On the design of integral buckle arrestors for offshore pipelines

Applied Ocean Research ELSEVIER

Applied Ocean Research 20 (1998) 9% 104

On the design of integral buckle arrestors for offshore pipelines S. Kyriakides, Research Center for Mechanics

T.-D. Park, T.A. Netto

of Solids, Structures and Materials,

The University of Texas at Austin, WRW I IO, Austin, TX 78712, USA

Abstract A common method of limiting the extent of damage induced by a propagating buckle to a deep water pipeline is to periodically install along the line thick rings welded between adjacent strings of the pipe (integral buckle arrestor). The rings locally increase the resistance to

collapse and, when properly designed, arrest an incoming buckle. The effectiveness of this type of local reinforcement as a buckle arrestor was studied through a series of full-scale experiments and through a numerical model. The model was first proven to be capable of accurately simulating the quasi-static crossing of such arrestors by a buckle and was then used to study the arresting efficiency of this device as a function of the pipe and arrestor geometric and material parameters. This paper briefly summarizes these results which are subsequently used to establish some bounds for the arrestor thickness and length and to develop empirical design formulae for such devices. 0 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction A major consideration in the design of deep water pipelines is collapse under the ambient external pressure. Given the pipe diameter and operating depth, proven models can be used to select the appropriate pipe wall thickness, yield stress and geometric tolerances to avoid collapse [l-3]. Unfortunately, this is not always sufficient because offdesign events such as impact of the line by a foreign body [5,4], local buckles induced by excessive bending [6,7] etc., can locally reduce the integrity of a pipeline and induce local collapse. This, in turn, can initiate a buckle which propagates at high speed and has the potential to destroy the whole structure. The propagation pressure (Pp), i.e. the lowest pressure at which such a buckle will propagate, can be as low as 15% of its coZlupse pressure (Pco) (see Kyriakides [8], Dyau and Kyriakides [9]). The pipe wall thickness could, of course, be increased so that Pp is higher than the ambient pressure ensuring that any local collapse does not propagate. This, however, may not be feasible due to prohibitive increases in the cost of the material and of the installation process. A more attractive alternative is the periodic placement of buckle arrestors along the line such that collapse, should it occur, is limited to the length of pipe between two arrestors. The arrestor spacing is usually decided by practical considerations particular to the project (e.g. repair procedures), but spacing of 60-180 m has been typical to date. Effective buckle arrestors are required to stop a buckle propagating at the pressure of the maximum water depth of 0141-l 187/98/$19.00 0 1998 Elsevier Science Ltd. All rights reserved PII SO141-1187(98)00007-S

the pipeline. They typically take the form of thick-walled rings which are either threaded on to the pipe (slip-on arrestor [lo-121) clamped on to it, wound on to it (spiral arrestor [ 13]), or threaded on to the pipe and welded (welded [10,14]). The integral arrestor consists of a ring with the same internal diameter but thicker than the pipe, welded between adjacent strings of pipe (see Fig. 1). It has no limits of performance and, thus, it is preferred for pipelines installed in moderately deep and deep waters (e.g. Bullwinkle, Auger, Mars projects in the Gulf of Mexico). The factors governing the performance of integral arrestors have recently been studied through combined experimental and analytical efforts [ 151. The problem can be summarized as follows. Given the geometric and material parameters of a pipeline, select the arrestor length, wall thickness, steel grade and profile which will arrest a propagating buckle initiated at the maximum project water depth. A set of full scale experiments were performed in which the arresting capacity of integral arrestors was established and the mechanisms governing their performance were identified. A finite element model of the engagement of an integral arrestor by an incoming propagating buckle was developed in parallel with the experimental work. The model was first validated by simulating numerically the physical experiments performed and was subsequently used to perform an extensive parametric study of the arrestor performance. These results are briefly summarized here and then used to establish some important bounds for the arresting efficiency and to develop empirical design formulae for such devices.

96

S. Kyriakides et al. / Applied Ocean Research 20 (1998) 95-104

‘t Fig. 1. Geometry

2. Experiments

th

of integral buckle arrestor used in the experiments.

and results Average Flowrate = 0.34 gallmin

Fifteen full scale experiments were performed under carefully controlled conditions, in which the arresting capacity of integral buckle arrestors of various dimensions was established. The experiments involved 4.5 in diameter seamless pipes with D/t values in the range of 2 l-22. The nominal steel grade was X-52 but the actual yield stresses ranged from 67 to 94 ksi (462-648 MPa). Arrestors of various lengths (L,) and wall thicknesses (h > t) were machined from two thicker seamless pipes with dimensions of 5 in O.D. X 0.5 in wall (127 X 13 mm) (Al) and 5.5 in O.D. X 0.75 in wall (140 X 19 mm) (A2). Their respective yield stresses were 68 and 44.8 ksi (469 and 309 MPa) (the ultimate stress was approximately 80 ksi (552 MPa) for both). The arrestor I.D. was made to match that of the welded pipes. Transition sections 0.5 in (13 mm) long with the same wall thickness as that of the pipes were machined at each end. These terminated into a radiused transition zone as shown in Fig. 1. The pipes and arrestors were welded as described in Park and Kyriakides [ 151. The hardness of the girth welds was made to be the same as that of the pipe and the arrestor. The triple pass slow welding used reduced the width of the heat affected regions in the pipe. These characteristics are necessary in order to avoid cracking of the welds when they are deformed by a propagating buckle.

Fig. 3. Pressure history recorded in experiment

involving arrestors 9 and 11.

The experiments were conducted inside a 4 m long pressure vessel with a pressure capacity of 9000 psi (620 bar). This allowed testing an assembly consisting of three pipe sections and two arrestors arranged as shown in Figs 2 and 3. One of the end pipe sections was pre-dented in order to initiate collapse at the desired location. The assembly was sealed, placed inside the vessel and pressurized with water (nearly volume controlled pressurization). The pressure was monitored with calibrated electrical pressure transducers. Fig. 3 shows the pressure-time history of a typical experiment and a schematic sequence of specimen collapse configurations. In this case, both arrestors were 5.623 in Pressure Gage

I-

Time (min)

Pressure

Seals

In

LL

Water Pump

Fig. 2. Schematic

of the high pressure test facility used

in

the

experiments.

S. Kyriakides et al. / Applied Ocean Research 20 (1998) 95-104

91

Interestingly, the major axis of the buckle developed in the third section of pipe was at 90” to the major axis of the collapsed pipe upstream (buckle &@). The experiment was terminated with the third section only partially collapsed. 0.6

.

* 8

0.4 .

2.1. Arrestor efficiencies

Q Q

0.2 7 1

o

Experiments

l

Analysis

2

The most appropriate measure of the effectiveness of buckle arrestors is the arresting eficiency, 7, defined in Kyriakides and Babcock [ 1 l]

3

-

PY-PD

h/t

P co -pP

1.0

“rl pe2.1

%s

0.6

0

i

p = 2.10

t

+ 8

0.6

8-s

0.4

0.2

9

b

0.5

1.0

0

Experiments

l

Analysis

1.5

-

2.0

:

L./D

Fig. 4. (a) Arrestor efficiency versus arrestor length (experiments and predictions). (b) Arrestor efficiency versus arrestor thickness (experiments and predictions).

(142.8 mm) long and had respective wall thicknesses of 0.419 in (10.64mm) and 0.557 in (14.15 mm). The first pressure peak (2506 psi, 172.8 bar) corresponds to the collapse of the dented section of pipe. The pressure valley that follows is associated with first contact of the collapsing walls and the subsequent brief pressure plateau of 945 psi (65.2 bar) represents steady-state, quasi-static propagation of the buckle. Arrestor 9 was engaged by the buckle approximately 4 min into the experiment. The buckle was arrested and the pressure started rising. As the pressure increased, the arrestor deformed by ovalizing in the same sense as the collapsed pipe. At a pressure of 3259 psi (224.8 bar) the arrestor had deformed sufficiently to allow enough ovalization through to the second section of pipe for it to collapse. The maximum pressure achieved is defined as the crossover pressure (Px) of this specific combination of pipes and arrestor. The buckle resumed steady-state propagation in the second section of pipe and engaged arrestor 11 at approximately the 9 min mark. At this point the pressure started rising once more and the sequence of events described for arrestor 9 was repeated. This arrestor had a thicker wall and, as a result, a significantly high pressure was required to cause the buckle to cross over (Px = 4726 psi, 326 bar). The crossover again resulted in a sudden pressure drop.

L, OI7J’l.

(1)

That is, a maximum effectiveness arrestor (7 = 1) is required to resist penetration for all pressures between the propagation pressure (Pp) of the pipe and its collapse pressure (PCo). By contrast, a crossover pressure equal to the propagation pressure of the pipe implies that r] = 0. Two major series of arrestor experiments were conducted using the procedure described above. The first series involved six arrestors machined from arrestor pipe Al. They had fixed thicknesses h = 2.lt and lengths which varied from 0.5D I L, I 2.00. Their arresting efficiencies are plotted against LJD in Fig. 4(a). The propagation pressure used to calculate 7 was the value measured experimentally. The collapse pressure was, by necessity, a calculated value (BEPTICO) using the measured geometric and material parameters of the section of pipe downstream of the arrestor @co). Clearly, the arresting efficiency increases with arrestor length but the relationship is nonlinear. Indeed, as the arrestor length increases, the efficiency tends to an asymptotic value which is lower than the collapse pressure of the pipe. That is, efficiency of 1.0 is not achievable for this particular value of arrestor thickness. The second series of experiments involved seven arrestors of fixed length L, = 1.250 and thicknesses that varied from 1.7t 5 h 5 3.3t. These were machined from arrestor pipe A2. The measured arresting efficiencies are plotted against h/t in Fig. 4(b). The efficiency is seen to be somewhat nonlinearly related to the arrestor thickness. In this case the arrestor with the maximum thickness achieved an arresting efficiency of 1.0. (Numerical values of the variables, of Px and 17can be found in Tables 1 and 4 in Park and Kyriakides [ 151). 2.2. Modes of buckle crossover Buckles were found to cross the arrestors through two modes which depended on the arrestor wall thickness and length. Thinner arrestors exhibited the jluttening mode described above and illustrated in Fig. 5(a). Higher efficiency arrestors exhibited a crossover mode which resulted in flipping the ovalization induced to the downstream pipe by 90”. As a result, the mode of collapse of the downstream pipe was orthogonal to the mode of collapse of the upstream

98

S. Kyiakides

et al. / Applied

Ocean

Research

20

(I 998)

95-104

(a)

W Fig. 5. (a) Example of flattening

mode of crossover.

Fig. 6. Geometry

of pipe/arrestor

(b) Example of flipping mode of crossover

assembly

analyzed

S. Kyriakides

et al. / Applied

t

D - = 22.3

0.8

t

0.6

0.4

0.2

0 (a)

3.1. Simulation

Fig. 7. (a) Simulated pressure-change in volume response of crossover of arrestor 3. (b) Sequence of deformed configurations corresponding to points in (a).

pipe as shown in Fig. 5(b). This will be called the flipping mode of crossover (more on the mechanism of this mode can be found in Park and Kyriakides [ 151). The two modes are identified in Fig. 4 as 8 and @. We observe that for the group of arrestors in Fig. 4(a), those with efficiencies of 0.688 and higher exhibited the flipped mode of crossover; arrestors in Fig. 4(b) with efficiency of 0.705 and higher also had this mode of crossover.

3. Analysis A finite

element

model

99

framework of ABAQUS) which can simulate numerically the crossing of an integral arrestor by a buckle propagating quasi-statically. Each arrestor was analyzed individually using the geometry shown in Fig. 6. It consists of an upstream pipe section of length L,, an arrestor of length L,, and a downstream section of pipe of length L2. The values of L, and L2 were similar to those of the experiments (see top of Fig. 3). Guided by the experimental observations, we assumed planes x1--x2, x2-x3 and x1-x3 to be planes of symmetry which reduced the section analyzed to the part of the schematic drawn in bold lines in Fig. 6. Since only one quarter of the cross section was analyzed, an imaginary rigid surface was placed along plane x,-x3 using rigid elements to simulate contact of the inner surface of the pipe wall. The boundary nodes at x, = (L, + L, + L2) were constrained to have zero radial and circumferential displacements but could move freely in the x1 direction (same condition as in the experiments). Details of the discretization scheme used can be found in Park and Kyriakides [ 151. The model was first used to simulate each of the physical experiments conducted using the actual geometric and material parameters of the pipes and arrestors. The materials of the pipes and arrestors were assumed to be J2-type, elastoplastic, finitely deforming solids which harden isotropically.

1 P P,

Ocean Research 20 (1998) 95-104

was

developed

(within

the

of arrestor crossover

The general characteristics of the calculated results can be outlined through an example involving arrestor 3. The structure was loaded by hydrostatic pressure. Collapse was initiated by prescribing some local, initial ovality to the pipe in the neighborhood of xi = 0. Fig. 7(a) shows the calculated P-b response (vO is the initial internal volume of the structure and 6v is the absolute value of the change of volume evaluated for each deformed configuration calculated). Fig. 7(b) shows a sequence of deformed configurations corresponding to points along the response identified by circled numbers. The first pressure peak represents the onset of collapse in the upstream section of pipe. Its value is strictly a function of the initial imperfection and is unimportant to the crossover pressure of the arrestor. With the pressure dropping, the collapse localizes as seen in configuration @ and 0. When the walls of the pipe comes into contact, the collapse is arrested locally and starts to propagate down the pipe. The length of pipe available is limited and, as a result, the presence of the arrestor is felt quickly. The pressure then takes an upturn and in the process the upstream pipe collapses further (see @ and @) and the extent of contact increases. The higher pressure and the deformation of the collapsed pipe adjacent to it cause the arrestor to ovalize. This, in turn, allows some ovalization to cross into the section of pipe downstream of the arrestor. Eventually, the external pressure and the ovalization in the second pipe section reach a critical combination and the downstream pipe collapses resulting in a precipitous

S. Kyriakideset al. / Applied Ocean Research20 (1998) 95-104

100

assumed to be X-65 grade steel with a smooth stress-strain response, which for strains less than 8% can be represented by the Ramberg-Osgood fit as follows: +[l+;(s)“-‘1

l

X-65 +

= 22.5

where E = 29.9 X lo3 ksi (206 GPa), oY = 57.97 ksi (400 MPa) and n = 10.7. The variables varied were the arrestor length and thickness and the pipe D/t.

1.5

m

1.0

=

0.75

A

0.5

l

0.25

(3)

4.1. Eficiency

as a function of arrestor thickness and length

I 1

2

3

4

5 -

Fig. 8. Calculated arrestor efficiency various arrestor lengths.

6

7 h/t

as a function of arrestor thickness for

decrease in pressure. The second pressure maximum in the response represents the crossover pressure of this arrestor (px). Its value is 3985 psi (274.8 bar) which is 3.4% lower than the measured value of 4126 psi (284.6 bar). This procedure was used to calculate the crossover pressures of the 15 arrestors tested. In all cases the difference between calculated and measured crossover pressures was less than 5% while the average absolute difference for the set was 2.29%. This level of performance by the model is considered to be excellent. Separate calculations were conducted to establish the collapse pressure (i’,o) of each section of pipe downstream of an arrestor using the computer program BEPTICO along with the appropriate geometric and material parameters. The propagation pressures (p,) of the pipes used in the experiments were also calculated individually using the analysis and procedures described in Dyau and Kyriakides [9]. The calculated values of i’x, p,o and p, (see Table 4 in Park and Kyriakides [15]) were used to establish predicted values of arrestor efficiencies using

(2) The calculated efficiencies are plotted in Fig. 4. The calculated and measured values of 7 are seen to be in good agreement. The predicted modes of crossover are indicated by the symbols + and +. It is reassuring to observe that except for two cases in the transition range between the flattening and flipping modes all other cases exhibited the same crossover modes as those of the experiments. 4. Parametric

study of arrestor efficiency

Having proved the dependability of the model, it was used to conduct an extensive parametric study of crossover pressures of integral buckle arrestors in order to supplement the experimental results and enrich the data base available for design. Here the pipe and arrestor material were both

The pipe geometry was first fixed to D = 4.5 in (114.3 mm) and t = 0.2 in (5.08 mm) and the arrestor length and thickness were varied extensively. This particular pipe with an assumed initial ovality of A0 = 0.1% has a collapse pressure of f,o = 4329 psi (298.6 bar) and a propagation pressure of P, = 990 psi (68.28 bar). Results for arrestor lengths of La/D = 1.5, 1.0,0.75,0.5,0.25 will be shown. For each of these values of L, the arrestor thickness h was varied to obtain several data points spanning values of 77from 0.1 to 1.0. The calculated efficiency is plotted versus h/t in Fig. 8 for each value of L,lD. As expected, as an arrestor becomes shorter, a thicker wall is required for it to achieve the same efficiency. The mode of crossover of each data point in Fig. 8 is also identified. It can be seen that all arrestors with efficiencies less than approximately 0.7 developed the flattening mode of crossover and those with higher efficiencies exhibited the flipping mode. Furthermore, in the transition from one mode to the other the arresting efficiency was not always monotonically increasing with the arrestor thickness h. Because of this behavior the calculated points are not joined with a smooth curve. For all arrestor lengths shown in Fig. 8 it was possible to estimate a thickness for which efficiency of 1.O is achieved. The arrestor with the minimum thickness that yields efficiency of 1.O is defined as the critical thickhess arrestor for the particular length and its thickness is designated as h,. The values of the critical thickness arrestors for seven arrestor lengths are listed in Table 1. Arrestors shorter than 0.250 were also considered. It was found that for narrow, fin-like arrestors an efficiency of 1.0 could not be achieved irrespective of the arrestor thickness chosen. The smallest arrestor length that yields an efficiency of 1.0 is given the name minimum arrestor length and is designated as L,,. Its value is expected to be affected by the Table I Critical arrestor Dlt = 22.5) L E

hc f

thicknesses

for

seven

arrestor

lengths

(X-65

pipe,

2.5

2.0

I .s

1.0

0.75

0.50

0.25

2.582

3.054

3.844

4.896

5.026

5.923

7.526

101

S. Kyriakides et al. / Applied Ocean Research 20 (1998) 95-104 1.5

hc9

&

T

1.0

7

t

Table 2 Collapse and propagation

t

5

pressures for pipe of three D/t

0 f

8,, psi (bar)

PCO.psi (bar)

17.0 22.5 34.0

1980(136.6) 990(68.28) 383(26.41)

6667(459.8) 4329(298.6) 1666(114.9)

0.5 3

1 0.0

0.0 3.cb -r

0.5

1 .o

1.5

2.0

Fig. 9. Critical arrestor thickness arrestor material volume.

2.5

l-,ID

versus arrestor length and corresponding

increases h,&, asymptotically approaches the value of the minimum thickness arrestor. The results indicate that for this combination of pipe and arrestor material, an arrestor with the minimum thickness (h,,) would behave as if it was infinitely long when it reaches a length of approximately 30. For completeness, the normalized material volume of each critical arrestor given by

pipe D/t and material

properties as well as by the properties of the arrestor material (if different from that of the pipe). For the present pipe L, was estimated to be 0.250. Another bounding geometric parameter is the lowest arrestor thickness which can yield efficiency of 1.0 (h,,). This is the lowest thickness that a long arrestor must have in order for its propagation pressure (Ph) to be equal to the collapse pressure of the pipe. Thus, h, can be evaluated from ~Pahn)

=

pco.

(4)

An arrestor of this thickness is called the minimum thickness arrestor (see also Power and Kyriakides [ 161). To establish h,, from Eq. (4) an accurate value for Pco must be available and, preferably, a closed form expression for Pr,. The best estimate of PC0 is one derived numerically (e.g. via BEPTICO). Less accurate, in general, predictions can be obtained by using one of the existing design formulas for collapse pressure (e.g. Murphey and Langner [ 11, Ju and Kyriakides [17]). No closed-form expression exists for the propagation pressure of pipes. A reasonable estimate can be obtained by using one of the empirical expressions for 8, such as the one suggested in Kyriakides and Babcock [ 181. In terms of the arrestor thickness, h, and diameter D, this expression is

f’pa-uo[a+B;] ($-)’

(5)

where the stress-strain response of the material is approximated by a bilinear fit with yield stress of u, and post-yield modulus of E’. A, B and 0 are constants determined empirically. (The bilinear approximation of the X-65 stress-strain response we used has u,, = 73.8 ksi (509 MPa) and E’ = 114 ksi (784 MPa).) Using Eqs. (4) and (5) the minimum thickness of an arrestor of efficiency 1.0 for an X-65 pipe with Dlt = 22.5 is h,, = 1.98t. The critical arrestor thicknesses for the seven arrestor lengths which appeared in Table 1 are plotted against L,lD in Fig. 9. Included in the figure are the bounding values of h,,lt calculated above and that of L,,lD which was estimated to 0.25. We observe that as the value of L$D

is included in Fig. 9. L,,, is the minimum length that a critical thickness arrestor must have for it to have an efficiency of 1.0 (assumed here to be 30). Clearly, the shorter arrestors are the most efficient from the point of view of minimizing the material used. However, arrestors which are very thick may be impractical due to difficulties they may cause during the installation and possibly during the operation of the pipeline. At the other extreme, arrestors longer than 2-2.5 pipe diameters are impractical because of the additional material and handling costs. 4.2. EfJiciency as a function

of pipe Dlt

The next variable to be varied was the pipe D/t. X-65 pipes with diameters of 4.5 in (114.3 mm) and D/t values of 34.0 and 17.0 were considered in addition to those with D/t = 22.5 already discussed. The calculated collapse and propagation pressures of the pipes were as given in Table 2. Arrestors of length L,/D = 0.5 and of various thicknesses were analyzed for each pipe so as to span efficiency values of 0.2 to 1.0. The arresting efficiencies calculated for the three pipe D/t values are plotted against h/t in Fig. 10. The modes of crossover are also identified in the figure in the usual manner. It is interesting to observe that the slope of the results exhibiting the flattening mode of crossover from the three sets are similar. The same can be said for the results with the flipping mode with this slope being shallower. Transition from the flattening to the flipping mode again occurs in the neighborhood of 71= 0.7.

5. Empirical

arrestor design equations

Clearly, a full scale FE calculation

of the type described

102

S. Kyriakides et al. / Applied Ocean Research 20 (1998) 95-104

to seek such a design formula for the integral arrestor using the experimental data enriched by the numerical results developed. In the quasi-static setting considered in this study, the crossover pressure will depend on the major problem variables, that is

1 f-i 0.8 t

=f(P,,E,a,,a,,,D,t,L,,h).

0.6

Px

0.4

From Buckingham’s n theorem this can be reduced to the following relationship between nondimensional variables

PX __=F PP

0.2

0

1

2

3

4

5

___t

(8)

We assume series:

6 h/t

(7)

that this can be expressed

as the following

Fig. 10. Calculated arrestor efficiency as a function of arrestor thickness for pipes of three D/t values.

(9) above remains a viable option. However, such an effort is computationally intensive and may not be the best option at the early stages of the design of a pipeline when the problem parameters are not well defined. Approximate, empirical design formulae are very useful for parametric studies which by necessity must be conducted before hard design parameters become available. Such formulae were successfully developed for the ‘slip-on’ [ 1 l] and ‘spiral’ [ 131 purely from experimental results. Here we use a similar procedure

rl

Since PxIPp 2 1 we deduce that A, = 1. In an effort to produce the simplest possible relationship we first neglect terms with powers n > 1 and consider the approximate relationship

The arresting efficiency

can now be deduced from Eqs. (l),

1 .o 0.8 Lower Bound Envelope

0.6 0.4 0 Experiments

. D/t= 17.0

0.2

. D/t = 22.5 A D/t = 34.0

0.0

(

0.2

0.1 )

Fig. 11. Arrestor efficiency

versus empirical

0.3

0.4

C1.6

0.5

[(ggg.gj’2pJ(~@P_ 1)]

X100

function of parameters

and lower bound construction

for design.

S. Kyriakides et al. / Applied Ocean Research 20 (1998) 95-104

the design of an effective integral buckle arrestor:

(2) and (10) to be:

rl=A,

103

/pco_l\ \PP ‘/

.

Qi (i = 15) are then chosen so as to produce the best correlation between all available data. Unlike the ‘slip-on’ and ‘spiral’ arrestors, it was not possible to fit all the data with just a single term of the series in Eq. (9). The data were found to exhibit a bimodal trend with n = 0.7 as the boundary. The relationship of the efficiencies of arrestors lower than this value is different from that of arrestors with efficiencies higher than 0.7. Furthermore, all arrestors in the first group exhibited the flattening mode of crossover and, with the exception of two cases, those in the second group exhibited the flipping mode. Although more complex data-fitting schemes may produce a better correlation of the data, in the interest of simplicity we decided to accept this bimodal trend and proceeded to select oi in a way that produced the best correlation of the data for n I 0.7 while simultaneously minimized the scatter in the data for 7~> 0.7. The results of this process produced the following equation

a. Calculate the collapse and propagation pressures of the pipeline. b. Select a steel grade of the arrestor. c. Calculate the thickness of the minimum thickness arrestor using Eq. (4). d. Select either the length of the arrestor such that L, > 0.250, or an arrestor thickness such that h > h,,. e. Use the problem variables in Eq. (12) to evaluate either the arrestor thickness, or its length for the desired efficiency. An appropriate safety factor should be applied either by the choice of Px designed for, or in the choice of the arrestor thickness (or length). f. Test your design by a dependable numerical model like the one discussed here, or preferably by a full scale test conducted as outlined in the experimental section (see Park and Kyriakides [ 151 for more details). It should be noted that like all empirical expressions of results of complex phenomena, Eq. (12) can be a dependable design tool provided that the parameters of the arrestor and pipe being designed do not deviate significantly from the range of variables of the data used to generate it. If the problem parameters deviate significantly from those of the present data base, new dependable data must be added to it and, if necessary, a new fit should be attempted before such an empirical design formula is used directly in design.

Acknowledgements All the data are plotted against the RHS of Eq. (12) in Fig. 11. The data for n I 0.7 have coalesced to produce a nearly linear relationship between n and the RHS of Eq. (12). The choice of A, = 667.6, drawn with a dashed line in Fig. 11, produces a correlation coefficient of 0.978 for the 26 data points with 1 5 0.7. For higher values of 7, where the flipping mode of crossover predominates, this expression is not valid. It is important to point out that in practice the amplitude of small initial geometric imperfections in the pipe and their orientation relative to the incoming buckle may play a role in this regime of n. For example, an initial ovality oriented with its major axis in the same plane as the incoming buckle will tend to delay the flipping mode of crossover (increase 7) while those oriented at 90” to the incoming buckle will tend to accelerate crossover (lower 9). Thus, in general, a conservative approach should be taken when designing arrestors in this higher efficiency regime. A lower bound envelope, such as the linear one drawn with a dashed line in Fig. 11, can be used to get an estimate of the key variables. These numbers can then be confirmed with a full scale numerical calculation of the type performed in this study, or preferably by a test involving the actual pipe and arrestor. The approach is also recommended when using Eq. (12) to design arrestors with 7~I 0.7. In summary, we recommend the following procedure for

The work reported was conducted with financial support from a consortium of industrial sponsors. The work of T-DP was also sponsored by Korea Telecom, that of TAN with a scholarship from CNPq of Brazil and that of SK by the Office of Naval Research under grant no. N-00014-91J1103. Any findings, conclusion and recommendations expressed herein are those of the authors and do not necessarily reflect those of the sponsors.

References [l] Murphey CE. Langner CG. Ultimate pipe strength under bending, collapse and fatigue. In: Proceedings of the 4th International. Conference on Offshore Mechanics and Arctic Engineering, Vol. 1, 1985, pp. 461-417. [2] Yeh M-K, Kyriakides S. On the collapse of inelastic thick-walled tubes under external pressure. ASME Journal of Energy Resources Technology 1986;108:35-47. [3] BEPTICO: Pipe Collapse under Bending, Pressure and Tension Loads. Special Purpose Computer Program, EMRL Report, No. 911 9, University of Texas at Austin. [4] Kyriakides S, Babcock CD, Elyada D. Initiation of propagating buckles from local pipeline damages. ASME Journal of Energy Resources Technology 1984; 106:79-87. [5] Park T-D, Kyriakides S. On the collapse of dented cylinders under external pressure. International Journal of Mechanical Sciences 1996;38(5 ):557-578.

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