizontal member, one element away from the intersection on the far side of the joint.

28

X. HUANG and J. W. HANCOCK IONON-OIMENSIONALISEO STRESS INTENSITY FACTOR EJ J-- il-u’)

B-

S\rQ 6-0

WELD PROFILE + II 0

ClRCULAR PROFILE STRAIGHT LSD PROFILE NO PROFILE LINE SPRING RESULTS /

0’

0.0

’

’ * ’ 0.2 0.5 NON-OiMENSlONALlS~O

*

’ 0,6 CRACK

’ 0.8 DEPTH ’

8

’ I.0

a/T

Fig. 3. The stress intensity factors of cracks in a plane strain T-butt joint under pure bending[l6]. The results are expressed in terms of J non-dimensional~ed by the maximum bending stress S and the plate thickness T.

element code by evaluating the J integral along several paths around the crack tip using the virtual crack extension method of Parks[19], In all representations of the cracked joint the elements were constrained to behave in plane strain conditions, and a pure couple was applied to the ends of the horizontal plate. Non-dimensionalised crack depths (a/T) between 0.2 to 0.8 were considered where a is the crack depth and T is the thickness of the horizontal plate. The corresponding non-dimensional stress intensity factors as determined by the two methods are shown in Fig. 3 in which the nominal stress S, is taken as the maximum fibre stress in the horizontal plate. The agreement is excellent for cracks in the range (a/T) 2 0.2, although it must be expected that the line-spring will not be effective for very short cracks in which the stress concentration due to the weld profile is important. However for deep cracks the details of the weld profile do not matter and the cracked joint can be efficiently represented by shell elements in conjunction with an elastic line-spring. The success of these simple preliminary calculations now allows attention to be focussed on the more difficult problem of semi-elliptical cracks in tubular welded joints. JOINT

GEOMETRY

The geometry of the tubular joint which was analysed is shown in Fig. 4. The symme~y conditions associated with axial loading allow the problem to be represented by one quarter of the whole joint as shown in Fig. 5. In this case, 208 elastic S-noded linear elastic shell elements provided by the ABAQUS[20] finite element code were used to model the structure. The part-through thickness cracks in the tubular welded joints have been modelled with between 3 and 5 six noded second order linear elastic line spring elements which provide between 7 and 11 calculation points for the stress intensity factors. For the un-cracked geometry, this resulted in a problem with 4128 degrees of freedom which was solved on a VAX 11/750 computer. Surface cracks represented by elastic line springs were introduced into the model at the site of maximum stress concentration. In the real structure the cracks are located at the toe of the weld which is approximately one plate thickness away from the centre of the wall. In the shell analysis the cracks, idealised as discrete line springs in the chord, were located one plate thickness away from the centre line of the chord-brace intersection. Despite the somewhat irregular shape of the cracks in real structures, they have been idealised for the purpose of analysis as semi-ellipses with a surface length of 2c and a depth of a as shown in Fig. 6. Two types of cracks must be distinguished: short cracks whose depth is less than 20% of the wall thickness, and cracks for which a/T 2 0.2. In the short crack regime a major cause of the evolving crack shape is the interaction and coalescence of small cracks in a stress field which is

SIFs of semi-elliptical

7

5

+

cracks in T-joint

0.79

I

p = $

= 0.71

y : $

z1L.L

&Z$-

=lO

Fig. 4. The geometry of the tubular joint.

Fig. 5. The finite element mesh of the tubular joint.

! t

I

c

Fig. 6. The notation of the surface crack.

29

30

X. HUANG and J. W. HANCOCK

.*

0.5

Q/c

.

0.L i

l

.

0.3 0.2 . . . .

. 0.1 1

18 O-0

H 1

0.2

%

I

.

0.L

I

0.6

.

t

.

043

I

1.0

O/T

Fig. 7. Surface crack development

in a tubular joint.

strongly influenced by the stress concentration of the weld profile. In the case of deeper cracks (a/T) ~0.2, there is usually a single dominant crack. The evolving shapes of the cracks compared to the wall thickness of the chord T, summarised in Fig. 7, have been extracted from the experiments of Dover et al.[S]. Although it is important to note that a wide range of crack shapes have been reported in the literature[21]. Dover et al.‘s data[8] has been represented as a simple relation

(2) On this basis, wall penetration (a/T = 1) occurs for a crack whose surface length (2~) is 9.22 times the wall thickness T. In order to examine the stiffness of the joint after the crack has penetrated the chord wall several somewhat arbitrary crack shapes have been analysed in which the surface length of the crack 2c is greater than 9.22 T, and in these cases eq. (2) has still been used to generate crack depths, but only for the part-through thickness section of the crack. In all cases the deepest point of the crack was maintained at the original site of the maximum stress concentration and this is also a simplification which is not always justified by the experimental results[22].

RESULTS The maximum stress concentration for the untracked joint was located at the saddle position and was determined by extrapolating the maximum stress in the chord to a position one half of the brace thickness away from the brace centre line at the chord brace intersection. This procedure is in accord with the recommended procedure of MacDonald and Wylde[6] and is intended to give the membrane and bending stresses at the weld toe. On this basis the maximum stress concentration occurs at the saddle point denoted by 4 = 90” in the coordinate system shown in Fig. 8 where the local hot spot stress S,, is 9.4 times the remote axial stress S, in the brace. This compares favourably with that determined from the parametric formula developed by Kuang[2] which gives a value of 9.6. Cracks near the chord-brace intersection of tubular welded joints are in general subject to mixed mode loading such that the three stress intensity factors Kr, Ku and Km exist around the crack perimeter, although mode 1 is usually the largest component. Figure 9 shows the Kr component at different positions around the edge of the developing crack. With the notation of Fig. 8, the deepest point of the crack is located at 4 = 90” which is the saddle point and at this location Kr is shown as a function of the non-dimensional crack depth (a/T) in Fig. 10. The Ku and Km components of the stress intensity factor around the perimeter of the crack are shown in Figs 11 and 12 with K IIIcorrectly reducing to zero on the plane of symmetry at 4 = 90”. The KI, KI, and KIII components can be combined through G, the strain energy release rate, which for linear elastic fracture mechanics is identical to J and is shown in Fig. 13

31

SIFs of semi-elliptical cracks in T-joint

Fig. 8. The definition of the angle (fi.

0.75

0.9 o-95

Fig. 9. Kr at different angular positions measured from the crown.

Ky.0 ShG 1.5 ---------

‘\

1.0

\

0.5 .

0

0.2

OA

CM

0 T

‘1

‘\ \

Q0

1.0

Fig. 10. The stress intensity factor Kr at the deepest point of the crack, non-dimensionali~d by the hotspot stress Sh and the maximum crack depth a, as a function of the non-dimensiona crack depth (a/T).

3.0,

KII

s,vrii .

Fig. 11. Kn, non-dimensionalised by the nominal stress in the brace S, and the maximum crack depth u, at different angular positions d, measured from the crown, EFM 30:1-C

X. HUANG and J. W. HANCOCK

32

4 0.9 q

1 .o

946

Fig. 12. Km, non-dimensionalised by the nominal stress in the brace S, and the maximum crack depth a, at different angular positions 4 measured from the crown.

0.6

0.75

0.9 0.95 70"

74"

78"

80"

86'

9@

'9

Fig. 13. J, non-dimensionalised by the nominal stress S, and the maximum crack depth a, at different angular positions 4 measured from the crown.

FH 2vn Ert

O-06' 0.0

o-2

OL

1 O-6

08

1.0

1.2

1-L

Fig. 14. The decay in joint stiffness with crack growth. The stiffness is non-dimensionalised to Young’s modulus and the dimensions of the joint given in Fig. 4.

with respect

33

SIFs of semi-elliptical cracks in T-joint

The stiffness of the whole joint under axial loading is given in an appropriately nondimensionalised form in Fig. 14. Here F is the axial force in the brace. E the Young’s modulus of the material, and u the corresponding average displacement of the end of the brace in the axial direction. As with the in plane loading case the stiffness of joint is maintained until the crack penetrates the chord wall at the original site of the maximum stress concentration. Subsequently the stiffness decays steadily as the through-thickness crack grows around the chord-brace intersection. DISCUSSION The stress concentration factor for the joint analysed agrees well with the parametric formulae of Kuang et a1.[2] and Wordsworth and Smedley[4] and this provides confidence for the fracture mechanics analysis. A comparison of fatigue crack growth rates in simple fracture mechanics specimens with those of semi-elliptical cracks in tubular welded joints has enabled Dover et a1.[9] to infer a K value at the deepest point of the crack. In general, cracks near the chord-brace intersection are subject to a mixed mode loading in which mode I is the dominant component, although the experimental technique does not enable the Kr, Krr and &I components to be distinguished. However as shown in Fig. 15 the comparison between the experimentally determined K and the Kr calculated in this study is very favourable particularly when it is considered that the simplifications involved in extracting the evolving crack shapes. In considering the ability of the line spring to model surface cracks in joints it must also be noted that the present analysis implicitly considers the case in which cracks grow normal to the shell surface. In fact experimental observation[22] often suggests that the cracks develop in a path which tends to curve under the chord brace intersection. The present results for a crack which is always normal to the surface (Fig. 10) show that as the crack becomes deeper, the Ku component becomes larger and probably accounts for the tendency of real cracks to deviate in such a way as to minimise the Ku component. The curvature of the crack through the thickness also leads to ambiguity in interpreting the available published experimental data from a.c. potential drop measurements in terms of (a/T). Nevertheless comparison of line spring calculations by Huang[l6] with 3D finite element solutions[23] in which detailed information the crack shape was made available show excellent agreement between the J values for both methods. The estimation of errors in finite-element calculations is a very important but somewhat vexed area. A comparison of the accuracy of line spring calculations with full three dimensional solutions of semi-elliptical cracks in flat plates has been made by Parks[24]. For semi-elliptic surface cracks with depths in the range 0.2 I (a/T) % 0.6 and with an aspect ratio a/c = 0.2, the line spring calculations agree to within 3.5% of the full three dimensional solutions of Raju and

HUANG PND HANCOCK F.E. T JOINT

0.25 -

I 0

I

0.25

I

0.5

0.75

I

1.0

+

Fig. 15. A comparison of the calculated and experimentally determined K values at the deepest point of the crack.

34

X. HUANG and J. W. HANCOCK

Newman[l l] at the deepest point of the crack. In general the agreement between line spring and three dimensional calculations was good around the crack circumference, even near the intersection of the crack front with the free surface where the line spring model has a poor physical basis. In applying the line spring concept to through cracks located at the weld toe of a T-butt joint under simple plane strain loading conditions, the present work has made a comparison of the line-spring method using shell elements and the virtual crack extension method using continuum elements[l6]. Despite of the inability of shell elements to model details of the weld profile, the two techniques generally gave closely similar results for deep cracks. For such joints under pure bending the deepest crack examined (a/T) = 0.8, agreement of better than 1.3% between the line spring and the continuum element solutions was obtained independent of the weld proflle. As with the plate calculations[ll, 241, the calculations for butt welds also show better agreement for deep than for shallow cracks, although even in the worst case with a crack depth (a/T) = 0.2, the agreement is within 7.1%. However as the crack depth approaches zero, shell analysis can never give a satisfactory account of a local stress field which is amplified by the stress concentration due to the weld profile. In order to make some estimate of the likely errors in the experimentally determined K values, a comparison has been made using the upper and lower bound fatigue crack growth rate data of Scott and Sylvester[25] and Gall and Hancock[26] for BS 4360 50D steel instead of merely the mean data. The use of upper and lower band fatigue crack growth data then produces upper and lower band (+ 2 standard deviation) values for K as shown in Fig. 15. When the likely experimental errors are considered the agreement between the calculated and experimentally determined values is seen to be good over the range 0.8 2 (a/T) 2 0.2. In the range (a/ T) L 0.8 the finite element results give significantly lower values than would be expected by extrapolating the experimental data. For such deep cracks the effect of any curvature in the crackpath will give the most pronounced difference between the calculations and the experiments. However, estimates of the plastic zone size from the experimental test indicate that the ligament is almost certain to become fully plastic at (a/T) = 0.8, and the crack is liable to extend by ductile tearing as well as fatigue. The experimental crack growth rate must thus be expected to be significantly higher than predicted by the Paris law and give K values which are too high in comparison with perfect elasticity. Under laboratory conditions the fatigue life of a tubular welded joint is usually determined by the inability of the test equipment to impose the required displacement as the stiffness of the cracked joint decays. The loss in stiffness, also represents a limit of useful load bearing life in a real structure. The current finite element analysis indicates that the stiffness of the joint is maintained until the crack penetrates the chord wall. Up to this point the loss in stiffness for a crack with dimensions (Q / T) = 1, (c / T) = 9.2 is less than 7%. However subsequently the stiffness decays markedly, although not as fast as in in-plane bending when the coalescence of adjacent cracks at the crown point leads to an abrupt decay of stiffness. Acknowledgemenrs-One of us (X. Huang) would like to acknowledge a grant from the British Council and to the Government of the People’s Republic of China. Thanks are also due to Hibbit, Karisson and Sorensen, Inc. for access to the finite element code ABAQUS.

REFERENCES B. Gibstein, Parametrical stress analysis of T joints. Europenn offshore Steel Research Select Seminar, Cambridge, Paper 26 (1978). 121J. G. Kuang, A. B. Potvin, and R. D. Leick, Stress concentration in tubular joints. Oflsshore Technology Conf., Houston, Paper number 2205 (1975). A. M. Clayton, Effect of weld profile on stresses in tubular T joints. U. K. 0. S. R. P. Interim Report 2103 (1977). ::3 A. C. Wordsworth and G. P. Smedley, Stress concentrations at unstiffened tubular joints. European qshore Steels Research Select Seminar, Cambridge, Paper 31 (1978). 151A. McDonald and J. F. Thomson, The fatigue strength of large scale welded tubular T-joints. European Ofshore Steels Research Select Seminar, Cambridge, Paper 34 (1978). _-. [6] A. McDonald and J. G. Wylde, Experimental results of fatigue tests on tubular welded joints. ConJ. l-attgue m Offshore Structural Steel, London, Paper 10 (1981). [7] T. Iwasaki, J. G. Wylde and G. S. Booth, Fatigue test on welded tubular joints in air and sea water. Znt. Conf. Fatigue and Crack Growth in Offshore Structures, London, Paper C138/86 (1986).

111M.

SIFs of semi-elliptical cracks in T-joint

35

[8] W. D. Dover, R. D. Hibberd and S. J. Holdbrook, A fracture mechanics analysis of the fatigue failure of T-joints subject to random loading. Proc. Symp. Zntegriry of offshore Sfrucfures, Glasgow, p. 3/l (1978). 191W. D. Dover. G. K. Chaudhurv and S. Dharmavasan. Exoerimental and finite element comuarison of local stress and compliance in tubular welded T joints. Znr. Coni S&Z in Marine Sbuctures, Paris, Paper 8.5 (1981). r101W. D. Dover and S. Dharmavasan, Fatigue fracture mechanics analysis of T and Y joints. Ojj’shore Technology Conf., Houston, Paper 4404 (1982). Dll J. S. Raju and J. C. Newman, Stress intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates. Engng Fracture Med. 11, 817-829 (1979). WI J. Hellot and R. C. Labbens, Results for benchmark problem 1, the surface flaw. Znr. J. Frachue 15, 197-205 (1979). [I31 M. Oore and D. J. Burns, Estimation of stress intensity factors for embedded irregular cracks subjects to arbitrary normal stress fields. J. Press. Vess. Tech. 102, 202-211 (1980). J. R. Rice and N. Levy, The part-through surface crack in an elastic plate. .Z. appl. Med. 39, 185-194 (1972). ::z; G. J. Desvaux, The line spring model for surface flaw, an extension to mode II and mode III. M.Sc. Thesis, Massachusetts Institute of Technology (1985). WI X. Huang, A fracture mechanics analysis of the fatigue reliability of tubular joints. Ph.D. Thesis, University of Glasgow (1987). N. S. Barsoum, On the use of isoparametric finite elements in LEFM. Znt. .Z. numer. Meth. Engng 10,25-37 (1976). t:i; N. D. Henshell and K. G. Shaw, Crack tip finite element are unnecessary. Znt. J. numer. Meh. Engng 9,495-509 (1975). r191D. M. Parks, A stiffness derivative finite element technique for determination of crack tip stress intensity factors. Znr. J. Fracture 10, 487-502 (1974). ABAQUS User’s Manual. Hibbitt, Karlsson and Sorensen, Inc., Providence Rhode Island (1982). :;9 A. M. Clayton, Assessment of UKOSRP crack growth data to investigate the remaining life of off shore structures following inspection. U.K.A.E.A. Report, ND-R-852 (R) (1982). P21 C. Noordhoek and A. Verheul, Comparison of the ACPD method of in-depth fatigue crack growth monitoring with the crack marking technique. Delft University Department of Civil Engineering Report (1984). D. Ritchie, Private communication, (1986). [ii3 D. M. Parks, The inelastic line-spring estimates of elastic-plastic fracture mechanics parameters for surfacecracked plate and shells. .Z. Press. Vess. Technol. 103, 246-254 (1981). WI P. M. Scott and D. R. V. Sylvester, The influence of mean tensile stress on corrosion fatigue crack growth in structural steel immersed in seawater. U.K.O.S.R.P. Report 3/02 (1977). WI J. W. Hancock and D. Gall, Fatigue crack growth under narrow and broad band stationary random loading. University of Glasgow Marine Technology Report (1985). (Receioed 18 May 1987)