stress analysis of drillstring threaded connection

Pergamon Engineering Failure Analysis, Vol 2, No. 1 pp. 1-30, 1995

Copyright 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

1350-6307/95 $9.50 + 0.00

1350-6307(95)00007-0

STRESS ANALYSIS OF DRILLSTRING THREADED CONNECTIONS USING THE FINITE ELEMENT METHOD

K. A. M A C D O N A L D and W. F. D E A N S

Department of Engineering, University of Aberdeen, Kings College, Aberdeen AB9 2UE, U.K.

(Received 25 January 1995)

Abstract--Stress analysis of the threaded connections in drillstrings and bottom hole assem- blies has contributed to the successful resolution of some downhole failures. The preload applied from joint make-up torque directly affects the static stress distribution within the connection--it also affects the local mean stress levels about which the stresses arising from fatigue loading oscillate. Considering a generic trapezoidal threadform, the relationship between the nominal applied load and the resulting peak and local elastic stresses at the critical thread roots are established using the finite element method. The distribution of peak stress in the connection is determined based on the axial preload arising from make-up, and how this distribution is modified by tensile and compressive axial loads. Starting with a procedure of mesh convergence and model validation, a two-dimensional axisymmetric elastostatic modelling approach is used. In all cases, the roots of the first loaded tooth in the pin and the last loaded tooth in the box are the sites of maximum peak stress as expected, the pin peak stress being the greater. However, considering the effects of fatigue loading by relating the individual preload and tensile load cases to local and peak stress ranges and mean levels demonstrates that the box becomes the critical component.

1. I N T R O D U C T I O N

Downhole failure of the threaded connections in drillstrings and bot tom hole assembly components by fracture and fatigue, although uncommon, nevertheless occurs with sufficient frequency [1] to focus attention on the detail design aspects of the connections. The pr imary factors influencing connection failure are the stresses at critical locations, and the material 's fatigue and mechanical properties. Knowledge of the applied loads during drilling in many circumstances carries with it great uncer- tainty. Material strength requirements, wellbore geometric constraints, hydraulic flow area considerations, and economic limitations have all led to the widespread, if not universal, selection of high-strength low-alloy (HSLA) steels for the majority of drillstring tubular applications [2]. Hence , as the loading regime and material selection are essentially either predetermined or uncertain, detail design aspects of threaded connections have a predominant role in offering control of fatigue and fracture performance.

The search for new hydrocarbon resources and the development of existing reserves are taking place in an economic climate characterised by a depressed oil price and a concerted industry-wide initiative to reduce costs. Such pressures have dictated modern industry trends towards deeper and deviated wells which place more stringent demands on the design and operat ion of the load-carrying components of the drillstring.

Conventional drillstring connections are geometrically complex and, in their threads, exhibit inherently severe geometr ic stress concentrations. The criticality with regard to fatigue that this infers is further exacerbated by the widely acknowledged differential distribution of the load in such joints. The body of knowledge regarding load distributions in threads dates f rom earlier this century [3] and has since been contributed to by many workers, whose collective efforts have been reviewed elsewhere [4]. Consideration of the stress concentrations in threads and notches has also been made [5, 6]. Estimating the fatigue behaviour of a connect ion-- in terms of

K. A. MACDONALD and W. F. DEANS

crack initiation life--is readily accomplished in two stages [7], where the stress concentration factor (SCF) resulting from the main body stresses and thread bending stresses is first evaluated, followed by determination of the material's fatigue strength reduction factor, itself based on the SCF and the notch sensitivity of the material. Finite element (FE) analysis readily provides the first part of this two-stage fatigue analysis process by determining the connection's general stress state and critical SCFs.

The objective of the present work is to provide stress analysis data in support of a wider ongoing research programme concerned with the development of predictive models of fatigue damage accumulation and limit state failure in drillstring threaded connections, particularly those in the bottom hole assembly, i.e. drill collars and stabilisers. Threadforms under consideration include standard API designs (V-0.038R) in addition to the trapezoidal type reported here.

1.1. Downhole failures

Fatigue has been recognised for some time as an important cause of failure in general drillstring connections [8, 9], but it is only in recent years that the serious problems arising from such failures have occurred with sufficient frequency to merit detailed attention to their cause and prevention [2, 10-14]. Recent North Sea data from a single operator puts costs associated with downhole separation of the drillstring at ca 10 million per annum [1]. Investigations of recent failures of drillstring components [1] have, in the case of the most common casualty connections, revealed a clear distinction between failure modes and the corresponding failure sites. Connections (Fig. 1) when failing primarily by fatigue, break consistently in the box at the root of the last engaged thread (LET) farthest from the seal face (Fig. 2). However, when the same connection fails entirely by ductile overload shear fracture, the site of failure is in the pin LET adjacent to the shoulder (Fig. 2). These observations are based on some 25 connection failures, where, subject to the difficulties presented by post-separation damage to fracture surfaces and equipment [1], no material or manufacturing deficiencies were apparent. In section, the morphology of fatigue cracks (Fig. 1) is characterised by an approximate 45 orientation to the thread flank at positions at and adjacent to the thread root, becoming more straight (transverse to the drillstring axis) once a distance of more than a few thread root radii away (Fig. 2). This behaviour is consistent with tooth bending stresses influencing the short crack, with body stresses becoming dominant at greater crack depths [15].

1.2. FE analysis o f large threaded connections

The severe SCFs in threaded connections give rise to stresses with high peak values and rapidly increasing gradients approching the thread roots. The accurate estimation of highly localised stresses in these regions is consequently an exacting computational task. The task is demanding to the extent that a sufficiently enriched three-dimensional FE model targeted to capture the peak stress and gradient would in practice be unsolvable with most generally available maniframe computing installations. Novel solution strategies have been formulated [16] whereby a coarse layer of anisotropic elements substitutes the threads, replicating the differential load distribution. The method does not directly yield the peak stresses giving the load distribution only-- the local stresses must be computed subsequently.

The oil and gas industry has widely adopted two-dimensional axisymmetric elastostatic FE analysis in the stress analysis of threaded connections in oilfield tubulars [17]. This arises primarily from the ability of the method to generate and solve meshes sufficiently refined to correctly compute the localised thread root stress field, other benefits accruing from ease of modelling and reduced solution time, although in some cases it is the only route possible to produce a solution. Elastic analysis is sufficient unless thread root plasticity is so extensive as to influence the overall load

rfenyvesHighlight

Stress analysis of drillstring threaded connections 3

Box connection

vNi ~x~

\

\ \ \ \

!

Pin connection

Pin LET

Z

R

Box LET

Drill collar Rotary shouldered

connection

Fig. 1. Drill collar showing the connection and its failure sites.

distribution by affecting the relative strains within the connection [15], and in any case is a recommended initial approximation. Axisymmetric models give a parallel thread representation of the joint and can accommodate axial and radial loading and boundary conditions, the latter seemingly representing a limitation for simulating drillstring components in bending. Some FE solver codes now offer axisymmetric elements capable of asymmetric loading [18] which facilitate non-uniform distributed loads and allow modelling of applied bending in two-dimensional axisymmetric models. Inevitably, however, such two-dimensional analyses fail to represent both the thread helix and the runout regions--important features of threaded connections.

Three-dimensional FE studies of threaded connections have been conducted [19, 20], investigating the effects of pitch, stress relief features and thread geometry

K. A. M A C D O N A L D and W. F. DEANS

modifications, with helpful results. However, the models are of necessity generally coarse and in some cases only a few threads are considered and with simplified thread profiles.

2. C O N N E C T I O N S T U D I E D

2.1. Geometry

The geometry of the connection studied is shown in Fig. 1. It is a 9 in. outside diameter drill collar connection but with a generic trapezoidal threadform dimen-

(a)

(b)

S c a l e c m

Fig. 2(a) and (b). Caption on p. 5.


(c)

Fig. 2. Drillstring service failures: (a) fatigue failure of a 6.625 in. diameter stabiliser pin; (b) overload failure of a pin from a crossover connection; (c) macrograph showing a fatigue crack at a thread root.

sioned and pitched to approximately match a standard API NC-61 V-profile threadform (V-0.038R) [21, 22]. An analysis of the NC-61 connection in standard form is included in this study and may be published at a later date.

2.2. Material

For the purposes of this numerical study, both the pin and box materials were assumed linear elastic with the general mechanical properties of steel, elastic modulus E = 207,000Nmm -2 and Poisson's ratio v = 0.29, and for the analyses involving plasticity, a yield point of 800Nmm -2 (116 ksi). In reality, the majority of ferritic drillstring materials are specified according to AISI 4145 H and AISI 4142 H, both HSLA steels in the quenched and tempered condition. Typical mechanical properties are given in Table 1.

3. NUMERICAL ANALYSIS METHOD

3.1. Computational facilities

Preparatory work and the FE analyses proper were performed using ABAQUS/ Standard, a general-purpose FE program [18], across a number of software releases:

Table 1. Typical specified mechanical properties for AISI 4145 H

Mechanical property Material specification

Charpy V-notch impact energy Hardness Minimum yield strength Ultimate tensile strength Reduction of area Elongation

41 J (30 ft-lbs) at RT 285-341 BHN 758.6 Nmm -2 (110 ksi) 965.5 Nmm -2 (140 ksi) 45% min 13% min


v. 4.9-v. 5.3, Presently, the software is mounted on the Aberdeen University Com- puting Centre's Sun SparcServer 1000 computer (four 50 MHz CPUs) running the SOLARIS v. 2.3 operating system, and accessed via SunSparc IPC workstations. The FE models were prepared using ABAQUS' s own model data definitions for the more simple models, but employed P3 /PATRAN [23], a graphics-based mesh-generating and results post-processing program, for the more refined and complex meshes. Results post-processing was carried out using both P3 /PATRAN and ABAQUS/Pos t .

Particular features of ABAQUS exploited in these analyses included geometric nonlinearity, frictional contact modelling using interface elements, automatic resolution of overclosed interfaces (shrink fitting) and the nonlinear asymmetric deformation capability for axisymmetric solid elements.

3.2. Axisymmetric modelling

The single most important approximation made in constructing the FE model is that of axisymmetry, where the geometry is considered a body of revolution about the Z-axis modelled in the R - Z plane (Fig. 1). This approximation significantly alleviates the computational load which would otherwise be associated with a three-dimensional analysis and has consequently been widely adopted by the oil and gas industry [17]. The validity of this assumption is inferred from corroboration between two-dimensional FE analyses and three-dimensional photoelastic studies performed on standard (ISO M30) nut and bolt connections [24]. The axisymmetric representation of the three-dimensional component does not, however, represent either the thread helix or the runout regions, simply modelling the connection as a series of parallel threads with a constant runout geometry. A further limitation exists in that conventional axisymmetric models are limited to axial loads, but bending loads are a major contributor to the stress state in a drillstring [8]. This apparent restriction is rationalised by arguments constructed around the remoteness of the connection wall from the neutral axis (small ratio of wall thickness to diameter) which leads to a limited stress gradient across the wall under applied bending loads, which consequently can be adequately approximated by a uniform membrane load (Fig. 3). This assumes no coupling between the regions of maximum tensile and compressive stress, disposed 180 apart, so that separate analyses of equivalent tension and compression loads are considered to give reliable results.

ABAQUS/Standard includes in its element library elements suited for the nonlinear analysis of initially axisymmetric components which undergo non-linear, asymmetric deformat ion--as occurs with tooljoints subjected to bending loads. Whereas the conventional axisymmetric continuum elements use standard isoparametric interpolation with respect to R and Z, such asymmetric-axisymmetric elements have ad- ditional Fourier interpolation with respect to 0. The use of such elements allows the recovery from a single model of stresses in a connection loaded in pure bending, or bending combined with axial load, and removes the simplification of replacing the actual non-uniform load with an idealised uniform load. However, the effects of thread helix and runout are still not considered.

4. MESH C O N V E R G E N C E

The extremes of mesh density can produce an incorrect solution if too coarse, and analysis costs disproportionate to the results if too fine. A fine mesh is needed in regions of high stress (and strain) gradient which occur at geometric discontinuities, where a coarser mesh will suffice in areas of constant stress or low stress gradient. Furthermore, element formulation is important in that, for a given problem, a linear displacement element enquires a finer mesh than a parabolic one which, in turn, needs a finer mesh density than a cubic element.

Because the effect of a stress concentration on the elastic stress field is local and

Stress analysis of drillstring threaded connections

outer fibre stress

stress differen wall t ickness across wall --,

inside diameter

Fig. 3. Through-wall distribution of stress in a pipe under a bending load.

outside diameter

. . . . longitudinal axis

dies away or diffuses with distance, a graduated mesh can be used in such areas [25], accommodating the transition from a fine mesh at the stress concentration to a coarse mesh in remote regions. On occasions where the local stress at a particular discontinuity is not of primary interest, but the stress at another site is, a coarse mesh can be used at the discontinuity and accurate stresses still obtained at the site of interest (with appropriate local mesh density) provided this site is sufficiently remote from the discontinuity, in doing so recognising that accurate stresses will not be obtained in the coarsely modelled region. Such an approach is useful provided the coarsely modelled region gives correct load paths, stiffness and boundary conditions.

Elements are typically defined in terms of the basic shape of the parent element, for example a square for a quadrilateral element, an isosceles triangle for a triangular element and so on. Complex geometries can pose difficulties in controlling element shape, increasingly distorted elements generally producing less accurate results. In general, more distortion can be accommodated without loss of accuracy with both higher-order elements and smaller stress gradients.

4.1. Mesh convergence study

With the aim of refining the mesh at the geometric discontinuity represented by the thread root, a series of small submodels were constructed covering a range of mesh densities local to the thread root (Fig. 4). The boundary conditions and loads were identical for each model: uniform axial pressure (stress) on one component; removal of the axial (Z-direction) rigid body mode on the other. Radial interference between the box and pin was also included by setting extreme typical manufacturing tolerances to give a radial interference of 0.0508 mm (0.002 in.). Due to their reliable performance [18, 24, 25] eight-noded, biquadratic interpolation, reduced integration, axisymmetric quadrilateral elements were used. Contact was modelled using interface elements between the mating thread surfaces, these elements allowing for closing and opening of contacting surfaces, small relative sliding and the modelling of friction by means of the classical Coulomb model [18]. Although a coefficient of friction of 0.09 was selected, being appropriate to drillstring connections [21], this coefficient was extremed in this study to investigate the sensitivity of the results to this assumption.


t J t l ~

~ - ~ H mesh

J-

) ~ mesh

Fig. 4. FE meshes used for mesh optimisation study.

A l t h o u g h ex t r ac t ed in i so la t ion f rom the full connec t ion , and desp i te its s impl ic i ty , the conf igura t ion o f the m o d e l is cons ide red a d e q u a t e l y r ep re sen t a t i ve o f the real case, in that the effect of the t h r ead notch , too th bend ing loads and body stresses are all inc luded.

The ax i symmet r i c F E mode l s were so lved for s tat ic load cond i t ions using gener ic elast ic ma te r i a l p rope r t i e s for s teel ( E = 2 0 7 , 0 0 0 N m m -2, v = 0.29) by, f irst ly, reso lv ing the rad ia l i n t e r fe rence b e t w e e n the pin t h r ead and box, and , secondly , app ly ing a subsequen t un i fo rm axial stress. The va r ia t ion of m a x i m u m pr inc ipa l stress* a r o u n d the roo t rad ius of the p in ' s e nga ge d t h r e a d is shown in Fig. 5 for a range of mesh densi t ies , where it is c lear that s ignif icant mesh r e f i ne me n t is n e e d e d to accura te ly r ep re sen t the stress d i s t r ibu t ion a r o u n d the t h r ead roo t radius. This peak stress occu r red at abou t 40 f rom the t h r e a d f lank (Fig. 6). The rad ia l ( th rough-wal l )

*From a rigorous theoretical standpoint, the maximum tensile stress at the thread root is located at an unloaded (free-surface) boundary and hence occurs at the site of the tangential in-plane principal stress at that point. This theoretical uniaxial stress state may not be recovered exactly by an FIE analysis where the maximum principal stress is potentially matched with a complementary but spurious non-zero minimum principal stress normal to the free surface. Taking the maximum principal stress difference in preference to the absolute maximum principal stress can ameliorate this marginal inaccuracy. However, such errors amounted to < 0.8% of peak values in the mesh optimisation study: hence, the maximum principal stress alone was used for convenience.

(a)

cO

1600

1200

800

400

-400


I I I I I | I

0 0.5 1 1.5 2 2.5 3

1 D

2

4 O

8 a

16

Distance, mm

(b) ~u

&

.,4

.M

@ n.

1500-

i000

500

A

< -500 , t w

0 1 2 3 4

box r'l

pin

Distance, mm

Fig. 5. Distribution of peak stress around the thread root: (a) mesh optimisation study (pin); (b) pin and box (mesh 4).

gradients of maximum principal stress approaching the thread roots (Fig. 7) are consistent for all mesh densities, only diverging close to the thread root where the stress gradient increases rapidly. Again, the more refined meshes capture the stress gradient: however, the maximum peak stress is only obtained with the most highly refined meshes. The accuracy and efficiency of the five meshes studied are clearly represented in Fig. 8, where the convergence of peak stress is evident, as is the associated computational cost. The results from a similar series of analyses using

1600

1200

.I

800

400

0

K. A. M A C D O N A L D and W. F. DEANS I0

Fig. 6. Contours of maximum principal stress showing the site of max imum stress along the thread root radius.

0 . 7 0 . 7 5 0 . 8 0 . 8 5 0 . 9 0 . 9 5 1

Normalised wall thickness, mm

Fig. 7. Stress distributions resulting from the different meshes.

16 []

8 :

4 O

2 & 1 -"

first-order quadrilateral elements are also given in Fig. 8, where the consistent superiority of the second-order elements is clear. Because the coefficient of friction used in the analyses ( f = 0.09) is simply selected from standards appropriate to drillstring connections [21], the effect of a higher value was also studied ( f = 0.5), but the difference in peak stress proved negligible (0.07%).

Mesh number 4 (Fig. 4), with eight elements defining the root radius, was selected as the optimised mesh for use in subsequent analyses based on its compromise between converged peak stress output and reasonable computational cost.


CAXSR stress 13 - - - 1 3 - - - CAXSR time

CAX4R stress & - ' - & - ' - CAX4R time

1600

1 4 o o

1200

~ lOOO

600

400

S I Q

$

800 ~ "//s/" --~

i I ,.o,A

200 ~.ii~ ~ ..41~. ~.

0 - - I I I 5 i0 15

8OO

700

600

5OO

4OO

'300

-200

-I00

0

2O

-r'l

O -r'l

r"4 O

r.t)

Number of elements

Fig. 8. Mesh optimisation results: dependence of peak stress and solution time on mesh refinement.

5. AXISYMMETRIC MODEL VALIDATION

5.1. Thread runout geometry

Representation of a three-dimensional threaded connection by a two-dimensional axisymmetric FE model inevitably fails to include the effects of thread helix and runout geometries. These geometric features combine in the critical regions of a connection--the pin and box LETs--to produce a nonuniform thread load distribution around the runout thread progressing from the first contact when partly formed to a fully developed inter-tooth load as the helix progresses and the thread chamfer diminishes, typically occurring in one complete revolution. The increased flexibility of the first partly formed thread over its fully formed neighbours initially encourages load shedding from the mating thread, until the thread becomes completely formed, attaining the peak thread root stress about one turn from the point of initial inter-tooth contact [26]. An attempt to model such behaviour in axisymmetry can be made by considering the thread runout geometry at several meridional sections, equi-spaced at angular sites. Although it is impossible to argue that such an approach embodies the actual stress state, the method has nevertheless been validated with experimental data for standard nut-bolt connections [24, 26], and as such is considered a viable approximation for use in the present study.

A typical FE mesh, from a series of three, representing different meridional planes through the whole connection, is shown in Fig. 9. The level of mesh refinement is identical at all threads--as the peak stress distribution for all threads is of interest-- and is based in the optimised mesh considered earlier. The typical model size was approximately 13,600 elements and 40,000 modes. These models were solved for resolved radial interference and axial tension loading, with the Z-direction rigid body

12 K. A. M A C D O N A L D and W. F. DEANS

/ _ . R[

Fig. 9. FE model of a full connection (perimeter plot) with pin LET: details of uniform and differentially meshed models.


mode restrained on the pin free end. Substructuring with superelement generation was not employed because ultimately the modes were intended to be solved for non-linear material behaviour and non-linear asymmetric-axisymmetric deformation. The differences in thread runout geometry gave rise to variations in peak stress (Fig. 10). In contrast to results from studies of standard V-form threads [24] the multi-meridional plane technique produces the maximum stress in the thread at a point close to the first attainment of full thread height on the contacting flank, i.e. where the thread is still not yet fully formed (Fig. 9). The peak stress occurs at this location due to the superposition of the maximum connector body stress and the

(a) ~ 1250

I000

.~ 7 5 0

500

250

0

0 1 2 3 4 5 6 7

(b) 1250 -

d i000 -

750 -

500 -

25O

- A

Tooth pitches from shoulder

- _ . : $ . . . . . .

I I I I I I I

8 9 i0 Ii 12 13 14 15

Tooth pitches from seal-face

Fig. 10. Thread peak stress distribution from three separate multi-meridional plane FE models with different thread runout geometries: (a) pin; (b) box.

14 K. A. MACDONALD and W. F. DEANS

maximum possible tooth load bending stress. Furthermore, the maximum pin and box stresses do not necessarily occur in the same meridional plane because of non-identical thread runout formation in a given model. Based on these results, a single axisymmetric geometry was chosen with similar states of runout evolution on both the pin and box, maximising the thread peak stresses in both.

5.2. Preload from make-up

Established drilling practice is to further tighten connections after assembly by applying a make-up torque. In addition to simply removing slack in a joint and providing resistance to downhole make-up arising from shock torsional loading and the reduced coefficient of friction of pipe dope at elevated downhole temperatures, this practice results in strong benefits from a live load carrying perspective. Once preloaded, axial tension is generated in the pin which is equilibrated by axial compression in the box. Seemingly, an alternative load path is created through the shoulder interface, bypassing the pin threads [15]: however, in reality applied loads are still transferred through the pin, causing it to stretch or shorten and the shoulder compression to release or increase. The complexities of modelling the relationship between applied torque and axial preload is sufficiently difficult that, in this study, use was made of prescribed axial preloads from drillstring standards. The recommended make-up torques used for connections are taken from drill collar guidance [21], where specified torques are intended to generate a minimum axial stress of 62,500 psi (431 N mm -2) in the weaker of the pin or box.

In FE modelling terms, overclosed interface elements were used at the seal-face- shoulder region which, when resolved as an interface fit, produce a compressive interfacial pressure which reacts in the threads as an axial preload. Furthermore, interface elements allow the shoulder interface pressure to vary in accordance with applied loads, mimicking the make-up preload mechanism in practice.

A series of identical FE models of the full connection with increasing amounts of shoulder overclosure were solved by resolving the overclosure. An overclosure of 0.3 mm produced-- f rom a linearisation procedure -- a membrane stress component of 467 N m m -2 located in the parallel section of the pin adjacent to the shoulder (Fig. 11), and was thus considered to compare well with the API guidance. This value of 0 . 3m m seal-face overclosure compares favourably with a value of 0.008in. (0.203 mm) used elsewhere for an analysis of a standard NC-46 connection [27]. The resultant peak stress distribution (Fig. 12), demonstrates that the pin LET and the other lower thread numbers* react with most of the axial preload.

5.3. Differential mesh density at thread roots

Building on the earlier explanation that coarse modelling of stress concentrations may not prevent a refined mesh elsewhere returning accurate results at the stress concentration of interest, a second mesh was prepared based on the same axisymetric geometry as in Sections 5.1 and 5.2 but, unlike its predecessor, it was targeted to give accurate results at the critical threads only (Fig. 9), with a more coarse mesh at the intermediate threads. This was done to provide an FE model of reduced size and attendant reduced computational load for use in analyses where only the critical threads will be of interest, as is the case with comparative studies. The original model size was dramatically reduced to about 4000 elements and 11,500 nodes. In solving the model for combined preload and axial load, comparison with the corresponding uniformly meshed model shows that the peak stress results at the intermediate threads

*Thread numbers counted from the shoulder.


1250

@

i000 o

o

750

500

,<

250

0

i

actual distribution

i

\

I I I I 0 0.2 0.4 0.6 0.8

linearisation

Normalised wall thickness

Fig. 11. Preload induced through-wall distribution of stress at the pin parallel section (0.3 mm shoulder-seal-face overclosure).

5000 -

d 4000 -

3ooo- -~I

i 2000 -.

m i000 -

box

D pin

O I , , , , , , , T7

0 2 4 6 8 I0 12 14 16

Tooth pitches from shoulder/seal-face

Fig. 12. Thread peak stress distribution solved for preload (0.3 mm seal face overclosure) from three runout geometries.

are significantly in e r ror (34 and 18% underpredic t ion for pin and box, respectively), but , more impor tant ly , the results at the critical threads are in g o o d ag reement (within 0 .41%). These results validate the non-uni formly meshed model , demonst ra t - ing that the coarsely model led region gives correct load paths, stiffness and bounda ry condit ions.

16 K.A. MACDONALD and W. F. DEANS

5.4. Non-linear asymmetric-axisymmetric elements

The fundamental premise of the axisymmetric models dicussed above is that drillstring loads can be adequately represented by uniform axial loads at level equivalent to the average membrane stress component across the pipe wall, or the extreme outer fibre stress (Fig. 3). This approach requires separate analyses of equal but opposite sign (tensile and compressive) axial loads to r~cover the peak stress distributions disposed at the 0 and 180 positions. Elements exist which allow the non-linear analysis of initially axisymmetric components which undergo non-linear, asymmetric deformation. In essence, such elements allow in-plane bending loads, assumed to be symmetrical about 0 = 0 , to be applied to axisymmetric models, overcoming the need for simplification of the stress distribution and for separate solutions. Results can be recovered at various angular positions depending on the number of Fourier modes employed, the minimum being two nodal planes at 0 and 180 , and the maximum being five nodal planes at 0, 45, 90, 135 and 180 . Bending loads can be combined with uniform axial loads to represent downhole loads comprised of a weight-on-bit or overpull component , and a bending component from a dog-leg or deviated wellbore. As the analysis is non-linear, a superelement solution strategy cannot be adopted. The analysis is still in two dimensions and the effects of thread helix and runout are still ignored.

To reduce the computational load, the differentially meshed model was employed with asymmetric-axisymmetric elements and solved for preload and an equivalent bending load (maximum outer fibre stress = uniform membrane stress from the axisymmetric analysis). Comparison with the corresponding results from the axisymmetric solution revealed significant differences in peak stress, particularly at the pin LET under preload and the box LET under preload plus bending. Comparison of the through-wall stress gradients at the straight section near to the pin shoulder showed that the asymmetric-axisymmetric elements returned lower stresses at near-surface positions than the plain asymmetric elements, although the general distributions were similar (Fig. 13). The same is true at the pin LET site, where the discrepancies at the extremities are amplified in the region of high stress gradient near to the thread root.

134

r~

~4

rO

1.4

c6

CAX8R

1

I I I I I

Normalised wall thlckness

Fig. 13. Differences in through-wall distribution of stress at the pin parallel section returned by plain axisymmetric (CAX8R) and asymmetric-axisymmetric (CAXA8R1) element types under preload only.

Stress analysis of driUstring threaded connections 17

5.5. Elastic-perfectly plastic material behaviour

In order to test the contention that an elastic analysis is sufficient unless thread root plasticity is so extensive as to influence the overall load distribution within the connection (by affecting the relative strains between threads), comparative analyses were performed with both elastic material behaviour, and with elastic-perfectly plastic behaviour using the Von-Mises yield surface. The elastic-plastic material model used an identical elastic modulus value of 207,000 N mm -2 and, beyond first yield at 800 N mm -2, assumed a constant value of yield stress (perfectly plasticity). Analysing a case of preload and 200 N mm -2 applied tension with the uniformly meshed model, the extent of plasticity at the pin LET was found to be small [~0.261 mm (Fig. 14)] with the through-wall stress gradients showing close agreement, only diverging once yielding is promoted in the high-stress region local to the thread root. The extent of plasticity at the next pin thread was smaller still, 0.070 mm, with all other pin and box thread roots remaining elastic. The non-uniform mesh was also analysed for elastic and elastic-plastic material behaviour with the results demonstrating excellent agreement with the uniform cases (Fig. 15). In comparing the merits of the two material models, it is clear that plasticity is sufficiently limited so as not to contribute significantly to the relative strains within the joint, and as the analysis costs of the elastic-plastic model are a factor of 5 greater than that of its elastic counterpart, it is concluded that the assumption of elastic material behaviour is justified in this case.

6. RESULTS AND DISCUSSION

The present interest lies in evaluating the stress distribution and stress concentration factors for a drillstring connection with a trapezoidal threadform under preload and preload plus uniform axial loads. Concerns over the discrepancies found between the preload cases modelled with axisymmetric and asymmetric-axisymmetric elements led to their suspension from this study, and has prompted closer examination of this

l 1 z

Fig. 14. Extent of plasticity at the pin LET for preload plus 200 N mm -2 axial tension (0.261 mm).

18

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m ~J m c~ O 0

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K. A. M A C D O N A L D and W. F. DEANS

-- Elast ic + $

Both meshes /~"

_ X Elas t i c -p las t i c . I l k . Differential mesh / ' ~

A - - E las t i c -p las t i c ~

I I I I I

() . .~ {) . .1 / . (~ () . ,q ]

Hormalised wall thickness

Fig. 15. Differences in through-wall distribution ot stress at the pin LET returned by plain axisymmetric (CAX8R) and asymmetric-axisymmetric (CAXA8R1) element types in uniform and differentially meshed models solved for preload plus 200 N mm 2 axial tension.

element type and its implementation. The results discussed here for the plain axisymmetric FE analyses are in the most part modelling make-up preload combined with uniform axial tension and compression loads.

In assessing the effect of a logical modification to the threadform, a comparative study considered the connection with matched thread root radii on both pin and box threads, and a modified thread with the root radii increased. These uniformly meshed models were analysed for axial tension without preload and the results clearly demonstrate the reduced thread root peak stresses in all the box threads (Fig. 16).

The main body of analyses considered the uniformly meshed model under preload (0.3 mm shoulder overclosure) combined with uniform axial tension and compression at nominal stress levels ranging from + 100 to + 800 Nmm-: , representing a range approximately 12.5-100% of assumed material yield strength. The primary results in terms of the distribution of thread root peak stress demonstrate that the preload effects dominate the overall distribution. The pin LET consistently shows the highest peak stress and through-wall stress gradient (Fig. 17). On first inspection the results for the box are unusual in that the first engaged thread (FET) exhibits a higher peak stress than the LET--a result apparently contradicting service experience. However, examination of the stress gradients at the box FET and LET positions (Fig. 18) reveals that, although the FET peak stress is indeed high,* the through-wall gradient of stress is largely compressive, only becoming slightly tensile once axial loads reach levels which promote shoulder separation (Fig. 19). In contrast, the box LET with a lower peak stress exhibits a consistently tensile through-wall gradient, confirming it as the known site of failure from service experience.

A vector representation of the thread root stress field demonstrates the variation of maximum principal stress direction (Fig. 20). The maximum principal stress direction is initially tangential to the root radius at the point of peak stress, but rotates to become axially aligned once more than a distance equivalent to a few root radii away

*The existence of this highly localised SCF at the box FET has been confirmed by metallographic examination, where loealised damage to the thread root has been observed.

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8 0 0 -

~ 600

E ~ 200


0 I I I I I I I 6 8 10 12 14 16 18

Tooth pi tches f rom seal-face

Fig. 16. Effect of enlarging the box thread root radius on the thread peak stress distribution (axial tension Nmm-~; no preload).

19

from the thread root. This gives rise to an initially curved crack propagation path near to the thread root which produces a characteristic lip feature on the failure surface. This fatigue crack morphology potentially provides an important point of reference for failure investigations as would be readily observed even when post- failure mechanical damage has taken place [1]. These results exemplify the competing mechanisms of tooth bending stress and the stress concentrating effect of the tooth notch on the body stresses [5, 15].

Tooth separation occurs at the higher tooth numbers under compressive loading, demonstrated by the tooth flank interface element openings (Fig. 21), where the spread of tooth disengagement from the box LET towards the pin LET at increasing compressive loads is clear. Radial interface displacements between the pin and box threads under preload and tension loading demonstrate the extent of radial expansion and contraction of the connection* (Fig. 21), the greatest changes from the preload state taking place at the connection extremities due to the high proportion of load transfer and increased flexibility arising from the taper at these sites.

6.1. SCFs

In preloaded connections, the representation of the stress state local to the critical thread root using classical SCFs is substantially dependent on how the SCF is defined. Evidenced by Fig. 12, the preload has a pronounced effect upon the stresses at low thread numbers while the preload remains in force. The notch local stress, defined as the sum of linearised bending and membrane through-wall stress components, displays non-proportional behaviour in the case of the pin where the LET local stress is clearly a nonlinear function of nominal pipe stress (Figs 22 and 23). The box LET local stress remains largely unaffected by the preload and consequently exhibits a proportional response to applied stress. The change in gradient of the pin local stress response at

*This relative radial displacement is measured between the crest of the box thread and the root of the pin thread, and is composed of both increases and decreases in radial displacement of the box and pin, respectively.

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7000 -

d 6000-

aa 5000 -

-,-I 4000 -

3000 -

< 2000 -

i000 -

600 MPa -- 400 MPa at 200 MPa

(a) 100 MPA [] preload

: i1200; MM~: -- -400 MPa i -600 MPa

. 4 ; 8 1 ~) ! 2 14 I

16

Tooth pitches from shoulder

~. 600 MPa -- 400 MPa at, 200 MPa - 100 MPa [] preload

~ 4000 1 m $ -100 MPa 1 at -200 MPa

- 4% ~ -400 MPa t~ ~ ~ 2 -600MPa / 3000]

2000

1i] II 2 4 6 8 i0 12 14 16

Tooth pitches from seal-face

Fig. 17. Effect of axial load oll the thread peak stress distribution: (a) pin: (b) box.

n.

S o

o

U

4J m

r6 .,-I

I000

800

600

400

200

0

-200

-400

-600


I I I I

0 0.2 0.4 0.6 0.8 1

[] FET

: LET

21

Normalised wall thickness

Fig. 18. Differences in through-wall distribution of stress at the box FET and LET sites (preload plus 200 N mm -2 axial tension).

1250

d i000

75O 0

500

I--4

250

tension []

compression

i l t I

200 400 600 800

Axial stress, MPa

Fig. 19. Effect of applied axial tension and compression on shoulder-seal-face interface pressure.

nomina l app l i ed s t resses of a p p r o x i m a t e l y ___450 N m m -2 (Fig. 22) def ines wel l the po in t s w h e r e the t o o t h p r e l o a d is o v e r c o m e in c o m p r e s s i o n and the shou lde r p r e l o a d is r e l e a s e d in tens ion . T h e va r i a t i on of no tch p e a k stress d e m o n s t r a t e s s imi lar b e h a v i o u r in tha t the box L E T re sponse is e f fec t ive ly l inear* bu t the pin L E T

22

\

{ {

3

K. A. M A C D O N A L D and W. F. D E A N S

I \ \ \ \ "" \ '

\

Fig. 20. Variation of maximum principal stress direction within the complex thread root stress field (preload plus 200 N m m - : axial tension).

response is evidently nonlinear (Fig. 22). Thus, the peak stresses cannot be expressed as constant geometric SCFs. In order to describe these peak stresses with respect to some reference stress within the connection, three appropriate definitions of SCF (Kt) can be used:

( i) K t relative to the nominal pipe membrane stress. Although the most con- veniently derived SCF, this classical geometric definition does not accommodate the potentially non-proportional relationship between the peak and nominal pipe stresses (Fig. 22) and fails to take account of the preload's profound effect upon the SCF.

(ii) Kt relative to the local membrane component of stress. Linearisation of the through-wall stress distribution at the critical thread (Fig. 23) to give the membrane stress component takes account of the body stresses and gives a better representation of the SCF.

(iii) Kt relative to the local membrane and bending components of stress. Although more onerous in terms of data reduction, a full linearisation of the through-wall stress distribution at the critical thread (Fig. 23) evaluates the thread root peak stress relative to the most effective measure of the notch nominal local stress.

Considering first the box LET, Fig. 24 gives the SCFs computed on the basis of the three above definitions as functions of nominal applied pipe stress. The SCF based on pipe stress is a weak function of applied stress, but when expressed with reference to the sum of local membrane and bending stress components it becomes effectively constant at a value of approximately 3.5 in tension, even when the preload mechanism is substantially overcome at pipe stresses above about 450 N mm -2. In compression, the box LET is no longer loaded by a meshing pin tooth but is simply subjected to compressive body stresses: consequently, the stress state alters accordingly and a lower SCF of 2.0 is the result.

*Under compressive load, the box LET disengages and is no longer loaded. In this condition, the stress state is characterised by the box body in compression and the site of compressive peak stress moves from the former site of the tensile peak stress to another position on the thread root.

(a)

0

,,~

Q)

~Z

0.35 -

0 . 3 -

0 . 2 5 -

0.2-

0.15 -

0.i-

0.05 -

0

0

Stress analysis of drillstring threaded connect ions

[] 100 MPa / A 200 MPa -- 400 MPa

I 2 4 6 8 i0 12 14 16

23

(b) 0.14 -

~ o . 1 2 - o

0.i-

@

0 . 0 8 - J.a in @

0.06 - o

0 0 0.04 - b

~Z

0 0.02 - 0

0

0


m preload t : 100 MPa l

/

200 MPa [ I

I 400 MPa ] /

! !

2 4 6 8 i0 12 14

i

16


Fig. 21. Effect of load on the thread flank and crest interface openings: (a) flank; (b) crest.

Identical but magnified effects are apparent for the pin LET in tension (Fig. 24), which might be expected as the preload mechanism has been shown to have most effect at this site. The SCF in tension with reference to pipe stress is profoundly affected by applied stress while the preload remains effective at applied stress levels below about 450 N mm -2. Computed on the basis of the linearised local bending plus membrane stresses, the pin SCF reduces to a linear and very weakly dependent function of applied stress, almost constant at a value of approximately 5.5. Similar behaviour persists into the compressive loading regime but, unlike the box LET, the pin LET remains initially engaged by the preload mechanism, with the pin SCF reducing once the preload is fully overcome at applied stresses below -450 N mm -2.

K. A . M A C D O N A L D a n d W . F. D E A N S

(a) 3.

r~

o~

c~

i g 08 n

~000

2 5 0 0

l

24

2 :. 0 0 I I I I 1 I

~':~: - 6 0 : ; 20: i 2 0 0 o~U-P'- 90@

,~ 2 0 0 0

1 5 0 0

i000

500 -

5 0 0 - O 0

-i000 -

(b)

Nominal applied stress, MPa

n pin

11. box

1 5 0 0 I i I I I I

- 9 0 0 . . . . . . . . -60,-: ~i,~ - 2 0 0 64r, : ,~, ,

N o m i n a l a p p l i e d s t r e s s , M P a

Fig. 22. Effect of appl ied load on LET: (a) peak: (b) l oca l stresses.

Once the dominant effects of preload are adequately considered, the SCFs for tension loading become effectively constant at 3.5 for the box LET and 5.5 for the pin LET (Table 2), reflecting the greater thread root radius of the box postulated in this study.

6 . 2 . Response of local stress to cyclic loading

Cyclic fluctuation of the applied pipe stress promotes an oscillatory response in the local stresses at the critical threads. These alternating through-wall stresses at the pin


- - peak stress

- - . i - - - -

Pb

linearised stress distribution Pm

Pm+Pb A t

25

A

1

Stress distribution at AA

Fig. 23. Stress linearisation procedure to give stresses used in the definition of SCFs.

and box L E T sites are the driving force for fatigue crack initiation and propagation behaviour, and are the foundations on which the notch peak stresses are based. Considering the axial tension load FE analyses, and the preload case, stress ranges and mean levels were computed for the local stresses at a stress ratio* of R = 0. Of particular note are the pin L E T responses in range and mean where clearly observable changes in slope take place at approximately 450 N mm -2 nominal pipe stress (Fig. 25). As with the peak stresses, this behaviour represents the point at which the preload is overcome and the seal-face-shoulder interface begins to open.

The box L E T has consistently higher local stress ranges than the pin L E T across the full extent of the applied stress ranges (Fig. 25). The difference between the two initially diverges but then converges with increasing applied stress range. Such behaviour occurs because the pin L E T receives most of the protection afforded by the preload mechanism [15, 28] whereas the box L E T receives virtually none and as such is exposed to the full effect of the applied stress range. The benefit afforded the pin is lost once the preload is overcome, accounting for the convergence of the pin and box responses at higher stress ranges. Notably, the limited stress range at the pin L E T is apparently obtained at the considerable expense of a consistently elevated mean level (Fig. 25) much higher than the mean level experienced by the box LET.

6.3. Response of peak stress to cyclic loading

The peak stresses are in fact idealised elastic stresses and as such these extremely high magnitudes do not occur in practice due to localised yielding. The effect of the stress concentration is to form a localised region of plastically deformed material which is best characterised by a strain parameter and not stress. In any case, it is alternating plastic strain that is acknowledged as the mechanism promoting the fatigue crack initiation process [29]. These points combine to make a peak stress representation of cyclic loading response unsuitable. However , for the straightforward case of pulsating tension at R = 0, it is nonetheless both justifiable and helpful to consider the range and mean levels of peak stress in order to examine the relative severity of the pin and box L E T sites under fatigue loading (Fig. 26). As with the local stresses,

*Stress ratio definitions: R = -1 is fully reversed tension-compression; R = 0 is pulsating tension from zero; R > 0 (positive R ratio) is tension-tension.

26 K.A. MACDONALD and W. F, DEANS


Table 2. SCFs in tension for a preloaded connection in a 9 in. diameter drill collar with a trapezoidal threadform

Nominal pipe stress Pin SCF Box SCF (Nmm -~)

Pipe Pm + Pb Pipe* Pm + Pb-t

100 50.7 6.5 7.0 3.7 200 27.1 6.1 6.3 3.5 300 18.9 5.8 6.0 3.5 400 14.6 5.5 5.9 3.4 500 12.2 5.5 5.8 3.4 600 11.1 5.1 5.8 3.4 700 11.1 4.9 5.7 3.4 800 11.0 4.9 5.8 3.4

*Constant value assumed = 5.5. ?Constant value assumed = 3.5.

results p resen ted here , the cor robora t ing service data refers to whole fatigue life and, as such, a combined and qualitative in terpreta t ion of the peak and local stress results is justified. In this respect , of the two critical threads, the box L E T displays the higher local and peak stress range but the lower mean level. The relative impact on connec t ion fat igue is then primari ly a funct ion o f the ferritic mater ia l ' s response to mean stress effects. For similar structural steels the stress range tends to be much m o r e significant than the m e a n level [30, 31]. Fat igue loading then induces a critical failure site at the box L E T under fatigue loading at R = 0, but, under static tension loading, the pin L E T has the greatest local and peak stress levels making it the ant icipated critical failure site when the joint is statically loaded (summarised in Table 3).

7. C O N C L U S I O N S

The stress distributions and stress concent ra t ion factors in a drillstring th readed connec t ion with a t rapezoidal t h read fo rm have been evaluated using the F E me thod

(a) 1400

1200

d i000

800

600

4OO

2OO

0

0 200 400 600 800

[] pin

& box

Nominal applied stress range, MPa

Fig. 25(a). Caption on p. 28.

28

(b) nJ

d m

m

0) N

i400

1200

i000

800

600

400

2 O0 -

0


I I I I

. , I 4 ; % f!, (3

pin

,I. box


Fig. 25. Effect of cyclic loading R = () on the LET local stress: (a) stress range; (b) mean level.

supported by a rigorous model validation exercise. The non-uniform distribution of idealised elastic peak stress was in agreement with existing analytical, experimental and numerical data for generic threaded connections. The classical stress concentration factor was found to be inconstant and a decreasing function of nominal applied load in tension. Allowing for the non-linear relationship between applied stress and tooth notch peak stress, and for the effects of make-up preload, constant stress concentration factors of 3.5 and 5.5 were derived for the box and pin L E T positions,

(a) 7000 ]

d 6000 ]

C

5000 1 m

4000

ooo 1 . / /

i001~- I I I I

0 200 400 600 800

[] pin

,L box


Fig. 26(a). Caption on p. 29.

(b) 7000

6000

5000

4000

3000

2000

i000


I I I I

200 400 600 800

O pin

A box

29


Fig. 26. Effect of cyclic loading at R = 0 on the LET peak stress: (a) stress range; (b) mean level.

Table 3. Stresses at critical threads under static and cyclic loading

Cyclic pipe stress range (0-400 N mm -2)

Static pipe stress (400 N mm -2) Peak stress Local stress*

Location Peak stress Local stress Range Mean Range Mean

Pin LET I 5857 1073 I 1156 5279 388 879 Box LET 2350 686 ~ 1244 ~ 351

*The sum of linearised membrane and bending stress components.

respectively, reflecting in par t the greater thread root radius of the box implemented in this study. The stress state at the thread root is character ised by a varying max imum principal stress direct ion which produces a characterist ic lip feature on the failure surface, potent ial ly providing an impor tan t point o f reference for failure

investigations. U n d e r cyclic loading in pulsating tension (R = 0), the p re loaded joint analysed

p roduced local and peak stress ranges marked ly reduced at the critical pin thread, but oscillating about high mean levels. In contrast , the critical box thread exhibited larger local and peak stress ranges but with lower mean levels. Overal l fatigue pe r fo rmance of a p re loaded connect ion is thus a compromise be tween decreased stress ranges and

increased mean levels. Pre load directly affects the static stress distr ibution within the connect ion making

the pin L E T the critical failure site for static loading, but it also affects the local and peak mean stress levels and stress ranges arising f rom fatigue loading, making the box L E T the expected critical site for fat igue loading at R = 0.

Acknowledgements--Particular thanks are due to D. M. R. Bell of the Aberdeen University Computing Centre for his expertise and assistance with the University's computing facilities, and to I. Mackinnon of the Department of Engineering for reprographic services.


REFERENCES

L. K. A. Macdonald, Engng Failure Analysis 1, 91-i 17 (1994). 2. M. B. Kermani, Proceedings of the International Conference on Environment Assisted Fatigue, Sheffield

(1988). 3. M. B. Kermani, Fatigue of Large Diameter Threaded Connections (edited by W. D. Dover, P. J.

Haagensen, S. Dharmavasan and G. Glinka), Howard Lee, London (1988). 4. E. A. Patterson and B. Kenny, Fatigue of Large Diameter Threaded Connections (edited by W. D.

Dover, P. Haagensen, S. Dharmavasan and G. Glinka), Howard Lee, London (1988). 5. R. B. Heywood, Proc. IMechE 193,384-391 (1948). 6. H. Neuber, Kerbspannungslehr (2nd edn), pp. 159-163, Springer, Berlin (1958). 7. E. A. Patterson, Fatigue Fractures Engng Mater. Struct. 13, 59-81 (1990). 8. A. Lubinski, JPT February, 175-194 (1961). 9. H. M. Rollins, AAODC Rotary Drilling Conference, Dallas, TX (21 February 1966).

10. M. C. Moyer and B. A. Dale, JPT May, 982-986 (1984). 11. S. D. Hampton, Proceedings qf the SPE/IADC Drilling Conference, New Orleans, LA. pp. 177-189,

SPE 16072 (1987). 12. B. A. Dale, SPE Drilling Engng December, 356-362 (1988). 13. C. E. Stromeyer, lnst NavalArchit. Trans. No. 60, 112-122 (1918). 14. B. A. Dale, SPE Drilling Engng September, 215-222 (1989). 15. D. A. Topp, Fatigue of Large Diameter Threaded Connections (edited by W. D. Dover, P. Haagensen,

S. Dharmavasan and G. Glinka), pp 1-15, Howard Lee, London (1988). 16. J. L. Bretl and R. D. Cook, Int. J. Numer. Meth. Engng 14, 1359-1377 (1979). 17. W. D. Dover, P. Haagensen, S. Dharmavasan and G. Glinka, Fatigue of Large Diameter Threaded

Connections, Howard Lee, London (1988). 18. Hibbitt, Karlson and Sorensen, Inc. ABAQUS User's, Examples, Theory and Verification manuals,

HKS, Providence, RI (1992). 19. H. C. Rhee, Proceedings of the OMAE 90, Houston, TX, 18-23 February, pp. 293-297 (1990). 20. A. Tafreshi and W. D. Dover, Int. J. Fatigue 15,429-438 (1993). 21. API Recommended Practice 7G (RP 7G) Recommended Practice for Drill Stem/Design and Operating

Limits" (14th edn), American Petroleum Institute, Washington, DC (1990). 22. API Specification 7 (Spec 7) SpeciFication [br Rotary Drill Stem Elements (38th edn), American

Petroleum Institute, Washington, DC (1994). 23. PDA Engineering, P3/PATRAN Release Notes, Release 1.3-2. Costa Mesa, CA (1994). 24. E. Dragoni, J. Offshore Mech. Arctic Engng 116, 21-27 (1994). 25. NAFEMS, A Finite Element Primer (2rid reprint) (1991). 26. E. A. Patterson and B. Kenny, J. Strain Analysis 21, 17-23 (1987). 27. J. E. Smith, Fatigue of Large Diameter Threaded Connections (edited by W. D. Dover, P. Haagensen,

S. Dharmavasan and G. Glinka), pp. 161-188, Howard Lee, London (1988). 28. A. Newport and G. Glinka, J. Engng Mech. 17. 1257-1273 (1991). 29. J. F. Knott, Fundamentals of Fracture Mechanics, Butterworths, London (1981). 30. S. J. Maddox, Fatigue Strength of Welded Structures (2nd edn), Abington Publishing, Cambridge

(1991). 31. T. R. Gurney, Fatigue of Welded Structures (2nd edn), Cambridge University Press, Cambridge (1979).

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