Effect of mean axial load on axial fatigue life of spiral strands

Int. J. Fatigue Vol. 19, No. 1, pp. 1-11, 1997 Copyright © 1997. Published by Elsevier Science Limited Printed in Great Britain. All rights reserved 0142- I 123/97/$17.00+.00

ELSEVIER

PII: S0142-1123(96)00052-7

Effect of mean axial load on axial fatigue life of spiral strands Morteza Alani* and M o h a m m e d R a o o f * * t *School of Engineering, University of Derby, Kedleston Road, Derby DE22 1GB, UK; **Civil and Building Engineering Department, Loughborough University, Loughborough, Leics LE11 3TU, UK (Received 7 March 1996; revised 24 June 1996; accepted 25 June 1996) A previously reported theoretical model for estimating the axial fatigue life to first outer (or inner) wire fracture of spiral strands is used to throw some light on the effect of mean axial load on the axial fatigue life of large diameter steel helical strands. Using theoretical parametric studies as applied to three realistic types of 127 mm diameter spiral strand constructions with lay angles of 12°, 18° and 24 ° which cover the full manufacturing limits, the plausibility of using an equivalent stress range based on the Goodman and/or Gerber approaches for plotting the axial fatigue data is critically examined. Despite previous claims, it is theoretically demonstrated that the use of a Goodman and/or Gerber transformation does not lead to a significant improvement in the fit of the fatigue data to a straight line and despite the additional computational efforts, such approaches do not provide any practical advantages over the traditional approaches which present the axial fatigue data in terms of stress range vs cycles to failure. The paper also examines the effect of mean axial load on the endurance limit of spiral strands and uses numerical studies to show that (within the practical range) endurance limit increases with increasing levels of strand mean axial load. Copyright © 1997. Published by Elsevier Science Limited. (Keywords: wire ropes; s p i r a l strands; fatigue; axial load; cyclic loading)

INTRODUCTION The safety of the many deep-water platform concepts is, among other considerations, strongly dependent on the reliability of the anchoring systems, which should have a high level of integrity, and whose costs of installation and replacement are very high. Since the mid-1970s there has been a significant increase in the size of steel cables (i.e. spiral strands and/or wire ropes) being considered by the offshore oil industry in addition to bridging applications: 127ram dia spiral strands were used for the catenary moorings of the (Exxon) Lena Guyed Tower and 137 m m dia spiral strands were employed on the (Woodside) North Rankin A platform. Moreover, 164 m m spiral strands were used /'or the Dartford cable stayed bridge in the UK. An important consideration in the design of cable structures is the fatigue resistance of cables under cyclic loading. All structures undergo cyclic loading of some form, although the variations in load magnitude for some structures might be low enough to be ignored during design. Once fatigue is addressed, the number of variables makes the interpretation of the extensive work by manufacturers and technical societies rather difficult, + Author for correspondence

and divergence of opinion on various aspects of cable behaviour under even closely controlled laboratory conditions is not uncommon. The need for a stress analysis which takes the interwire contact forces and slip between wires into account is obvious, and it is this aspect of the problem which has been given much attention in recent years. In a fairly recent series of publications, the second author and his associates have put a substantial analytical and experimental effort into assessing contact forces and the associated relative displacements between wires, taking full account of frictional effects, in large diameter spiral strands and wire ropes, where a spiral strand is composed of a group of helically laid wires with a common axis. The term rope, on the other hand, is applied to an assembly of a number of helical strands in one or more helical layers over a central core which may in itself be made up of a fibre or an independent wire rope, with the choice being largely dependent on the use for which the rope is intended. The theoretical basis of the work has been fully discussed elsewhere in the context of the work of others, in particular Costello and his associates.l For the present purposes much attention will be devoted to the influence of mean axial load on the axial fatigue life of multi-layered spiral strands.

2

M. Alani and M. Raoof

BACKGROUND Among a number of detailed theoretical and experimental studies of various characteristics of spiral strands, Raoof 2'3 has addressed the axial fatigue behaviour of large diameter multi-layered spiral strands where, using results from a previously reported orthotropic sheet concept, 4'5 an insight is given into the pattern of normal (Hertzian) stresses throughout realistic strand constructions. Based on such information in conjunction with available axial fatigue data on single component wires, an analytical model is developed which is capable of predicting strand axial fatigue life from first principles. The so-developed model is capable of throwing some light into various cable characteristics of practical importance. The match between experimental data on some substantial spiral strands with diameters ranging from 39 to 127 mm and theoretical predictions has been shown to be very encouraging, e,3,6 The theory enables one to predict both internal and/or external first wire fractures which may either happen at the end terminations or in the free field (i.e. sufficiently away from the ends so that the end effects are minimal). Experimental constant axial fatigue loading on cables is generally in the form of a nominally sinusoidal stress perturbation with an amplitude S'a -- - 21( S. . . . . - - S m i n ) , superimposed on a steady mean axial stress Sin. The results relating to the number of fatigue cycles to failure, N, are usually presented in terms of two variables, for example, S.... - N (or S a - N, where the stress range S~ = Smax- Sm~,) curves for various values of the ratio R, which is defined as R=(S~,,/S .... ). In this context, rather than plot a separate curve for each value of R, Birkenmaier 7 used a single plot of S~ - N for all values of R, but kept the upper stress, S ...... constant throughout his tests which were conducted on parallel wire tendons. In connection with spiral strands, Hobbs and Ghavami 8 used two different Sm~, stresses of 10% and 30% of the nominal ultimate strength, S,, with rather arbitrary Sm and R values covering a large range of strand fatigue lives, and reported near independence of the fatigue life and the mean axial load. Their conclusion, however, is not substantiated by the experimental findings of Yeung and Walton 9 and Chaplin and Potts ~° who suggest that axial fatigue life of cables is not only a function of stress range but also (albeit to a lesser degree) the mean axial load. In the absence of any other information, Ref. (9) has suggested the use of a Goodman treatment of the fatigue data (through which 'least square' lines are drawn), in order to interpolate or extrapolate into regions for which no data exists. In view of space limitations a critical review of the literature on axial fatigue characteristics of steel cables is not presented here: some exhaustive recent surveys have recently been reported by Raoof3,1~ and Chaplin and Potts t° and based on their findings it has become obvious that the definition of cable fatigue 'failure' is, at least for the present, rather arbitrary, and the use of the fairly well established techniques of Goodman and/or Gerber for displaying axial fatigue data of plain specimens (as opposed to helically laid cables) under various mean loads in connection with rope fatigue analysis is not well understood. References (9) and (10) have, in the absence of

MBL

o ,...2

. . . . . . . .

J-

-

Smax

Seq

/

I I

Line of zero load

I I

MBL

Figure 1 Concept of 'equivalent' load range applied to fatigue assessment of wire ropes--alter Ref. (9)

any other more reliable technique, used equations of the fbrm Sa

S~q

S.

(I)

-

,u , {'mt",u in replotting some of the limited available data on spiral strand axial fatigue life, with a view to reducing some of the scatter in composite plots of the experimental results. In Equation (1), S~ is the range of applied axial stress (i.e. twice the amplitude) and Sm is the mean. S~q is an equivalent stress analogous to those defined by Goodman and Gerber: in Equation (l), Goodman used n = l and Gerber took ii = 2. Obviously, Equation (1) attempts to obtain a single curve for all possible combinations of Sm and S,,, for each given strand construction with the provision that the transformation is done to a range with a minimum load of zero, 9 Figure 1. Figure 2 presents some experimental substantiation of the concept for 38 mm 1 x 172 and 51 mm 1 x 139 spiral strands tested at different mean axial loads [2(L40% of maximum breaking load (MBL) and load ranges of 10-28% MBL] as reported by Yeung and Walton 9 who used Equation (1) with n = 1 for calculating the equivalent load range (expressed as a percentage of MBL). lOC 9C 8(1 7G ~ " 6C

-

o 38 mrn 1 xl72 strand o 51 m m 1 x 1 3 9 s t r a n c

N 5c ~ 40 ~ 30

~ 20

Oo o

o

°

~ 13 -q 6 5 10 5

i

t

i

106

107

108

L09

Endurance (cycles)

Figure2 Fatigue performance of 38 mm 1x 172 and 51 mm I x 139 spiral strands in tension fatigue--after Ref. (9)

Mean axial load and fatigue life As discussed by Chaplin, the tests at Reading University as presented by Chaplin and Potts, 1° indicate the benefits of such an approach for six strand ropes in tension-tension fatigue tests. In their work, the load range was expressed as a percentage of the actual breaking load (ABL) of the rope as they followed the findings of Fleming 12 who suggested that by using ABL (as opposed to MBL as given in, for example, the manufacturer's catalogue), improved correlations may be obtained when comparing results obtained from different ropes. Figures 3a and 3b present plots of load range [expressed as a percentage of the ultimate breaking load (UBL)] and the equivalent load range [with n in Equation (1) set equal to 1] vs cycles to failure for the tension-tension fatigue performance of a 19 mm six strand rope as reported by Chaplin and Potts: these figures are intended to illustrate the effect of plotting in terms of an equivalent load range to reduce spread caused by mean load. As noted from Figure 3b, the scatter in the fatigue results (when presented in terms of the equivalent load range) is found to somewhat increase with decreasing levels of the equivalent load range (i.e. increasing fatigue life). However, no explanation has been offered by Chaplin and Potts as to why this happens and one cannot (in the absence of, say, a sound theoretical model) ascertain as to whether (a) 100

3

this finding is due to the peculiar nature of the plotting technique or whether experimental uncertainties due to factor(s) such as the definition of fatigue failure, length or construction of the test specimens, etc., may be responsible for such scatter. Recently, Bridon Ropes made the construction details for three realistic types of 127 mm spiral strands available to the second author. In particular, three different levels of lay angles (12 °, 18 ° and 24 °) were used for designing these strand constructions with each strand having the same lay angle in all its layers and their other geometric factors (such as number and diameters of the helical wires) kept very nearly the same. As fully discussed elsewhere]3 J5 following extensive theoretical and experimental work, the lay angle is the primary factor which controls the strand overall structural characteristics with the other geometrical factors being of secondary importance. It should also be noted that the 12-24 ° range of presently adopted lay angles covers the full practical range of this parameter as currently used by cable manufacturers. The purpose of the present paper is to report results based on theoretical parametric studies which use these three types of spiral strand constructions in order to throw some light on the advantages (and/or disadvantages) of using Equation (1) (with n = l or 2) in plotting spiral strand axial fatigue data and make a theoretical attempt to examine as to whether (in the absence of any experimental uncertainties) the Goodman and/or Gerber type approaches may, indeed, prove of value in catering for the variations in the mean axial load under rope axial fatigue conditions. Moreover, the effect of mean axial load and lay angle on the magnitude of endurance limit will be examined on a theoretical basis. As a prerequisite to this, however, a description will be given of the salient features of the strand axial fatigue theoretical model as previously reported by Raoof. 23 THEORY

10

. . . . . . . . 10 4

'

.

.

.

.

.

.

.

1

.

l0 5

. . . . . . . .

l0 6

10 7

Cycles to failure

(b) 100

2 ÷

Using the orthotropic sheet model 4'5 it is now possible to obtain reliable estimates of interwire contact forces (and stresses) throughout multi-layered structural strands. Experimental observations suggest that individual wire failures are largely located over the trellis points of interlayer contact and it is now believed that this is as a result of high stress concentration factors in these locations. Once the maximum effective Von-Mises stress O"m,x, over trellis points of contact for a given strand mean axial load is calculated, the stress concentration factor, K, is defined as j6 --r O" max

K, -

(2)

.m

10

. . . . . . . . 10 4

" 10 5

. . . . . . . .

i 10 6

-

,

• '. . . . 10 7

Cycles to failure

Figure3

Tension-tension results for 19 m m six strand rope illustrating the effect of plotting in terms of an equivalent load range to reduce spread caused by mean load--after Ref. 10: (a) life plotted against actual load range; (b) life plotted against equivalent load range

where ~r' is the nominal axial stress in the wire (corresponding to the strand mean axial load), which may be calculated using the method developed by Raoof and Hobbs 5. Using the so-obtained values of K~ in conjunction with axial fatigue data on single wires, a theory has been developed which predicts the axial fatigue life of strands (under constant amplitude cyclic loading) from first principles. The model is based on the observation that for

4

M. Alani and M. Raoof

carbon steel wires the fatigue stress-number of cycles plot (S-N curve) possesses an endurance limit, S', below which no damage occurs. Traditionally the magnitude of S' is compared to the ultimate wire strength, S~,: tests on single galvanised wires suggest an approximate value of S' = 0.27 Su~t. The reduced magnitude of the endurance limit, Se, which takes interwire contact and fretting plus surface conditions and size effects etc. into account, may be defined as S~ = K~ K b S'

(3)

where Kb = (1/KD, and K, is a constant. The so-obtained values of the parameter S~, then, are used to produce the S-N curves for fatigue life to first outer (or inner) wire fractures in spiral strands using the S-N curves available in the literature for axial fatigue life of individual wires of a given grade. 2'3 Extensive experimental verification of the above theoretical model has been reported elsewhere, 2"3'6 and the match between large scale axial fatigue data for spiral strands with diameters ranging from 39-127 mm and theoretical predictions has been very encouraging. There is little point in repeating such correlations here. However, in order to demonstrate the role of the parameter Ka, perhaps it is worth presenting the correlations between theory and axial fatigue test data for a 51 mm 1 x 139 strand (with the test data obtained from Bridon Ropes) which is shown in Figure 4, where fatigue life is defined as the number of cycles to first outer layer wire fracture. In Figure 4, the scatter in the experimental data is covered by the K~ factor in the range 0.5 ~< K, ~< 1.0. All the initial wire failures in this strand occurred in the free field away from the epoxy resin end terminations. However, as discussed elsewhere, w for the end terminations to have no effect on the wire fractures remote from the ends, the minimum length of test specimens must be around 10 lay lengths with the wire fractures occurring within the central region which extends by 2.5 lay lengths on either side of the middle of the test specimen (i.e. within the central portion with a length of 5 lay lengths). It then follows that due to the total length of the tested 51 mm O.D. strands being significantly less than 10 lay lengths, even for the wire fractures away

40

- Theory 30.

* Experiment

E~ *

Ka = 1 . 0

0 5 '~ o

lO

51 mm O.D. strand

10 6

10 7

Life to first outer layer wire fracture (cycles) Figure 4 Comparison of theory and experimental fatigue data for 51 mm O.D. strand--after Ref. 2

from the ends, certain test data points in Figure 4 have been influenced by the detrimental end effects with the correlations between the theory and such test data suggesting K~=0.5 as an appropriate factor in the presence of end effects. Otherwise, for wire fractures which happen away from the ends and, in addition, are not influenced by end effects, one may assume Ka = 1.0. RESULTS

Tables 1-3 give the construction details for 127 mm dia spiral strands used in the theoretical parametric studies. The assumed Young's modulus for galvanized steel wires E = 2 0 0 k N m m -2, and the Poisson's ratio for the wire material u = 0.28. The UBL of the strands was assumed to be the same, equal to 13,510kN, while the tensile ultimate strength of the wire material Suit= 1 5 2 0 N m m -2. For the purposes of theoretical parametric studies, four values of strand mean axial strains S'~ = 0.001, 0.002, 0.002867, 0.004 were assumed which cover the usual practical working ranges for structural applications. Axial fatigue life was defined as the number of cycles to first wire fracture. Figure 5 presents theoretical plots of equivalent load range (expressed as a percentage of UBL) against axial fatigue life to first outer layer wire fracture, for all the three 127 mm O.D. spiral strand constructions with lay angles of c~= 12 °, 18.01 ° and 24 ° . The values of equivalent load range in these plots are calculated by using Equation (1) with an assumed value of n = 1. The assumed value of the parameter K, = 0.5 with the data presented for four different values of the strand mean axial strain S'I =0.001, 0.002, 0.002867 and 0.004. An examination of the plots in Figure 5 clearly suggests that for decreasing levels of the strand mean axial strains, S',, the individual plots of equivalent load range versus axial fatigue life (corresponding to a given strand lay angle) undergo a clockwise rotation with the combined plots covering a wide range of S'~ values (for given values of c~ and K,,) exhibiting increasingly wider scatter as the magnitude of equivalent load range is decreased. Figure 6, on the other hand, presents plots of load range (as a percentage of UBL) against the number of cycles to first outer layer wire fracture for the strand with a lay angle c~= 12 ° and Ka=0.5, where (similar to plots in Figure 5) for decreasing levels of the strand mean axial strains, S',, the individual plots of load range vs axial fatigue life (for a given lay angle) are also found to undergo a clockwise rotation. In other words, the endurance limit is found to increase with increasing values of the strand mean axial strain, S'.. This observation is consistent with the previous findings of Raoof, 2 and, indeed, as shown in Figure 7, the endurance limit for a given wire material and with fatigue life defined as number of cycles to the first wire fracture, depends on the type of construction, and, for a given strand, increases with increasing level of mean axial strain. This is because of the geometrically non-linear nature of the problem of wire flattening at the trellis points of contact between neighbouring wires of the various layers. As shown in Figure 8, because of the geometrically non-linear nature of the contact between crossed cylinders (wires), the magnitudes, of the maximum trellis Von-Mises contact stresses cr ......

Mean axial load and fatigue life Table 1

5

Construction details of the 127 mm O.D. spiral strand with c~ = 12 °

Layer

Number of wires

Lay direction

Wire diameter D (ram)

Lay angle (°)

1

56

RH

6.6

12.0

2 3 4 5 6 7 8

50 44 38 32 26 20 14 7+7 7

LH LH RH LH RH LH RH ---

6.6 6.6 6.6 6.5 6.5 6.5 6.6 4.0 and 5.2 5.2

12.0 12.0 12.0 12.0 12.0 12.0 12.1 8.57 and 8.03 4.92

1

--

7.1

--

Core

Table 2

Construction details of the 127 mm O.D. spiral strand with ~ = 18.01 °

Layer

Number of wires

Lay direction

Wire diameter D (mm)

Lay angle (degrees)

1 2 3 4 5 6 7 8

54 48 42 36 31 25 19 14 7+7 7

RH LH LH RH LH RH LH RH ---

6.55 6.55 6.55 6.55 6.55 6.55 6.55 6.30 3.90 and 5.10 5.25

I

--

7.00

18.01 18.01 18.01 18.01 18.01 18.01 18.01 18.01 13.07 and 12.20 7.62 --

Core

Table 3

Construction details of the 127 mm O.D. spiral strand with a = 24 °

Layer

Number of wires

Lay direction

Wire diameter D (ram)

Lay angle (°)

1 2 3 4 5 6 7 8

54 48 42 36 30 24 18 14 7+7 7 I

RH LH LH RH LH RH LH RH ----

6.4 6.4 6.5 6.5 6.6 6.6 6.8 6.1 3.9 and 5.1 5.25 7.00

24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 17.89 and 16.75 10.58 --

Core

for various trellis points of inter-layer contact throughout a 184ram dia spiral strand (with construction details given elsewhere TM) increase with increases in the magnitude of strand mean axial load, Tm (expressed as a percentage of maximum breaking load), in a nonlinear fashion with the rates of increase in the values of ~' .... decreasing with increases in Tin. It then follows that with the values of wire mean direct stresses ~' increasing linearly with increases in Tm, the magnitudes of the stress concentration factors Ks as defined by Equation (2) decrease with increases in Tin. With K, being a constant factor, then, the magnitudes of wire endurance limits in the presence of interwire contact, Se [as defined by Equation (3)] will increase with associated increases in the magnitude of externally applied strand mean axial load, Tm. The plots in Figure 7 cover the three strand constructions with lay angles of c~= 12 ° , 18.01 ° and 24 ° and Ka values of

0.5 and 1.0 and these figures show that the endurance limit expressed as a percentage of UBL decreases with increasing levels of lay angle. The plots such as those in Figure 7 are of particular importance in offshore applications, where the small-amplitude forces are the ones with the highest number of occurrences and a knowledge of small-amplitude/long life behaviour is necessary for these applications. As explained previously, the theoretical model is also capable of predicting the number of cycles to first inner wire fracture within any layer of a spiral strand and, indeed, as shown elsewhere 2,j8 inner wire fractures are theoretically shown to happen earlier than the outer layer first wire fracture. Figure 9 present plots of endurance limit vs mean axial load (expressed as a percentage of UBL) for the strands with three levels of lay angle c~= 12 ° , 18.01 ° and 24 ° and for two values of K, = 0.5 and 1.0 with the fatigue life defined

6

M. Alani and M. Raoof 1000 o 0.001 = S'I

(a) I0-

o 0.002

For the most outer layer

o 0.002867 100

0.004

o,za O o

~

o

Oo o

o

a

o

n=l

10

o

o

6o

o

= 12° U.B.L. = 13510 kN Ka = 0.5 Suit = 1520 N/ram 2 ~ ~

1 1E+04

-'D. ,-2 8r~

m

-- 4

li+05

li+06

li+07

IE+08

~

Alpha = 24.00 ° ~ = 18.01 ° = 12.00 ° Ka = 1.0

Fatigue life to first outer layer wire fracture ( c y c l e s ) ,..j 100 ,-n

o

UlUU~ =

S',

,%

o 0.002

;0

'~

6¢~°°°ooOO o

° ~ l0

o

~ 0.004 o o

n=l = 18.0t ° U.B.L. = 13510 kN Ka = 0.5 Suit = 1520 N/ram 2

..q

~ o

60

Mean axial load (% of U . B . L . )

o 0.002867 e-

• ,~

o

g

(b)

o

o

20-

For the most outer layer

1 1E+06 IE+07 1E+08 1E+04 1E+05 Fatigue life to first outer layer wire fracture ( c y c l e s ) /

~100.

I0.

o 0.001 = S' 1 o 0.002

ao

o 0.002867

2~%~o ~:ac~° 10

~ 0.004

0

n=l = 24 ° U.B.L. = 13510 kN Ka = 0 . 5 Suit = 1520 N / m m 2

g

A

o

0

o 0.002 ,o

o 0.002867 a 0.004

too 0

©

1

[]

12 °

U . B . L . = 13510 kN K a = 0.5 Suit = 1520 N/mm 2

1E+05

1E+06

= 24.0(Y ~- - ~ , - - = 18.01 ° * = 12.00 ° Ka = 1.0

;o

3'o

20

so

60

Mean axial load (% of U . B . L . )

[] 0.001 = S't

~=

a

o o

100 -

10-

h

og

0

Figure 5 Theoretical plots of equivalent load range vs fatigue life with n = 1 a n d K ~ = 0 . 5 for the three spiral strand constructitms for 0.001 ~ S'; ~< 0 . 0 0 4

o

p

0

1 1E+06 IE+07 IE+08 1E+04 1E+05 Fatigue life to first outer layer wire fracture ( c y c l e s )

p~

l

L~

0000

0

6 no

IE+07

IE+08

Fatigue life to first outer layer wire fracture ( c y c l e s ) Figure 6 Theoretical plots of load range vs fatigue life tbr the spiral strand with lay angle a = 12 ° f o r 0 . 0 0 1 <~ S'~ ~< 0 . 0 0 4

as the number of cycles to first wire fracture in the innermost layer. Similar to the plots in Figure 7, for a given lay angle (i.e. strand construction) and grade of wire, endurance limit is found to increase with increasing level of strand mean axial strain with

Figure 7 Theoretical plots of endurance limit vs mean axial l o a d f o r the three spiral strand constructions with fatigue failure criterion defined as number of cycles to first outer wire fracture: (a) K:, = 0.5: (b) K:, = 1.0

increases in the magnitude of the lay angle leading to a decrease in the magnitude of endurance limit. A comparison of the plots in Figure 5 and the experimental results of Chaplin and Potts m as reproduced in Figure 3b (which relate to a 19 mm six strand rope) is instructive: in Figure 3b the test data presented in terms of an equivalent load range (calculated on the basis of a simple Goodman transformation) also show significantly more scatter for decreasing values of the equivalent load range, which is in line with the theoretical predictions. The question, however, is raised as to whether the use of a Goodman and/or Gerber transformation does indeed lead to a significant improvement in the fit of the fatigue data to a straight line as previously claimed by Chaplin and Potts m based on a comparison between the plots of the raw test data in terms of the actual load range as reproduced in Figure 3a and their Goodman transformation procedure which has led to the plots reproduced in Figure 3b. Figure lOa presents the theoretical plots of load range vs number of cycles to first outer layer wire fracture for the three spiral strand constructions covering a wide range of strand mean axial strains 0.001 ~< S'1 ~< 0.004, based on an assumed value of K,, = 1.0, with the least square straight line fit to data

Mean axial load and fatigue life

3400 E Z 3000 E

~

2600

~

7

(a) I 0

10& 11

Alpha = 24.00 ° = 18.01 ° = 12.00°

"-2". ,2

9&10

Ka = 0.5

7&8 6&7 4&5

3 220o

.g

e-

;~ 1800

/ /

//

/

/

/

/ 1

& 2 = Trellis contact patch between

layers E = 1400 E

0

l0

20

30

40

5()

60

Mean axial load (% of U.B.L.) /

184 mm O.D. strand

(outer & inner)

(b) 1b

2b

30.

. 40.

. 50

1o

Cable mean axial load (% M.B.L.) ,-d t~

Figure

8 Variations of maximum trellis Von-Mises contact stresses, ~r',,~a~, with changes in the magnitude of strand mean axial load, T,,, for various interlayer trellis points of contact in a 184 mm dia strand--after Ref. 18

for each strand also included in the plots. Similar plots of equivalent load range versus fatigue life based on a Ka value of 1.0 and assumed values of n = 1 and 2 in Equation (1) are presented in Figures lOb and 10c, respectively. Visual examination of Figures lOa-c strongly suggests that the use of the concept of equivalent load range with n = 1 or 2 does not result in any reduction in the scatter of theoretical data points about the fitted straight line. This is, perhaps, best demonstrated by an examination of the values of the correlation factors, R, for the fitted lines as presented in Table 4: unlike previous claims in the literature the values of R in the last column of Table 4 show less degrees of scatter in the theoretical data points (i.e. a better correlation) for the plots which are based on the Table 4

Lay angle (°) t2.00 18.01 24.00 12.00 18.01 24.00 12.00 18.01 24.00 12.00 18.01 24.00 12.00 18.01 24.00

e-

Alpha = 24.00 ° = 18.01 ° • = 12.00° . .L Ka = 1.0 10

20

30

4'0

50

60

Mean axial load (% of U.B.L.) Figure 9 Theoretical plots of endurance limit vs mean axial load for the three spiral strand constructions with fatigue failure defined as number of cycles to first wire fracture in the innermost layer: (a) K~ = 0.5; (b) K~, = 1.0

Details of theoretical data for all three spiral strand constructions based on K~ = 1.0 Location of first wire fracture used in the definition for axial fatigue life

K,

Type of assumed cyclic axial load (% UBL)

1.0

Load range

--

Outer layer

1.0

Equivalent load range

1.0

Outer layer

1.0

Equivalent load range

2.0

Outer layer

1.0

Load range

--

Innermost layer

1.0

Equivalent load range

2.0

Innermost layer

Correlation factor R 117.24 108.32 95.081 172.61 149.66 120.28 131.54 117.85 99.803 124.01 112.46 102.47 139.91 121.66 106.50

-15.170 -14.691 -13.147 -22.559 -20.519 -16.747 -17.084 -16.046 -13.830 -17.445 -16.152 -14.901 -19.738 -17.511 -15.500

0.955 0.962 0.964 0.888 0.94 I 0.941 0.948 0.965 0.960 0.947 0.943 0.941 0.947 0.945 0.940

M. Alani and M. Raoof

8 (a)

(b)

100 •

0.001

= S'

0.002 0.002867 0.004

,.d

o ,.d

1oo.

= S'1 0.002 0.002867 O.004

• 0.001

,..A

n=l ~ = 12,00 ° U.B.L. = 13510 KN Ka = 1.0 Suit = 1520 N/mm 2 "" 10 ........ , ........ , ........ ' 108 104 105 106 107 F a t i g u e life to first outer l a y e r w i r e fracture (cycles)

= 12.00 ° . U.B.L.=13510KN " "~ Ka = 1.0 Suit = 1520 N/ram 2 10 ........ , ........ , ....... ~ ...... 104 105 106 107 108 F a t i g u e life to first outer l a y e r w i r e fracture ( c y c l e s )

~100

100" ,.d

~10 o o .1

(x = 18.01 ° U.B.L. = 13510 KN Ka = 1.0 Suit = 1520 N/mm 2

n=l c~ = 18.01 ° U.B.L. = 13510 KN Ka = 1.0

• o.ool =s') 0.002 0.002867 0.004

, o.ool =s'~ 00o2 0.002867 0.004

Suit = 1520 N/ram 2

1

104 105 106 107 108 F a t i g u e life to first outer l a y e r w i r e fracture (cycles)

104 105 106 l07 108 F a t i g u e life to first outer l a y e r w i r e fracture (cycles)

100 100

• •

"

~

0.001

=

S't

~

0.002 0.002867 0.004

0.001

= S'~

0.002 .002867 0.004

..d

E 1o 10

2 0 ,.d

= 24.00 ° U.B.L.= 13510 KN Ka = 1.0 Suit = 1520 N/mm 2 1

104 105 106 107 108 F a t i g u e life to first o u t e r l a y e r w i r e f r a c t u r e (cycles)

n=l = 24.00 ° U.B.L. = 13510 KN Ka = 1.0 Suit = 1520 N/mm 2

g

•

I

108 104 105 106 107 F a t i g u e life to first outer l a y e r wire fracture (cycles)

F i g u r e 10 S - N plots for the three spiral strand constructions covering 0.001 <~ S'~ ~< 0.004 and K~, = 1.0, based on different approaches with fatigue life defined as n u m b e r of cycles to first outer layer wire fracture: (a) cyclic load expressed as load range; (b) cyclic load expressed as e q u i v a l e n t load range with n = 1; (c) cyclic load expressed as e q u i v a l e n t load range with n = 2 (continued opposite)

load range (as opposed to the calculated values of the equivalent load range based on either n = 1 or 2). Even if one changes the criterion for axial fatigue failure to that corresponding to the number of cycles to first wire fracture in the innermost layer, data in Table 4 which is based on n = 2 and/or the plots in Figures l l a and b still do not show any reduction in the scatter of data points by using such an approach and, indeed, the correlation factors, R based on the plots associated with the load range concept are found to be higher in magnitude than those of the plots associated with the concept of equivalent load range. Similar data to those in Table 4, but with K, = 0.5 for the three strand constructions, are given in Table 5 with fatigue failure criterion defined as number of cycles to first outer layer wire fracture. Yet again, the correlation factors, R, based on the use of the load

range are found to be as useful (for all practical purposes) as those based on the concept of equivalent load range with either n = 1 or 2. Finally, the fitted straight lines to all the theoretical data points are expressed as y = ax + b

(4)

The above equation corresponds to an individual strand construction, with each line covering the full assumed range of S'~, with the values of the constants a and b for each individual fitted line also given in T a b l e s 4 and 5. CONCLUSIONS A theoretical model, which has been backed by extensive large scale test data as already reported elsewhere,

Mean axial load and fatigue life (c)

100 • 0 . 0 0 1 = S'

0.002 0.002867 0.004

2

n=2 • ot = 12.00 ° a -,,,,,~ U.B.L. = 13510 KN ° "-,. Ka = 1.0 • Suit = 1520 N/mm 2 I0 ........ , ........ , ....... "It ........ 104 105 106 107 108 F a t i g u e life to first outer l a y e r w i r e fracture (cycles)

~'100

10 n=2

= 18.01 ° U.B.L.= 13510KN Ka = 1.0 Suit = 1520 N/mm 2 |

........ 104

, 105

........

• O.OOl=S'~ 0.0o2 0.002867 0.0o4 , 106

........

w 107

- ....... 108

F a t i g u e life to first outer l a y e r w i r e fracture ( c y c l e s ) 100 •

,1 •~

S' l 0.002 0.002867 0.004 0.001 =

"" 10

2 n=2 a = 24.00 ° U.B.L. = 13510 KN Ka = 1.0 Suit = 1520 N/mm 2 1

104

105

106

107

10

Table 5

has been used to throw some light on the effects of mean axial load on the axial fatigue life of realistic multilayered structural strands. Numerical studies have been carried out on three different 127 mm O.D. spiral strands covering the full manufacturing limits of the lay angles, a, within the range 1 2 ° ~< ce ~ 24 ° with other geometrical properties of these realistic strand constructions such as their number of wires and their diameter in individual layers kept very nearly the same. Theoretical parametric studies suggest that for a given wire material, due to the geometrically nonlinear nature of the wire flattening at the critical trellis points of interlayer contacts, endurance limit (for a given lay angle) increases with increasing levels of strand mean axial load within the practical range, with different magnitudes of lay angle exhibiting significantly different levels of endurance limit. Indeed, increasing the lay angle is theoretically shown to lead to a reduction in the magnitude of endurance limit for both the outer (or inner) wires in a spiral strand. This finding may have significant practical implications in, say, offshore floating platform applications where the small-amplitude forces are the ones with the highest number of occurrences and small-amplitude/long life axial fatigue behaviour is of importance. The paper also presents a critical theoretical examination of certain previous claims in available literature as regards the potential advantages of using Goodman and/or Gerber transformations to cater for the variations in the mean axial load under axial fatigue conditions. Despite previous claims, it is theoretically demonstrated that the use of the concept of equivalent load range based on Goodman and/or Gerber transformations does not lead to a significant improvement in the fit of the fatigue data to a straight line. Indeed, it is shown that the traditional use of the stress range in plotting the fatigue S-N curves for a wide range of mean axial loads (for a given spiral strand construction) leads to at least the same degree of scatter as those plots based on Goodman and/or Gerber transformations.

108

F a t i g u e life to first outer l a y e r w i r e fracture ( c y c l e s ) Figure

9

ACKNOWLEDGEMENT The second author acknowledges the long-standing and friendly cooperation of Bridon Ropes Personnel, Doncaster, UK.

Continued

Details of theoretical fatigue data for all three spiral strand constructions based on Kd = 0.5

K~

Type o f a s s u m e d cyclic axial load (% U B L )

n

Location of first wire fracture used in the definition for axial fatigue life

12.00 18.01 24.00

0.50

L o a d range

--

Outer layer

122.59 111.83 101.15

-17.097 -16.005 -14.710

0.953 0.948 0.940

12.00 18.01 24.00

0.50

E q u i v a l e n t load range

2

Outer layer

138.31 121.07 105.16

-19.349 -17.368 -15.306

0.952 0.950 0.940

12.00 18.01 24.00

0.50

E q u i v a l e n t load range

I

Outer layer

135.497 124.539 128.070

-19.008 -18.006 -18.136

0.964 0.946 0.963

L a y angle (°)

Correlation factor R

M. Alani and M. Raoof

10 (a) 100

(b) •

0.001

S' l

0.002 0.002867 0.004

~

,.d

=

100---'2 -1

10.

= S'1 0.002 0.002867 0.004

• 0.001

'~

10 n=2 ~ = 12.00 ° U.B.L. = 13510 KN Ka = 1.0 Sult = 1520 N/mm2

c~ = 12.00 ° U.B.L. = 13510 KN Ka = 1 . 0 Suit = 1520 N/mm 2

o

,.d

1 1

104 105 106 107 108 Fatigue life to first i n n e r layer wire fracture (cycles)

104 105 106 107 108 Fatigue life to first tnner layer wire fracture (cycles)

100 ~

lO0

= S' 1 0.002 0.002867 0.004

•

•

0.001

~

r~

• 0.001

,-d

~lO

= S'1~

0.002 0.002867 0.004

~ 10 o et = 18.01 ° U.B.L. = 13510 KN Ka = 1.0 Suit = 1520 N/mm 2

o

n=2 c~ = 18.01 ° U.B.L. = 13510 KN Ka = 1,0 Suit = 1520 N/ram2

1

104 105 106 107 108 Fatigue life to first ~nner layer wire fracture (cycles)

1

104 105 106 107 108 Fatigue life to first inner layer wire fracture (cycles)

100. •

"

~

100

0.001 = S' 1 0.002 0.002867 0.004

,.d

•

0.001 = S' I 0.002 0.002867

0.004

~a 10"

e~

*~ lO e a = 24.00 ° U.B.L. = 13510 KN Ka = 1.0 Sult = 1520 N/mm2

O

1

,

........

,

. . . . . . .

........

,

" °

=

~

U.B.L. = 13510 KN Ka = 1.0 Suit = 1520 N/mm2

• ,

n= 2 c~ = 24.00

.....

104 105 106 107 108 Fatigue life to first t n n e r layer wire fracture (cycles)

"

1

10 4

10 5

10 6

10 7

10 8

Fatigue life to first inner layer wire fracture (cycles) Figure 11 S-N plots for the three spiral strand constructions covering 0.001 <~ S~' <~ 0.004 and K, = 1.0, based on different approaches with fatigue life defined as number of cycles to first wire fracture in the innermost layer: (a) cyclic load expressed as load range; (b) cyclic load expressed as equivalent load range with n = 2

REFERENCES 1 2 3

4

5

Costello, G.A. 'Theory of Wire Rope', Springer-Verlag, New York, 1990 Raoof, M. J. Engng Mech. ASCE 1990, 116, 2083-2099 Raoof, M. "Axial fatigue life prediction of structural cables from first principles', Proc. The Institution of Civil Engineers, Part II, Vol. 89, March, 1991, pp. 19-38 Hobbs, R.E. and Raoof, M. Interwire slippage and fatigue prediction in stranded cables for TLP tethers. 'Behaviour of Offshore Structures' (Eds Chryssostomidis, C. and Conner, J.J.), Hemisphere Publishing/McGraw-Hill, New York, Vol. 2, 1982, pp. 77-99 (Proc. 3rd Int. Conf. Behav. Offshore Structures, 1982, M.I.T., Cambridge, MA, USA) Raoof, M. and Hobbs, R . E . J . Engng Mech. ASCE 1988, 114,

8 9

10

11

12

1166-1182

6

7

Alani, A. and Raoof, M. 'Axial fatigue characteristics of large diameter spiral strands'. Proc. 5th Int. Offshore Polar Engng Conf. (Eds J.S. Chung, M. Sayed, R.E. Hobbs and D.R. Yoerger), The Hague, The Netherlands, Vol. II, June, 1995, pp. 260-265 Birkenmaier, M. 'Fatigue resistant tendons for cable-stayed construction'. IABSE Proc. P-30/80 1980, pp. 65-79

13 14 15

Hobbs, R.E. and Ghavami, K. Int. J. Fatigue 1982, 69-72 Yeung, L.C.T. and Walton, J.M. 'Accelerated Block Tension Fatigue Testing of Wire Ropes for Offshore Use', Organisation Internationale pour l'Etude de I'Endurance des Cables (OIPEEC), Round Table Conference, National Engineering Laboratory, Glasgow, June, 1985, pp. 2.5.1-2.5.14 Chaplin, C.R. and Potts, A.E. 'Wire Rope Offshore--A Critical Review of Wire Rope Endurance Research Affecting Offshore Applications', University of Reading Research Report for Department of Energy, HMSO Publication OTH91341, 1991 Raoof, M. 'A Critical Review of Draft API Recommended Practice 2FP1 Regarding Fatigue Life Estimation of Moorings', Proc. llth Int. Conf. Offshore Mech. Arctic Engng, ASME, Calgary, Canada, Vol. III, Part B, June, 1992, pp. 521-532 Fleming, J.F. Fatigue of Cables. Research Report No. SE TEC CE 74-079 for the American Iron and Steel Institute, Project No. 1201-311, School of Engineering, University of Pittsburgh, June, 1974 Raoof, M. J. Strain Analysis, Inst. Mech. Engineers, 1991, 26(3), 165-174 Raoof, M. and Huang, Y.-P. J. Engng Mech. ASCE, 1992, 118, 2335-2351 Raoof, M. 'Effect of Lay Angle on Various Characteristics of

Mean axial load and fatigue life

16 17 18

Spiral Strands', Proc. 6th Int. Offshore Polar Engng Conf. (Eds J.S. Chung, M. Sayed, R.E. Hobbs and D.R. Yoerger), Los Angeles, CA, USA, Vol. IL May, 1996, pp. 189-196 Knapp, R.H. and Chiu, E.Y.C.J. Energy Resources Techn., ASME, 1988, 110 (March) 12-18 Raoof, M. and Hobbs, R.E. Int. J. Fatigue 1994, 16, 493-501 Raoof, M. 'Cable Fatigue Prediction in Offshore Applications', Proc. lOth Int. Conf. Offshore Mech. Arctic Engng, ASME, Vol. III, part B, Stavanger, Norway, June, 1991, pp. 403411

R Sa

Correlation factor Range of applied axial stress

Smin Smax

Minimum axial stress Maximum axial stress Mean axial stress Nominal ultimate stress Endurance limit Reduced endurance limit for the wire material taking interwire contact into account Strand mean axial strain Equivalent load range Wire ultimate strength Strand mean axial load Poisson's ratio for wire material Mean wire nominal axial stress Maximum Von-Mises contact stress

( = Sma x - Smin)

Sm S~ S' So

NOMENCLATURE E

K, K. N n

Steel Yo_ung's modulus --p p :

O" max/O"

SP i Seq SUIt

A constant (0.5 ~< Ka ~< 1.0)

Tm

= 1/K~

p

Number of cycles to failure Goodman (n = 1) or Gerber constant (n = 2)

O'tmax

\,

11