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Stress concentration factors for tubular Y- and T-joints
Int d Fatigue 12 No 1 (1990) pp 13-23
Stress concentration factors for tubular Y- and T-joints
A. K. Hellier, M.P. C o n n o l l y and W . D . D o v e r
A systematic study of stresses in tubular Y- and T-joints has been conducted in which nearly 900 thin-shell finite-element analyses were performed. These cover a wide range of joint geometries under axial loading, in-plane bending and out-ofplane bending. For each mode of loading, and for both the chord and brace sides of the intersection, semi-empirical equations are derived which relate the stress concentration factors at selected locations to a parametric function of the joint geometry. Equations are also obtained for the angular location of the hot-spot stress site around the intersection. The accuracy of these parametric equations is then assessed by comparing the predicted values with results from steel model tests and also with the predictions of other previously published equations. Key words: tubular joints; bending stress; membrane stress; stress concentration factors; design Steel offshore structures used for the extraction of oil and gas are composed of tubular joints welded together to form a three-dimensional space frame. Such a structure is susceptible to localized fatigue failure at the welded intersections as a result of the high stresses in these regions together with the large number of stress cycles experienced during its operational life. In order to ensure that the structure is adequately designed against fatigue failure it is necessary to be able to predict the stresses around the welded intersection where cracks are likely to initiate and grow. As a consequence of this, much research effort has been directed towards the determination of stress concentration factors (SCFs) for simple tubular joints, and these results have often been expressed as
parametric functions of the joint geometry for different modes of loading. 1-3 Tubular joints are conventionally designed using a lower bound stress-life (S-N) curve approach. S-N curves, established experimentally from large-scale fatigue tests on tubular joints, have been successfully used together with parametric stress equations for a number of years to ensure that tubular joints satisfy the fatigue design requirements, measured in terms of the total number of cycles to failure at a given applied stress range. Of late, however, there has been an increasing realization that a need exists to perform remaining life calculations on cracked joints. These are based on fracture mechanics techniques which utilize the stresses
Notation al, a2, a3,
coefficients in regression analysis
d4~ a5~ a6~ e t c
d D K~.~0
Ks KHS Kfis Kfis L R2 SCF t
T tCx~ ~y~ /4z 0~
T
external diameter of brace external diameter of chord SCF at crown position (~=0) SCF at crown position (~= 180°) SCF at saddle position SCF at hot spot SCF at positive hot spot (IPB) SCF at negative hot spot (IPB) chord length product moment correlation coefficient stress concentration factor brace wall thickness chord wall thickness displacements in x, y and z directions geometric ratio (=2LID)
0 tp ~x, ~y ¢~HS
~fis ¢Pfis chord brace hot-spot stress crown saddle IPB OPB
geometric ratio (=d/D) geometric ratio (=D/2T) geometric ratio (=t/T) brace angle angle around intersection Loof node rotations position of hot spot position of positive hot spot (IPB) position of negative hot spot (IPB) main tubular member tubular attachment to main tubular member peak stress in chord (or brace) weld toe regions at 0° or 1800 (see Fig. 2) weld toe regions at 90 ° or 270 ° (see Fig. 2) in-plane bending out-of-plane bending
0142-1123/90/010013-11 (~ 1990 Butterworth & Co (Publishers) Ltd Int J Fatigue January 1990
13
acting on the anticipated crack path and the material crack growth behaviour. For this purpose, not only is information required on the absolute maximum value of stress at the intersection (the 'hot-spot' SCF), but also the distribution of SCFs around the intersection as well as the stress variation through the thickness must be available. The problem of predicting the relative proportions of through-thickness bending and membrane stresses in tubular Y- and T-joints has been addressed in an earlier paper 4 published by us in this issue. This present paper is concerned with obtaining parametric expressions for the SCFs at all locations of significance on both the chord and brace sides of the intersection. Since the same finite-element meshes were used for both studies, the equations resulting from each arc directly compatible and may be used together. Also, although parametric equations already exist for SCFs in tubular Yand T-joints, those presented in this paper arc the only comprehensive set which include values at the crown, saddle and hot-spot positions, as well as equations to describe the location of the hot-spot stress site. It is anticipated that all this information will be required in order to fit characteristic formulae for the stress distributions around the intersection adequately. The objective of this work, then, is to provide a full predictive capability for the stresses acting on the anticipated crack plane in a tubular joint in service. All the equations are to be incorporated into a software package s which may be used to conduct rapid fracture mechanics analyses and remanent life calculations on cracked tubular members.
Finite-element analyses Owing to the complex geometrical nature of tubular joints, analytical solutions for their stress distributions are not feasible. Instead these must be determined by numerical methods such as finite-element analysis or by experimental tests on strain-gauged steel or acrylic models. A finite-element approach was adopted in this study, and since a detailed description of this has already been given in a previous paper 4 it will only be summarized here.
Element type The stress analyses were performed with the 'PAFEC' finiteelement package 6 using semi-Loof thin-shell elements. These elements are not capable of transmitting shear forces, but displacements normal to the elements and rotations about their edges are allowed. The elements used were mostly quadrilaterals with eight nodes, denoted '43210', together with a smaller number of their triangular six-noded counterparts known as type '43110'. A diagram of the eight-noded element is shown in Fig. 1. All eight nodes possess three translational degrees of freedom ux, u), and u~, and a rotational degree of freedom also exists at each of the Loof nodes marked with an 'x' in Fig. 1. These rotations are tangential to the element side but are considered for the purposes of input and output as ~. and %, rotations associated with the nearest midside node. In a shell analysis of tubular joints, the junction between brace and chord is modelled as the intersection of their midplanes. This is because shell elements are really two dimensional, having thickness only in a mathematical sense (necessary to define the element stiffness). Furthermore, the weld cannot be incorporated in a model which consists purely of shell elements. For both these reasons the SCF values obtained from thin-shell finite-element analyses are subject to some error, and are usually found to overestimate equivalent 14
Fig. 1 Eight-noded semi-Loof thin-shell element
steel model results, with the error on the brace likely to be greater than that on the chord.
Mesh generation The wide range of joint geometries covered by.' this study necessitates the use of an automatic mesh generator. This must be capable of producing fine elements near the intersection, where the stresses are changing very rapidly, and coarser elements in regions where the stresses are more evenly distributed. Also the elements should not be excessively elongated or otherwise distorted. A suitable program was developed at UCL (Ref. 7) and modified for the present work so that it would reliably generate meshes for tubular Y- and T-joints having widely differing geometric parameters cl, [3, y, T and 0, with an absolute minimum of user input. The mesh generation program was written in FORTRAN to run on a MicroVAX II machine and produces a data filc suitable for immediate analysis by PAFEC. A typical Y-joint mesh comprising 1478 nodes and 477 elements is shown in Fig. 2. It took approximately five minutes of CPU time to generate and a further 90 CPU minutes for the analysis to be performed. Only one half of the tubular joint needed to bc meshed and further details of the procedure used and mesh validation may be found in Ref. 4.
Boundary conditions The boundary" conditions and applied loads are sumrnarizcd diagrammatically in Figs 3a,b respectively. All degrces of freedom (u,, Uy, u~, q:~, %) were fixed at the chord ends, where te, corresponds to the displacement in the x direction
./J.
.JJ
I
SaddJe
Fig. 2 Typical e x a m p l e of finite-element mesh used to model t u b u l a r joint (c~ = 8.97, !3 = 0.6, ~, = 14.5, T = 0.8, 0 = 60°)
Int J Fatigue
January
1990
AI
a
BB
u z c~x £by
U x Uy
in offshore structures. Values for the brace angle 0 of 36 °, 45°, 60°, 75° and 90° were used. The minimum value of O was restricted to 35 ° since it was not possible to generate topologically sound meshes having smaller brace angles. For each load case and angle 0 approximately 60 analyses were conducted. The nominal stress for each joint and loading mode was calculated automatically by the mesh generation program and written to a data file. This was subsequently read, along with the PAFEC stress output file, by a postprocessor program and used to calculate SCF values at the crowns, saddle and hot spot(s) on both brace and chord. The SCFs were obtained by dividing the numerically greatest principal stress at a given point around the intersection by the appropriate nominal stress for that mode of loading. Nominal stresses were calculated by dividing the total applied load by the cross-sectional area of the brace for axial loading. For moment loading, the nominal stresses were derived from simple beam theory using a moment arm measured from the brace end along its outer surface to the (q0=0) crown position for IPB, and to the saddle position (~=90 °) for OPB. ~=0 refers to the crown acute angle location for a Y-joint (see Fig. 2). In all cases it is assumed that the free length of brace or chord is at least three times the diameter of the relevant member.
Parametric equations
Axial
IPB
OPB
b Fig. 3 Details of: (a) boundary conditions; (b) modes of loading used for finite-element joint analyses
and ¢Px and ~y are the Loof node rotations. Note that these are not rotations about the x and y axes (refer back to Fig. 1). Under axial loading or in-plane bending (IPB), symmetry exists about the xy plane containing the chord and brace centre-lines. A half-joint mesh may therefore be used provided the displacements uz and rotations q~x and % are restrained at all the nodes which lie on this plane. Since the restraints are identical for both these modes of loading, they were analysed consecutively as two load cases of the same finiteelement run, without the need for recomputing the element stiffness matrix. For out-of-plane bending (OPB) this symmetry about the xy plane no longer exists. However, it was found that results of acceptable accuracy could be obtained by using a half-joint mesh with the displacement components Ux and uy fixed at all points on the xy plane. 4
Extraction of results The tubular joints analysed in this study encompassed the following ranges of geometries, expressed in terms of the dimensionless parameters el, 13, ~/, "r and the brace angle 0: 6.21 ~< ~x ~< 13.10
(1)
0.20 ~< [3 ~< 0.80
(2)
7.60 ~< ~/~< 32.0
(3)
0.20 ~< $ ~< 1.00
(4)
35 ° ~< 0 ~< 90 °
(5)
With the exception of a, which will be discussed in the next section, these include the majority of tubular joints used
Int J Fatigue January 1990
From the raw database of SCF values, parametric equations were obtained for the SCF at a number of locations around the intersection, both chordside and braceside, under axial loading, in-plane bending and out-of-plane bending. The parametric equations were derived using a statistical regression package known as 'MINITAB' (Ref. 8) which is capable of performing multiple regression and correlation analysis. The methodology used in deriving the equations was as follows. (a) The variations of the SCF at the point in question were plotted for each brace angle 0 as a function of the parameters c~, [3, ~/and • in order to determine the best forms of the terms required, and also to ascertain if any cross-correlation existed between the terms. (b) A first attempt at the equation was made using the simple form
SCF = al cl'~ [3"3y'4 ,r~5 0'~
(6)
where al to a6 were determined from the regression analysis. (c) Equation (6) was then modified by using other (eg exponential) terms, and numerous regressions performed until a suitable equation with a large product moment correlation coefficient was obtained. (d) Having obtained the basic form of the final equation, the exponent for the oe-variation (a2 in Equation (6)) was adjusted to reflect the observation that increasing o~ (ie the chord length) beyond the limit of 13.10 for this study is known to have little effect on SCF values. By taking into account this far-field behaviour it has been deemed possible to remove the upper validity limit on o~for these equations. A summary of the parametric equations obtained together with their correlation coefficient R 2 is given in Table 1. A value of R 2 = 100% would imply that the fitted equation explains all the variations in SCF. It is apparent from Table 1 that a correlation of better than 90% has been achieved in most cases. All the equations derived in this study are set out
15
Table 1. Summary of the parametric equations obtained Type of loading
Chord/ Quantity R 2 (%) Equation
brace
Axial Axial Axial Axial Axial Axial Axial Axial Axial Axial
Chord Chord Chord Chord Chord
K° K(!8° Ks Kas ~PHS
97.2 97.4 99.2 98.8 76.0
Brace Brace Brace Brace Brace
K°
85.5 (A6)
K~8°
93.2
Ks
98.8 (A8)
KHs
~HS
IPB IPB IPB IPB IPB IPB IPB IPB IPB IPB IPB IPB IPB IPB OPB OPB OPB OPB OPB OPB OPB OPB OPB OPB
Chord Chord Chord Chord Chord Chord Chord Brace Brace Brace Brace Brace Brace Brace Chord Chord Chord Chord Chord Brace Brace Brace Brace Brace
K° K18° Ks KGs KHS ~s q;HS K° K18° Ks KGs KHs ~;4s ~HS K° K~8° Ks KHS ~HS Kc° K~8° Ks KHs ~HS
96.7 84.9 99.4 98.1 -99.5 98.4 --86.4 91.9 -89.7 91.2 ----98.8 98.8 ---96.4 96.8 --
(A1) (A2) (A3) (A4) (A5) (A7) (A9) (A10) (All) (A12) (A13) (A14) (A15) (A16) (A17) (A18) (A19) (A20) (A21) (A22) (A23) (A24) (A25) (A26) (A27) (A28) (A29) (A30) (A31) (A32) (A33)
fully in Appendices 1-6. A consequence of this approach is that for some particular geometries more than one equation can apply. These equations should give very similar answers. If the calculated values are slightly different one should take the hot-spot value in preference to crown or saddle values, or if choosing between two crown or saddle values take the highest calculated value. For axial loading, equations were fitted to the SCF variations at all significant locations, namely the two crown positions (at ~p=0 and ~p=180°), the saddle point (~p=90°) and the hot-spot stress site where the greatest value of SCF occurs. An equation was also developed to describe the angular location of the hot spot, ~pHS.This information is used more extensively in the development of parametric equations describing the distribution of stresses around the connection. The equations will be reported in the third paper in this series (Ref. 21, published in this issue). Since the finite-element results are sampled at a limited number of nodes around the intersection, the raw data for ~PHSare discrete, and a degree of circumspection is required in the use of this equation. A similar set of equations was also derived for out-of-plane bending with the exception of the crown positions. Here the SCFs on both chord and brace are very small in most cases, and were therefore taken to be zero. Also, the hot-spot locations in
16
general lie close to the saddle point and so for simplicity were always taken to occur at ~p-90 °. The parametric equations derived for in-plane bending are more numerous than those for either of the other load cases. This is due to the presence of two hot spots under this type of loading, one having a positive SCF and the other having a negative value. For a T-joint these are equal and opposite. For the loading direction shown in Fig. 3, the positive hot spot lies at or near the ~p-0 crown position while the negative hot spot has a ~p-value of 180° or thereabouts. The positive hot spot for a Y-joint is usually, but not invariably, of greater magnitude than the negative onc. However, when dealing with a mixed-mode loading condition and, in particular, attempting to fit a stress distribution curve to Y-joints under in-plane bending, knowledge of the magnitudes and locations of both hot spots is necessary. Simplified expressions were considered sufficient to descrii~e the hot-spot locations in this case. An attempt will now be made to assess the accuracy ot these parametric formulae by comparing them with other parametric equations from the literature and finally with results obtained from steel model tests.
Comparison with other equations As a check on the general applicability of the parametric equations given in Appendices 1-6 a series of comparisons are now made, where possible, with existing parametric formulae given in the literature. All of these formulae predict SCF values for both the chord and brace sides of the intersection under each of axial, IPB and OPB loading. The particular equations considered are as follows. (a) Kuang's formulae for T- and Y-joints.' These predict hot-spot SCFs and are based on thin-shell finite-element (FE) analyses of 46 different geometries. (b) Wordsworth and Smedley's formulae for T- and Yjoints.-' These were obtained from the results of an unspecified number of acrylic model tests and give the SCF at the saddle and crown positions. (c) Gibstein's formulae for T-joints) These arc for hot-spot SCFs and were derived from a study of 17 geometries, again using thin-shell FE analysis. Comparison of the 'UCL' equations (which are based on thin-shell FE analyses of 291 T- and Y-joint geometries, each subject to all three modes ot loading) with these other parametric formulae is not intended to be exhaustive, but rather to give a flavour of the relative variations in hot-spot SCF when one of the geometric parameters is allowed to change. Fig. 4 shows the variation in chord SCF for all the equations as a function of [3 for axially loaded T-joints. It can be seen that the UCL equation is a curve of similar shape to the others but predicts values which are nearest those given by the highest of the existing curves, lying closest to the Kuang equation at low !3 and approaching Smedley's equation at high 13. The peak value in the SCF is at a slightly lower value of [3 for the UCL equation when compared with Gibstein and Smedley, but higher when compared with Kuang. The chord SCF variation with y for T-joints under inplane bending is shown for all four equations in Fig. 5. All the curves show a steady increase in SCF as y becomes larger. Of the previously published formulae, Smedley's equation always yields the highest value in this case, and the UCL equation predicts slightly higher values than Smedley.
Int J Fatigue January 1990
15_
10
UCL
..... .....
12
-
.
Kuang Smedley -
O,
.,~: ~ " , ~ I .~'
te,n
..'5- / ~° ~ UCL Kuang Smedley Gibstein
. . . .
..... ~-=
o
1,,,lILl,li, 0
0.2
0.6
0.4
0.8
1.0
0 0
0.2
0.4
0.6
0.8
.0
T Fig. 4 C h o r d SCF v a r i a t i o n w i t h 13 f o r axially loaded T-joints (cx = 24.0, ~/ = 16.0, T = 0.5)
UCL ..... Kuang ..... Smedley m . m Gibstein
--
,O@e
i
i
i
i
I
6
i
i
n
J
I
o
i
n
12
n
J
I
=
18
i
J
n
I
24
=
a
a
I
30
Fig. 5 Chord SCF v a r i a t i o n w i t h 3, f o r IPB-Ioaded T-joints (cx = 24.0, ~ = 0.5, T = 0.5)
Fig. 6 shows the chord SCF variation with T for OPBloaded T-joints. The UCL equation gives a linear increase in SCF with increasing $ in similar fashion to the existing equations, but is a shade more conservative than the highest prediction obtained using any of these. The graphs given in Figs 4-6 show firstly that the UCL equations exhibit similar trends to those displayed by the established parametric formulae, and secondly that the UCL formulae predict SCF values which appear to be consistently higher than those given by these other equations. This tendency toward conservative predictions is an important attribute for design purposes. The FE-based studies of Kuang, Gibstein and UCL show similar trends but always differ. This could be a consequence of the elements chosen and the number of cases studied.
Comparison with steel models In order to test the relative accuracy of the parametric equations in predicting hot-spot SCFs, the UCL equations together with the Kuang, 1 Wordsworth and Smedley, 2 and Gibstein, 3 equations are compared with steel model test results. These steel model results are normally obtained by carrying out large-scale loading tests on strain-gauged tubular
Int J Fatigue January 1990
Fig. 6 Chord SCF v a r i a t i o n w i t h ~ f o r OPB-Ioaded T-joints (cx = 24.0, 13 = 0.5, 3' = 16.0)
joints and measuring the changes in strain due to the applied loads. The experimental procedures required to determine the so-called extrapolated weld toe stresses have been well defined and are summarised in Ref. 9. A body of data is therefore available which relates the measured SCFs to the tubular joint geometric parameters. Unfortunately, many of these tests were conducted on 'unrealistic' joints, in the sense that their geometric parameters have values unlikely to be found in practice, and also lie outside the limits of the parametric equations. In attempting to assess the accuracy of the various parametric equations, a database of comparable steel model results was compiled and is given in Tables 2, 3 and 4 for the axial, in-plane bending and out-of-plane bending cases respectively. Many of these were obtained from Ref. 9 but do not include results where (x < 5.0, ~ > 32.0 or 13 = 1.0. Tubular joint geometries with cl < 5.0 were ignored since it was felt that for these cases the end conditions applied to the chord became too influential. Joints having - / > 32.0 are not common in practice and also lie beyond the validity range of the parametric equations. The 13 = 1.0 case (where both brace and chord are of the same diameter) has a very complex intersection geometry which places it in a category of its own as far as numerical analysis is concerned. Although quite often found in offshore structures, this geometry is beyond the scope of the present study and has therefore been excluded from consideration. The remaining results are now considered for axial loading, in-plane bending and out-of-plane bending in turn.
Axial loading The comparisons between the steel model results and the SCF values produced by the UCL, Kuang, Wordsworth/Smedley and Gibstein formulae are given in Table 2 for axial loading. All results are for the chord unless otherwise stated. The total number of steel model results is 24, and for convenience the comparisons between the predicted and actual values are expressed as the ratio of predicted SCF to actual SCF for each of the four formulae. A ratio of 1.0 indicates an exact fit. From this table it can be seen that the UCL formula predictions are generally conservative, with only one result having a ratio less than 1.0 (Ref. 13, Spec. No 44). The Kuang and Gibstein equations underpredict the steel model results seven and five times respectively, whereas the results from the Smedley formulae are generally similar to those for the
17
Table 2. Comparison between predicted and measured SCFs for axially loaded T- and Y-joints Ref. SpecNo imen No 10
1 4 11 13
11
c,
13
10.0 10.0 10.0 10.0
0.50 0.50 0.25 0.50
e Steel (deg) SCF
UCL/ steel
Kuang Kuang/ Smedley Smedley/ Gibstein Gibstein/ steel steel steel
0.50 0.50 0.39 0.50
90 90 90 90
5.7 6.7 4.7 7.7
7.5 7.8 5.7 7.8
1.32 1.16 1.21 1.01
6.3 6.6 5.4 6.6
1.11 0.99 1.15 0.86
7.5 8.0 5.0 8.0
1.32 1.19 1.06 1.04
6.1 6.5 4.0 6.5
1.07 0.97 0.85 0.84
II
5.0 0.46 20.0 1.00
60
11.0
16.8
1.53
17.0
1.55
17.3
1.57
--
--
12
1(c) l(b)
6.9 0.66 23.1 0.91 6.9 0.66 23.1 0.91
45 45
8.6 6.5
10.8 8.7
1.26 1.34
9.7 8.4
1.13 1.29
11.2 8.1
1.30 1.25
---
---
13
19 41 42 43 44
0.86 1.00 0.28 0.47 0.25
90 90 90 90 90
11.4 12.0 3.3 4.8 3.7
14,5 18.0 3.7 7.0 3.2
1.27 1.50 1.12 1.46 0.86
12.6 16.1 3.0 6.7 2.9
1.11 1.34 0.91 1.40 0.78
12.9 16.1 4.5 5.9 3.1
1.13 1.34 1.36 1.23 0.84
12.8 16.1 2.8 4.9 2.1
1.12 1.34 0.85 1.02 0.57
14
UCL TW2,4 TW3
7.3 0.71 12.0 1.00 7.3 0.71 12.0 1.00 7.3 0.71 12,0 1.00
90 90 90
10.0 10.5 10.3
13.7 13.7 13.7
1.37 1.30 1,33
10.8 10.8 10.8
1.08 1.03 1.05
11.7 tl.7 11.7
1.17 1.11 1.14
12.0 12.0 12.0
1.20 1.14 1.17
15
CP1 CP2 CP3 CP4
7.3 7.3 7.3 7.3
1.00 1.00 1.00 1.00
90 90 90 90
9.3 13.4 13.7 13.1
13.7 15.6 15.6 15.6
1.47 1.16 1.14 1.19
10.8 12.4 12.4 12.4
1.16 0.93 0.91 0.95
11.7 13.9 13.9 13.9
1.26 1.04 1,01 1.06
12.0 14.0 14.0 14.0
1.29 1.04 1,02 1.07
16 SHl(c) SHl(b)
7.3 0.71 14.4 0.80 7.3 0.71 14.4 0.80
90 90
9.3 6.1
12.0 10.5
1.29 1.72
9.3 10.2
1.00 1.67
11.2 8.1
1,20 1,33
10.4 7.8
1,12 1.28
17 GCI(c) GCI(b)
7.2 0,71 14.3 0.79 7.2 0.71 14.3 0.79
90 90
8.7 6.0
11.7 10.4
1.34 1.73
9.0 10.0
1.03 1.67
11.0 7.9
1,26 1.32
10.1 7.7
1,16 1,28
45
5.2
6.1
1.17
5.6
1.08
6.1
1.17
--
--
18
Y
10.5 5.0 5.0 5.0 5.0
13.0
0.53 0.50 0.50 0.24 0.24
0.71 0.71 0.71 0.71
13.4 14.3 14.3 14.3
UCL
13,4 14.4 14.4 14.4 14.4
12.0 14.3 14.3 14.3
0,48 15.9 0.63
(c) = chord, (b) = brace. All others are chord values.
Table 3. Comparison between predicted and measured SCFs for IPB-Ioaded T- and Y-joints Ref. SpecNo imen No
(x
13
~/
T
e Steel (deg) SCF
UCL
UCL/ steel
Kuang Kuang/ Smedley Srnedley/ Gibstein Gibstein/ steel steel steel
9
1
10.0
0.50 13.4 0.50
90
1.1
2.6
2.36
1.9
1.73
2.4
2.18
2.1
1.91
12
1(c) l(b)
6.9 6.9
0.66 23.1 0.91 0.66 23.1 0.91
45 45
3.3 2.7
5.7 4.0
1.73 1.48
3.6 2.8
1.09 1.04
4.5 3.8
1.36 1.41
---
---
13
41 42 43 44(b)
5.0 5.0 5.0 5.0
0.50 0.50 0.24 0.24
1.00 0.28 0.47 0.25
90 90 90 90
4.9 1.3 1.5 2.0
5.2 1.6 2.2 2.2
1.06 1.23 1.47 1.10
3.6 1.2 1.9 2.4
0.73 0.92 1.27 1.20
4.3 1.6 2.2 1.8
0.88 1.23 1.47 0.90
4.5 1.2 2.0 1.8
0.92 0.92 1.33 0.90
14
UCL
7.3
0.71 12.0 1.00
90
3.0
5.0
1.67
3.2
1.07
3.7
1.23
4.0
1.33
0.71 0.71 0.71 0.71
90 90 45 45
3.1 2.0 2.7 2.5
4.2 3.3 4.0 3.6
1.35 1.65 1.48 1.44
2.9 2.5 2.4 2.3
0.94 1.25 0.89 0.92
3.4 3.2 3.0 2.9
1.10 1.60 1.11 1.16
3.3 2.3 ---
1.06 1.15 --
17 G C I ( c ) 7 . 2 GCI(b) 7.2 GC2(c) 7.2 GC2(b) 7.2
14.4 14.4 14.4 14.4
14.3 14.3 14.3 14.3
0.79 0.79 0.79 0.79
--
(c) = chord, (b) = brace. All others are chord values.
18
Int J F a t i g u e J a n u a r y 1990
T a b l e 4. C o m p a r i s o n
between
predicted
Ref. No
Specimen No
e<*
13
~/
•
19
23/1(c) 23/2(c) 23/3(c) 23/1(b) 23/2(b) 23/3(b) 24/1(c) 24/2(c) 24/3(c) 24/1(b) 24/2(b) 24/3(b)
-------------
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3
0.39 0.39 0.39 0.39 0.39 0.39 0.28 0.28 0.28 0.28 0.28 0.28
90 90 90 90 90 90 90 90 90 90 90 90
19
703/1 703/2 703/3 704/1 704/2 704/3
-------
0.53 0.53 0.53 0.53 0.53 0.53
13.4 13.4 13.4 13.4 13.4 13.4
0.86 0.86 0.86 0.51 0.51 0.51
14
UCL
7~3 0.71
16 20
SHI(c) SH2(c) SHl(b) SH2(b)
17
GCI(c) GCI(b)
and measured
0 Steel (deg) SCF
SCFs for OPB-Ioaded
T-joints
Kuang Kuang/ S m e d l e y S m e d l e y / Gibstein Gibstein! steel steel steel
UCL
UCL/ steel
2.3 2.9 2.6 2.0 2.2 1.9 1.9 1.6 1.8 1.7 1.6 1.8
3.2 3.2 3.2 2.8 2.8 2.8 2.3 2.3 2.3 2.4 2.4 2.4
1.39 1.10 1.23 1.40 1.27 1.47 1.21 1.43 1.28 1.41 1.50 1.33
2.2 2.2 2.2 2.9 2.9 2.9 1.6 1.6 1.6 2.4 2.4 2.4
0.96 0.76 0.85 1.45 1.32 1.53 0.84 1.00 0.89 1.41 1.50 1.33
2.2 2.2 2.2 2.4 2.4 2.4 1.6 1.6 1.6 2.0 2.0 2.0
0.96 0.76 0.85 1.20 1.09 1.26 0.84 1.00 0.89 1.18 1.25 1.11
2.1 2.1 2.1 2.3 2.3 2.3 1.4 1.4 1.4 2.0 2.0 2.0
0.91 0.72 0.81 1.15 1.05 1.21 0.74 0.88 0.78 1.18 1.25 1.11
90 90 90 90 90 90
8.0 7.7 8.4 5.4 5.0 5.4
9.6 9.6 9.6 5.7 5.7 5.7
1.20 1.25 1.14 1.05 1.14 1.05
7.5 7.5 7.5 4.7 4.7 4.7
0.94 0.97 0.89 0.87 0.94 0.87
9.5 9.5 9.5 5.6 5.6 5.6
1.19 1.23 1.13 1.04 1.12 1.04
9.5 9.5 9.5 5.2 5.2 5.2
1.19 1.23 1.13 0.96 1.04 0.96
12.0 1.00 90
8.5
10.0
1.17
7.1
0.84
11.9
1.40
10.5
1.24
7.3 7.3 7.3 7.3
0.71 14.4 0.80 90 0.71 14.4 0.80 90 0.71 14.4 0.80 90 0.71 14.4 0.80 90
9.7 10.6 6.2 6.0
9,8 9.8 8.2 8.2
1.01 0.92 1.32 1.37
7.0 7.0 7.5 7.5
0.72 0.66 1.21 1.25
11.4 11.4 8.2 8.2
1.18 1.08 1.32 1.37
9.6 9.6 7.4 7.4
0.99 0.91 1.19 1.23
7.2 7.2
0.71 0.71
9.1 5.1
9.6 8.1
1.05 1.59
6.9 7.4
0.76 1.45
11.2 8.2
1.23 1.61
9.4 7.3
1.03 1.43
14.3 0.79 90 14.3 0.79 90
(c) = chord, (b) = brace. All others are chord values. *a assumed to be 7.0 where no value given.
UCL formulae, with only one underprediction. The UCL formulae are a little more conservative than Smedley's but occasionally give a fairly high level of overprediction (for example a ratio of 1.73 for Ref. 17, Spec. No GCI(b)). In-plane
bending
Table 3 contains the comparisons between the steel model results and the SCF values predicted by the UCL, Kuang, Wordsworth/Smedley and Gibstein formulae for in-plane bending. The total number of steel model results is 12 in this case. Once again the UCL equations are the most conservative, providing predictions which are consistently higher than the steel model results. The Kuang, Smedley and Gibstein equations underpredict the steel model results 5, 2 and 3 times respectively. Out-of-plane
bending
The steel model results and the SCFs obtained from the UCL, Kuang, Wordsworth/Smedley and Gibstein formulae are compared in Table 4 for out-of-plane bending. There are 25 results for this mode of loading, all of which were obtained from tests on T-joints. It is apparent that in every case except one (Ref. 16, Spec. No SH1 (c)) the UCL equations overpredict the steel model results. For the other equations the numbers of cases where the SCF is underestimated are considerable, with 15, 5 and 10 underpredictions for the Kuang, Smedley and Gibstein equations respectively.
Int J F a t i g u e J a n u a r y
1990
Summary
The total number of steel model results given in the SCF database of Tables 2-4 for axial loading, in-plane bending and out-of-plane bending is 61. Fig. 7 shows the SCF values predicted by the UCL equations plotted against the experimental results for all these cases together, and serves to confirm the inherently conservative nature of the equations. If all the results for the three modes of loading are now broken down into intervals depending upon the ratios of predicted SCF to actual SCF for each equation in turn, then the histograms of Fig. 8 are obtained. The tendency of the UCL formulae to overpredict SCF values is again clear to see, with a mere 3% of the values being underpredictions. This should be contrasted with the Kuang, Wordsworth/Smedley and Gibstein formulae which underpredict 44%, 13% and 34% of the results respectively. Of these, the Wordsworth/Smedley equations are generally considered the most appropriate for design 9 because of their relatively low level of underprediction. On this basis, the UCL equations proposed in this paper would appear to offer an even more reliably conservative prediction of SCF when designing a tubular joint. It can be seen that under some circumstances the equations will predict a SCF of zero. In these cases consideration should be given to using a minimum value instead of zero. The minimum value would be determined from previous design/ service experience.
19
20
16 LL
12
I.,3
CJ
8 Q.
4
o
,,
, I ,,, 4
,I
,,,
, I ,,,
, I,,,,
12 Experimental
15
8
20
SCF
Fig. 7 SCF predicted by UCL equations plotted against experimental SCF for all modes of loading
Conclusions A thin-shell finite-element study has been performed in order to obtain stress concentration factors for a wide range of tubular Y- and T-joint geometries. Parametric formulae or simple expressions have been derived for the SCF at all
40
locations of significance on both the chord and brace sides of the intersection under axial loading, in-plane bending and out-of-plane bending. Equations have also been obtained which describe the positions of the hot-spot stress sites around the intersection. It is anticipated that these might be used in conjunction with characteristic distribution formulae to provide a full description of the SCF variation around a tubular joint. The equations for the hot-spot SCF are found generallx to overestimate the measured SCFs from steel model tests. They have also been compared with existing parametric formulae due to Kuang, Wordsworth/Smedley and Gibstein, and found to be generally more conservative than anx of these. It is therefore concluded that the hot-spot S('F equations presented in this paper are the most reliable in predicting a conservative value of SCF which could bc used in design.
Future work Several areas may be identified in which further work could profitably be undertaken. The first of these, which is considered in the next article in this issue, 2~ is the generation of characteristic stress distribution formulae for each mode of loading which utilize the equations presented in this paper as input parameters, thus providing a full description of the SCF variation around the intersection. This should enable
UCL
--
30 - -
40
-
30
-
I
Smedley
I I I
>. L) r"
>~ u r-
20
--
U"
20
U_
U_
10
10
0
a
C 40 -
40
30I
Gibstein
Kuang 30
>, (J
>, ~J c"
20
20
EY
LL
LL
10
10
/ / / / A
0
0 b
0.75
I .0
Predicted
I .25
SCFltest
0.75
I .5 result
d
1.0
1.25
Predicted SCF/test
1.5 result
Fig. 8 SCF histograms for all load cases combined: (a) UCL formulae; (b) Kuang's formulae; (c) Wordsworth and Smedley's formulae; (d) Gibstein's formulae
20
Int J Fatigue January 1990
more refined fracture mechanics analyses of cracked tubular joints to be performed using a two-dimensional stress field on the anticipated crack plane. Secondly, it would be desirable to extend the scope of this study to encompass higher values of 13 (the ratio of brace diameter to chord diameter) up to and including 13 = 1.0. Future studies to produce similar sets of parametric equations for X- and K-joints would also be worthwhile.
Acknowledgements The authors wish to acknowledge the Marine Technology Directorate Ltd, SERC and Sponsors of the Cohesive Fatigue Programmes 1985-9 for providing some of the financial support needed for this work.
Parametric equation for SCF at the chord crown position (~0=0) under axial loading:
51.°"°6313-°2°7°1/25(°"988-°'133/°)
K ° = 0.575 exp(0.66513 s + 0.0204~/+ 1.64 sin 0 - 0.469[35) (A1) (0 is in radians). Parametric equation for SCF at the chord crown position (lp=180 °) under axial loading: =
Klc8° = 1 9 . 0 5 5-0"035[3 (0"213/02 - 00439)5(0'48 + 000589/05) x exp(-0.00148~ 2 - 1.25 sin 0 + 0.00055~2/[3 - 1.09132"r)
(A7)
(0 is in radians). Parametric equation for SCF at the brace saddle position under axial loading: Ks = 0.538 0f°°28650'963 x exp(-0.987133 - 4.67~/-°.s + 4.87 sin 1/2 0 - 0.1335/[3)
(AS)
Parametric equation for SCF at the brace hot-spot stress site under axial loading:
Appendix 1
K~:8°
(0 is in radians). Parametric equation for SCF at the brace crown position (lp=180 °) under axial loading:
3.94 (~-0"03913--0'10505(0"906 + 0"055/03)exp(0.327134 + 0.0177~ - 0.05 sin 0 - 0.422[3"0 (A2)
(0 is in radians). Parametric equation for SCF at the chord saddle potation under axial loading: Ks = 2.51 5°°3725112 exp(-1.96134 - 11.3/~/+ 2.47 sin 0 + [35)
X exp(--1.18613 s -- 4.2811-y -°.s + 1.987 sin 0 + 0.427[35)
(Ag)
Parametric equation for position of brace hot-spot stress site under axial loading: ¢4~HS --
58.6 500309[30-05585(0.14402
-
-
0.404)
x exp(1.01/~/+ 0.343/sin 0 - 0.0503[3/5) (A10) (0 is in radians, IPHS is in degrees). Validity ranges: 6.21 ~< 5 0.20 ~< 13 ~< 0.80 7.60 ~< ~/~< 32.0 0.20 ~< 5 ~< 1,00
(A3)
Parametric equation for SCF at the chord hot-spot stress site under axial loading: KHS = 9.92 50'029513 exp(--1.3813 s's -- 5.53~/-°-s + 2.15 sin 0 -- 0.05695/[3)
KHS = 4.40 50"°s6850 ss3
35 ° ~< 0 ~< 90 °.
Appendix 3 Parametric equation for SCF at the chord crown position (lp=0) under in-plane bending:
(A4) Kc° = 2.84 5 - ° °°ss~/°-2345(T M - o.171o)exp[(0.00213,
Parametric equation for position of chord hot-spot stress site under axial loading: lPHS = 193.7 5--0"041513 (0.267 -- 0"093502)5(0"2710 -- 0.491) × exp(15.5/'y 2 - 0.77 sin 0 + 0.000028~/2/[32 - 0.00932/[35)
(A5)
- 0.0373)/[32 - 0.0179/sin 0] (All) (0 is in radians). Parametric equation for SCF at the chord crown position (~=180 °) under in-plane bending: K~cs° = - 3 . 8 4 5 5--O'O29'yO'4075(O'966 -- 0.07190)
(A12)
× exp[(0.00166~/- 0.0399)/[32 - 0.682/sin 0]
(0 is in radians, 0HS is in degrees). Validity ranges:
(0 is in radians). Parametric equation for SCF at the chord saddle position under in-plane bending:
6.21 ~< 5 0.20 ~< [3 ~< 0.80
Ks = 0.0
7.60 ~< -¢ ~< 32.0
(A13)
Parametric equation for SCF at the chord positive hot spot under in-plane bending:
0.20 ~< -r ~< 1.00
Kfts
35 ° <~ 0 ~< 90 °.
=
2.31 5°'°°33~/°'326'r (1"24 - 0.1870)
(A14)
x exp[(0.00154~/- 0.0323)/[32 - 0.0248/sin 0]
Appendix 2 Parametric equation for SCF at the brace crown position (~p=0) under axial loading: g°
=
1.46 50'0098130'214/05 (0.623
-
-
Int J Fatigue January 1990
/iris = - 8 . 9 3
0'0811/02)
x exp(0.87213 s - 0.00104~/2 + 1.47 sin 0 + 0.000474~/2/[3 - 1.73135)
(0 is in radians). Parametric equation for SCF at the chord negative hot spot under in-plane bending: 0 f - - 0 " 0 1 7 6 ' ~ / 0 ' 4 5 5 (0"94 --
0.04410)
(A15)
× exp[(0.00153~/- 0.0386)/[32 - 1.69/sin1/20] (A6)
(0 is in radians).
21
Parametric equation for position of chord positive hot spot under in-plane bending: + -- (y/0.41)]~ 2 ~PHS
(A16)
(~P~s in degrees). Parametric equation for position of chord negative hot spot under in-plane bending: ~PHS = 180 -- (y/0.45)[372
(A17)
(q~fis is in degrees). Validity ranges :
7.60 ~< 7 ~< 32.0 0.20 ~< T <~ 1,00 35 ° ~< 0 <~ 90 °.
K ° = 0.0
7 . 6 0 ~< y ~< 32.0 ~ ~<
0.20 ~< 13 ~< 0.80
Parametric equation for SCF at the chord crown position (q~=0) under out-of-plane bending:
0.20 ~ [3 ~< 0.80
~
6.21 ~- c~
Appendix 5
6.21 ~ cx
0.20
Validity ranges:
(A25)
Parametric equation for SCF at the chord crown position (~p- 180°) under out-of-plane bending:
1.00
35 ° ~< 0 ~< 90 °.
K ~ ° = 0.0
Appendix 4 Parametric equation for SCF at the brace crown position (~p=0) under in-plane bending:
Parametric equation for SCF at the chord saddle position under out-of-plane bending: Ks = 0.315 (t. 0"O54y(hI2
K ° = 1.89 O¢--0"065%/-0"2142/O2T0"186~O'298
-
(A26)
O'O939/O2)'1 -
× exp(-0.001391134 + 0.654 sin 2 0)
× e x p ( - 5 . 8 8 / y + 1.81/sin 0 0.0534[33y - 0.084818/7)
(A18)
(0 is in radians). Parametric equation for SCF at the brace crown position (lp=180 °) under in-plane bending:
(0 is in radians). Parametric equation for SCF at the chord hot-spot stress site under out-of-plane bending: I~THS
:
0.255 (~0.O13y(t.24
0.224/0)T
x exp(-0.00135/iB 4.' + 0.923 sin 0) K{~°
=
- 7 . 0 2 ~-O'O4S~O'141/O2yO'167/0270"143
-
× exp(--0.015/132 -- 164.4/73 -- 0.524/sin20 0.02541327) (A19) -
Parametric equation for SCF at the brace saddle position under in-plane bending: Ks : (-,//60.0)(1/0 2)
(A20)
(0 is in radians). Parametric equation for SCF at the brace positive hot spot under in-plane bending:
(A27)
(A28)
(0 is in radians). Parametric equation for position of chord hot-spot stress site under out-of-plane bending: ~Hs = 90.0
(A29)
(tpHs is in degrees). Validity ranges: 6.21 ~< c~ 0.20 ~< 13 ~< 0.80 7.60 ~< 7 m 32.0
K~ts = 0.332 O~O'OOS3~/(0"432 O"161/02)70"296 --
× exp(--0.00436/lg 2 + 1.4/sin 0)
(A21)
(0 is in radians). Parametric equation for SCF at the brace negative hot spot under in-plane bending: Kfis
=
0 . 2 0 ~< T ~< 1.00 35 ° ~< 0 ~< 90 °.
Appendix 6 Parametric equation for SCF at the brace crown position (~p=0) under out-of-plane bending:
- 9 . 4 0 0c°°277(°°~82 + 0.~3o) × exp(-0.0289/132 - 3.59/7 - 0.633/sin 0 + 0.00558y/[8)
(A22)
K~
: 0.0
(A30)
(0 is in radians). Parametric equation for position of brace positive hot spot under in-plane bending:
Parametric equation for SCF at the brace crown position (%0=180°) under out-of-plane bending:
(A23)
Parametric equation for SCF at the brace saddle position under out-of-plane bending:
~P'~s = (y/0.35)[ 32
(lpfis is in degrees). Parametric equation for position of brace negative hot spot under in-plane bending: ~PF~s= 180 - (7/0.45)!B72 (~pHs is in degrees),
22
(A24)
K{~° : 0.0
(A31)
Ks = 0.0788 O£°'°91~-°'132/°'~°'9°aT°'957 × exp (-0.211/18 t2 + 2.62 sin 0)
(A32)
Parametric equation for SCF at the brace hot-spot stress site under out-of-plane bending:
Int J Fatigue January 1990
KHS = 0.103 ~x°°°s ~--0'227/8'~/0'848'1"0"469 x exp(-0.232/131-2 + 2.59 sin e)
Brandi, R. 'Behaviour of unstiffened and stiffened tubular joints' ibid paper TS6.1
12.
Kratzer et al. 'Schwingfestigkeitsuntersuchungen an Rohrstrukturelement-modellen von offshore-plattformen' Report BMFT-FB (M81) (June 1981)
13.
Irvine et al. 'Tubular joint fatigue data obtained at the National Engineering Laboratory' UKOSRP Report 4/02 (D. En., 1986) Dover, W.D. and Wilson, T.J. 'Fatigue fracture mechanics assessment of tubular welded T-joints' Dept. of Energy Contract OT/F/917, Final Report (Dept. of Energy, July 1983-June 1985)
(A33)
(O is in radians). Parametric equation for position of brace hot-spot stress site under out-of-plane bending: • HS = 90.0
11.
(A34)
(~ is in degrees). Validity ranges:
14.
6.21 ~< o~ 0.20 ~< 13 ~< 0.80
15.
Dover, W.D., Peat, C. end Sham, W. P. 'Random load corrosion fatigue of tubular joints and tee butt welds' Dept. of Energy Contract TA 93/22/94 , Final Report (Dept. of Energy, December 1987)
16.
Dover, W.D., Holdbrook, M.S.J., Hibberd, R.D. and Charlesworth, F.D.W. 'Fatigue crack growth in T-joints: out-ofplane bending', lOth Ann. Offshore Technology Conf., Houston, Texas, May 8-11 1978, paper OTC 3252 Dover, W.D., Cheudhury, G.K. end Dharmavasan, S. 'Experimental and finite element comparisons of local stress and compliance in tubular welded T and Y joints Int. Conf. on Steel in Marine Structures, Paris, 1981, paper No 4.3
7.60 ~< ~/~< 32.0 0.20~< $~<1.00 35 ° ~< 0 ~< 90 °. References 1.
2.
Potvin, A.B., Kuang, J.G., Leick, R.D. and Kahlick, J.L. 'Stress concentration in tubular joints' Soc Petroleum Engrs J (August 1987). Wordsworth, A.C. and Smedley, G.P. 'Stress concentrations at unstiffened tubular joints' European Offshore
17.
18.
Gibstein, M.B. end Moe, E.T. 'Numerical and experimental stress analysis of tubular joints with inclined braces' Int. Conf. on Steel in Marine Structures, Paris, 1981, paper No 6.3
19.
Irvine et al. 'Tubular joint fatigue data obtained at The Welding Institute', UKOSRP Report 4/03 (D. En., 1986)
20.
Holdbrook, S.J. 'The application of linear elastic fracture mechanics to fatigue crack growth in tubular welded joints' PhD Thesis (University College London, 1980) Hellier, A.K., Connolly, M.P., Kare, R.F. end Dover, W.D. 'Prediction of the stress distribution in tubular Y- and Tjoints' /nt J Fatigue 12 1 (1990) pp 25-33
Steels Res. Semin., Cambridge, UK, November 1978 (Welding Institute, Cambridge, 1978) Paper 31
3. 4.
Gibstein, M.B. 'Parametric stress analysis of T-joints' ibid Paper 26 Connolly, M.P., Hellier, A.K., Dover, W.D. and Sutomo, J. 'A parametric study of the ratio of bending to membrane stress in tubular Y- and T-joints' Int J Fatigue 12 1 (1990) pp 3-11
5.
S E R CProgramme on Fatigue Crack Growth in Offshore Structures, July 1987-June 1989 (UCL Internal Report, London, 1989)
6.
PAFEC Data Preparation User Manual Version 6.1 (PAFEC Ltd, Nottingham)
7.
Dharmavasan, S. 'Fatigue fracture mechanics analysis of tubular welded Y-joints' PhD Thesis (University College London, 1983)
8.
M/N/TAB Reference Manual (Minitab Inc., PA, 1988)
9.
UEG Design of Tubular Joints for Offshore Structures 'Local behaviour of simple welded joints' (UEG, London, Vol. 2, part C)
10.
Dijkstra, O.D. et al. 'Fatigue strength of tubular X and T joints (Dutch) tests' Steel in Marine Structures, Paris, October 1981, paper TS8.4
Int J F a t i g u e J a n u a r y 1990
21.
Authors
W.D. Dover is with The Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, UK, where this work was carried out. A.K. Hellier is now with The School of Materials Science and Engineering, University of New South Wales, Kensington, NSW 2033, Australia. M.P. Connolly is now with General Electric, Aircraft Engine Business Group, Cincinnati, Ohio 45215, USA.
23
Stress concentration factors for tubular Y- and T-joints
A. K. Hellier, M.P. C o n n o l l y and W . D . D o v e r
A systematic study of stresses in tubular Y- and T-joints has been conducted in which nearly 900 thin-shell finite-element analyses were performed. These cover a wide range of joint geometries under axial loading, in-plane bending and out-ofplane bending. For each mode of loading, and for both the chord and brace sides of the intersection, semi-empirical equations are derived which relate the stress concentration factors at selected locations to a parametric function of the joint geometry. Equations are also obtained for the angular location of the hot-spot stress site around the intersection. The accuracy of these parametric equations is then assessed by comparing the predicted values with results from steel model tests and also with the predictions of other previously published equations. Key words: tubular joints; bending stress; membrane stress; stress concentration factors; design Steel offshore structures used for the extraction of oil and gas are composed of tubular joints welded together to form a three-dimensional space frame. Such a structure is susceptible to localized fatigue failure at the welded intersections as a result of the high stresses in these regions together with the large number of stress cycles experienced during its operational life. In order to ensure that the structure is adequately designed against fatigue failure it is necessary to be able to predict the stresses around the welded intersection where cracks are likely to initiate and grow. As a consequence of this, much research effort has been directed towards the determination of stress concentration factors (SCFs) for simple tubular joints, and these results have often been expressed as
parametric functions of the joint geometry for different modes of loading. 1-3 Tubular joints are conventionally designed using a lower bound stress-life (S-N) curve approach. S-N curves, established experimentally from large-scale fatigue tests on tubular joints, have been successfully used together with parametric stress equations for a number of years to ensure that tubular joints satisfy the fatigue design requirements, measured in terms of the total number of cycles to failure at a given applied stress range. Of late, however, there has been an increasing realization that a need exists to perform remaining life calculations on cracked joints. These are based on fracture mechanics techniques which utilize the stresses
Notation al, a2, a3,
coefficients in regression analysis
d4~ a5~ a6~ e t c
d D K~.~0
Ks KHS Kfis Kfis L R2 SCF t
T tCx~ ~y~ /4z 0~
T
external diameter of brace external diameter of chord SCF at crown position (~=0) SCF at crown position (~= 180°) SCF at saddle position SCF at hot spot SCF at positive hot spot (IPB) SCF at negative hot spot (IPB) chord length product moment correlation coefficient stress concentration factor brace wall thickness chord wall thickness displacements in x, y and z directions geometric ratio (=2LID)
0 tp ~x, ~y ¢~HS
~fis ¢Pfis chord brace hot-spot stress crown saddle IPB OPB
geometric ratio (=d/D) geometric ratio (=D/2T) geometric ratio (=t/T) brace angle angle around intersection Loof node rotations position of hot spot position of positive hot spot (IPB) position of negative hot spot (IPB) main tubular member tubular attachment to main tubular member peak stress in chord (or brace) weld toe regions at 0° or 1800 (see Fig. 2) weld toe regions at 90 ° or 270 ° (see Fig. 2) in-plane bending out-of-plane bending
0142-1123/90/010013-11 (~ 1990 Butterworth & Co (Publishers) Ltd Int J Fatigue January 1990
13
acting on the anticipated crack path and the material crack growth behaviour. For this purpose, not only is information required on the absolute maximum value of stress at the intersection (the 'hot-spot' SCF), but also the distribution of SCFs around the intersection as well as the stress variation through the thickness must be available. The problem of predicting the relative proportions of through-thickness bending and membrane stresses in tubular Y- and T-joints has been addressed in an earlier paper 4 published by us in this issue. This present paper is concerned with obtaining parametric expressions for the SCFs at all locations of significance on both the chord and brace sides of the intersection. Since the same finite-element meshes were used for both studies, the equations resulting from each arc directly compatible and may be used together. Also, although parametric equations already exist for SCFs in tubular Yand T-joints, those presented in this paper arc the only comprehensive set which include values at the crown, saddle and hot-spot positions, as well as equations to describe the location of the hot-spot stress site. It is anticipated that all this information will be required in order to fit characteristic formulae for the stress distributions around the intersection adequately. The objective of this work, then, is to provide a full predictive capability for the stresses acting on the anticipated crack plane in a tubular joint in service. All the equations are to be incorporated into a software package s which may be used to conduct rapid fracture mechanics analyses and remanent life calculations on cracked tubular members.
Finite-element analyses Owing to the complex geometrical nature of tubular joints, analytical solutions for their stress distributions are not feasible. Instead these must be determined by numerical methods such as finite-element analysis or by experimental tests on strain-gauged steel or acrylic models. A finite-element approach was adopted in this study, and since a detailed description of this has already been given in a previous paper 4 it will only be summarized here.
Element type The stress analyses were performed with the 'PAFEC' finiteelement package 6 using semi-Loof thin-shell elements. These elements are not capable of transmitting shear forces, but displacements normal to the elements and rotations about their edges are allowed. The elements used were mostly quadrilaterals with eight nodes, denoted '43210', together with a smaller number of their triangular six-noded counterparts known as type '43110'. A diagram of the eight-noded element is shown in Fig. 1. All eight nodes possess three translational degrees of freedom ux, u), and u~, and a rotational degree of freedom also exists at each of the Loof nodes marked with an 'x' in Fig. 1. These rotations are tangential to the element side but are considered for the purposes of input and output as ~. and %, rotations associated with the nearest midside node. In a shell analysis of tubular joints, the junction between brace and chord is modelled as the intersection of their midplanes. This is because shell elements are really two dimensional, having thickness only in a mathematical sense (necessary to define the element stiffness). Furthermore, the weld cannot be incorporated in a model which consists purely of shell elements. For both these reasons the SCF values obtained from thin-shell finite-element analyses are subject to some error, and are usually found to overestimate equivalent 14
Fig. 1 Eight-noded semi-Loof thin-shell element
steel model results, with the error on the brace likely to be greater than that on the chord.
Mesh generation The wide range of joint geometries covered by.' this study necessitates the use of an automatic mesh generator. This must be capable of producing fine elements near the intersection, where the stresses are changing very rapidly, and coarser elements in regions where the stresses are more evenly distributed. Also the elements should not be excessively elongated or otherwise distorted. A suitable program was developed at UCL (Ref. 7) and modified for the present work so that it would reliably generate meshes for tubular Y- and T-joints having widely differing geometric parameters cl, [3, y, T and 0, with an absolute minimum of user input. The mesh generation program was written in FORTRAN to run on a MicroVAX II machine and produces a data filc suitable for immediate analysis by PAFEC. A typical Y-joint mesh comprising 1478 nodes and 477 elements is shown in Fig. 2. It took approximately five minutes of CPU time to generate and a further 90 CPU minutes for the analysis to be performed. Only one half of the tubular joint needed to bc meshed and further details of the procedure used and mesh validation may be found in Ref. 4.
Boundary conditions The boundary" conditions and applied loads are sumrnarizcd diagrammatically in Figs 3a,b respectively. All degrces of freedom (u,, Uy, u~, q:~, %) were fixed at the chord ends, where te, corresponds to the displacement in the x direction
./J.
.JJ
I
SaddJe
Fig. 2 Typical e x a m p l e of finite-element mesh used to model t u b u l a r joint (c~ = 8.97, !3 = 0.6, ~, = 14.5, T = 0.8, 0 = 60°)
Int J Fatigue
January
1990
AI
a
BB
u z c~x £by
U x Uy
in offshore structures. Values for the brace angle 0 of 36 °, 45°, 60°, 75° and 90° were used. The minimum value of O was restricted to 35 ° since it was not possible to generate topologically sound meshes having smaller brace angles. For each load case and angle 0 approximately 60 analyses were conducted. The nominal stress for each joint and loading mode was calculated automatically by the mesh generation program and written to a data file. This was subsequently read, along with the PAFEC stress output file, by a postprocessor program and used to calculate SCF values at the crowns, saddle and hot spot(s) on both brace and chord. The SCFs were obtained by dividing the numerically greatest principal stress at a given point around the intersection by the appropriate nominal stress for that mode of loading. Nominal stresses were calculated by dividing the total applied load by the cross-sectional area of the brace for axial loading. For moment loading, the nominal stresses were derived from simple beam theory using a moment arm measured from the brace end along its outer surface to the (q0=0) crown position for IPB, and to the saddle position (~=90 °) for OPB. ~=0 refers to the crown acute angle location for a Y-joint (see Fig. 2). In all cases it is assumed that the free length of brace or chord is at least three times the diameter of the relevant member.
Parametric equations
Axial
IPB
OPB
b Fig. 3 Details of: (a) boundary conditions; (b) modes of loading used for finite-element joint analyses
and ¢Px and ~y are the Loof node rotations. Note that these are not rotations about the x and y axes (refer back to Fig. 1). Under axial loading or in-plane bending (IPB), symmetry exists about the xy plane containing the chord and brace centre-lines. A half-joint mesh may therefore be used provided the displacements uz and rotations q~x and % are restrained at all the nodes which lie on this plane. Since the restraints are identical for both these modes of loading, they were analysed consecutively as two load cases of the same finiteelement run, without the need for recomputing the element stiffness matrix. For out-of-plane bending (OPB) this symmetry about the xy plane no longer exists. However, it was found that results of acceptable accuracy could be obtained by using a half-joint mesh with the displacement components Ux and uy fixed at all points on the xy plane. 4
Extraction of results The tubular joints analysed in this study encompassed the following ranges of geometries, expressed in terms of the dimensionless parameters el, 13, ~/, "r and the brace angle 0: 6.21 ~< ~x ~< 13.10
(1)
0.20 ~< [3 ~< 0.80
(2)
7.60 ~< ~/~< 32.0
(3)
0.20 ~< $ ~< 1.00
(4)
35 ° ~< 0 ~< 90 °
(5)
With the exception of a, which will be discussed in the next section, these include the majority of tubular joints used
Int J Fatigue January 1990
From the raw database of SCF values, parametric equations were obtained for the SCF at a number of locations around the intersection, both chordside and braceside, under axial loading, in-plane bending and out-of-plane bending. The parametric equations were derived using a statistical regression package known as 'MINITAB' (Ref. 8) which is capable of performing multiple regression and correlation analysis. The methodology used in deriving the equations was as follows. (a) The variations of the SCF at the point in question were plotted for each brace angle 0 as a function of the parameters c~, [3, ~/and • in order to determine the best forms of the terms required, and also to ascertain if any cross-correlation existed between the terms. (b) A first attempt at the equation was made using the simple form
SCF = al cl'~ [3"3y'4 ,r~5 0'~
(6)
where al to a6 were determined from the regression analysis. (c) Equation (6) was then modified by using other (eg exponential) terms, and numerous regressions performed until a suitable equation with a large product moment correlation coefficient was obtained. (d) Having obtained the basic form of the final equation, the exponent for the oe-variation (a2 in Equation (6)) was adjusted to reflect the observation that increasing o~ (ie the chord length) beyond the limit of 13.10 for this study is known to have little effect on SCF values. By taking into account this far-field behaviour it has been deemed possible to remove the upper validity limit on o~for these equations. A summary of the parametric equations obtained together with their correlation coefficient R 2 is given in Table 1. A value of R 2 = 100% would imply that the fitted equation explains all the variations in SCF. It is apparent from Table 1 that a correlation of better than 90% has been achieved in most cases. All the equations derived in this study are set out
15
Table 1. Summary of the parametric equations obtained Type of loading
Chord/ Quantity R 2 (%) Equation
brace
Axial Axial Axial Axial Axial Axial Axial Axial Axial Axial
Chord Chord Chord Chord Chord
K° K(!8° Ks Kas ~PHS
97.2 97.4 99.2 98.8 76.0
Brace Brace Brace Brace Brace
K°
85.5 (A6)
K~8°
93.2
Ks
98.8 (A8)
KHs
~HS
IPB IPB IPB IPB IPB IPB IPB IPB IPB IPB IPB IPB IPB IPB OPB OPB OPB OPB OPB OPB OPB OPB OPB OPB
Chord Chord Chord Chord Chord Chord Chord Brace Brace Brace Brace Brace Brace Brace Chord Chord Chord Chord Chord Brace Brace Brace Brace Brace
K° K18° Ks KGs KHS ~s q;HS K° K18° Ks KGs KHs ~;4s ~HS K° K~8° Ks KHS ~HS Kc° K~8° Ks KHs ~HS
96.7 84.9 99.4 98.1 -99.5 98.4 --86.4 91.9 -89.7 91.2 ----98.8 98.8 ---96.4 96.8 --
(A1) (A2) (A3) (A4) (A5) (A7) (A9) (A10) (All) (A12) (A13) (A14) (A15) (A16) (A17) (A18) (A19) (A20) (A21) (A22) (A23) (A24) (A25) (A26) (A27) (A28) (A29) (A30) (A31) (A32) (A33)
fully in Appendices 1-6. A consequence of this approach is that for some particular geometries more than one equation can apply. These equations should give very similar answers. If the calculated values are slightly different one should take the hot-spot value in preference to crown or saddle values, or if choosing between two crown or saddle values take the highest calculated value. For axial loading, equations were fitted to the SCF variations at all significant locations, namely the two crown positions (at ~p=0 and ~p=180°), the saddle point (~p=90°) and the hot-spot stress site where the greatest value of SCF occurs. An equation was also developed to describe the angular location of the hot spot, ~pHS.This information is used more extensively in the development of parametric equations describing the distribution of stresses around the connection. The equations will be reported in the third paper in this series (Ref. 21, published in this issue). Since the finite-element results are sampled at a limited number of nodes around the intersection, the raw data for ~PHSare discrete, and a degree of circumspection is required in the use of this equation. A similar set of equations was also derived for out-of-plane bending with the exception of the crown positions. Here the SCFs on both chord and brace are very small in most cases, and were therefore taken to be zero. Also, the hot-spot locations in
16
general lie close to the saddle point and so for simplicity were always taken to occur at ~p-90 °. The parametric equations derived for in-plane bending are more numerous than those for either of the other load cases. This is due to the presence of two hot spots under this type of loading, one having a positive SCF and the other having a negative value. For a T-joint these are equal and opposite. For the loading direction shown in Fig. 3, the positive hot spot lies at or near the ~p-0 crown position while the negative hot spot has a ~p-value of 180° or thereabouts. The positive hot spot for a Y-joint is usually, but not invariably, of greater magnitude than the negative onc. However, when dealing with a mixed-mode loading condition and, in particular, attempting to fit a stress distribution curve to Y-joints under in-plane bending, knowledge of the magnitudes and locations of both hot spots is necessary. Simplified expressions were considered sufficient to descrii~e the hot-spot locations in this case. An attempt will now be made to assess the accuracy ot these parametric formulae by comparing them with other parametric equations from the literature and finally with results obtained from steel model tests.
Comparison with other equations As a check on the general applicability of the parametric equations given in Appendices 1-6 a series of comparisons are now made, where possible, with existing parametric formulae given in the literature. All of these formulae predict SCF values for both the chord and brace sides of the intersection under each of axial, IPB and OPB loading. The particular equations considered are as follows. (a) Kuang's formulae for T- and Y-joints.' These predict hot-spot SCFs and are based on thin-shell finite-element (FE) analyses of 46 different geometries. (b) Wordsworth and Smedley's formulae for T- and Yjoints.-' These were obtained from the results of an unspecified number of acrylic model tests and give the SCF at the saddle and crown positions. (c) Gibstein's formulae for T-joints) These arc for hot-spot SCFs and were derived from a study of 17 geometries, again using thin-shell FE analysis. Comparison of the 'UCL' equations (which are based on thin-shell FE analyses of 291 T- and Y-joint geometries, each subject to all three modes ot loading) with these other parametric formulae is not intended to be exhaustive, but rather to give a flavour of the relative variations in hot-spot SCF when one of the geometric parameters is allowed to change. Fig. 4 shows the variation in chord SCF for all the equations as a function of [3 for axially loaded T-joints. It can be seen that the UCL equation is a curve of similar shape to the others but predicts values which are nearest those given by the highest of the existing curves, lying closest to the Kuang equation at low !3 and approaching Smedley's equation at high 13. The peak value in the SCF is at a slightly lower value of [3 for the UCL equation when compared with Gibstein and Smedley, but higher when compared with Kuang. The chord SCF variation with y for T-joints under inplane bending is shown for all four equations in Fig. 5. All the curves show a steady increase in SCF as y becomes larger. Of the previously published formulae, Smedley's equation always yields the highest value in this case, and the UCL equation predicts slightly higher values than Smedley.
Int J Fatigue January 1990
15_
10
UCL
..... .....
12
-
.
Kuang Smedley -
O,
.,~: ~ " , ~ I .~'
te,n
..'5- / ~° ~ UCL Kuang Smedley Gibstein
. . . .
..... ~-=
o
1,,,lILl,li, 0
0.2
0.6
0.4
0.8
1.0
0 0
0.2
0.4
0.6
0.8
.0
T Fig. 4 C h o r d SCF v a r i a t i o n w i t h 13 f o r axially loaded T-joints (cx = 24.0, ~/ = 16.0, T = 0.5)
UCL ..... Kuang ..... Smedley m . m Gibstein
--
,O@e
i
i
i
i
I
6
i
i
n
J
I
o
i
n
12
n
J
I
=
18
i
J
n
I
24
=
a
a
I
30
Fig. 5 Chord SCF v a r i a t i o n w i t h 3, f o r IPB-Ioaded T-joints (cx = 24.0, ~ = 0.5, T = 0.5)
Fig. 6 shows the chord SCF variation with T for OPBloaded T-joints. The UCL equation gives a linear increase in SCF with increasing $ in similar fashion to the existing equations, but is a shade more conservative than the highest prediction obtained using any of these. The graphs given in Figs 4-6 show firstly that the UCL equations exhibit similar trends to those displayed by the established parametric formulae, and secondly that the UCL formulae predict SCF values which appear to be consistently higher than those given by these other equations. This tendency toward conservative predictions is an important attribute for design purposes. The FE-based studies of Kuang, Gibstein and UCL show similar trends but always differ. This could be a consequence of the elements chosen and the number of cases studied.
Comparison with steel models In order to test the relative accuracy of the parametric equations in predicting hot-spot SCFs, the UCL equations together with the Kuang, 1 Wordsworth and Smedley, 2 and Gibstein, 3 equations are compared with steel model test results. These steel model results are normally obtained by carrying out large-scale loading tests on strain-gauged tubular
Int J Fatigue January 1990
Fig. 6 Chord SCF v a r i a t i o n w i t h ~ f o r OPB-Ioaded T-joints (cx = 24.0, 13 = 0.5, 3' = 16.0)
joints and measuring the changes in strain due to the applied loads. The experimental procedures required to determine the so-called extrapolated weld toe stresses have been well defined and are summarised in Ref. 9. A body of data is therefore available which relates the measured SCFs to the tubular joint geometric parameters. Unfortunately, many of these tests were conducted on 'unrealistic' joints, in the sense that their geometric parameters have values unlikely to be found in practice, and also lie outside the limits of the parametric equations. In attempting to assess the accuracy of the various parametric equations, a database of comparable steel model results was compiled and is given in Tables 2, 3 and 4 for the axial, in-plane bending and out-of-plane bending cases respectively. Many of these were obtained from Ref. 9 but do not include results where (x < 5.0, ~ > 32.0 or 13 = 1.0. Tubular joint geometries with cl < 5.0 were ignored since it was felt that for these cases the end conditions applied to the chord became too influential. Joints having - / > 32.0 are not common in practice and also lie beyond the validity range of the parametric equations. The 13 = 1.0 case (where both brace and chord are of the same diameter) has a very complex intersection geometry which places it in a category of its own as far as numerical analysis is concerned. Although quite often found in offshore structures, this geometry is beyond the scope of the present study and has therefore been excluded from consideration. The remaining results are now considered for axial loading, in-plane bending and out-of-plane bending in turn.
Axial loading The comparisons between the steel model results and the SCF values produced by the UCL, Kuang, Wordsworth/Smedley and Gibstein formulae are given in Table 2 for axial loading. All results are for the chord unless otherwise stated. The total number of steel model results is 24, and for convenience the comparisons between the predicted and actual values are expressed as the ratio of predicted SCF to actual SCF for each of the four formulae. A ratio of 1.0 indicates an exact fit. From this table it can be seen that the UCL formula predictions are generally conservative, with only one result having a ratio less than 1.0 (Ref. 13, Spec. No 44). The Kuang and Gibstein equations underpredict the steel model results seven and five times respectively, whereas the results from the Smedley formulae are generally similar to those for the
17
Table 2. Comparison between predicted and measured SCFs for axially loaded T- and Y-joints Ref. SpecNo imen No 10
1 4 11 13
11
c,
13
10.0 10.0 10.0 10.0
0.50 0.50 0.25 0.50
e Steel (deg) SCF
UCL/ steel
Kuang Kuang/ Smedley Smedley/ Gibstein Gibstein/ steel steel steel
0.50 0.50 0.39 0.50
90 90 90 90
5.7 6.7 4.7 7.7
7.5 7.8 5.7 7.8
1.32 1.16 1.21 1.01
6.3 6.6 5.4 6.6
1.11 0.99 1.15 0.86
7.5 8.0 5.0 8.0
1.32 1.19 1.06 1.04
6.1 6.5 4.0 6.5
1.07 0.97 0.85 0.84
II
5.0 0.46 20.0 1.00
60
11.0
16.8
1.53
17.0
1.55
17.3
1.57
--
--
12
1(c) l(b)
6.9 0.66 23.1 0.91 6.9 0.66 23.1 0.91
45 45
8.6 6.5
10.8 8.7
1.26 1.34
9.7 8.4
1.13 1.29
11.2 8.1
1.30 1.25
---
---
13
19 41 42 43 44
0.86 1.00 0.28 0.47 0.25
90 90 90 90 90
11.4 12.0 3.3 4.8 3.7
14,5 18.0 3.7 7.0 3.2
1.27 1.50 1.12 1.46 0.86
12.6 16.1 3.0 6.7 2.9
1.11 1.34 0.91 1.40 0.78
12.9 16.1 4.5 5.9 3.1
1.13 1.34 1.36 1.23 0.84
12.8 16.1 2.8 4.9 2.1
1.12 1.34 0.85 1.02 0.57
14
UCL TW2,4 TW3
7.3 0.71 12.0 1.00 7.3 0.71 12.0 1.00 7.3 0.71 12,0 1.00
90 90 90
10.0 10.5 10.3
13.7 13.7 13.7
1.37 1.30 1,33
10.8 10.8 10.8
1.08 1.03 1.05
11.7 tl.7 11.7
1.17 1.11 1.14
12.0 12.0 12.0
1.20 1.14 1.17
15
CP1 CP2 CP3 CP4
7.3 7.3 7.3 7.3
1.00 1.00 1.00 1.00
90 90 90 90
9.3 13.4 13.7 13.1
13.7 15.6 15.6 15.6
1.47 1.16 1.14 1.19
10.8 12.4 12.4 12.4
1.16 0.93 0.91 0.95
11.7 13.9 13.9 13.9
1.26 1.04 1,01 1.06
12.0 14.0 14.0 14.0
1.29 1.04 1,02 1.07
16 SHl(c) SHl(b)
7.3 0.71 14.4 0.80 7.3 0.71 14.4 0.80
90 90
9.3 6.1
12.0 10.5
1.29 1.72
9.3 10.2
1.00 1.67
11.2 8.1
1,20 1,33
10.4 7.8
1,12 1.28
17 GCI(c) GCI(b)
7.2 0,71 14.3 0.79 7.2 0.71 14.3 0.79
90 90
8.7 6.0
11.7 10.4
1.34 1.73
9.0 10.0
1.03 1.67
11.0 7.9
1,26 1.32
10.1 7.7
1,16 1,28
45
5.2
6.1
1.17
5.6
1.08
6.1
1.17
--
--
18
Y
10.5 5.0 5.0 5.0 5.0
13.0
0.53 0.50 0.50 0.24 0.24
0.71 0.71 0.71 0.71
13.4 14.3 14.3 14.3
UCL
13,4 14.4 14.4 14.4 14.4
12.0 14.3 14.3 14.3
0,48 15.9 0.63
(c) = chord, (b) = brace. All others are chord values.
Table 3. Comparison between predicted and measured SCFs for IPB-Ioaded T- and Y-joints Ref. SpecNo imen No
(x
13
~/
T
e Steel (deg) SCF
UCL
UCL/ steel
Kuang Kuang/ Smedley Srnedley/ Gibstein Gibstein/ steel steel steel
9
1
10.0
0.50 13.4 0.50
90
1.1
2.6
2.36
1.9
1.73
2.4
2.18
2.1
1.91
12
1(c) l(b)
6.9 6.9
0.66 23.1 0.91 0.66 23.1 0.91
45 45
3.3 2.7
5.7 4.0
1.73 1.48
3.6 2.8
1.09 1.04
4.5 3.8
1.36 1.41
---
---
13
41 42 43 44(b)
5.0 5.0 5.0 5.0
0.50 0.50 0.24 0.24
1.00 0.28 0.47 0.25
90 90 90 90
4.9 1.3 1.5 2.0
5.2 1.6 2.2 2.2
1.06 1.23 1.47 1.10
3.6 1.2 1.9 2.4
0.73 0.92 1.27 1.20
4.3 1.6 2.2 1.8
0.88 1.23 1.47 0.90
4.5 1.2 2.0 1.8
0.92 0.92 1.33 0.90
14
UCL
7.3
0.71 12.0 1.00
90
3.0
5.0
1.67
3.2
1.07
3.7
1.23
4.0
1.33
0.71 0.71 0.71 0.71
90 90 45 45
3.1 2.0 2.7 2.5
4.2 3.3 4.0 3.6
1.35 1.65 1.48 1.44
2.9 2.5 2.4 2.3
0.94 1.25 0.89 0.92
3.4 3.2 3.0 2.9
1.10 1.60 1.11 1.16
3.3 2.3 ---
1.06 1.15 --
17 G C I ( c ) 7 . 2 GCI(b) 7.2 GC2(c) 7.2 GC2(b) 7.2
14.4 14.4 14.4 14.4
14.3 14.3 14.3 14.3
0.79 0.79 0.79 0.79
--
(c) = chord, (b) = brace. All others are chord values.
18
Int J F a t i g u e J a n u a r y 1990
T a b l e 4. C o m p a r i s o n
between
predicted
Ref. No
Specimen No
e<*
13
~/
•
19
23/1(c) 23/2(c) 23/3(c) 23/1(b) 23/2(b) 23/3(b) 24/1(c) 24/2(c) 24/3(c) 24/1(b) 24/2(b) 24/3(b)
-------------
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3
0.39 0.39 0.39 0.39 0.39 0.39 0.28 0.28 0.28 0.28 0.28 0.28
90 90 90 90 90 90 90 90 90 90 90 90
19
703/1 703/2 703/3 704/1 704/2 704/3
-------
0.53 0.53 0.53 0.53 0.53 0.53
13.4 13.4 13.4 13.4 13.4 13.4
0.86 0.86 0.86 0.51 0.51 0.51
14
UCL
7~3 0.71
16 20
SHI(c) SH2(c) SHl(b) SH2(b)
17
GCI(c) GCI(b)
and measured
0 Steel (deg) SCF
SCFs for OPB-Ioaded
T-joints
Kuang Kuang/ S m e d l e y S m e d l e y / Gibstein Gibstein! steel steel steel
UCL
UCL/ steel
2.3 2.9 2.6 2.0 2.2 1.9 1.9 1.6 1.8 1.7 1.6 1.8
3.2 3.2 3.2 2.8 2.8 2.8 2.3 2.3 2.3 2.4 2.4 2.4
1.39 1.10 1.23 1.40 1.27 1.47 1.21 1.43 1.28 1.41 1.50 1.33
2.2 2.2 2.2 2.9 2.9 2.9 1.6 1.6 1.6 2.4 2.4 2.4
0.96 0.76 0.85 1.45 1.32 1.53 0.84 1.00 0.89 1.41 1.50 1.33
2.2 2.2 2.2 2.4 2.4 2.4 1.6 1.6 1.6 2.0 2.0 2.0
0.96 0.76 0.85 1.20 1.09 1.26 0.84 1.00 0.89 1.18 1.25 1.11
2.1 2.1 2.1 2.3 2.3 2.3 1.4 1.4 1.4 2.0 2.0 2.0
0.91 0.72 0.81 1.15 1.05 1.21 0.74 0.88 0.78 1.18 1.25 1.11
90 90 90 90 90 90
8.0 7.7 8.4 5.4 5.0 5.4
9.6 9.6 9.6 5.7 5.7 5.7
1.20 1.25 1.14 1.05 1.14 1.05
7.5 7.5 7.5 4.7 4.7 4.7
0.94 0.97 0.89 0.87 0.94 0.87
9.5 9.5 9.5 5.6 5.6 5.6
1.19 1.23 1.13 1.04 1.12 1.04
9.5 9.5 9.5 5.2 5.2 5.2
1.19 1.23 1.13 0.96 1.04 0.96
12.0 1.00 90
8.5
10.0
1.17
7.1
0.84
11.9
1.40
10.5
1.24
7.3 7.3 7.3 7.3
0.71 14.4 0.80 90 0.71 14.4 0.80 90 0.71 14.4 0.80 90 0.71 14.4 0.80 90
9.7 10.6 6.2 6.0
9,8 9.8 8.2 8.2
1.01 0.92 1.32 1.37
7.0 7.0 7.5 7.5
0.72 0.66 1.21 1.25
11.4 11.4 8.2 8.2
1.18 1.08 1.32 1.37
9.6 9.6 7.4 7.4
0.99 0.91 1.19 1.23
7.2 7.2
0.71 0.71
9.1 5.1
9.6 8.1
1.05 1.59
6.9 7.4
0.76 1.45
11.2 8.2
1.23 1.61
9.4 7.3
1.03 1.43
14.3 0.79 90 14.3 0.79 90
(c) = chord, (b) = brace. All others are chord values. *a assumed to be 7.0 where no value given.
UCL formulae, with only one underprediction. The UCL formulae are a little more conservative than Smedley's but occasionally give a fairly high level of overprediction (for example a ratio of 1.73 for Ref. 17, Spec. No GCI(b)). In-plane
bending
Table 3 contains the comparisons between the steel model results and the SCF values predicted by the UCL, Kuang, Wordsworth/Smedley and Gibstein formulae for in-plane bending. The total number of steel model results is 12 in this case. Once again the UCL equations are the most conservative, providing predictions which are consistently higher than the steel model results. The Kuang, Smedley and Gibstein equations underpredict the steel model results 5, 2 and 3 times respectively. Out-of-plane
bending
The steel model results and the SCFs obtained from the UCL, Kuang, Wordsworth/Smedley and Gibstein formulae are compared in Table 4 for out-of-plane bending. There are 25 results for this mode of loading, all of which were obtained from tests on T-joints. It is apparent that in every case except one (Ref. 16, Spec. No SH1 (c)) the UCL equations overpredict the steel model results. For the other equations the numbers of cases where the SCF is underestimated are considerable, with 15, 5 and 10 underpredictions for the Kuang, Smedley and Gibstein equations respectively.
Int J F a t i g u e J a n u a r y
1990
Summary
The total number of steel model results given in the SCF database of Tables 2-4 for axial loading, in-plane bending and out-of-plane bending is 61. Fig. 7 shows the SCF values predicted by the UCL equations plotted against the experimental results for all these cases together, and serves to confirm the inherently conservative nature of the equations. If all the results for the three modes of loading are now broken down into intervals depending upon the ratios of predicted SCF to actual SCF for each equation in turn, then the histograms of Fig. 8 are obtained. The tendency of the UCL formulae to overpredict SCF values is again clear to see, with a mere 3% of the values being underpredictions. This should be contrasted with the Kuang, Wordsworth/Smedley and Gibstein formulae which underpredict 44%, 13% and 34% of the results respectively. Of these, the Wordsworth/Smedley equations are generally considered the most appropriate for design 9 because of their relatively low level of underprediction. On this basis, the UCL equations proposed in this paper would appear to offer an even more reliably conservative prediction of SCF when designing a tubular joint. It can be seen that under some circumstances the equations will predict a SCF of zero. In these cases consideration should be given to using a minimum value instead of zero. The minimum value would be determined from previous design/ service experience.
19
20
16 LL
12
I.,3
CJ
8 Q.
4
o
,,
, I ,,, 4
,I
,,,
, I ,,,
, I,,,,
12 Experimental
15
8
20
SCF
Fig. 7 SCF predicted by UCL equations plotted against experimental SCF for all modes of loading
Conclusions A thin-shell finite-element study has been performed in order to obtain stress concentration factors for a wide range of tubular Y- and T-joint geometries. Parametric formulae or simple expressions have been derived for the SCF at all
40
locations of significance on both the chord and brace sides of the intersection under axial loading, in-plane bending and out-of-plane bending. Equations have also been obtained which describe the positions of the hot-spot stress sites around the intersection. It is anticipated that these might be used in conjunction with characteristic distribution formulae to provide a full description of the SCF variation around a tubular joint. The equations for the hot-spot SCF are found generallx to overestimate the measured SCFs from steel model tests. They have also been compared with existing parametric formulae due to Kuang, Wordsworth/Smedley and Gibstein, and found to be generally more conservative than anx of these. It is therefore concluded that the hot-spot S('F equations presented in this paper are the most reliable in predicting a conservative value of SCF which could bc used in design.
Future work Several areas may be identified in which further work could profitably be undertaken. The first of these, which is considered in the next article in this issue, 2~ is the generation of characteristic stress distribution formulae for each mode of loading which utilize the equations presented in this paper as input parameters, thus providing a full description of the SCF variation around the intersection. This should enable
UCL
--
30 - -
40
-
30
-
I
Smedley
I I I
>. L) r"
>~ u r-
20
--
U"
20
U_
U_
10
10
0
a
C 40 -
40
30I
Gibstein
Kuang 30
>, (J
>, ~J c"
20
20
EY
LL
LL
10
10
/ / / / A
0
0 b
0.75
I .0
Predicted
I .25
SCFltest
0.75
I .5 result
d
1.0
1.25
Predicted SCF/test
1.5 result
Fig. 8 SCF histograms for all load cases combined: (a) UCL formulae; (b) Kuang's formulae; (c) Wordsworth and Smedley's formulae; (d) Gibstein's formulae
20
Int J Fatigue January 1990
more refined fracture mechanics analyses of cracked tubular joints to be performed using a two-dimensional stress field on the anticipated crack plane. Secondly, it would be desirable to extend the scope of this study to encompass higher values of 13 (the ratio of brace diameter to chord diameter) up to and including 13 = 1.0. Future studies to produce similar sets of parametric equations for X- and K-joints would also be worthwhile.
Acknowledgements The authors wish to acknowledge the Marine Technology Directorate Ltd, SERC and Sponsors of the Cohesive Fatigue Programmes 1985-9 for providing some of the financial support needed for this work.
Parametric equation for SCF at the chord crown position (~0=0) under axial loading:
51.°"°6313-°2°7°1/25(°"988-°'133/°)
K ° = 0.575 exp(0.66513 s + 0.0204~/+ 1.64 sin 0 - 0.469[35) (A1) (0 is in radians). Parametric equation for SCF at the chord crown position (lp=180 °) under axial loading: =
Klc8° = 1 9 . 0 5 5-0"035[3 (0"213/02 - 00439)5(0'48 + 000589/05) x exp(-0.00148~ 2 - 1.25 sin 0 + 0.00055~2/[3 - 1.09132"r)
(A7)
(0 is in radians). Parametric equation for SCF at the brace saddle position under axial loading: Ks = 0.538 0f°°28650'963 x exp(-0.987133 - 4.67~/-°.s + 4.87 sin 1/2 0 - 0.1335/[3)
(AS)
Parametric equation for SCF at the brace hot-spot stress site under axial loading:
Appendix 1
K~:8°
(0 is in radians). Parametric equation for SCF at the brace crown position (lp=180 °) under axial loading:
3.94 (~-0"03913--0'10505(0"906 + 0"055/03)exp(0.327134 + 0.0177~ - 0.05 sin 0 - 0.422[3"0 (A2)
(0 is in radians). Parametric equation for SCF at the chord saddle potation under axial loading: Ks = 2.51 5°°3725112 exp(-1.96134 - 11.3/~/+ 2.47 sin 0 + [35)
X exp(--1.18613 s -- 4.2811-y -°.s + 1.987 sin 0 + 0.427[35)
(Ag)
Parametric equation for position of brace hot-spot stress site under axial loading: ¢4~HS --
58.6 500309[30-05585(0.14402
-
-
0.404)
x exp(1.01/~/+ 0.343/sin 0 - 0.0503[3/5) (A10) (0 is in radians, IPHS is in degrees). Validity ranges: 6.21 ~< 5 0.20 ~< 13 ~< 0.80 7.60 ~< ~/~< 32.0 0.20 ~< 5 ~< 1,00
(A3)
Parametric equation for SCF at the chord hot-spot stress site under axial loading: KHS = 9.92 50'029513 exp(--1.3813 s's -- 5.53~/-°-s + 2.15 sin 0 -- 0.05695/[3)
KHS = 4.40 50"°s6850 ss3
35 ° ~< 0 ~< 90 °.
Appendix 3 Parametric equation for SCF at the chord crown position (lp=0) under in-plane bending:
(A4) Kc° = 2.84 5 - ° °°ss~/°-2345(T M - o.171o)exp[(0.00213,
Parametric equation for position of chord hot-spot stress site under axial loading: lPHS = 193.7 5--0"041513 (0.267 -- 0"093502)5(0"2710 -- 0.491) × exp(15.5/'y 2 - 0.77 sin 0 + 0.000028~/2/[32 - 0.00932/[35)
(A5)
- 0.0373)/[32 - 0.0179/sin 0] (All) (0 is in radians). Parametric equation for SCF at the chord crown position (~=180 °) under in-plane bending: K~cs° = - 3 . 8 4 5 5--O'O29'yO'4075(O'966 -- 0.07190)
(A12)
× exp[(0.00166~/- 0.0399)/[32 - 0.682/sin 0]
(0 is in radians, 0HS is in degrees). Validity ranges:
(0 is in radians). Parametric equation for SCF at the chord saddle position under in-plane bending:
6.21 ~< 5 0.20 ~< [3 ~< 0.80
Ks = 0.0
7.60 ~< -¢ ~< 32.0
(A13)
Parametric equation for SCF at the chord positive hot spot under in-plane bending:
0.20 ~< -r ~< 1.00
Kfts
35 ° <~ 0 ~< 90 °.
=
2.31 5°'°°33~/°'326'r (1"24 - 0.1870)
(A14)
x exp[(0.00154~/- 0.0323)/[32 - 0.0248/sin 0]
Appendix 2 Parametric equation for SCF at the brace crown position (~p=0) under axial loading: g°
=
1.46 50'0098130'214/05 (0.623
-
-
Int J Fatigue January 1990
/iris = - 8 . 9 3
0'0811/02)
x exp(0.87213 s - 0.00104~/2 + 1.47 sin 0 + 0.000474~/2/[3 - 1.73135)
(0 is in radians). Parametric equation for SCF at the chord negative hot spot under in-plane bending: 0 f - - 0 " 0 1 7 6 ' ~ / 0 ' 4 5 5 (0"94 --
0.04410)
(A15)
× exp[(0.00153~/- 0.0386)/[32 - 1.69/sin1/20] (A6)
(0 is in radians).
21
Parametric equation for position of chord positive hot spot under in-plane bending: + -- (y/0.41)]~ 2 ~PHS
(A16)
(~P~s in degrees). Parametric equation for position of chord negative hot spot under in-plane bending: ~PHS = 180 -- (y/0.45)[372
(A17)
(q~fis is in degrees). Validity ranges :
7.60 ~< 7 ~< 32.0 0.20 ~< T <~ 1,00 35 ° ~< 0 <~ 90 °.
K ° = 0.0
7 . 6 0 ~< y ~< 32.0 ~ ~<
0.20 ~< 13 ~< 0.80
Parametric equation for SCF at the chord crown position (q~=0) under out-of-plane bending:
0.20 ~ [3 ~< 0.80
~
6.21 ~- c~
Appendix 5
6.21 ~ cx
0.20
Validity ranges:
(A25)
Parametric equation for SCF at the chord crown position (~p- 180°) under out-of-plane bending:
1.00
35 ° ~< 0 ~< 90 °.
K ~ ° = 0.0
Appendix 4 Parametric equation for SCF at the brace crown position (~p=0) under in-plane bending:
Parametric equation for SCF at the chord saddle position under out-of-plane bending: Ks = 0.315 (t. 0"O54y(hI2
K ° = 1.89 O¢--0"065%/-0"2142/O2T0"186~O'298
-
(A26)
O'O939/O2)'1 -
× exp(-0.001391134 + 0.654 sin 2 0)
× e x p ( - 5 . 8 8 / y + 1.81/sin 0 0.0534[33y - 0.084818/7)
(A18)
(0 is in radians). Parametric equation for SCF at the brace crown position (lp=180 °) under in-plane bending:
(0 is in radians). Parametric equation for SCF at the chord hot-spot stress site under out-of-plane bending: I~THS
:
0.255 (~0.O13y(t.24
0.224/0)T
x exp(-0.00135/iB 4.' + 0.923 sin 0) K{~°
=
- 7 . 0 2 ~-O'O4S~O'141/O2yO'167/0270"143
-
× exp(--0.015/132 -- 164.4/73 -- 0.524/sin20 0.02541327) (A19) -
Parametric equation for SCF at the brace saddle position under in-plane bending: Ks : (-,//60.0)(1/0 2)
(A20)
(0 is in radians). Parametric equation for SCF at the brace positive hot spot under in-plane bending:
(A27)
(A28)
(0 is in radians). Parametric equation for position of chord hot-spot stress site under out-of-plane bending: ~Hs = 90.0
(A29)
(tpHs is in degrees). Validity ranges: 6.21 ~< c~ 0.20 ~< 13 ~< 0.80 7.60 ~< 7 m 32.0
K~ts = 0.332 O~O'OOS3~/(0"432 O"161/02)70"296 --
× exp(--0.00436/lg 2 + 1.4/sin 0)
(A21)
(0 is in radians). Parametric equation for SCF at the brace negative hot spot under in-plane bending: Kfis
=
0 . 2 0 ~< T ~< 1.00 35 ° ~< 0 ~< 90 °.
Appendix 6 Parametric equation for SCF at the brace crown position (~p=0) under out-of-plane bending:
- 9 . 4 0 0c°°277(°°~82 + 0.~3o) × exp(-0.0289/132 - 3.59/7 - 0.633/sin 0 + 0.00558y/[8)
(A22)
K~
: 0.0
(A30)
(0 is in radians). Parametric equation for position of brace positive hot spot under in-plane bending:
Parametric equation for SCF at the brace crown position (%0=180°) under out-of-plane bending:
(A23)
Parametric equation for SCF at the brace saddle position under out-of-plane bending:
~P'~s = (y/0.35)[ 32
(lpfis is in degrees). Parametric equation for position of brace negative hot spot under in-plane bending: ~PF~s= 180 - (7/0.45)!B72 (~pHs is in degrees),
22
(A24)
K{~° : 0.0
(A31)
Ks = 0.0788 O£°'°91~-°'132/°'~°'9°aT°'957 × exp (-0.211/18 t2 + 2.62 sin 0)
(A32)
Parametric equation for SCF at the brace hot-spot stress site under out-of-plane bending:
Int J Fatigue January 1990
KHS = 0.103 ~x°°°s ~--0'227/8'~/0'848'1"0"469 x exp(-0.232/131-2 + 2.59 sin e)
Brandi, R. 'Behaviour of unstiffened and stiffened tubular joints' ibid paper TS6.1
12.
Kratzer et al. 'Schwingfestigkeitsuntersuchungen an Rohrstrukturelement-modellen von offshore-plattformen' Report BMFT-FB (M81) (June 1981)
13.
Irvine et al. 'Tubular joint fatigue data obtained at the National Engineering Laboratory' UKOSRP Report 4/02 (D. En., 1986) Dover, W.D. and Wilson, T.J. 'Fatigue fracture mechanics assessment of tubular welded T-joints' Dept. of Energy Contract OT/F/917, Final Report (Dept. of Energy, July 1983-June 1985)
(A33)
(O is in radians). Parametric equation for position of brace hot-spot stress site under out-of-plane bending: • HS = 90.0
11.
(A34)
(~ is in degrees). Validity ranges:
14.
6.21 ~< o~ 0.20 ~< 13 ~< 0.80
15.
Dover, W.D., Peat, C. end Sham, W. P. 'Random load corrosion fatigue of tubular joints and tee butt welds' Dept. of Energy Contract TA 93/22/94 , Final Report (Dept. of Energy, December 1987)
16.
Dover, W.D., Holdbrook, M.S.J., Hibberd, R.D. and Charlesworth, F.D.W. 'Fatigue crack growth in T-joints: out-ofplane bending', lOth Ann. Offshore Technology Conf., Houston, Texas, May 8-11 1978, paper OTC 3252 Dover, W.D., Cheudhury, G.K. end Dharmavasan, S. 'Experimental and finite element comparisons of local stress and compliance in tubular welded T and Y joints Int. Conf. on Steel in Marine Structures, Paris, 1981, paper No 4.3
7.60 ~< ~/~< 32.0 0.20~< $~<1.00 35 ° ~< 0 ~< 90 °. References 1.
2.
Potvin, A.B., Kuang, J.G., Leick, R.D. and Kahlick, J.L. 'Stress concentration in tubular joints' Soc Petroleum Engrs J (August 1987). Wordsworth, A.C. and Smedley, G.P. 'Stress concentrations at unstiffened tubular joints' European Offshore
17.
18.
Gibstein, M.B. end Moe, E.T. 'Numerical and experimental stress analysis of tubular joints with inclined braces' Int. Conf. on Steel in Marine Structures, Paris, 1981, paper No 6.3
19.
Irvine et al. 'Tubular joint fatigue data obtained at The Welding Institute', UKOSRP Report 4/03 (D. En., 1986)
20.
Holdbrook, S.J. 'The application of linear elastic fracture mechanics to fatigue crack growth in tubular welded joints' PhD Thesis (University College London, 1980) Hellier, A.K., Connolly, M.P., Kare, R.F. end Dover, W.D. 'Prediction of the stress distribution in tubular Y- and Tjoints' /nt J Fatigue 12 1 (1990) pp 25-33
Steels Res. Semin., Cambridge, UK, November 1978 (Welding Institute, Cambridge, 1978) Paper 31
3. 4.
Gibstein, M.B. 'Parametric stress analysis of T-joints' ibid Paper 26 Connolly, M.P., Hellier, A.K., Dover, W.D. and Sutomo, J. 'A parametric study of the ratio of bending to membrane stress in tubular Y- and T-joints' Int J Fatigue 12 1 (1990) pp 3-11
5.
S E R CProgramme on Fatigue Crack Growth in Offshore Structures, July 1987-June 1989 (UCL Internal Report, London, 1989)
6.
PAFEC Data Preparation User Manual Version 6.1 (PAFEC Ltd, Nottingham)
7.
Dharmavasan, S. 'Fatigue fracture mechanics analysis of tubular welded Y-joints' PhD Thesis (University College London, 1983)
8.
M/N/TAB Reference Manual (Minitab Inc., PA, 1988)
9.
UEG Design of Tubular Joints for Offshore Structures 'Local behaviour of simple welded joints' (UEG, London, Vol. 2, part C)
10.
Dijkstra, O.D. et al. 'Fatigue strength of tubular X and T joints (Dutch) tests' Steel in Marine Structures, Paris, October 1981, paper TS8.4
Int J F a t i g u e J a n u a r y 1990
21.
Authors
W.D. Dover is with The Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, UK, where this work was carried out. A.K. Hellier is now with The School of Materials Science and Engineering, University of New South Wales, Kensington, NSW 2033, Australia. M.P. Connolly is now with General Electric, Aircraft Engine Business Group, Cincinnati, Ohio 45215, USA.
23