On the Propagating Buckle and Its Arrest Citation Kyriakides, Stelios (1980) On the Propagating Buckle and Its Arrest. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/c81x-nk02. https://resolver.caltech.edu/CaltechETD:etd-10292003-134840 Abstract If an offshore pipeline locally buckles in the presence of sufficiently large external pressure, a propagating buckle is initiated. The buckle propagates along the pipeline until it encounters a region of adverse conditions -- low pressure or an arresting device. The lowest pressure at which a buckle propagates is defined as the Propagation Pressure. An experimental study of this quantity is presented and a semiempirical expression derived by examining various models of the phenomenon. The dynamics of the Propagating Buckle are also examined and a parametric study of the steady state velocity of Propagation is carried out. A systematic way of empirically deriving the parametric dependence of arresting devices is presented and experimental results of two such arrestors are discussed. During this study a unique "flip-flop" mode of propagation was discovered and studied. An explanation of the phenomenon is also attempted. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Aeronautics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Babcock, Charles D. Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 14 April 1980 Funders: Funding Agency Grant Number Caltech UNSPECIFIED US. Dept. of Energy UNSPECIFIED American Petroleum Institute UNSPECIFIED Record Number: CaltechETD:etd-10292003-134840 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-10292003-134840 DOI: 10.7907/c81x-nk02 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 4300 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 30 Oct 2003 Last Modified: 18 Dec 2024 00:28 Thesis Files Preview PDF (Kyriakides_s_1980.pdf) - Final Version See Usage Policy. 14MB Repository Staff Only: item control page

ON THE PROPAGATING BUCKLE AND ITS ARREST Thesis by Stelios Kyriakides In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1980 (Submitted April 14, 1980) Li © 1980 STELIOS KYRIAKIDES All Rights Reserved 1ii DEDICATION This work is dedicated to my parents whose encouragement, guidance and sacrifices I could not have done without. iv ACKNOWLEDGEMENTS I would first like to express my sincere thanks and appreciation to Professor Chuck Babcock for his guidance, encouragement and friendship during the course of this in- vestigation. The advice and suggestions of other members of the Engineering Staff is also appreciated. Special thanks to the graduate students of the Solid Mechanics group of GALCIT for providing a warm atmosphere to work in. I would also like to thank my friends Rohan Abeyiaratne and John Psycharis for their help in numerous ways. I am also grateful to the technicians of GALCIT and to Doug Brisky, Charles Narty and Sam Chang who patiently assisted in the experimental work. Mrs. Betty Wood, Mr. Harry Hamaguchi, Mrs. Marlys Ricards, Mrs. Maureen Matteson and Mrs. Karen Valente are all thanked for all they've done in preparing this manuscript. Finally, the financial assistance of the California Institute of Technology, the American Petroleum Institute and the Department of Energy is gratefully acknowledged. Vv ABSTRACT If an offshore pipeline locally buckles in the presence of sufficiently large external pressure, a propagating buckle is initiated. The buckle propagates along the pipeline until it encounters a region of adverse conditions -- low pressure or an arresting device. The lowest pressure at which a buckle propagates is defined as the Propagation Pressure, An experimental study of this quantity is presented and a semi- empirical expression derived by examining various models of the phenomenon. The dynamics of the Propagating Buckle are also examined and a parametric study of the steady state velocity of Propagation is carried out. A systematic way of empirically deriving the parametric dependence of arresting devices is presented and experimental results of two such arrestors are discussed. During this study a unique "flip- flop"' mode of propagation was discovered and studied. An explanation of the phenomenon is also attempted. Vi TABLE OF CONTENTS ACKNOWLEDGEMENTS ABSTRACT NOMENCLATURE 1. INTRODUCTION (a) (b) (c) (d) Ce) Causes of Local Buckles Buckling of Circular Pipe Under External Pressure Initiation of a Propagating Buckle Buckle Propagation Propagating Buckle Arrest 2. ON THE PROPAGATION PRESSURE 2.1 2.2 2.3 2.4 Introduction Experimental Determination of the Propagation Pressure 2.2.1 Experimental Set Up 2.2.2 Experimental Procedure 2.2.3 Experimental Results-Discussion Theoretical Modeling 2.3.1 One-Dimensional Model (a) Collapse Load of a Circular Ring Under External Pressure (b) Large Deformation Collapse Analysis of an Inelastic Inextensional Ring (c) The Problem 2.3.2 A Three-Dimensional Model Conclusions Page iv ix 10 ll 11 12 12 L5 17 19 20 21 26 26 41 43 vil DYNAMICS OF THE PROPAGATING BUCKLE 3.1 Introduction 3.2 Experimental Determination of the Velocity of Propagation 3.2.1 Velocity Measurements-Results 3.3. High Speed Photographic Observation of a Propagating Buckle 3.3.1 Experimental Results 3.4 Parametric Dependence of the Velocity of Propagation 3.5 Conclusions THE FLIP-FLOP MODE OF COLLAPSE 4.1 Introduction 4.2 High Speed Photographic Observation of the Formation of a Flip-Flop BUCKLE ARREST 5.1 Introduction 5.2 The Problem Parameters 5.3 Experimental Procedure (a) Quasistatic Experiments (b) Dynamic Experiments 5.4 Experimental Results (a) Quasistatic Results (b) Dynamic Results (c) Gap Effect 5.5 The Spiral Buckle Arrestor 5.5.1 Parametric Study of the Spiral Arrestor Efficiency 51 54 56 58 61 63 63 66 71 71 74 77 77 79 80 80 80 84 85 87 viil Page 5.5.2 Experimental Results 88 5.6 Conclusions 90 ENCES REFER 92 APPENDIX A 97 APPENDIX B 98 APPENDIX C 100 FIGURES 104-188 jah) ”“ n aoe 8 og Mo Mm mom fb o" Sl = Pr ep QO col gO UO Vv Ss DO VY VW WS YH ° ° th oF oO i) Fh ae) ae) ix NOMENCLATURE Buckle Profile Length Pipe Diameter, O.D. Young's Modulus Post-Yield Slope of Bilinear o-e Curve Secant Modulus Tangent Modulus Gap Size Normalized Curvature Arrestor Thickness Horizontal Force at any point in Ring Normalized Horizontal Force Arrestor Spacing Arrestor Length Moment Normalized Moment Yield Moment Pressure Buckling Pressure of Pipe under External Pressure Collapse Pressure Inelastic Buckling Pressure Initiation Pressure Lowest Pressure at which Flip-Flop Occurs Arrestor Crossover Pressure Arrestor Dynamic Crossover Pressure Propagation Pressure <a <a a aie = sf & PL x KP KF = S PR ba Md | <| Normalized Pressure Pipe Radius Normalized Ring Mid-Plane Coordinate Pipe Thickness Time Vector (H, V, M, 8) Buckle Velocity Yoolo Vertical Force at any Point on Ring Normalized Vertical Force Amplitude of Initial Imperfection External Work done on Ring Plastic Strain Energy in Pipe Kinetic Energy of Pipe Kinetic Energy of Fluid Work Done by External Pressure Cartesian Coordinates Normalized Cartesian Coordinates E/E! Displacement of Ring at y(1) Arrestor Efficiency Inelastic Correction Factor for Buckling Pressure Arrestor Dynamic Efficiency Arrestor Static Efficiency Angle between Normal and x-axis xi Curvature Curvature at First Yield Poisson Ratios Pipe Material Density Fluid Density Yield Stress, Pipe (stress at « = 0.005) Yield Stress, Arrestor L. INTRODUCTION In recent years the seas, off the continents, and the oceans beyond are being looked upon with ever increasing interest as main sources of fossil energy for the future (30% of total U.S. oil production by 1983). As exploration of the continental shelf continues the deeper waters beyond are being leased by the government and exploration of oil and gas in depths of up to 5000 ft. (1.5 km) is already under way. This endeavor has demanded a whole new technology to be established rather quickly and in many cases industry is hard pressed to meet the demand. Existing technology was modified to suit the new field conditions. In many cases restricted methods are used in order to obtain quick answers to new problems. . Offshore pipeline design, construction and opera- tion is one of the new industries which had to be devel- oped. Based on the experience obtained from pipelines laid in shallow waters as early as the 1940's, a number of pipe- laying methods have been developed. Ref. 1.1 gives an up to date summary of these. Pipelaying off a barge, an early favorite, still remains the most commonly used method. References 1.2-1.6 give extensive studies of the equilibrium of a pipe during such laying operations (fig. 1.1). Although a few problems still exist, such as the stinger contact problem and elastic-plastic pipe material behavior, this problem area is fairly well understood. In recent years the development and deployment of barges equippedwith reels [1.1, 1.7] have made this type of process much faster. During such a pipelaying operation (fig. 1.1) the pipe is under the combined effect of bending, pressure, tensional and buoyancy loads. The tension applied at the barge usually controls the bending stresses in the pipeline. If tension is lost either due to sudden ship movement or malfunctioning of the tensioner the bending stresses in the sag bend area (area of maximum moment) can exceed the buckling limit and cause a bending buckle. Bending buckles are local in nature affecting a region less than ten diameters long (fig. 1.2a,b). If the external pressure at this point is high enough the local buckle transforms itself into one that propagates, driven by the pressure (fig. 1.2c). Once initiated this buckle continues propagating, completely flattening the pipe behind it. This continues until an area of low pressure is reached, where propagation cannot be sustained or until some mechanical obstacle is encountered which hinders further flatten- ing of the pipe. In any case, it is clear that practically the whole pipeline is potentially in danger. It is important to make clear that the role of the bending moment is important only until the initial local buckle is formed which initiates the Propagating Buckle. In fact, a buckle can be initiated from any type of local dent or damage. For example, once the pipe becomes operational, many external factors can cause damage to the pipeline even though the pipeline is not under the effect of any substantial bending moments. Earthquakes, sea bottom instabilities and sea floor currents can all cause local buckles or damage to the pipeline. Another common cause of damage is heavy equipment or debris falling onto the pipe. Ship anchors falling onto the pipe or becoming hinged to it is another serious danger. Other ocean energy extraction schemes such as Ocean Thermal Energy Conversion (OTEC) and Kelp Farming involve suspended pipes up to 15,000 ft. (4.57 km) in length [1.8] (fig. 1.3). Through these pipes, large quantities of water are pumped from deep down in the sea to the surface, providing the temperature difference for a heat engine or for nutrients for growing kelp. The pipes are subjected to loads due to cur- rents, to pumping pressure differences and motion induced by the floating structure to which they are attached. Buckling of these structures can occur from the combined effect of bending and external pressure. This can create the condi- tions necessary for a Propagating Buckle to be initiated, Although this problem has been observed primarily in offshore pipeline applications it is by no means restricted to these, In fact, any long cylindrical structure under external pressure can be Propagating Buckle critical pro- vided the potential danger of local dents, buckles or other damage occurring, exists. One rather interesting example is described in ref. 1.9. In that case a small diameter gas a pipeline is placed inside an existing large diameter one. (See fig. 1.4.) In some circumstances water fills the space | between the two pipes. During the winter this water freezes and for a range of pipe diameter ratios the expansion of ice can cause buckling and complete flattening of the pipe. Ina case like this buckling always occurs at the weakest point along the pipeline. The result is a long pipe with a section that is damaged. This acts as an initiating point for a Propagating Buckle. It is driven by the pressure provided by the expanding ice. The rate of propagation depends on the rate of ice expansion. The problem of the Propagating Buckle and its potentially catastrophic results were first noticed by a number of dif- ferent researchers in the early seventies. The first pub- lished report on the problem came from researchers at Battelle in July 1973 [1.10]. Industry responded by sponsor- ing a multiyear research program for tackling this and other related buckling problems of offshore pipelines [1.11]. Quite independently Palmer,in a short article [1.12],made a first attempt at quantifying the so called Propagation Pressure. His experimental and theoretical work were not taken far enough to produce a useful formula for this critical quantity. In 1976 (ref. 1.13) ‘the Battelle group expanded on their earlier report. Although practi- cally all aspects of the problem were touched, the restric - tive nature of these projects caused the fundamental under- lying mechanics problems to be slighted, In all fairness ~5- many of the problems involved are rather difficult in nature involving large deformation plasticity, post buckling effects as well as hydrodynamic effects. However, the above reports served the very useful purpose of making industry conscious of the problem. This report is a record of a systematic effort, which started in the summer of 1976, to better understand the mechanics of the problem in all its stages. Experiments on model pipes carried out in controlled environments were conducted in order to decide the importance of the various parameters on the exhibited phenomena. With these results established, modelling of some of the problems involved was attempted with the general objective of obtaining approxi- mate but useful expressions for the various critical quantities. . In order to help understand the various aspects of the problem and the mechanical processes involved in initiat- ing, propagating and arresting a buckle,the problem has been subdivided into 6 sub problems. Each of these is briefly described below. (a) Causes of Local Buckles As already mentioned a Propagating Buckle is always initiated from some local damage or dent on _ the pipe. Clearly the best way of avoiding the problem is to make sure that the conditions that cause these local imper- fections are restricted. As far as the pipelaying opera- tions are concerned this goal has been achieved to a great -~- 6 - extent due to the addition of a number of new pipelaying methods for special applications [1.1] and the perfection of existing techniques. The advent of dynamically positioned laying vessels (barges, reel ships and semi-submersibles) has also helped reduce accidents due to sea roughness. Itis safe to say that today most buckling problems during offshore pipe- laying are caused by human error. In addition, better trenching and dredging techniques [1.14-1.16] coating the pipe with protective coating, securing the pipe to the ocean floor, [1.17], better sea bottom survey methods, etc., have all contributed to increasing the safety of the pipeline during its operation. (b) Buckling of a Circular Pipe Under External Pressure Since offshore pipelines are under the influence of a variety of compressive loads there exist conditions under which the pipe becomes buckling critical. For the purposes of this report the simplifying assumption that the pipe is under external pressure loading only, will be made. In such a case the buckling pressure of an infinitely long pipe is given by the classical result: p = 2E_ (5) (1.1) This equation assumes purely elastic buckling. For typical steel alloys (4% ~ 50 x 10° psi) this formula is good for D/t = 25. For pipes having diameter: thickness ratios smaller than the above an elastoplastic buckling correction factor for 1.1 as calculated by ref. 1.18 can be used. P =n Pp cp yoe (1.2) -e[h+3.f] Ny “ E LG" GE S (c) Initiation of a Propagating Buckle As already mentioned a Propagating Buckle will only occur if a local damage exists from which it can be initiated. The pressure at which such a local imperfection transforms itself into a propagating one is called the Initiation Pressure (P,). If a pipe under purely lateral pressure is considered, it has been shown experimentally that a Propa- gating Buckle prefers a "dogbone" type of collapse. This is independent of the way the buckle was initiated. The Initia- tion Pressure then can be looked at as the pressure at which an imperfection, having a general shape, transforms itself into the correct profile which collapses the pipe into the dogbone shape. It is clear that the value of pressure at which this happens depends on the characteristics of the specific im- perfection considered, These include the type of imperfec- tion (e.g,, local dent of the type introduced by bending buckles or some kind of random imperfection) and its magni- tude (i.e., length and amplitude). The relation of the geometry of the imperfection to the required profile of a Propagating Buckle is another factor. For example, a longi- tudinally extending and symmetrically placed dent of small amplitude, can have a lower initiation pressure than an irregular and awkwardly oriented dent of much bigger ampli- tude. If one considers the more complicated case of pres- sure, moment and tension interacting, then the additional problem of the local stress state has to be considered. It became clear that since an infinite number of initial dents can be defined, the problem of quantifying the Initia- tion Pressure in a systematic way (analytical or experimen- tal) is enormous. On the other hand,one is faced with the real demand for this knowledge in the field. Some kind of rule that will enable an evaluation of the seriousness of a given situation where a dent exists on a pipeline is neces- sary in order to make a decision about the course of action to be taken. Stopping a laying operation or the pumping process are extremely costly decisions. Any advice that might help in evaluating the seriousness of the damage is very useful, If engineering judgment and experience derived from recorded mishaps are used, the imperfections to be considered can be restricted to a manageable small number and in such a case the parametric dependence of the Initiation Pressure can be studied in a systematic way with the hope that the results derived will apply to a high percentage of the cases encountered. Such a study has not been one of the objectives of the reported work but represents a long term objective of the investigator. (d) Buckle Propagation Once a buckle is initiated it propagates down the pipe flattening it. Typical pictures of Propagating Buckles on three different size pipes are shown in fig. 1.5. It is clear that the damage is extensive and terminal. Due to the large deformation, cracking at the sharp corners sometimes occurs. This can be very hazardous because any oil in the pipe at the time of the accident spills into the ocean with adverse effects to the environment. In addi- tion if this occurs during the laying operation flooding of the pipe occurs which can cause loss of the pipeline. Any analy- sis of the above problems has to involve the nonlinear material behavior as well as nonlinear large deformation kinematics. As mentioned by ref. 1.10 there exists a critical pressure below which a Propagating Buckle cannot be sustained. This is given the name Propagation Pressure (P) and an em- pirical value for this pressure is given by ref. 1.10 as t\4ss5- Py = 340(5) (1.3) The validity of this expression‘is tested experimentally in chapter 2 and an effort to derive a more general expres- Sion is also made. It is important to note that for each pipe P, s Pps PQ* (1.4) For buckles initiated at pressures higher than Po the buckles were recorded to propagate at substantial velocities. *At least for D/t<100 and metallic pipes. - 10 - The dynamics of this propagation and the dependence of the velocity on the various parameters of the problem is studied in chapter 3. Chapter 4 deals with a new mode of propagation W termed the "flip-flop" discovered during the process of this investigation. (e) Propagating Buckle Arrest Accepting the potentially catastrophic results of the Propagating Buckle one would like to restrict its effects to as small a section of pipe as possible. This is done by using strengthening devices called Buckle Arrestors so that if an accidental buckle occurs the damage is restricted to a sacrificial length of the pipeline. The final chapter deals with the design of such devices. - jl - 2. ON THE PROPAGATION PRESSURE 2.1 Introduction When a circular cylindrical pipe is subjected to ex- ternal pressure there exists a pressure above which a Propa- gating Buckle can occur. Above this pressure plastic defor- mations of a local nature that might arise from a variety of sources, such as local buckles or local dents, can cause a Propagating Buckle. The magnitude and configuration of these local imperfections required to cause such a buckle depends primarily on the operating pressure. The Lowest pressure at which a Propagating Buckle can be sustained is defined as the pipe's Propagation Pressure, (P): As shown in fig. 2 the Propagation Pressure is well below the critical buckling pressure of the same pipe under external pressure, at least for pipes with D/t of interest for pipeline applica- tions (25 < D/t < 100). Thus buckle propagation can occur for pressures anywhere between Po < P< Poe For pressures closer to the buckling pressure, any kind of local imperfec- tion that will cause local buckling will automatically cause a Propagating Buckle. For pressures closer to the Propaga- tion Pressure a Propagating Buckle is more difficult to initiate. Propagating Buckles prefer a dogbone type of collapse. This is independent of the way the buckle is initiated. As a result, an n = 2 imperfection is much more likely to cause a Propagating Buckle, at pressures closer to Po than other modes of imperfection are. Investigation of the problem of buckle initiation has not been one of the -~ 12 - objectives of this study. In this section a concentrated effort is made to understand the governing parameters that determine the value of the Propagation Pressure once it is initiated. The problem is simplified even further by avoid- ing the perplexing issue of propagation under combined load- ing such as bending, pressure and tension. Instead, propaga- tion under purely external pressure conditions is considered. The problem is first examined experimentally and then some simplifying models are considered in order to help ex- plain some of the experimental results. 2.2 Experimental Determination of the Propagation Pressure 2.2.1 Experimental Set-up The experiments were conducted in a specially built pressure tank shown in fig. 2.1. This tank has a diameter of 8 in. (.2m) and a length of 12 ft. (3.61m). It is made from three different segments each one of which can be used independently. The first is a 2 ft. (.6m) long initiator section shown in fig. 2.2. A pneumatically operated knife edge is part of this assembly and was used for initiating a buckle at one end of the pipe under test. The pneumatic cylinder was operated through a solenoid valve remotely and used up to 125 psi (8.62 bar) air pressure. This device was essential for the dynamic tests, described in chapters 3 and 4 where it was necessary that a buckle be initiated at a specific pressure. Two more sections can be added to the initiator section making the tank 6 or 12 ft. (1.8 or 3.6m) in total length. ~ 13 - The tank was pressure tested to 1000 psi (69.0 bar) but has an operating pressure of 800 psi (55.2 bar). Fig. 2.3 shows a 4 £t. (1.2m) long transparent section which was designed, fabri- cated and used for observation of the Propagating Buckle. It is made of a floating (no axial stresses present) inner cylindrical acrylic transparent tube turned to the design diameter and repolished. This was slid into a steel casing which was honed allowing only 0.01 in. (.25mm) between the two cylindrical surfaces. Two "0"-rings provided the neces- sary sealing. Through this design the hoop stresses were taken primarily by the acrylic tube and the axial stresses wholly by the steel casing. The discontinuity at the window was designed as shown in the figure to reduce stress concen- trations. This section was pressure tested to a pressure of 240 psi (16.6 bar) but had a working pressure of 200 psi (13.8 bar). It was used primarily for experimental visualization as it restricted the range of D/t that could be tested. The tank was periodically pressure tested for possible fail- ure of the acrylic due to the cyclic loads that the tank was subjected to. A release safety valve was added to the system when this section of the pressure tank was in opera- tion. The complete tank assembly is shown in fig. 2.1. The tank can be pressurized with a totally liquid medium, a totally gaseous medium or it can be partly filled with a liquid and pressurized by a gas. In our case all three -~ 14 - situations were encountered; the liquid was water and the gas, air or just nitrogen. For quasistatic experiments, the tank had to be com- pletely filled with water in order to allow control over the speed of propagation. To achieve this the tank was posi- tioned at an inclination of 2° to the horizontal. The test procedure was as follows. The tube to be tested was pre- pared and placed into the tank. One end of the tube was directly connected to the atmosphere in order to ensure the pressure inside the tube was atmospheric as the buckle propagated (decreasing its volume). The end flanges were secured to place. "0''-rings between the end flanges and the slip-on flanges of the tank provided the necessary seals. Referring to fig. 2.1 with valves D, E, and F closed and A, B, and C open, water was pumped into the tank. As water filled the tank the air escaped through B-C. Valve B is positioned at the highest possible point ensuring that all the air escaped out of the tank through B-C. Water was pumped until it started escaping through exhaust C. Valve C was first closed and then valve A. A Teledyne S-216-C single piston air driven pump with a pressure ratio of 10 was used for pressurizing the tank (fig. 2.1). This pressurizing procedure was as follows. With valves A, C, and D closed, valve E was opened. Valve F was used to control the rate of pumping and to monitor the pres- sure. The pressure in the tank could be read from a cali- brated pressure gage. In addition a pressure transducer - 15 - signal was recorded on a chart recorder giving a continuous time-pressure reading. For experiments in air the tank was pressurized with compressed air or nitrogen from high pressure cylinders through valve D with A, E and F closed. For dynamic experiments in water the tank was partially filled with water leaving an air gap at the top. The system was then pressurized with compressed air. The air gap was big enough to allow near constant pressure experiments. (Pressure difference from the beginning to the end of the experiment was 2-3%; for the case of the tank being completely filled with air the pressure difference was less than 1%- these apply to the biggest diameter pipes used in dynamic experiments 1.5 in. (.38m) OD). 2.2.2 Experimental Procedure As defined earlier the Propagation Pressure is the low- est pressure at which a buckle will propagate. A series of experiments were carried out on a number of aluminum alloy as well as on stainless steel pipes to find an experimental value for the Propagation Pressure of each. The geometric characteristics of the pipes tested can be found in Table 2.1. Longitudinal tensile test specimens were cut from each of the pipes tested and the stress-strain behavior of the pipe material was obtained up to a maximum strain of about 1%. Typical results from these tests are shown in fig. 2.4. The experimental procedure was as follows. A pipe was sealed and placed in the tank as shown in fig. 2.1. A large - 16 - local initial imperfection was inflicted to the pipe either externally or by the knife edges available in the initiator section. The tank was completely filled with water andslowly pressurized by the water pump. A typical pressure time history of how the experiment progressed is shown in fig. 2.5. For T oi ts the pressure rises to a maximum value Ps called the Initiation Pressure of the specific imperfec- tion. For Ty <T< To the initial imperfection deforms plastically and rapidly until at To it has taken the geometry of the profile of a Propagating Buckle. A typical profile of a quasistatically Propagating Buckle is shown in fig. 2.5. For To < T< Ty, the buckle slowly propagates in a dogbone type of collapse. (Typical crossections of collapsed pipes are shown in fig. 2.6.) The rate of propagation depends wholly on the rate water is pumped into the tank, this was usually left constant for T, < T< Ty. The buckle continues to propagate at relatively constant pressure until it starts "feeling" the effect of the end plug, at about 15 pipe diameters away from the end. At this point the constraining effect of the plug makes the pressure rise and this usually signalled the end of the experiment. Due to the fact that the pump used is a single piston pump, small variation in pressure occurred in the experiment. These variations were of periodic nature and the mean pres- sure was very easily obtained. This was taken to be the Propagation Pressure. As shown later the Propagation Pressure is very sensitive - 17 - to pipe thickness variations as well as to yield stress variations. Wall thickness irregularities of about max {At| = 7% exist in all drawn tubes. As a result sometimes the Propagation Pressure was position dependent. Careful measurement of wall thickness at different points along the pipe and at different points around the perimeter partly negated this error. Other sources of error were instrumenta- tion inaccuracies and human error. The Propagation Pressure results are tabulated in Table 2.1. 2.2.3 Experimental Results-Discussion Accepting the fact that this phenomenon is primarily governed by rather large plastic deformations, in a first attempt to plot the results in some logical fashion, the assumption that the whole stress-strain behavior of the material can be characterized by one parameter only, i.e., the yield stress, is accepted (see ref. 2.1). For comparison reasons the yield stress 9, is taken to be the same as ref. 2.1, i.e., do = fo] e€ = 0.005} This definition has been standardized by API for steel pipes. From fig. 2.4 it can be seen that for steel, this definition gives 4-8% higher values for the yield stress compared to the .2% offset strain criterion. On the other hand in the case of aluminum alloys the first criterion gives 5-15% lower values than the latter one. The Propagation Pressure normalized by the yield stress - 18 - is plotted on log-log scales, vs D/t in fig. 2.7. Theresults are fitted with a straight line (least squares fit). This straight line has a slope of -2.281. The small scatter of the results is due to experimental errors described above but probably the biggest contribution is due to discrepancies in the determination of the yield stress. As mentioned earlier the yield stress was obtained from tests carried out on specimens cut along the axis of the pipes. In the case of drawn tubes the properties in the transverse direction are different by as much as + 5%. Due to the size of the pipes used, obtaining the properties in the transverse direction was very involved so the compromise solution of longitudinal specimens was accepted. Using the slope obtained from fig. 2.7 the results are plotted on natural scales in fig. 2.8. On the same figures results from ref. 2.1 are also plotted. These were carried out using 2 in. (.05l1m) and 12 in. (.305m) OD steel pipes of different thicknesses. The slope of the fitted straight line is -2.254. Comparing the steel results with those from aluminum alloys, it is evident that the slope is practically the same (1% difference) but the two straight lines are shifted in the log-log plane. Two experiments carried out by the author on stainless steel pipes (SS-304) which has similar o - ¢« behavior as the steels used by ref. 2.1 are also shown in fig. 2.8. They are in good agreement with the other steel results reinforcing the suspicion that a fundamental difference exists between the two sets of results. - 19 - The equations of the fitted curves for the two sets of results are as follows: P 8B _ t Z=a (5) (2.2.1) A 7 8 Aluminum Alloys 11.3 2.281 Steel Alloys 14.5 2.254 Very clearly, replacing the complete stress-strain behavior by one parameter only is not a very good approximation. A more realistic assumption seems to be Pp. EL ty | | (2.2.2) So E(52)(5) where E is the Young's modulus of the material and g is a strain hardening parameter which could depend on the stress state. The effect of the elastic deformation and strain hardening will be examined in conjunction with some simpli- fied models of the phenomenon, which follow. 2.3 Theoretical Modeling Having obtained some knowledge about the phenomenon, effort was made to somehow model it in some simplified way that included the main mechanical parameters of the problem. Remseth, Holthe, Bergan and Holland in ref. 2.2 looked at the problem of combined bending and external pressure acting on a pipe and the associated initiation and propaga- tion of a buckle. Due to the combined loading considered -~ 20 - most of their effort went into initiating the buckle which as will be explained later depends very much on the load time history used or in deformation terms it depends on the initial imperfection. This is because in practical sense all the bending moment does is induce an initial local imper- fection. Once the buckle was initiated their solution became unstable and no data, of usable value, for the propagation pressure are presented in their paper. Possible modifications to their loading and procedure are suggested at the end of this chapter. Although solution of the problem numerically, through a large deformation shell numerical analysis with plasticity included, is feasible this route was avoided due to the com- plicated numerical work necessary. Instead it was decided to look at various simplified models of the problem with a varying degree of complexity. The purpose of these models was first to understand the governing physical processes that go on when the buckle quasistatically propagated a dis- tance x; second, to look in more detail at the structure of £ in equation (2.2) and third, to obtain, if possible, an approximate theoretical expression for Po 2.3.1 One-dimensional Model As a first approximation to the problem, collapse of a circular ring (or infinitely long pipe) under external pres- sure is examined. At first this is done through the limit load placticity theory and at a more advanced stage the problem is examined numerically. (a) Collapse load of a circular ring under external pressure, Consider a ring of radius R, thickness t, under external pressure P. The limit theorems of plasticity will be used to establish bounds on the collapse load of this ring. (See refs. 2.3, 2.4, and 2.5.) (i) Upper Bound Theorem Let the ring material be idealized to a rigid (or elastic)-perfectly plastic one. Consider the kinematically admissible deformation field shown in fig. 2.9a. Four hinges are placed as shown and the ring quadrant between the hinges are made rigid. The system becomes a mechanism whose motion is restricted to the one shown in fig. 2.9b. The moment at the hinges is assumed to be the full plastic moment = Oot? Mo FIG. 2.9. A KINEMATICALLY ADMISSIBLE DEFORMATION FIELD FOR A COLLAPSING RING ~ 22 - If the hinges are rotated as shown in fig. 2.9b through an angle 6 the internal energy dissipated is given by The volume change is given by AV = 2R*{1 - sin (GF - 6)} or (2.3.2) - oR? § | av= aR {1- - 9) +29 -(S)71 J Since 8 e=1- sin t (- _R Z, 75 (2.3.3) | Thus the external work done is w= Pav = 2PpR? {1 - (1 - Dlr + 2% -(C)* FP 2.3.4) The upper bound limit theorem says that "the structure must collapse if for any kinematically admissible deformation the rate at which the external forces do work is equal (or bigger) to the rate of internal dissipation." Thus for no collapse from (2.3.1), (2.3.3) and (2.3.4) no. -1f1 = 8/R Note: from (2.3.5) = ] = P= 7 Go ri] o> + 0 = pr @ rad] Or Thus mt (t\? Poollapse “4 oo(#) (ii) Lower Bound Theorem The same assumptions are made for the ring as in (i). Due to the double symmetry consider the equilibrium of only one quadrant as shown in fig. 2.10. = —_ a MK Mo FIG. 2.10. STATICALLY ADMISSIBLE STRESS FIELD H, V, and Mo define a statically admissible force field if the ring is in equilibrium. — ( ; :) (2.3.6) M, = 5 CPR? - HR (1 - si © (2 ~ | (x) The lower bound theorem states that if the internal forces equilibriate the external forces and the material is at yield or below yield the structure will not collapse, Note +0> P + & (2.3.7) | o> 2 + L> P > 5008) Tf we thus combine the results of (2.3.5) and (2.3.7) and A] o> define collapse when . = 1, then 1 t\? 1 t 700 (R) < Pf collapse ©“ 4 °° (i) (2.3.8) Expressions (2.3.5) and (2.3.7) are plotted as functions of A (1 = 2 x): in fig. 2.11. The problem of collanse of R a circular ring under external pressure is solved numerically in the next section. The post buckling displacement of point A is also plotted on fig. 2.11. (Note a = 10*.) Excluding the behavior for A < .1 the equilibrium path of the statically admissible path described above is very close to the exact one. For A > .1 the error in pressure is 1.5 < Ap < 2.5%, but is always positive. According to the lower bound theorem the error should be negative. It is thought that since the error is so small this discrepancy is probably due to trunca- tion and roundoff errors in the numerical scheme used to solve the problem exactly. The kinematically admissible mechanism gives a clear upper bound at all points. If we associate the propagation pressure with the col- lapse pressure and remember that we have assumed a rigid ~ 25 - perfectly plastic material behavior from (2.3.8), we can say that AN 1 1 (E)’ 1 tT o(E) — 5 Oole)] < P OO Sele) | (1 _ yt2 2 0 R —_ p - (1 _ tae R ‘ i i | (2.3.9) WIN | <A < 1.047 | P= a(t) Oo As expected this model does not give us a good representa- tion of the propagation pressure. The material o-e behavior is represented by only one parameter oo) and the power of (R) is only 2. Of course this is to be expected due to the fact that the third dimension of the problem has been neglected. If we take the maximum value for A then in fig. 2.12 we see the difference between the experimental and the predicted result. As a next step we look at the exact solution of the problem of the collapse of a ring, numerically. Examining the collapsed cross sectional configuration of a buckled pipe in fig. 2.6 we see that the configuration of the actual cross section is not very different from what assumed, if the region close to the plastic hinge is neglected. In fact the four quadrants remain practically undeformed on collapse. This leads us to the next problem where the effects of including the complete o-c behavior (Elastic-Linear strain hardening) to the problem of ring collapse under external pressure are examined numerically. - 26 - (b) Large Deformation Collapse Analysis of an Inelastic Ring Under External Pressure In the last section the collapse analysis of a ring was examined assuming a rigid perfectly plastic material and using theorems of limit load plasticity. It has already been men- tioned that the effect of including the complete o-e behavior in the problem is probably critical. Since the complete problem will not be solved it was thought prudent to examine the effect this has on at least the ring model which, after all, includes the fundamental mode of deformation that the pipe undergoes (Strip Theory Approximation). For this reason the large deformation response of an inextentional ring under external pressure is examined. The nonlinear equilibrium equations are used and the material behavior is approximated by an elastic linear strain hardening o-e curve symmetric about the zero strain axis. The problem is solved numerically and the deformations are followed until complete collapse of the ring. A parametric study of the problem is carried out in order to examine the dependence of the collapse load on the various parameters of the problem. (c) The Problem Consider a circular ring of radius R and thickness t, as shown in fig. 2.13. The ring can have an initial imperfec- tion. For convenience this will be taken to be of the form W; = W, cos 206 (2.3.10) - 27 - Equilibrium Equations The nonlinear equilibrium equations can be obtained by considering an elemental segment of the ring as shown in . fig. 2.14. dH ds = ~P cos @ dV _ . dM Hl dg H cos @ + V sin 8 H and V are the horizontal and vertical forces whereas M is the moment as defined by fig.2.14. S is the coordinate along the midsurface of the ring and 8 is the angle the normal at any point makes with the x-axis. Geometry Only inextensional deformations of the ring will be considered, asa result dx dg = sin 6 (2.3.12) x = cos @ where (x, y) defines a point on the deformed ring relative to the Cartesian frame shown in fig. 2.14. Constitutive Behavior The main purpose of this exercise is not to fit any stress strain curve exactly, but to see how the response of the ring depends on some well defined material parameter. - 28 - For this reason a material with o-e curve as shown in fig. 2.15 was chosen. The symmetry of the O-€ curve about the zero strain axis should be noted. The inelastic moment curvature relationship for such a material is derived in Appendix A and is given by 2 (2.3.13) = dee _ do dS «dS This is the nonlinear curvature displacement relation- . 8 . ees ship for a curved beam. ave is the initial curvature. Ko = fe fm = Wt} Fe (2.3.13b) . K With these definitions the moment curvature relationship becomes c? + (3(@ - 1) - oo? - x(a - 1) = 0 ? ; (2.3.14a) M=m/<e Let the required root of this cubic be G = G(a, M) (2.3.14b) Note that G(1, M) = M. (2.3.12b), (2.3.14a), (2.3.14b) = (2.3.15) - 29 - Because of symmetry about the x-axis and y-axis only one quadrant of the ring need be considered. The boundary conditions then are: H(O) = 0 0 (5R) = 1/2 0.4.16) 6(0) = 0 v(5R) = 0 - Non-dimensionalizing equations (2.3.11), (2.3.15) and (2.3.16) appropriately, one obtains: dH ds -Q cos 6 0 <s< 1 w = -Q sin 0 a = H cos 6 + V sin 6 is ~ “as > “e) (#) c@ # ds ds ae (SE Gla, M) (2.3.17) G1, M) = M G(o, M) = M for M< 1 with B. C. H(O) = 0 8(0) = 0 a(1) = 1/2 V(1l) = 0 In addition (2.3.12) becomes * = - sin Ss) O<s<«l s SSS (2.3.18) aie, SiR tt cos 8 - 30 - x(l) = 0 y(0) = 0 where s = 2S/mR , x= 2x/mR , y = 2y/mR , (2,3,19) M = 6M/oot® | H = 30HR/oot? V = 30VR/oot? | g = 3n2(8}(2 2 t 5) (2.3.17) can be expressed in vector form as: du ds = £(s, u) , — O<s<l u= CH, V, M9) , | (2.3.20) u,(0) = 0 , u,(l) = 0 , u,(0) = 0 , u,(1) = 1/2 (2.3.20) describes aset of four nonlinear ordinary differential equations in the form of a two point boundary value problem. Newton's iterative numerical method is used to solve for u. The interval s€|o, 1| is discretized into N discrete points where 26 < N < 50, depending on the situation. The convergence criterion used was as follows: <10°* (v4) (v) max | u. - u, | j J J The numerical method used is described in more detail in ref. 2.6. The solution of the elastic small deformation linearized (9) problem --ref. 2.7 -- was used as initial guess u_ to start ew 31 - the iteration. The calculation of the initial guess is shown in Appendix B: The pressure parameter Q was increased by AQ and the solution at Q was used as initial guess for Q+AQ, i.e., u (Qu (Q + AQ) (i) Linearly Elastic Case (a = 1) Linearly elastic material behavior is obtained if a = 1 in (2.3.17). The procedure described above is used to find the different configurations as a function of pressure (see fig. 2.16). The ring was assumed to have collapsed when (y(1), 0) = (0, 0). Figure 2.16 shows how the displacement (non-dimensional) of the point (y(1), 0) varies with pressure. The same study was done for five different values of the im- perfection. For zero imperfection the following procedure was used for solution: Define uU” = w(Q, we) the vth iteration value of u 7 pressure Q and imperfection wo R- Suppose that on the vth iteration the solution converged. Using this definition (v) (0) u (1.5, .009) + ui (1.5, 0) v) 0) u (1.5, 0) + u (1.4, 0) a’ (1.4, 0): + £9 .3, 0), ete. Of course once the first point on the curve was obtained then the same procedure was followed as before. The problem of collapse of an elastic ring with no imperfection has been studied in detail by ref,2.8, where the -~ 32 - existence of solution is proved both above and below P/P. =], In the same reference Newton's method was used to find the deformed shape as a function of pressure. For collapse as defined above, P/P. = 1,68. This compares with a value of 1.75 found in our case. The error depends both on the mesh as well as on the convergence criterion. More recently Budiansky and Sills [2.9] looked at the "initial post-buckling" of a ring with no imperfection, under external pressure. They showed that the buckling pressure is not imperfection sensitive. This was in disagreement with the results of ref. 2.10. The complete pressure-displacement curves clearly agree with the conclusion of Budiansky and Sills. Pressure vs. the volume change curves have also been drawn and have the same nature as those of fig. 2.1.6 as a result they are not included. (ii) Linear Strain Hardening Case (a > 1) T£ a>1 then G = G(a, M) is used in equations (2.3.17) where G is the solution of (2.3.14a). For most metals of interest ae(50, 250). If this range of a is considered then the pressure-displacement response changes drastically in nature. A distinct limit load appears and the ring becomes unstable beyond this point. Using load increments (AQ) the solution had difficulty converging for points around the limit point. Arbocz in ref. 2.11, facing the same problem but using the double shooting technique, successfully modified his equations so that increments of deformation were prescribed - 33 - instead of load. This enabled him to extend the curve beyond the limit point. The same idea is used in this case but a different constraint is used. From (2.3.18) d _ ae = cos 6 , y(O) = 0 Suppose we add the constraint that y(1) = A , where A is to be prescribed each time. Since displacement cannot be prescribed simultaneously, and traction at a point, Qis treated as a new unknown to the problem, Since Q = const dQ _ de = 0 Equations (2.3.17) then become ce = - Q cos 86 sv = - Q sin 6 O<s<l _ (2.3.21) a = H cos 8+ V sind d9 _ d0_ _ W(®)(q)eca if) GQ, ) == ds s t)\E , , - _ Gia, My =M M<i1 oa = cos 6 dQ _ g - 34 - with B. C's, H(O) = 0 6(0) = 0 6(1) = 7/2 V(1) = 0 y(0) = 0 y(l) =A (A prescribed each time) In addition dx ds - sind, x(l)=0 . (2.3.22) A separate program was written for equations (2.1). They were solved in exactly the same manner, The procedure for the complete collapse of the tube was as follows: Using increments of Q (i.e., eqns. (2.3.17)), the pressure was taken up to point k (see fig. 2.17), asmall distance below the limit point. The solution usually diverged at this point. The last converged solution (i.e., solution at pointk) was used as initial guess for the next program where increments of displacement A were prescribed. Figure2.17 shows in detail a typical example of the curve around the limit point. Increments of displacement were used until complete collapse of the ring. Figure2.16 shows how the pressure displacement graphs change with the introduction of linear strain hardening. A distinct limit point is present in each case, The maximum pressure reached in each case is taken to be the buckling pressure. The post-buckling behavior is unstable. On ~ 35 - collapse some residual resistance to pressure remains in the tube. The minimum pressure always occurs for A < 2/n (collapse condition). Figure 2.18 depicts a quadrant of the ring at different stages of collapse. The spread of plasticity along the length of the ring is also shown. The region of maximum moment , i.e., at s = 0 goes plastic first, followed by the region at s = 1. As the deformation continues the two regions spread inwards as shown in the figure. Fig. 2.19 shows a typical collapse sequence of a complete ring. Figure 2.20 shows how the moment at s = 0 (point of maximum moment for all values of A) varies with displacement. It should be noted that no unloading occurs at any value of A. This is true everywhere along the ring. This fact helps simplify the numerical analysis considerably as no unloading capability had to be added to the program. The variation of forces at s = 0 and 1 with displacement are shown in fig. 2.21, Since these are the main reacting forces for the ex- ternal pressure they follow the pattern of the pressure behavior with displacement A. The moment distributions around the perimeter of a ring quadrant just before buckling and on final collapse are shown in fig. 2.22. Similar plots for the horizontal and vertical forces are shown in fig. 2.23. - 36 - (iii) Parametric Analysis The purpose of carrying out the above analysis was two fold. First to find the dependence of the collapse load on the various parameters of the problem and second to find the buckling pressure of a tube with initial imperfection, as a function of the initial imperfection and D/t. The latter is needed for analyzing the the "Flip-Flop" mode of dynamic. propa- gation described in chapter 4. It so happens that the buckling pressure is a direct by-product of the analysis required for the first problem. Collapse pressure will be defined as the pressure at which the two points of maximum displacement finally touch. (See fig. 2.19). A parametric study of the collapse load has been carried out to determine its dependence to the various parameters of the problem. For the inelastic case, buckling pressure is defined as the maximum value of pressure reached, or as the maximum pressure for which re = 0. Figure 2.16 shows how the collapse behavior varies with different values of initial imperfec- tion. Although the initial imperfection affects very much the buckling load, it has practically no effect on the post buckling behavior and absolutely no effect on the collapse load. It is of interest to compare this behavior with that of the elastic case, where of course the post buckling be- havior is stable. Obviously the buckling load in this case is not imperfection sensitive. The initial imperfection has minimal effect on the collapse load. Varying - 37 - E changes the buckling load. As a result the prebuckling be- havior is affected. The post buckling behavior very quickly converges to exactly the same values for all values of E. As a result the collapse load does not depend on E. Figure 2.25 shows how the post buckling behavior is affected by the value of a. a = 1 represents the elastic case and a = 10" represents what is practically an elastic- perfectly plastic material behavior. It can be seen that the effect of strain hardening on the buckling load is insignifi- cant. On the other hand the post buckling behavior, which after all takes place totally in the plastic regime (at least for the ring parameters considered) , is critically affected by strain hardening. Figure 2.26 shows how the final collapse shape is affected by the same parameter. For bigger values of a the radius of curvature at the pointed ends becomes smaller increasing the possibility of cracking at these points due to material failure. Such cracking leads to flooding of the pipe (Wet Buckle) which is highly undesirable. This subject obviously warrants further investigation. The next parameter to be varied is D/t where the values of D/t from 26 to 100 were examined. This is the region of interest for pipeline application. Also for D/t < 25 the effect of membrane stresses cannot be ignored. A study was done for four different values of a and the results are plotted on log-log scales in fig. 2.27. Straight line results are obtained indicating a power law dependence - 38 - of the collapse pressure on this parameter. The powers 8 of t/D were as shown below. Gi 8B 50 2.207 80 2.162 200 2.108 10000 2.05 It is important to note that although good agreement exists in the powers of these curves they are shifted in the log-log space giving quite different collapse load for the same D/t. Comparing these powers with those obtained from the experiments (from fig. 2.7) we find them a little smaller. (Steel B = 2.25; AX B = 2.28.) It is important to keep in mind though that this analysis can only be considered as a strip theory representation of the full three-dimensional problem. Although the power of the (t/D) parameter is not exactly the same the shift in the curves observed in the experiments is repeated in the analytic results. For the standard ring (D/t = 35.7, oo/E = 4.2 x 10 °) the value of a was varied further in order to see in detail its effect on the collapse pressure. For the imperfection considered (wo/R = 8 x 10 *) the collapse process is always unstable if a > 1.2. The value of P/P, at collapse varies from 1.73 in the elastic case to 0.145 in the elastic-perfectly plastic case (a = 10*). The dependence of the collapse Pressure P,.; on a is shown in fig. 2.28. Clearly its effect ~ 39- is substantial. Materials of interest (i.e., aluminum and steel alloys) have ace(50,200) which as shown in fig. 2.28 is the region of most variation of the collapse pressure. The next parameter to be varied was o,/E and the results are shown in fig. 2.29. Varying the yield stress changes the point of instability initiation along the elastic collapse curve. It is interesting to note that only a small increment in pressure is necessary for collapse to occur once the first (outside) fiber of the ring yields. The post buckling path is also affected and consequently the collapse pressure P is affected. The normalized collapse load is plotted vs. the normalized yield stress in fig. 2.30. Clearly the collapse load varies linearly with the yield stress parameter at least for the range of this parameter examined (range of interest for pipeline applications). Summarizing some of the above results: It has been shown that the collapse pressure does not depend on E but on the other hand it is heavily affected by E/E' and o,/E. It can thus be concluded that the ring collapse pressure P has a parametric behavior as follows: woh. £(E) (8) (2.3.24) Oo Oo Pool /o is plotted vs. E'/o9 in figs. 2.31 (a) and (b) for two ranges of E'/o,). Approximating this curve with a straight line the following expression is obtained for the collapse pressure of the inelastic ring. - 40 - a E + o(=-}]() (2.3.25) fe) oO where a = 4.6 B = 2.2 As already mentioned a one dimensional model suchas the one presented above cannot by itself represent the behavior of the Propagating Buckle. In particular, the profile through which the pipe transforms itself to a dogbone type of cross section undergoes bending and stretching deformations in the longitudinal sense; and these cannot be considered by a one-dimensional model. Even so, it is clear that the one-dimensional model represents by far the dominant effect. For this reason it is thought prudent to generalize conclusions about its behavior to the behavior of the complete problem. For instance, the one-dimensional problem predicts a linear dependence of the collapse load on the yield stress 0, (actually OU/E but E does not affect the collapse load). This of course is in agreement with result (2.2.2) drawn from the experimental results. A second important conclusion is that the collapse load of the ring is strongly affected by the strain hardening parameter E'/o,. A third important conclusion is that due to the nature of the solution the unstable part of the load- displacement curve is practically independent of the value of E. In fact the collapse load is completely independent of E. In addition to the above, as expected, the magnitude of the initial imperfection has no effect on the collapse load. - 41 - 9.3.2 A three-dimensional model In section 2.3.1(a) a limit analysis on a one-dimensional ring model was carried out which gave upper and lower bounds to the collapse pressure of that model. In addition to the single dimension simplification an inextensionality condition was also used. In this section the problem is examined through a more realistic geometry by considering the energy balance of a slowly propagating buckle front. The geometry of a typical buckle front is shown in fig. 1.5. The idealized geometry is shown in fig. C.1; this geometry represents a kinematically admissible one. The strain energy required by the material when the buckle. moves forward a unit distance is calculated. The following assumptions are made (a) The kinematically admissible geometry of fig. C.1 is assumed. (b) The material is assumed to have a rigid perfectly plastic material behavior. (c) Circumferential deformations are assumed to be inextensional. (d) Strain energy going into bending and membrane defor- mations of longitudinal fibers are not neglected. As shown in fig. C.1 the buckle front has a length a. It represents a transition region from a circular to a dogbone eross section of the pipe and is divided into regions A and B. Each region is assumed to undergo a uniform extension and the curvature does not vary in the circumferential direction. The longitudinal fibers of each region are assumed to have a cosine z dependence. This dependence was found to be realistic enough ~ 42 - when compared to real life buckle fronts. From these assump- tions the total strain energy required for the buckle front to move a unit distance forward can be calculated as U = U(oo, t, R, a)/unit pipe length flattened . The above expression then includes contributions from circumferential and longitudinal bending strain energies as well as energy gone to longitudinal membrane deformations. The detailed calculations of U can be found in Appendix C. The external work done can be calculated in the same way as the one dimensional problem and can be given by W = W(P, R) = 2PR/Unit pipe flattened From Appendix C 95 2 3 2 U = Sir ot? + e T (2) Oot? + (1 - oe) i (2) ooRt U=W => P=% oo(£) [1 + 3 (8) +3 - /2)07(B) (8) (23,26) To find the dependence of P on R/t a relationship between R/t and R/a is necessary. It has not been possible to find such an expression theoretically; as a result the experimental values of fig. 4.3are used in order to compare the values calculated from (2.3.26) with the experimental results for Pp. It should be noted that the numerical values of this calculation are restricted to A&-6061-T6 for which the measurements were made. The results are plotted in fig. 2,12, This provides a reasonably good upper bound for the value of Po The error is of the order of 15% and is bigger for bigger D/t'’s. The dominant term in - 43 - expression (2.3.26) is the first one (Circumferential bending). The second and third terms increase the value of P/o»o by about 8%. Due to the fact that this is an upper bound neglecting the second and third terms brings the answer closer to the experi- mental results, especially for lower D/t's. 2.4 Conclusions The value of the pipe propagation pressure (P) was found experimentally for both aluminum and steel alloys. Trying to find a parametric dependence for Po it was noted that it is proportional to oo and to some power of (t/R). However, describing the complete material o-e behavior of the pipe by only o9 was shown to be inadequate. From limit plasticity analysis of a one-dimensional ring model, upper and lower bounds were found for the collapse pressure which can be associated -- as a first approximation -- to the propagation pressure. A more extensive three- dimensional model indicated that including longitudinal fiber bending and membrane deformations increased the predicted upper bound by 8%. However, these models suffer from the inherent problem of single parameter characterization of the pipe material behavior. The above models, even though crude, have shown that the circumferential bending represents the dominant deformation mechanism. With this in mind the behavior of the ring model was considered using the nonlinear equations of equilibrium and large deformation kinematics (small strains). This was done for a class of pipe materials having elastic-linear -~ 44 - strain hardening material behavior. A parametric study of the collapse pressure was carried out and it was found that com- pared to the experimental results relatively good agreement was found as far as the parameters R/t and o)/E are concerned, but in addition the collapse pressure depends on E'/o,, in a simple algebraic form shown in (2.3.25). Based on the above results the Propagation Pressure is assumed to have the same dependence on E'/oo as in (2.3.25). The values of (a) and (b) can be calculated by assuming that for the aluminumalloys used in the experiments a = 80 and for the steel alloys a = 200. The Propagation Pressure expression can then be approximated as: 2.25 P - [ 10.7 + 0.54 ()]( ) (2.4.1) 10) oO C D - 45 - 3. DYNAMICS OF THE PROPAGATING BUCKLE 3.1 Introduction In the preceding chapter buckle propagation under quasi- static conditions was discussed. Although that study is very important in establishing a criterion for the Propagation Pressure of the pipe, a physically more relevant examination of the problem cannot ignore the dynamics of the problem. It has already been stated that the Propagating Buckle phenomen.n can only occur at pressures P such that P< P< P p- — "ec If a buckle is initiated at any pressure above PL then it will propagate at a velocity U which is a function of the various parameters of the problem. It becomes clear then that quanti- fying the velocity is important in establishing the correct arrest conditions when trying to design arresting devices. An attempt to quantify the velocity of propagation is given in ref. 3.1. An expression is presented which gives the velocity as a function of the parameters of the problem. This expression was derived (personai communication with T.G. Johns) by considering a steady state propagation of an elastic, small displacement axisymmetric disturbance in an infinitely long circular pipe. This model has two important drawbacks. First it does not take into account the inertial effects of the fluid and secondly it is restricted to elastic deforma- tions. As established in chapter 2, the Propagating Buckle phenomenon is characterized above all by its dependence on the post yield stress-strain behavior of the pipe material. - 46 - Neglecting this in modeling the phenomenon is inappropriate. In what follows, an effort is made to treat the problem in a more general manner by including important parameters neglected in ref. 3.1. The objective is to obtain empirical relations for the parametric dependence of the velocity. Again the experimental path of investigation is followed. The main parameters of the problem are shown in fig. 3.1. In addition to those considered in the quasistatic case, structural as well as hydrodynamic inertial parameters are included. Some of the problems considered are for instance the variation of the buckle profile length with pressure the examination as to whether a steady state propagation con- dition is reached and the determination of the acceleration time to the steady state condition. High speed photography is used in order to observe the propagation and to investi- gate whether hydrodynamic phenomena such as separation or cavitation are present. With the guidance of these experi- mental results an effort is made to obtain a useful parametric relation with an energy balance type of approach. 3,2 Experimental Determination of the Velocity of Propagation These experiments were the first to be carried out and the equipment used was not optimal. Most of the experiments were carried out in the high pressure test facility shown on fig. 3.2. It con- sists of a high pressure tank (3000 psi-207 bar-working pressure) having internal diameter of 22 in.(.56m) andheight of 10.5 ft (3.2m). Access into the tank was provided by a 2.5 in (.064m) opening at the bottom end and two 1.25 in(.032m) openings _ 47 - at the top end. These restricted the pipe dimensions to be tested to maximum length of 8 ft (2.44m) and maximum diameter of 1.5 in (0.038m). The systemwith its hydraulic connections was pressure tested to 2000 psi (138 bar). The facility can operate with wet or dry pressurizing medium (air or water). For experiments in water the tank was filled with water but pressurized with compressed air. A suf- ficiently large air pocket was left to insure nearly constant pressure experiments. (Pressure drop from the beginning to the end of the experiment was 2-3%.) The pipe to be tested was placed into the tank through an opening in the floor. Lead weights were hung on the bottom of the pipe to keep it under the water surface at all times. The tank was filled with water to the necessary level and then pressurized with compressed air supplied from compressed air (or nitrogen) bottles (max. pressure 2200 psi (152 bar)). Apressure regulator kept the pressure at the required level. Pressure gages on the feed line provided pressure reading. In addition a pressure transducer, suitably calibrated, was connected to a digital voltmeter and a strip chart recorder provided continuous pressure reading. For experiments in air the tank (capacity 25 cuft (.7 cum)) was pressurized solely by air supplied from pressurized air bottles. Depending on the specific operating pressure, up to four bottles were used to pressurize the tank and the pres- surization process could take as long as one hour. At the latter stages of these experiments an air compressor was ~ 48 - added to the system but its maximum pressure was only 250 psi, (17.2 bar). The pressure monitoring systemwas the same as above. The pipes were plugged at both ends. In some experiments the pipe was connected to the atmosphere to ensure (nearly) constant pressure experiments. Due to the tank construction it was practically impossible to inflict an imperfection ex- ternally in order to start the buckle; as a result an imper- fection had to be machined on the pipe, providing a weaker section that could cause local buckling at a predetermined pressure, With careful machining this imperfection could be designed to buckle with an accuracy of + 5%. The machined imperfection was usually about 4-5 diameters long (fig. 3.3b), placed about 5-6 diameters from one end. The buckling pressure of the imperfection was calculated using expression 17 from ref. 3.2. In cases where accuracy of more than + 5% error was necessary the following technique was used. An 1 in. plug of a lower melting point alloy (melting point 150°F (66° C)) was molded inside the tube under the imperfection. Aheating coil was placed around the imperfection as shown in fig. 3.3b. The pressure in the tank was taken up to the required level and kept constant. Current was passed through the heating coil. The plug melted removing the lateral support it was providing to the weakened pipe section, allowing it to locally buckle under the exact external pressure required, thus initiating a buckle. Considering that the process had to be carried out inside the tank with a wet environment it is easy to see that it was a very inefficient process. At the first opportunity the ~ 49 - facility described in section 2.2.1 was built. Due to the design of this facility, it was relatively easy to inflict a damage to the pipe at a specific pressure externally. The details of the design and operation of this facility are explained elsewhere. The experimental efficiency of the dynamic experiments was increased by a factor of 5 and all dynamic experiments apart from the ones described in this section were carried in this improved facility. The velocity of propagation was measured by means of four strain gages attached externally to the tube at measured intervals. As the propagating buckle passed through the location of a strain gage a large response signal was obtained. In cases where the pipes were immersed in water, a special insulation had to be used to insulate the gages from the water (Micro-measurement M-Coat F). This insulation provided practically no resistance to bending or stretching and it did not affect the smooth propagation of the buckle in any sig- nificant way. The signals were amplified (Astro DATA 810 DC differen- tial amplifiers) and subsequently recorded on a four channel variable speed HP 3100 tape recorder. The recording pro- cedure was as follows. When the pressure was within about 10% of the precalculated buckling pressure of the machined initiator, the instruments were switched on and the recorder started. The pressure was then allowed to rise slowly until buckling of the initiator occurred. This in turn caused a propagating buckle. The tape recorder and chart recorder were switched off. The exact pressure at which the measure- ments were taken could be found from the pressure transducer which indicated a sudden drop in pressure in the tank. (When the tube collapses the volume it displaces becomes approximately 30% of its original volume.) The recorded strain gage signals were played back at 15 times slower speed. The output of the recorder was displayed on a scope as well as on a high frequency response light pen galvanometer chart recorder (up to 80 in/sec paper feed). Through this procedure the time base of the signals could be stretched as much as necessary decreasing the measurement error. The period between two signals was measured and the average velocity between the two points could be calcu- lated. Figure 3.4 shows a typical visicorder output and a schematic of the corresponding tube with the strain gages in place. Typically, the first of the four strain gages was placed 6-8 diameters downstream of the initiator section. The second and third were at intervals of 2 ft (.15m) apart and the fourth was placed about 15 diameters from the other end plug. Thus, three values of velocity could usually be obtained although in some cases the elastic wave travelling ahead of the buckle, or the buckle itself could shear off one or two of the gages or break one of the electrical connections. In such a case fewer values of velocity were obtained. Usually the velocity obtained from the first two gage signals was taken to be the velocity of propagation. If the pipe interior - 51 - was connected to the atmosphere then the three values of velocity were usually within a 5% difference band with the first value being the biggest and the third the smallest (this was mainly due to the decrease in tank pressure caused by the decrease in displaced volume by the tube). In cases where the pipe was not connected to the atmosphere then the internal pressure at the moment the buckle reached the gage could be approximated by 1 Y P= Pa [s> oy] g : where L = Length of Pipe Ly = Distance of gage from point of buckle initiation Pot = Atmospheric Pressure y = Specific gas constant (Note: Adiabatic change of state is assumed.) Thus the internal pressure when the buckle reaches the second gage can be calculated to be about 3 psi higher than the atmospheric pressure. The maximum error produced by this factor in the cases examined was about 3%. However, the error produced to the value of velocity obtained by the third and fourth gages was considerably higher, as a result when- ever this value of velocity had to be used corrections had to be made to the pressure to compensate for the above error. 3.2.1 Velocity measurement -- results. The characteristics of the pipes tested are tabulated in table 3-1. In the first series of experiments the velocity of - 52 _ propagation was measured as a function of pressure for a cons- tant D/t (= 35.7). This was carried out for both air and water to see the significance of the fluid structure inter- action. Figure 3.5 shows the experimental results. The ex- periments covered the whole range of pressure between the propa- gation pressure (115.8 psi (10.7 bar)) and the buckling pressure (488 psi (33.7 bar)). Due to inadequacies of the facility usedit was not possible to carry out experiments below PIP, = 1.95. This was mainly due to the fact that near Po whether an initial dent propagates or not depends heavily on the type of dent inflicted on the pipe. Specifically it was found that diagonally opposite longitudinal dents tend to help the initiation of the buckle as opposed to transverse dents. At pressures above about 435 psi (30 bar) anew mode of propagation was discovered; this has been called the "Flip-Flop" mode of collapse and is discussed in detail in chapter 4. By examining the time intervals between the different gages along the tube it was established that the buckle, in a relatively short time (6-8 diameters), accelerates to steady state. Provided that all other variables are kept constant, it continues travelling at a constant velocity. As the results indicate, the velocity is dependent on pressure as well as on the density of the pressurizing medium. The velocity of the propagating buckle in water is less than half of that in air, indicating the considerable importance of the fluid structure interaction problem. The same data have also been plotted against a pressure parameter stemming - 53 - from the energy balance calculations that follow. Using this parameter the data fall on a straight line (see fig. 3.6), but the experimental lines fail to go through the origin. The reason for this is still unexplained. Reference 3.1 makes an attempt to calculate the velocity of propagation (details are not presented). The prediction of this reference is plotted on fig. 3.6. Comparing this analysis with the experi- mental results, the theory is inadequate as it neither pre- dicts the correct pressure dependence nor is it sensitive to pressurizing medium density changes. Additional results for different D/t are shown in fig. 3.7. It is of interest to note that for thicker pipes the buckle travels faster in both air and water. Note also that the various straight lines are parallel to each other. The dependence of velocity on D/t for constant P/P, (= 1.7) is shown in fig. 3.8. During the course of this investigation it was noticed that the length of the strain gage signals varied substan- tially with velocity. For one series of experiments (speci- men 2 of table 3-1) careful measurement of the signal lengths was carried out and the length of the buckle profile was calculated as follows. Referring to fig. 3.4, let the length of a typical signal (usually the average value of the first two gages was taken) be T sec. It is known that in Ti sec. a distance X, ft is traveled by the buckle. Thus the length of the profile is given by ~ 54 - It should be noted that the length of the signal depends on the orientation of the buckle with respect to the gage posi- tion. Since in these early series of experiments no control was available as to the orientation of the buckle, some of the results had to be disregarded. Figure 3.9 shows a plot of the profile length as calculated vs. a pressure parameter. This pressure parameter is proportional to the velocity (different constants of proportionality for air and water). Although the quantitative value of these results is question- able, qualitatively they indicate that the profile "sharpens" with higher velocity. This conclusion is tested in the next section. 3.3 High Speed Photographic Observation of a Propagating Buckle Although quasistatic observation of the propagating buckle had been observed both by the author as well as other investigators, no effort had been made to observe a buckle propagating under dynamic conditions. Such an endeavor was worthwhile for a number of reasons. For instance, nothing was really known about the fluid behavior at the buckle front. The possibility of cavitation or other fluid effects required direct observation. The problem of the "flip-flop" mode of collapse needed direct observation in order to verify a theory as to its mechanism. Another problem requiring further experimental examination was the velocity dependance of the length of the profile of the propagating buckle. For these reasons and others the transparent test facility described earlier was designed and fabricated. - 55 - Figure 3.10 represents a schematic of the circuit used to initiate and photograph a propagating buckle. With switches a and b closed and the knife edge retracted the camera was started. At a prearranged time, toe after the camera was started, the microswitch would automatically close, activat- ing the solenoid valve which operated the piston and the knife edge would move down, locally crushing the tube. Thus a buckle would be initiated. The buckle very quickly would accelerate to its steady state condition and enter the camera viewing field at a constant velocity. By this time the camera would be running at the required speed capturing the phenomenon. From the above description it can be seen that the process required reasonably good timing for a successful experiment. As a result some side experiments had to be carried out to determine the order of magnitude of the time elapsed from the moment of triggering to the moment the buckle entered the camera viewing field. This was done as follows. A micro- switch was connected to the pneumatic cylinder and arranged to open when the piston moved lmm down. This microswitch triggered a scope as well as a counter. A strain gage placed on the pipe in the middle of the camera viewing field gave a signal when the buckle reached that point. This amplified signal was recorded on the scope and triggered off the counter. Thus two measures of the necessary period were obtained. This time was found to be about 40 msec for thespecified tube usedinthe experiments. (Pressure was 180 psi (12.4bar)). This - 56 - time transformed into feet of film had to be taken into con- sideration when arranging the delay system of the camera in order to obtain the correct framing rate in the period of interest. Numerous tests were also carried out in order to test the uniformity as well as the amount of light necessary for the photography. 3.3.1 Experimental results. A total of three films of the propagating buckle were obtained. The main parameters of each experiment are listed in table 3.2, below. Pres. P D Frames/| U ft/sec a Medium Po _t sec (m/sec) D 350 1 Water 878 50 3,250 (107) 5 372 2 Water .902 50 4,500 (113) 5 . 726 3 Air . 890 50 11,500 (221) 4 TABLE 3.2. EXPERIMENTAL PARAMETERS OF PHOTOGRAPHIC EXPERIMENTS Two experiments were run with water as a pressurizing medium in order to investigate the possibility of cavitation. Figure 3.11 shows the results of experiment 2. No cavitation is visible. The velocity of propagation was calculated by knowing the width of the viewing field and the frame rate. It should be noted that in the cases where the tank was filled - 57 - ° with water, due to its cylindrical construction the ratio of the two principal magnifications was about 1.5. As a result the linear dimensions of the pipe are disproportionate. The velocities calculated here are in good agreement with those of fig. 3.7, where direct measurements were made by the strain gage signal method. As far as can be told from such a small viewing section (16 in. wide) the velocity of propagation does not change during the experiment, reinforcing previous results. By carefully looking at the pictures it can be seen that somewhere ahead of the profile the diameter of the pipe is bigger than the undisturbed diameter at the right hand side, This indicates ovalization with major axis perpendicular to that of the buckle. More will be mentioned about this in the next chapter. The film from experiment 1 has nothing to add to the above and is not included, Figure 3.12 shows the film obtained from experiment 3. This experiment was carried out in air, as a result the velocity is much higher. Again the value of velocity is in good agreement with the value calculated from fig. 3.7. The major axis of the buckle in this case is in the viewing plane. As a result this picture complements the previous one. As can be seen from the figure the profile is much sharper than the one from a quasi-static buckle (see fig. 3.13). The lengths of the profiles are measured in both cases and are tabulated in table 3.2. <A difference is noted from air to water. From fig. 4.3 the value of the a/D for the quasi-static case is found to be 7.75. This indicates that the profile length -~ 58 - decreases with velocity. No effort to further quantify this was made. 3.4 Parametric Dependence of the Velocity of Propagation As established in the experiments a buckle propagating in a constant pressure environment achieves steady state con- ditions (velocity, U) rather quickly (6 - 8 diameters). It would be useful if the parametric dependence of the velocity could be found. The complete problem involves all the non- linear effects described earlier. An energy balance type of analysis is attempted with the objective of obtaining some useful parametric relationships. | For the purpose of this analysis consider a buckle propa- gating at velocity U (fig. 3.1). The pipe has a diameter D, a thickness t and is made of a material with a stress-strain behavior that can be approximated by a bilinear one with parameters E, E' and oo, these having their usual meaning. The pipe material density is p. The surrounding medium is assumed to be inviscid, incompressible and of density Pe: As the buckle propagates with a velocity U, the structure absorbs a certain amount of energy in a given time interval AT. During this time the buckle moves forward a distance UAT. The energy absorbed by the structure is produced by the work done by the external pressure. In addition to the strain energy going into deforming the structure kinetic energy is important to the structure as well as the surrounding fluid. As energy balance gives a relation between velocity and the other parameters of the problem. The various terms of this - 59 energy balance are discussed below. As the buckle passes through a given location, the pipe deforms from a circular to a collapsed state (dogbone). If the material is assumed to be rigid-perfectly plastic, then the only characteristic material property is the yield stress oo. Assuming that only plastic bending deformations occur, a four hinge mechanism analysis ag in section 2.3.1 gives the plastic strain energy absorbed to be W ~ dot? UAT PL This term is appropriate provided that the pipe goes from its original circular shape to a developable shape without stretch- ing and plastic work is confined to the yield hinges. In actuality, stretching and bending of the longitudinal fibers do occur. As a result the plastic work is a function of the length (a) of the buckle profile and the pipe diameter (D). In addition, as shown for the quasistatic case in (2.3.25), the strain energy is also a function of E'/o,. The expression can then be written as = 2 Pa, E’) Wor, = Oot For = BD’ oo UAT. (3.1) The pipe wall will be accelerated as the buckle pro- &resses. This imparts a kinetic energy to the pipe which is assumed to be dissipated when the pipe collapses and impacts the other side. The amount of kinetic energy depends upon the shape of the buckle, however the parametric dependence can be expressed as follows: yn ap 2yeR Da NVuap Wp = pt?UFep (2, a \uar (3.2) where 0 is the pipe mass density. Likewise, energy goes into accelerating the surrounding fluid medium, which is assumed to follow the pipe deformations at all times. It is assumed that the kinetic energy imparted to the fluid is subsequently dissipated. The fluid kinetic energy term then can be written as = 2772 a Wy = 9 gD°U Fyp ( $) var (3.3) The work done by the surrounding medium consists of the pressure times the change in volume that occurs when the buckle moves UAT forward. This is given by Wpp = %PD*UAT (3.4) The energy balance is then given by: W = W pr 7 Wp t W xp + Wpy, (3.5) (3.1 - 3.4) + (3.5) = -oot2F,, + %PD? = U* pt* Fea + ae ws) PL 0-D*F Fe (3.6) KP If the propagation velocity is zero then the pressure is equal to the propagation pressure (P) - . _ D a E | cy’ i.e. PS = 200Fp, ( & DD Ge Db . (3.7) If the complicating effect of the profile length dependence on velocity is neglected then 5 = f (2) . Then from (2.4.1) -~ 61 - and (3.7) 125 F,( 2. Be) a; ;fro.7 + 0.5e(E)| 5) - G® (3.7) + (3.6) # Ui! Pe PL e Oo (3.9) Go Pp F (2) + £ (D\? (2) p KP \t re) t KF\ tC The general parametric dependence of the velocity can be seen in (3.9). For is given by (3.8). In the case where experi- ments are carried out in air = ~» 0(10°*), thus the second term in the denominator can be neglected. In that case an expression for F,, can be obtained by using (3.8) and fig. 3.8. The proportionality of the velocity to Ve - 1 was shown in fig. 3.7, where the experimental results are shown to behave in the predicted way. However, in all cases tested the straight lines failed to pass through the origin, indicating a different behavior for buckles propagating at pressures close to the propagation pressure. Both expression (3.9) as well as the experimental results in fig. 3.7 show the dependence of velocity on the pressurizing medium properties. 3.5 Conclusions Experiments have shown that a buckle propagating in a constant pressure environment rapidly reaches a constant velocity. The acceleration distance to this steady state was - 62- found to be of the order of 6-8 diameters. The parametric dependence of this velocity has been discussed through an energy balance. The velocity has been shown to be proportion- al to Poel . The experimental results showed this pro- P portional ity satisfactory over the range of pressure and in the experiment. However, this dependence could not be tested for buckles propagating very close to the propagation pressure. The velocity of propagation has also been shown experimentally to be influenced by the density of the pressurizing medium. For instance, differences of more than a factor of 2 were noticed in velocities of buckles propagating in water and air. High speed filming of the propagation in water did not reveal (at least for the cases tested) any cavitation occurring during propagation of the buckle. However, it was noticed that the length of the profile of the propagating buckle was velocity dependent. For buckles initiated at pressures close to the pipe buckling pressure a new mode of collapse and propagation was observed. This is described in detail in the next chapter. While the analysis and experiments described above do not lead to a conclusive solution of the dynamic problem, they do give the important parametric relations which should help in formulating an analysis of the problem. The completion of this analysis will probably dictate a numerical approach. - 63 - 4. THE "FLIP-FLOP" MODE OF COLLAPSE 4.1 Introduction "Flip-Flop'' is the name given to a new mode of collapse discovered in the process of carrying out the dynamic prop- agation experiments described previously. A picture of a pipe which suffered this collapse is shown in fig. 4.1. As can be seen from the figure the pipe underwent a number of changes in the orientation of the collapse mode. The buckle was always initiated by some means at one end of the pipe. Since the pipe was initially undisturbed, the phenomenon was always considered as a change in the mode of propagation. In other words the buckle once initiated starts travelling down the tube until after some distance it changes the orientation of collapse by exactly 90°. It again propagates a certain distance and repeats the change in collapse orien- tation returning to the mode in which it was originally travelling. The process is repeated a number of times until the end of the pipe is reached. Early in this investigation Dr. Martin Mikulas” pointed out that when a circular pipe is flattened locally, some distance away from the damaged point ovalization occurs with major axis perpendicular to the major axis of the flattened area. The same effect was noted in a series of papers on the collapse of short tubes under point loads by Reid, Johnson and others (ref. 4.1). Yuan obtained the same effect when examining the problem of point load on an infinitely long at “Visiting Associate in Aeronautics, 1977-78 - 64 - pipe [4.2, 4.3]. Although Yuan's problemwas restricted to small displacement linearized shell theorv, ovalizationwas ob- served some distance away fromthe point of load application. The reason for this kind of behavior is thought to be the fact that through such a deformation the local stretching in the tube is minimized. To examine the magnitude of this ovalization in the plastic deformation regime a series of experiments were carried out, where long tubes were subjected to local deformations by knife edge indentors as shown in fig. 4.2. The deformation of the cylinder along its principal axes was measured carefully for different values of indentation. The procedure was re- peated until the two pipe walls were touching at the point of maximum indentation. The results from one of these experi- ments are shown in exaggerated form in fig. 4.2. At a distance of about four diameters away from the point of indentation, the pipe ovalized with the major axis perpendicular to that of the indentation deformation. The region of ovalization was about five diameters in length with the maximum ovaliza- tion occurring about five diameters away from the point of indentation. From the analysis of ref. 4.2 it was found that the Length to which ovalization extends, in the elastic, small deformation analysis was much bigger, however, the point of maximum ovalization was almost exactly the same as in the experiment. In light of this and the results of fig. 4.2 it was concluded that the position of maximum ovalization is not highly dependent on the magnitude of deformation. - 65 - The same experiments were repeated using an indentor with a different contact diameter. It was found that although the area close to the indentor exhibited different deforma- tions the dimensions of the ovalized region remained unaf- fected. It was thus concluded that some of these results could be transferred to the case of the Propagating Buckle. A series of experiments were carried out where a buckle was allowed to propagate quasistatically halfway down the pipe and then was stopped. The profiles of the buckles were subse- quently measured along the two principal directions. The dimensions of the pipes and the measured results are given in table 4-1. The nominal length of the profile as defined in table 4-1 is plotted vs. D/t in fig. 4.3. These results were be used in calculating the plastic work done during propaga- tion. The maximum imperfection resulting in the ovalized region is plotted against D/t in fig. 4.4. The mean ovaliza- tion over the same region is also given in this figure. As previously mentioned the profile of propagation becomes sharper as the velocity increases. From the simple tests with the different radii indentors it was found that the magnitude of ovalization was independent of the indentor radius. As a result it is assumed that the magnitude of the ovalized region measured from the static experiments is ap- proximately the same as that which occurs in the dynamic case. The mean size of the imperfection induced by the buckle, calculated over a length of three diameters of the affected region, is given in table 4-1 and can be found as a function of D/t in fig. 4.4. As previously discussed, Timoshenko's simple formula (ref. 4.4) for calculating the buckling pressure of an im- perfect ring compares very well with more accurate analysis. Using this criterion for buckling and assuming that the imperfection measured above is of the type R = Ryo + wo cos 26 the pressure at which the imperfect section in front of the Propagating Buckle collapses canbecalculated. This calculated value is then assumed to be the lowest pressure at which the "Flip-Flop" mode can occur. | "Flip-Flops" were obtained in anumber of experiments with two different pipes. The geometric and material parameters of the pipes used are tabulated intable 4.2. The calculated values for the minimum pressure at which flip-flop is predicted to occur (Pee) and pressures at which flip-flovs occurred(P¥,) experimentally are giveninthe table. All the observed pressures are above the calcu- lated minimum pressure. This criterion was used in order to es- tablish the operating pressure of the experiments that are described next. 4,2 High Speed Photographic Observation of the Formation of a "Flip-Flop" A theory as to the mechanism of the "Flip-Flop" was described inthe preceding section. What follows is an experimental verifi- cation of this theory. This was done through high speed photographic observation of a buckle at the instant of "Flip-Flop". The trans- Parent test facility and the photographic technique described in the dynamic experiments were used to observe the phenomenon. In addition to the timing problems described in the dynamic propagation - 67 - experiments the problemof calculating the position wherea "'Flip- Flop" node would occur, a vriori, had to be faced. The simple theory described above and the repeatability of experiments made the correct determination of the node position possible. The same tube used for the other dynamic experiments was used in this case. After initiation the buckle would propagate down the tube, undergo one "Flip-Flop" and then enter the camera viewing field where it would undergo another flip-over. By this time the camera would be running at the required speed, capturing the phenomenon. Three different successful experiments were carried out where the "Flip-Flop" was photographed. Each has a different framing rate and different degree of clarity and detail. Figure 4.5 shows a series of pictures obtained at 192 psi and 4000 fr/sec. The pictures were originally 16mm but were cut down and rearranged as shown. The film used was Kodak 2476 with an ASA rating of 200. The quality of the pictures is satisfactory, although some light nonuniformity is present in the pictures. As can be seen, the buckle entered the camera viewing field from left to right. The total width of the viewing fieldwas .4m(16 in). The velocity of the incoming buckle was calculated to be about 24lm/s (790 ft/sec). fromthe first tothe fifth frames the severe ovalizationin front of the buckle can be observed. The point of maximum ovalization travelled at the same speed as the buckle front and remained in all frames about 5 diameters in front. In the fifth frame the ovalized section became unstable and in the sixth and - 68 _ seventh frames the collapse of the ovalized region is shown, The incoming buckle is thus stopped and a new one having its major axis at 90° to the old one starts propagating. Although the phenomenon can be seen clearly from these pic- tures it was felt necessary to increase the frame rate in order to observe the phenomenon in more detail. Figure 4.6 shows a more elaborate sequence of pictures obtained in the second experiment. This film was taken at 15,000 fr/sec. The frames were 8mm. The film used was Kodak PAN 2484 with an ASA rating of 800. In spite of the film sensitivity, the light requirement was too high for the sources available. The pipe was coated with a special illum- inating coating to help the situation. The light uniformity was carefully checked with a lightmeter but the right hand side of the picture still is under-illuminated. Due to the high film speed the pictures are more grainy. In spite of its imperfections, this sequence of pictures provides far more detail of the event than the previous one. It can, for instance, be seen that the buckle ahead of the node takes some time to accelerate and develop itself into the usual profile of propagation. The velocity of the incoming buckle was calculated to be 259m/s (850 ft/sec). This compares well with that givenin fig.3.7. Due tothemore frames available this value of the velocity is more acceptable than the one calculated from fig. 4.5. The mean pressure for this experi- ment was 192 psi (13.2 bar). The third picture was taken at 190 psi and 13,000 fr/sec ~ 69 - per second. The film in this case was Eastman 4-X Neg 7224, ASA 400. This film was originally underexposed. An effort was made through photographic tricks to salvage it. The results can be seen in fig. 4.7. The incoming buckle cannot be seen very clearly but a better description of what happens after the ovalized section collapses is shown. The initia- tion of the new buckle and its development can be seen in a sequence of frames. The time taken for the ovalized section to collapse and for the buckle to progress to its steady state propagation profile can be calculated to be about 400 usecs or 6-8 diameters distance. After this distance the buckle profile has completely developed and a new section of pipe ovalizes in the opposite direction. This ovalized section is about 4-6 diameters away from the buckle front. Thus it can be concluded that a node will occur every 12-16 diameters. This was indeed the case in as many as six experiments on the same tube, out of which only three were photographically recorded. Figure 4.8 shows a static picture of one of the nodes after the dynamic filming was completed. It can be clearly seen that the two buckles are circumferentially exactly 90° out of phase. In a separate experiment four strain gages were placed at distances of 6, 7, 8 and 9 diameters away from the knife edge, The knife edge triggered a microswitch with the slightest downward movement. All five signals were recorded. From the gage signals it was found that the velocity of Propagation reached a maximum value after about seven 70 diameters. It was thus confirmed that the profile takes 6-8 diameters to accelerate to its maximum velocity. Although pictures were obtained only for pipes of D/t = 50, the above conclusions can be carried over, with reasonable success, to D/t = 35.7 (the second pipe for which the "Flip-Flop" was obtained). -71- 5. BUCKLE ARREST 5.1 Introduction In the preceding chapters the emphasis was concen- trated in better understanding the problem of the Propagating Buckle and its mechanics. In this chapter an effort is made to look at the problem from a more practical angle. With the results established so far, in mind, it is clear that one way of avoiding the problem completely is to base the design of a pipeline on its propagation pressure (2.4.1). However this is considered wasteful due to the substantial increase in thickness that is required (factor of 2-3). A second way of dealing with the problem is to accept the fact that buckles will always somehow be initiated and look for a method of restricting the damage, that the pipeline suffers, to a minimum. This requires the use of some kind of a buckle arresting device. Buckle arrestor is the name given to any. device which locally reinforces the pipe and prevents the buckle from propagating any further. In their most common design, these devices are installed at regular intervals along the pipeline and if properly designed they restrict the damage to the length £, of pipe, between two arrestors (fig. 5.1). A designer then has to choose a balance between a more economical pipeline and the possibility of damage occur- ring to a length of pipe 2. Some of the factors that should be considered in deciding the value of £ are the total length of the pipe- ~ 72 - line, the diameter, thickness and unit length cost of the pipe, the water depth at which the pipeline is laid, as well as the cost of retraction and replacement of a unit length of damaged pipeline. Table 5.1 gives an example where arrestors were used every 150 m (490 ft.) [5.1]. A number of types of arresting devices have been sug- gested to date [5.2, 5.3, 5.4]. They vary in shape, cost, ease of application, etc. Reference 5.2 describes a device which is temporarily placed on the inside of the pipe ta prevent any local deformation. The device, which operates in a similar way as a “pig", is always kept at a critical distance away from the ship so that it protects the pipe- line downstream from its position. The main disadvantage of this device is that it only gives temporary protection to the pipeline as it cannot be left in place during the operation of the pipeline. It is also difficult to use in the case of pipelaying off a reel lay barge, References 5.3 and 5.4 describe devices which are permanently placed on the outside of the pipe. They have the purpose of locally strengthening the pipe by increasing its section modulus so as to prevent a propagating buckle from crossing over the arrestor. Some of these devices are schematically shown in fig. 5.2 (a-d). The "Slip-On" arrestor is a close fitting ring slipped on the pipe as shown in fig. 5.2-a. The "Grouted" arrestor (fig. 5.2-b) is a similar device but with a gap between the pipe and the arrestor; this is filled with grout--usually -~ 73 - cement. Figure 2-c represents another version of the Slip- On but in this case the ends of the ring are welded to the pipe either all around or in "stitch" form. The "Heavy walled cylinder" arrestor (fig. 5.2-d) consists of a length of pipe having usually the same internal diameter as the pipe but having a thicker section. This is welded between two sections of pipe. Each of these devices has relative merits and problems associated with its use (ease of application, cost of installation, ete.). However, one common disadvantage of these arrestors is the fact that they are difficult to use in the case of a continuous pipelaying process such as one where the pipe is laid from a shipboard reel onto which the pipeline has been prewound (see fig. 5.1). This is because in such a case no end exists over which the sleeves can be slipped on or to which the heavier sections can be welded. Another disadvantage with these arrestors is that due to their construction, bending stress concentration points can be created during the laying process due to the thick- ness discontinuity at the end of the arrestor. This can be designed against but at the expense of rather extensive machining. Figure 5.2-e shows a new arresting device suggested by the authors which due to its design does not have these disadvantages. It is called the "Spiral" arrestor and it can be simply described as a rod closely wound to the pipe for a number of turns and welded at the ends to keep it in -~ 74 - place and prevent it from expanding while a buckle propa- gating in the pipe is arrested. It can be wound onto the pipe at any stage of the operation (before the pipe is wound onto the reel, during laying before the tensioner and straightener, or after these devices). Due to the relative ease of application it can also be used in laying by traditional lay barges as well as in cases where the pipe- line is coated with concrete. Although arresting devices have been in use for some time now, their design was based on very limited tests. A number of tests have been carried out by investigators at Battelle [Ref. 54] but the parametric dependence of arrestor performance may not have been explored. In what follows an effort is made to improve this situation by putting the problem on the correct basis through dimension- al analysis and establishing an experimental methodology for studying the arrestor efficiency. The slip-on and spiral arrestors are studied through this method both under quasistatic as well as dynamic conditions. Empirical expressions for the arrestor efficiency are derived in each case, and these can be used for designing the device. 5.2 The Problem Parameters The objective of this study is to find the parametric dependence of the arrestor efficiency. The slip-on arrestor will be used as an example for developing the methodology for tackling the problem but the arguments hold true for any arrestor, with the appropriate changes in the parameter in expression (5.2.4). A slip-on arrester is a close fitting ring slipped over the pipe with the purpose of arresting an incoming propa- gating buckle. The goal is to design this ring in an effi- cient and effective way, given a set of field conditions. Figure 5.3 schematically shows the various parameters of the problem. The pipe is characterized by two critical pres- sures: (a) Propagation Pressure (Py): This is defined as the lowest pressure at which an initiated buckle will propagate. From (2,4.1) this expression is given by E’ t 2.28. P= [20.7 + 0.54 (| (5) (5.2.1) (b) Buckling Pressure (Pd: This is the pressure at which a long pipe under external pressure becomes unstable (buckles). A classical result for this pressure (for elastic buckling) is p = 7,25 (5) (5.2.2) ce (l-v?7) \D The assembly of pipe and arrestor are characterized by one critical pressure known as the Crossover Pressure (P,). This is defined as the pressure at which an incoming propagating buckle penetrates the arrestor and continues propagating. An arrestor efficiency is defined and can be used as a basis for comparing the merits of various arrestor designs; it is given by Pi/P -1 a * POPP . (5.2.3) Clearly an arrestor which allows a buckle, propagating at Po» to go through has an efficiency of zero. On the other hand, one that has a crossover pressure equal to P, has an efficiency of l. It is required that the crossover pressure (and the efficiency) be found as a function of the various parameters of the problem. The problem again involves large deforma- tions in the plastic regime, post-buckling considerations as well as contact area problems. Due to these formidable complications an experimental path was again chosen for analyzing the problem. If the study is restricted to the quasistatic problem, then all intertial parameters can be neglected, which simpli- fies the problem. The crossover pressure can then be | expressed as Pos FCP,» E, do, 0§, t, D, L, h), where the parameters in f£( ) are considered to be the main parameters of the problem. From Buckingham's nm theorem this can be expressed as pont (Es ge Po Oo Oo ah) autem ct} ) , (5.2.4) If it is assumed that this can be expressed in the form of an infinite power series then, ~ J7 - PL. . 4. \a 1\ G2 a3 oan a5 |jn ene (ZY (MEL ELEY LP oz From physical considerations - > 1, thus Ay = 1. If powers of n > 1 are neglected* then (5.2.5) becomes P 0) owes 3 4 (5.2.3), (5.2.6) => i ‘ 3 4 5 P | A series of experiments have been carried out to find the unknown a; for efficiencies ranging from 0 to l. 5.3 Experimental Procedure The experiments were conducted in the same facility described earlier (2.2.1) and shown in fig. 2.1. The trans- parent section of this facility was also used in order to observe the engagement of the buckle to the arrestor by photographic means. (a) Quasistatic Experiments In the case of the quasistatic buckle arrest, cont- rol over the speed of propagation was required so the tank was completely filled with water. An air driven piston water pump was used for pressurizing the tank. A buckle was initiated at one end of the pipe and allowed to propagate at *The validity of this assumption will be shown later. - 78 - a controlled speed (very small) by monitoring the amount of water pumped into the tank. For the quasistatic experiments a number of arrest- ors (4-6) were arranged on 3.66 m (12 ft.) long pipe speci- mens. Enough distance was allowed between arrestors to avoid interference between them (usually 15-20 diameters). A pressure transducer provided continuous readings of the pressure in the tank. Figure 5.4 shows schematically the output from the transducer for a typical experiment. In order to initiate the propagating buckle the pressure was slowly increase until propagation started from the damaged section of the pipe. The pressure then dropped back to the propagation pressure. On reaching the first arrestor, the buckle was stopped and the pressure increased. At some pressure the buckle suddenly (usually) penetrated the arrestor and continued propagating on the other side of the arrestor at the propagation pressure of the pipe. This continued until the buckle encountered another arrestor. The highest value of pressure reached in each case was taken to be the Crossover Pressure of the anestorn. The propagation pressure of each pipe was also recorded. In the case of the slip-on arrestors, these were machined from solid stock to tolerances of 0.013 mm (0.0005 in.) in the thickness and 0.025 mm (0.001 in.) in the length. Stress-strain curves were obtained for both the pipe and arrestor materials used. The inside diameter of the arrestor was always kept less than 0.051 mm (0.002 in.) larger than ~ 79 _ the maximum diameter of the cross section of the pipe. The pipes had typical imperfections in the radius of the order of AR/R = .006. Thickness imperfections in the pipe could only be measured in the mean and were taken into considera- tion. The nominal dimensions of the pipes used are listed in Table 5.2. (b) Dynamic Experiments As already shown in Chapter 3, for pressures above the propagation pressure, the buckle travels at substantial velocities. It therefore becomes imperative to examine the influence of the dynamics on the crossover pressure. The static efficiency of the arrestor was first measured and subsequently tests were carried out to find the efficiency under dynamic conditions. Experiments were car- ried out both with water and air as the pressurizing medium. In the case of water, the tank was filled with water so the pipe was completely covered. An air gap was left in the tank and the system was pressurized with compressed air. The air gap ensured nearly constant pressure conditions during the experiment (pressure drop from the beginning to the end of the experiment was 2-3%). In the case of the tank being completely filled with air, the drop in the pressure was less than 1%. In each case a number of pipes were fitted with the Same arrestor and tested one by one starting from the static crossover pressure and increasing (or decreasing) the pres- sure in each case by 0.34-0.69 bar (5-10 psi). At the desired pressure a buckle was initiated at one end of the pipe and it was observed whether the propagating buckle penetrated the arrestor. The arrestor was placed far enough from the initiation point (20-30 diameters) to ensure that the buckle reach steady state conditions before engaging the arrestor. If the buckle was arrested, a new experiment was conducted with higher pressure. This was repeated until the buckle finally penetrated the arrestor. The dynamic crossover pressure was taken to be the mean between the last pressure at which the buckle was stopped and the pressure at which it penetrated. Thus an error of 0.14-0.34 bar (2-5 psi) or 1-2% is possible in all the recorded dynamic crossover pressures. 5.4 Experimental Results: (a) Quasistatic Results The first series of experiments was conducted varying the thickness of the arrestor and keeping all other parameters constant. The crossover pressure normalized by the propagation pressure is given as a function of the arrestor thickness normalized by the pipe thickness in fig. 5.5 for the range of pressures from Pp to Pe. ft should be noted that once the buckling pressure is reached, the pipe ahead of the arrestor buckles independently, no further increase in the crossover pressure is possible. For this set of experiments an arrestor having a thickness of 3.5 times that of the pipe takes the crossover pressure up to the buckling pressure. Plotting these results on log- 81 log scales, the power of this parameter is found to be 3. The same results are plotted vs. (h/t)? in fig. 5.6 and direct proportionality is observed. A similar set of tests was conducted for arrestors of different yield stresses. Four different values of o$/oo were used. The results are shown in fig. 5.7, vs. (h/t)?*. If the same results are plotted vs. the parameters (oo/o0) (h/t)? then the data coalesce to one straight line indicating direct proportionality with the (op/o0) parameter (fig. 5.8). In another set of experiments, the arrestor length was varied keeping all other parameters constant, Going through the same procedure as above it is found that the unknown power for the (L/t) parameter is 1.5. These results are shown in fig. 5.9. This set of experiments was repeated for different (D/t). The results are plotted vs. (L/t)?° in fig, 5.10. Note that if D/t is varied then the maximum crossover pres- sure (buckling pressure) of each pipe changes. If the ordi- nate is changed to the arrestor efficiency n and the abscissa to (t/D) (L/t)?*% then the results coalesce. A final set of experiments in this series was carried out with E/oo varied. Tests were carried out with pipes made from: i} Al - 6061-T6 (oo = 42.0 x 10° psi, E = 10.0 x 10° psi) Al - 2024-T3 (oo i} 55.0 x 10°? psi, E = 10.2 x 10° psi) SS - 304 (coo = 49,5 x 10°? psi, E = 27,6 x 10® psi) - 89 - Figure 5,11 shows the results. Since no appreciable differ- ence was observed, it is concluded that the crossover pres- sure is insensitive to this parameter. Using the above empirical results the efficiency of a slip-on arrestor can be expressed as _ ,. fos\ fun (t\o/f Pe _ The results of fig. 5.8 and fig. 5.10 are plotted in fig. 5.12 as a function of this parameter. There exists some scatter in the results which can be attributed to inad- equate parametrization, experimental error or other uncon- trolled factors. But the proportionality of this parameter to n is quite obvious, indicating that neglecting higher powers of this parameter is acceptable. The results given in ref. 5.4 are within this scatter band for n < 0.5 but show considerable disparity for higher efficiencies. (b) Dynamic Results The results from these sets of experiments are tabulated in Table 5.3. For each pipe (D/t = 35.7 and 50,0) arrestors of three different static crossover pressures were tested. Each one was tested both in air and in water. The normalized static and dynamic crossover pressures as well as the static and dynamic arrestor efficiencies are tabulated. The velocity at which the buckle was travelling on engaging the arrestor was found from fig. 3.7. For both the air and water experiments, the dynamic crossover pressure was higher than the static one. (As a result "py > Nop) The dynamic - 83 - vs. the static arrestor efficiencies are plotted on fig. 5.13, for both air and water. Comparing the results for the same arrestor in air and in water, one notes that the increase in efficiency is higher for buckles travelling in air; this difference being quite substantial in some cases. It should be noted that buckles travel much faster in air than in water. The increase in arrestor efficiency is much higher for the higher arrestor efficiencies. For low effici- ency arrestors (ny < 0.4) very small difference was noted. In previous work on dynamic arrest (ref. 5.4) it was reported that there was very small difference between the dynamic and static crossover pressures. Since the work of ref. 5.4 was carried out with low efficiency arrestors, this is consistent with the present results. However, for high efficiency arrestors, considerable difference exists between dynamic and static crossover pressures for the slip- on arrestor. Figure 5.14 shows a statically and a dynamically arrested buckle. In the dynamic case, the buckle, which propagates in a dogbone cross section if unrestricted, pene- trated the arrestor much further and tried to push its way through the arrestor in a doubly symmetric mode. Figure 5.15 shows buckles that penetrated the arrestors statically and dynamically. Without the inertial effects the buckle usually penetrates in a U-shaped mode (single symmetry mode). In the dynamic case this becomes impossible due to the deep penetration of the doubly symmetric mode and a completely - 84 _ different mode results for which a higher pressure is neces- sary. These observations are restricted to the slip-on arrestor. Whether inertial effects have the same effect on other arrestor configurations has not been investigated. It should be noted that for both the quasistatic and dynamic cases a second mode of penetration exists. It is shown on fig. 5.15c. It has been called the Flip-Flop mode of penetration due to its resemblance to the Flip-Flop mode of collapse. Like the Flip-Flop mode of collapse it occurs for pressures very close to the buckling pressure of the pipe. In such a case a small ovalization occurs ahead of the arrestor once the buckle is arrested. The axis of ovalization is at 90° to the axis of propagation and if the pressure is high enough this is enough imperfection to cause the pipe to buckle locally. (c) Gap Effects In comparing some of the results of ref. 5.4 with the ones presented above, it was found that considerable difference existed in the case of arrestors of efficiency larger than 0.75. Although some relatively large sized arrestors (which relative to the pipe could almost be con- sidered rigid) were used, no arrestor reported in this ref- erence had an efficiency of 1.0. This indicated that there existed an upper limit to the arrestor efficiency which was completely independent of the arrestor dimensions. This is contrary to the results presented above. In an effort to investigate the reason for this disagreement, an additional -~ 85 - series of experiments were carried out which included a gap between the arrestor and the pipe. In each case the bending rigidity of the arrestors was kept constant (EI = const.) but the gap was varied. The results of these experiments are shown in fig. 5.16. Five different arrestor sizes were tested. In all cases the gap size affected the crossover pressure. The effect was more pronounced in the cases of arrestors with higher efficiency. Cross plotting from fig. 5.16 the effect of gap size on the crossover pressure can be obtained (fig. 5.17). Please note that these results hold only in the case examined and should not be used for design purposes. In the light of these results it is thought that the gap size may be the main reason for disagreement between the results presented here and in ref. 5.4. It should be pointed out that as pipes are inher- ently imperfect gaps will always exist for this type of arrestor. Careful manufacturing of the arrestor can mini- mize the problems but this will remain a definite disadvan- tage for this arrestor. 5.5 The "Spiral" Buckle Arrestor As already mentioned, the spiral arrestor consists of a rod wound tightly and closely onto the pipe to form a num- ber of turns. The ends of the rod are secured, as by weld- ing, to the pipe (fig. 5.18a) to the rod itself (fig. 5.18b) or to the rod and pipe simultaneously, thus forming an obstacle which a propagating buckle must overcome. The turns may also be individual and unconnected (fig. 5.18c). Due to - 86 - the relatively small bending rigidity the device has, in the pipe's longitudinal direction, the pipe can be bent to rela- tively small radii without any arrestor induced restriction. Thus the arrestor can be applied before the pipe is wound onto the reel or at any stage during the laying operation. The latter can be achieved since this device does not require a free pipeline end for installation. In the case of the traditional section by section laying process, the arrestor can be prewound on some of these sections which can be welded in place at the required intervals. If the pipe has to be coated with concrete then the arrestor can be placed underneath the concrete as shown in fig. 5.19, thus providing a smooth outside surface which can be important for the tensioner. If the pipe does not have a concrete coating but some kind of corrosion coating instead, then the arrestor can be placed after the coating and it can also be coated. The arrestor rod can be of any cross sectional shape although shapes maximizing the bending rigidity of the cross section of the rod should be preferred as they are more efficient. Some possible cross sectional configurations are shown in fig. 5.20. Of these, the circular cross section is considered the easiest and most economical to use. This was therefore chosen to carry out a series of experiments to verify the effectiveness of the arrestor and obtain some design specifications. _ 87 5.5.1 Parametric Study of the Spiral Arrestor Efficiency As in the case of the slip-on arrestor, the experi- mental path is again followed in order to parametrize the arrestor efficiency. The methodology established in Section 5.2 will be used here as well. The crossover pressure can be written as Po Oo 0 t t ¢t (5.5.1) where the symbols are as defined in fig. 5.2.1. According to 5.2.5-5.2.7 the efficiency can then be approximated by E Q1 oh a2 D Q3 L a4 x as h Q6 P n = Ai— — = = — s— -li. (5.5.2) Oo Jo t t t Po Ai (a constant) together with the unknown powers Os have to be found experimentally. The experimental procedure was the same as for the slip- on arrestor. One of the parameters shown to be important in the case of the slip-on arrestor was the gap between the arrestor and the pipe. Care was taken to avoid this by applying some tension to the rod during the winding process, thus making sure of good contact. From preliminary experi- ments it was found that the way the end of the rod was welded did not have an effect on the crossover pressure. On the other hand welding it to the pipe sometimes caused breakage of the weld when it was forced to follow the large deformation through which the pipe undergoes as it buckles. Because of these reasons the ends of the rod were usually welded to the - 88 last complete turn of the spiral. This technique was used in all experiments. All the pipes used were A&-D-6061-T6. Four different thicknesses of drawn rod were specially made and used, hav- ing the same material properties (same grade and heat treat- ment). For the slip-on arrestor it has been shown that the quasistatic efficiency underestimates the efficiency under dynamic conditions. A number of tests were conducted in order to test the validity of such an assumption as far as the spiral arrestor is concerned. Exactly the same behavior was observed for all experiments. As a result, further work on the subject was not considered necessary and the quasi- static crossover pressures for the spiral arrestor are con- sidered to be conservative for the dynamic conditions, 5.5.2 Experimental Results Figure 5.22 shows a spiral arrestor stopping a buckle and fig. 5.33 shows a buckle that was forced through the arrestor. As in the case of the slip-on arrestor, the mode of penetration is usually the U-type mode. For low effici- ency arrestors the arrestor usually follows the contour of the flattened pipe and for arrestors with efficiency close to 1 the flip-flop mode of penetration described in chapter 4 was observed. In the first series of experiments all the parameters were kept constant and the pitch of the arrestor winding was varied. It was found that the closer the winding, the -~ 89 - greater the efficiency. As a result, all further experi- ments were wound with no spacing between each turn. In this case the arrestor length L is equal to Nh and the fifth parameter in 5.5.1 is a product of the fourth and sixth, i.e., it is no longer an independent quantity; it can thus be dropped from 5.5.2. Figure 5.24 shows how the crossover pressure of a specific arrestor was affected by the arrest- or's pitch. It is considered safe to generalize the result and say that the drop in efficiency is small (less than 10%) if the pitch is kept less than half of the rod diameter. In the next experiment the number of turns (N) in an arrestor was varied with all other parameters constant. The results are shown in fig. 5.25, from which the crossover pressure (and the efficiency) is shown to be directly proportional to N (thus as = 1). In another experiment the rod diameter (h) was varied. The results appropriately plotted in fig. 5.26 indicate that ae = 3. It should be noted that this value is in agreement with that found for the power of the corresponding thickness parameter (h/t) of the slip-on arrestor. From the tests discussed so far it is clear that a great similarity exists between the behavior of the slip-on and the spiral arrestors. The method of resisting penetration and the mechanism of the crossover are much the same. In fact the spiral arrestor can be considered as a series of narrow slip-on rings slipped onto the pipe with no spacing between them (in our case). Due to these similarities between the ~ 90 - two arrestors it was considered reasonable to expect the dependence of the efficiency D/t, E/oo and o0/oo9 expres- sion (5.5.2) to be the same as in the corresponding expression for the slip-on arrestor (5.2.7). With this assumption a1 = 0, a2 = 1 and a3 = -l. Accepting the above values of Os all the experimental results are plotted on an efficiency vs. parameter plot in fig. 5.27. The results thus plotted exhibit a linear behav- ior indicating that the assumptions made in the dimensional analysis in (5.2.6) were reasonable. Some scatter in the results is present and can be attributed to possible small differences in the material behavior of each tube, differ- ences in the tension that was applied in winding the differ- ent arrestors and differences in the size and orientation, with respect to the buckle direction, of the end welds. How- ever the scatter is acceptable for design purposes. Like the slip-on arrestor, this arrestor's performance exhibits a bifurcation behavior when the crossover pressure reaches the buckling pressure. Any further reinforcement of the arrestor above this pressure does not help because the pipe buckles in front of the arrestor independently. 5.6 Conclusions A relatively extensive parametric study of the crossover pressure and efficiency of the slip-on arrestor has been carried out. A new arrestor design termed the spiral arrestor has been suggested and was shown to behave in very - g1 - much the same way as the slip-on arrestor. It has the advantage of not restricting the pipe in longitudinal bend- ing. In addition, this device can be used in continuous pipelaying operations such as the case of pipelaying off a reel. Empirical design formulas have been derived for both these devices. The empirical method established can very easily be used for testing other arrestor designs as well, It has been shown that with careful design arrestor effici- ency of 100% can be achieved. Any gap between the arrestor and the pipe was shown to decrease the arrestor efficiency. Finally, for these two types of arrestors, quasistatic design criteria always underestimate the arrestor effici- encies under dynamic conditions. It should be noted that for each pipe a limit pressure exists above which a buckle will always penetrate a rigid infinitely long arrestor. For the materials considered this pressure was always higher than the pipe buckling pressure. However, it is possible that with the right choice of pipe material this maximum efficiency pressure may be lower than the buckling pressure. In such a case arrestors cannot achieve an efficiency, as defined by (5.2.3), of 100%. The parametric dependence of this limit pressure still remains to be found. -~ 92 _ 1.1 Brown, R.J. "New Technology Reduces Deepwater Construc- tion Cost", Pipe Line Industry, July 1979, pp. 39-42. 1.2 Powers, J.T. and Finn, L.D. "Stress Analysis of Offshore Pipelines During Installation". 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Numerical Analysis, 14, pp. 91-111, 1977. Timoshenko, S. "Working Stresses for Column and Thin- Walled Structures", Trans. ASME, Applied Mechanics Division, Vol. 1, vp. 173-183, 1933. Tadjbakhsh, I. and Odeh, F. "Equilibrium States of Elastic Rings", Journal of Mathematical Analysis and Application, 18, pp. 59-74, 1967. Sills, L.B. and Budiansky, B. "Postbuckling Ring Analysis". Brief Notes. Transactions of the ASME Journal of Applied Mechanics, Vol. 45, March 1979, p. 208. Rehfield, L.W. "Initial Post Buckling of Circular Rings Under Pressure Loads", AIAA Journal, Vol, 10, 1975, pp. 1358-1359. wll - 95 - Arbocz, J. "On the Problem of Limit Point Calculations", Mesloh, R., Johns, T.G., Sorenson, J.E. ''The Propagat- ing Buckle", Battelle, Columbus, Ohio. BOSS 76, Proceedings, Vol. l. NASA SP-8007. "Buckling of Thin-Walled Circular Cylinders", Sept. 1965, Revised Aug. 1968. Reid, S.R., Johnson, W., Watson, A.R. "Large Deforma- tions of Thin-Walled Circular Tubes Under Transverse Loading", I, II, III, Intern. Journal of Mech. Science, 1976. Yuan, Shau Wen, "Thin Cylindrical Shells Subjected to Concentrated Loads", Quarterly of Applied Mathematics, Vol. 4, No. 1, 1946, pp. 13-26. Yuan, S.W. and Ting, L. "On Radial Deflections of a Cylinder Subjected to Equal and Opposite Concentrated Radial Loads", J. of Appl. Mech., Vol. , 1957, pp. 278-282. Timoshenko, S. "Working Stresses for Columns and Thin Walled Structures", Trans. ASME, Applied Mechanics Division, Vol. 1, pp. 173-183, 1933. Hayes, D.J. and Lux, M.D. "Full Scale Crack Arrest Test and Arrestor Device Performance for the Flags Gas Line", SPE 8220, 54th Annual Fall Technical Conference and Exhibition of the SPE of AIME, Sept. 1979. Lochridge, J.C. and Gibson, T.L. "Method of Arresting the Propagation of a Buckle in a Pipeline", U.S, Patent -~ 96 _ 3,747,356, July 24, 1973. Broussard, D.E., Ayers, R.R. and Walker, Jr., G.E. "Mitigation of Propagating Collapse Failures in Pipe- lines due to External Load", U.S. Patent 3,768,269, Oct. 30, 1973. Johns, T.G., Mesloh, R.E. and Sorenson, J.E. '"Propagat- ing Buckle Arrestors for Offshore Pipelines", OTC 2680, May 1976. Ives, Jr., G., Construction Editor, Pipe Line Industry, “Loop Construction Program on Schedule for 1980 Start- up", Pipe Line Industry, Oct. 1979, pp. 35-37. "Mediterranean Sea Gas Line Crossing Under Construction", Pipeline Industry, Nov. 1979, pp. 80-87. Lallier, L. and Tegon, A. Total Oil Marine, "A Technical Review of the Frigg Pipelines Construction", OTC 2915, 1976, pp. 303-314. Walker D.B.L., B.P. "A Technical Review of the Forties Field Submarine Pipeline", OTC 2603, 1976, pp. 819-829. APPENDIX A Moment-Curvature Relationship Thin Inextentional Ring The stress-strain curve assumed is shown in fig. A.1. Consider a section of a curved ring as shown below. The ring is assumed to have unit length. C 7,.| te ~ x Fe F etna _ ,(d80 _ de\_ _ de, _ de ca c(Gse- gn te k= GS oS (A.1) ®@ and s are as defined in fig. 2.14. Stress-strain Only one stress and one strain are present in the problen. For | c | < t/2 . Oo = tEK (A, 2) _ _ : 1 t/2 < | | stl © = 69 + E'(e - €) = «(1 - =) + E cK t/2 t/2 M = 2{ frccar + J[ev@-2) t+ Eck | ac} (A.3) 0 t/2 . 2 2 t Define ky = Fe K = Soo => p74 (A.4) (A.4) + (A.3) oo M = Sgt" | & (5) - 2(1 - =) (Ss) + p(t - :)] APPENDIX B Calculation of Initial Vector ul? The problem of the response of a circular ring with an initial imperfection under external pressure was solved by Timoshenko in ref. 2.7. His analysis of course is restricted to small deflections and linearly elastic solids. In the notation of chapter 2 for the elastic case the moment at any point is given by M=- a Q (Re) ( r Pre.) cos 7S , (BL) Looking at the equilibrium of a quadrant of the ring the forces H and V can be derived as follows: wl H = - PR sin 6 (B.2) V = PR cos 6 oF 2 T or H = - 7 Q sin 7 Ss | (B.3) T <I II | ee) Q ie) n Ny n ~ 99 _ . : fo. os . Thus, the inital guess u” is given by: Ho) f_ £ Q sin 5 s uo =) NO? = 4 2 Q cos 5 Ss >(B.4) MO 5 Ss (0) _ 4 9 (Wo 1 10s 1 L067] [7 2 (x) (r= prez) cos ms - 100 - APPENDIX C Calculation of Strain Energy in Buckle Profile Consider the kinematically admissible assumed geometry for the buckle front shown in fig. C.1l. The strain energy required (gone to plastically deform the pipe) for the buckle to move a unit length forward can be separated into three categories as follows: (I) Strain energy due to circumferential bending. Four hinges are placed on the pipe's circumference as shown below. The ring quadrants between the hinges are made rigid. The moment in the hinges is assumed to be 2 Mo = SE prey (C1) For perfectly plastic material v' = 1/2, thus (1) = 2 M = ope (C-2) The strain energy required to take a unit length of the pipe from the undisturbed to the collapsed configuration can be found to be + 8Mo7- = S10 ot? (C-3) - 1lOl - (II) Strain energy due to longitudinal fiber bending. (a) Region A Consider first region A as shown in fig. C,2(mote that there are two of these). Assume that its geometry is given by: T fs Ro} b biz — 5 FIG.C.2 ASSUMED GEOMETRY OF REGION A. 6 —_ 1 ZT where a 72 {cos = 1) (C-4) Then the elemental longitudinal bending strain energy, OUpr A is given by _ d*6§ Thus bz 2a a = d*6 _ bdo Usa 7 of fe Moa dz = a MC. (C-5) bz fe) But - 35 dot? 69 =R , b= TR, My = Z . (C-6) Thus, strain energy per unit length is given by Upra = 5(8) t*/ unit pipe length (C-7) (b) Region B. Let the geometry of region B be as follows: - 102 - FIG. C.3 ASSUMED GEOMETRY OF REGION B where th = 5 (cos an - 1) In a similar manner as region A the bending strain energy is given by Co 2a a 2 Uni a =f ax fom8 dg = Moy, , (C-9) Cz ° fe) 2a oot? Where Co = (/2-1) R= YR , My = 0 C= TR. (C-10) vN Thus the strain energy is given by U _ wy R? 2 : F 1 h C-10 BLB 7 De q Cot?/ unit pipe lengt (c-10) (III) Strain energy due to longitudinal membrane stretching. Again assume that the strain in each region is equal to that of the longest fiber of that region. (a) Region A Let the deformed length of the biggest longitudinal fiber be a' then from fig. C.2. a , 1 /dé\? : te) (C-4) > (C-12) => at = a _ 1 (bor \? eg = 32 = EGF) (C-13) Thus the stretching energy/unit pipe length is given by 2F ey ] wh = ma , 69 =R (C-14) _ 1? (R\? : . Usta = tela o oRt/unit pipe length. (C-15) (b) Region B In a similar manner _ 2] q - Usrn = = Ao o¢ gt | (C-16) ; 2 Where ty =7Ra oo, €, = i (327) , bo = (2-1) R= yR (C-17) (C-17) + (C-16) => 7? 2 R 2 ° ° Vcr = 76Y (8) ooRt/unit pipe length. (C-18) Thus the total strain energy per unit length of pipe flattened is given by U= Uzge + [Uspa + Usp | * Usa + Usa (C-3), (C-7), (C-11), (C-15), (C-18) —~ (C-19) aa 2 3 2 & mot? + (1 + nz (4) oot? + (1 +y)Fe (Boort U Hl y = v2 -1l (C-20) 104 - NOILVYSdO ONIAVT 3NINSdId log i ODER eee nee tae 400)4 uD89O pueqbog—_,_ Bul} ed Iq ee ee — eee yideg DOH uolpBbodosy 496uns \—— — 48udisue) —/ PUuBqIBACO eBuog Ap q s x= (a) Bending buckle on pipe (At) with D/t=35.7. (b) Diamond bending buckle on pipe (At) with D/t=39.3 (c) Bending buckle initiated a propagating one. FIG, 1,2 BENDING BUCKLES, - 106 - OTEC | 753 hr Binding Strap A-A Polyethylene Cold Water Pipe Pipe Weight FIG.1.8O0TEC COLD WATER SUPPLY SYSTEM f Outside Steel Pipe Ice \ op: Inner Plastic Pipe FIG. 1.4 GAS LINE ARRANGEMENT PRONE TO BUCKLING - 107 - Propagating buckle on a 12 in, pipe. (b) Propagating buckle on a 32 in. diameter pipe. (c) Propagating buckle on model pipe (1.25 in. diameter) FIG. 1.5 PROPAGATING BUCKLES CN DIFFERENT SIZE PIPES. 108 (ISd OO8 A3YNSS3Yd ONIMYOM) AIGWASSY ANVL JYNSSaydI'2’ols i" 9 1NO puy U} VAD PVAAAVALA SAV AVAUV AVAL ANASAAAAANLAAYN p24 pp AAAS AA ‘4 JO) 0K\r == i dwind 48}OomM ¢snoUxy WAS lees 5 jaoddns 4 | P e4Nsseid 48pi00ey4 F4DYD OL yUDL sunesoug_/ edid 4epuljAd o1jDwNeUd ebo9 eunsseid O- ysnoyxg <~e Q ul atv @A001D Bury, 0+7 - 109 - ebud)4 uQ-dijg—y NOILOSS YOLVILIN] JO NOISSQ G3 tivigsd2'2‘9l4 SS —— Nt yn yudL ZZ 1 a edig—* LL ebp3 1-9] () y ha VLLL LL) 4 Ajquessy |D98g diwoUkg—| — =| depulpAy a oDwneug BINJONAYS juoddng sepuyjk5 —~7] L pdiy pessesdwoy LU piousjos 12 110 NSISS0 GNV AISWASSVNOILOSS ANVL LNSYVdSNVYLE 2?’ Ol4 -—_- ~ LN eqn, IjAsoy yussodsuday LID ILIAD TIT ILI IIIT I TRITIATED AT a A 212 dd 4 GAL ELZ ILI LILI III IIT III IIIT ITA ILLIA ILI IIIT ALI I TIT TL LLL LT IAI IL ILOILO ALI bod u 0, _— PIOM YO4HS Buisp9 \eeig / Lk leeele DS PISM JOljuesejyWNd419 MOPUIM VrV — male, /* _— 8did Gd eqny 4euu] djAuoV juesOdsudsy -~ Bulso9 j8ejs a - lil - 70- S-X-60 | 60- i SS-304 | i | L aL2024 50- / . ify x | ‘ & | — A ~ - 6 204 . i / L-6061-T | @ ! 1 / = | Wp) / ia ! | fj | i / | 1 / | 107 Hl / | 1 / | i 0! 0.2 ol4 o'6 o's 'o — > % Strain (%éE) FIG.2.4TYPICAL STRESS-STRAIN BEHAVIOUR OF ALUMINUM AND STEEL ALLOYS 112 SYNSS3Yd NOILVOVdONd JO NOILVNINYSL3AG IVINSWINSdxX3d SO DILVNSHOSG'2'9Ol4 eu! | —— a ‘h “4 rn 0; 1 | 84NSseig uolyobodo1g | ' A pan F4% | | \ | | | aN eunsseid) Nv —-4ly | ! UOI}D1j1UI | | ! | | | i SBnig Pua, | | ! ~=f] 3 <i eisyudsow jy += > i —Il OL =f] =< A 7—l UO}}09j;sOd Wy |OI JU] POyoijjuy A} }OUsG}xX3 Wa eansseag ——— - 113 - / \ D=1.5in. / \ t= 0.49in. \ XN ve ™™ — _— eee ~~ Yo o\ / \ D=1.25in. / \ t =0.035in. / \ / aN 7 NX 7 / \ D=1.125in. / \ t =0.035in. \ / \ / “ 2 FIG.2.6 CROSSECTIONS OF REPRESENTATIVE COLLAPSED TUBES 114 1/Q —=—— OS SNIT LHOIVYLS V HLIM GSLlLliad 31V9S 9071-907 NO (3/0) SA 3YNSSAYd NOILVOVdONd A3SZINVWHON2 2 9Ols Ov Of Oe Ol OO! 06 O8 O24 OS Lf q i LJ 1884 SSBIUIDJS © Ao}ly 1884S a Koy iv © | i i i Loree hg n it Ol Lily Ke) g OlX Ol 2o 115 SSAYND MV Y53MOd HLIM G3LLIS 3/0 SA SAYNSS3Yd NOILVOVdONd Q3ZINVWHYON SZ OI4 vqQ=—— , OO! OS o8 O2 09 OS Ov Of 02 are 99 1891S SSejUIDIS e@ er SAO|]Y 1884 L 9 Sko|y ‘ly © - Ol - 116 - Statically Admissible Mechanism ~Kinematically Admissible Mechanism ‘ | (C Exact Solution(Elastic-Perfectly Plastic) \ \ i T T T O A . . FIG.2.1ILUPPER AND LOWER BOUND PATHS FOR THE COLLAPSE PATH COMPARED TO THE EXACT SOLUTION 117 T3GOW-€ WONS ASNIVLEO SV 4, yO4 -QNNOS Y3ddNadl25l4 vQ—- Oo! O6 O8 Oz O9 OS Ov of O02 l t f ; ' —-.- 26 ~ mame S}INSOY |DJUBWIUedxy ) \ UO!}DIPB1q JEPOW Q-¢ luo Bbuipueg joijueseywnouig —--——- Lg - 118 - H + AH FIG.2.15 ELASTIC-LINEAR STRAIN HARDENING MATERIAL BEHAVIOUR ASSUMED 119 - ------ Elastic Inelastic 1.8 L6r L4pr FIG.2.16 COMPLETE POST BUCKLING BEHAVIOUR IN ELASTIC AND INELASTIC CASE - 120 - Po © Increment of P ry © Increment of A Limit Point FIG.2I7DETAILED DRAWING OF LIMIT POINT ONIY ONISAVIIOD JO GNVYLVNO NO NOLLNGINISIO OlLSVIANI sI'2‘9ls x= rr T — 4 : . i | L___. nr a ee ON ‘ o veo . an eee G2I'0 See. \ pee ' Sree LE ee _ N { Lec Ue \ veo L210 \ wooo L FINN einer ------""" 920 rt NS rd | j = tN. xf \ eee" 19-0 d Y + 'k ae { ~ 9148D}8u] ---- ee pe es 9148013 —— 8'0 re) 0 dy, - 122 - FIG.2.19 COLLAPSE SEQUENCE OF A CIRCULAR RING - 123 - M(1) ¢ ; _ S M (0) apo ty i M (0) 2 | —_ 1 —y T T 1 I wl 2 3 4 5 6 —~ A FIG2. 2O0VARIATION OF MOMENT AT S=O WITH DISPLACEMENT AT S=l - 124 - FIG.2.21 VARIATION OF END FORCES WITH DISPLACEMENT - 125 - Pe =0.176 (Collapse R Condition) = +0804 (Prebuckling Cc Condition) FIG.2.22 MOMENT DISTRIBUTION JUST: BEFORE BUCKLING AND ON | COLLAPSE - 126 - — 1s M V \ O yt SS | 30 7 \ a PL “Pp, * 0-804 (Prebuckling 20 - Condition) | A 10 ae ™. SN ~™ J #701176 ™ C (Collapse Condition) ~ __ ne . ™ ~ _—- SON qT T T T — T 2 4 6 8 1.0 FIG2.23FORCE DISTRIBUTION JUST BEFORE BUCKLING AND ON COLLAPSE - 127 - Poa So 7 a p-E= 3xl0 a. I7.86 | e _ 7 W, 15 - T — rT T —T —T 1 JA .2 3 4 5 6 7 FIG.2.24 POST BUCKLING BEHAVIOUR FOR DIFFERENT A VALUES OF E - 128 - az {2 a= 15 Ler al L i | i _t O A 2 3 4 ae) 6 FIG.2.25 VARIATION OF POST BUCKLING BEHAVIOUR WITH STRAIN HARDENING PARAMETER & 130 p3jO SANIVA INSYSdSI0 YOS (4/0) JO NOILONNS V SV BYNSS3Yd 3SdV1IOO ONIN 22°? Ol 1/0——— OO!106 08 OL O89 OG Ov o¢ 02 Ol Ll l j 1 | i Lt _1 1'O -—O'] pOl=oD 0 00¢=0D Og=0d _ OGsD Ae) QS - Ol * 709, 1.47 127 - 131 - \ Elestic Case u 17.86 ma la =42x10° | a SS ee —+— Perfectly Plastic Case 2 | T v rr T | 100 200 300 ——a OF FIG.2.28 VARIATION OF COLLAPSE PRESSURE WITH a ——>p|~ - 132 - R 1786 t a= 50 = 100x100 —___70x10"" 55x!lo 4 45x1074 35x10 7 FIG.2.29 VARIATION OF POST BUCKLING BEHAVIOUR WITH YIELD STRESS - 133 - Pool Pc " oi O 0.54 —__ ma +|z i 5 oo o 0.45 0.3- 0.27 0.1 7 T T T 7 : 6 6) 10 O 20 40 fe) 8 fe) 2 104 FIG.2.30 VARIATION OF COLLAPSE LOAD WITH YIELD STRESS - 134 - <i =17.86 | | T T TT T T T @) 2 4 6 8 {@) I]? E! 207 Peol To { 15 - 10 - -i786 t 5 ! —y T T if : T re) 50 100 150 200 250 E (b) %o FIG.231 VARIATION OF COLLAPSE PRESSURE WITH = 135 - 44, -3 D t P co, xlO 3 p oO + P,/onx!1O in (mm) in (mm) D/t psi (bar) psi (bar ) pro 1.001 035 243 42. IT ies.43) (889) 28.6 (16.75) (2.895) 5.765 1.503 .049 225 43 2 | (38.18) (1.245) 30.67 (15.51) (2.965) 5.233 1. 126 .036 178.5 43 3 | (28.6) (9144) 31.25 (12.31) (2.965) 4.156 1254 *| 10365 262 52 4 | (31.85) (9271) 34.36 (18.00) (3.585) 5.038 1251 **) 9355 163.4 50 5 | (31.78) (9017) 35.24 1.27) (3.447) 3.268 6 1.253 .035 35.8 157.4 53 2.970 (31.85) (.889) (10. 85) (3.654) 1.376 .036 128 45 7 | (34.95) (91.44) 38.22 (8.825) (3.103) 2.844 1.003 021 74.6 50 8 (25.48) (.5 334) 47.76 (5.143) (3.447) 492 1.500 .029 57.3 42 9 | en (.7366) 51.72 (3.951) (2.895) 1.364 1.375 020 32.9 43 10 (34.93) .508 68.75 (2.268) (2.965) 765 2.500 .035 29.6 49 We} (63.5) (889) 71.43 (2.041) (3.378) 604 2.003 022 17.5 42.5 l2 | (50.88) (.5588) 91.05 (1.207) (2.93) Ale 2.750 029 15 41.5 13 (69.85) (7366) 94.85 (1.034) (2.861) - 36! + +t Op = > On005 * KK All TABLE 2.1 CHARACTERISTICS OF PIPES TESTED AND PROPAGATION PRESSURE RESULTS 1 bar = 10° N/m ~ 14.5 psi SS-304 AL-2024-T3 others A-6061-T6 136 - SYSLIWVY Vd W3780ud- NOILVOvVdOud DINVNAG le‘9l4 ~T o 137 ALINOVS LSSL 3YNSSAYd HOIH 2’ “OlS Buippoo 7 edig 4o4 BuiuedO 400j)4 ZitllL2iL LL LALA LL LA LLL CLL LLL LLLL LLL LE L LLL LL LLL LLL 4Epsooay OL s}ouBisg e6bp9 j , Buluedo, g°'2— r ‘ul "bS/q| OOOE OL dy yudL e609 ulp4sjg — 40401414] sBuruedo, GZ"] OML A 9,01 8804S pe;ODU;WO-7 — | diy peziunssesg & JopojnBey eunsseid y ¥ Cham) WAG \ 49onpsudis] B4nSs eid <— - su01j9euUU0D o1pnD4apAY 138 uoljojnsu] . ? 109 Bulyoey—"s is a” Bnid Burjysoddng uedoieg NOILOAS YOLVILINI SO NOISSG €€ ‘Old (4) (9) eh Add i jr smpoy edig ——*" Bid pug —_, WN 139 SWSHOS LNSN3SYNSVAW ALIOOISAP Ee Old 40401}1U} 8)49Ng V SY S10 ‘uUMOG peuIN| Uu0!}9eS | it £&x \ ae 2X a ID 'X a eBoy uldajs + he 29 £9 I v9 seB09 ulodujs wos sjouBIig pessed0jd ~ 140 - 5.7 155.8 psi =3 D t P einsseig Buljyong @SdDI]OD 30 BpoW dojy-—dijy 4 eee ee eee eed in Water OO ee L4r Pp 12k vf Yo = 979 ft/sec 0.6F 0.46r 0.87 0.2F /p > OF Qe oO FIG.3.5PROPAGATING BUCKLE VELOCITY VS PRESSURE - 141 - Ler DO. ' = 35.7 Pp= 155.7 psi LO-r af fp = 979 ft/sec U V Sofp t oak 0.6- 0.47 O0.er FIG.3.6 PROPAGATION VELOCITY VS PRESSURE PARAMETER ~ 142 - 0.8 - 0.6 - 0.47 In Water 0.27 i J T UJ T O 0.2 0.4 0.6 0.8 1.0 FIG.3.7 PROPAGATION VELOCITY FOR DIFFERENT D/t - 143 - Fo / % a = 979 ft/sec 0.6+ 0.4F 0.2 + O wd. 1 a Bae i 1 “soak 20 :- 25 , 30 35 40 45 50 + + FIG.3.8 PROPAGATION VELOCITY AS A FUNCTION OF PIPE GEOMETRIC PARAMETERS - 144 - | a 1 o In Air © In Water D. = 35.7 Fy 155.8 psi Buckling Pressure —= _ bee ba FIG.3.9 BUCKLE a PROFILE LENGTH VS PRESSURE - 145 - ah Triggering Circuit | internal Microswitch Oo Hycam S . Valve olenoid . Pneumatic Cylinder Knife Edge y Pipe —~nl? p ‘3 yan 3 SS) FIG. 31AOTRIGGERING CIRCUIT AND SOLENOID OPERATED PNEUMATIC CYLINDER USED FOR INITIATING BUCKLE - 146 - co : ree ae He FIG. 3.11 DYNAMIC BUCKLE PROPAGATION--FRONT VIEW (4500 fr/sec, P/P, = .902, U = 372 ft/sec, Water) FIG. 3.12 DYNAMIC BUCKLE PROPAGATTON--PLANE VIEW (11,500 ft/sec, P/P. = .890, U = 726 ft/sec, Air) - 148 - ' p,* P, mm (in) | mm (in) D/t Pp Cc bar (psi) bar (psi) 38.1 1.27 15.5 62.2 | (1,50) (.050) 30.0 (225) (902) 3 31.8 0.89 35.7 10.8 33.7 (1.25) (.035) (156) (488) 31.8 0.74 6.55 20.4 3 (1.25) (029) 44.6 (95) (296) 25.4 0.51 0 4.97 12.9 4 (1.00) 1.020) 50. (72) (187) TABLE 3.1 PROPERTIES OF PIPE USED IN DYNAMIC PROPAGATION EXPERIMENTS * Experimental Results E=6.90x10° bar (l0x10® PSI) v= 0.3 Oy * 2.41x:10°bar (35x 10°PSI) Fig. 4,la A buckle that propagated and then came to a stop. Note the "dogbone" collapse mode. Fig. 4.1b The "flip-flop" compared to the normal collapse mode. NOILVONSGNI 1V901 OL 3NG NOILVZINVAO Adld 2vSls GeO'O= + OG21=0 ebpy ej!uy { a Vv - 151 - 1/Q HLIM HLON3S1 31is0ud SO NOLLVINWAE'b' 91d a 06 08 OL 09 Os Ob Of 02 4 i i j 7s i. > 9 a“ ~ o-~ ° “a re a a“ 91-1909 -1V eo” va - 9 a oO a“ a - “G L a“ ° 6 a“ a“ a“ a“ LO r Ol ~ fe) ' { og a D ro } } VOHLIM YSLSAWVYEVd NOILOSSYSdWI SO NOILVIYWAVYOIS qs OO! O06 O08 OL 09 OS Ov oe O¢ ! j i L i I i l 2 152 x ,Ol oe a oe Cee cf OE os ne ate 2 s C Cs ee ‘UU ae Nea BASS fe ae FIG. 4.5 FLIP-FLOP MODE OF COLLAPSE (4000 fr/sec, P/P, = .934, U = 790 ft/sec, Air) FIG. 4.6 FLIP-FLOP MODE OF COLLAPSE (15,000 fr/sec, P/P. = .934, U 850 ft/sec, Air) FIG. 4.7 FLIP-FLOP MODE OF COLLAPSE (13,000 fr/sec, P/P, 925, Air) - 156 - t Wom _ — | - i _ ~ — F ) + D t in in D/t 2/0 o/p |W../D (mm) (mm) om | 1.126 036 (28.6) (914) 31.3 4.9 7.6 0027 1. 250 035 (31.8) (885) 35.7 6.0 8.0 .0036 1.00I O22 . 0.0 9.0 0050 (25.4) (559) 47.7 I 1.500 029 (38.1) (737). 51.7 6.0 9.3 .0053 1.375 .020 (34.9) (508) 68.7 8.0 10.5 .0038 2.500 035 (63.5) (.889) 71.4 4.5 11.2 .0060 2.003 | .022 (50.9) (559) 91.1 7.7 12.7 0057 1.741 029 (44.2) (737) 94.5 03 10.2 .0058 TABLE 4.1 MEASURED DIMENSIONS OF PROFILE OF PROPAGATING BUCKLE - 157 - D t p Pe (in) mm (in) mm D/t_ ff _ff 1.25 035 94 (31.8) (.89) 35.7 .887 .96| 995 1.00 O21. 909 (25.4) (51) 47.6 .875 934 .953 E=10’ ps| %= 42,000 PSI * Experimental Pressures At Which Flip-Flop Was Observed TABLE 4.2 FLIP-FLOP EXPERIMENTS 158 39Y¥VE AVI 13354 V 3SO NOILWYSdO ONIAVT 3did I'S‘dI3 e000! "gr aeete JO}Seddy ~_ GY edid CL >|» (a) (b) (c) (d) (e) - 159 - WITTE ee ~ ~ - - Slip-On Arrestor WITT TTT Corr —_ - ~ ~ ~ Grouted Arrestor ILZ77 TTT MLITITZIET Ds —— — = - _ - Welded Arrestor MOLL ALTTT ITT TTT __ _ 7 _ a Heavy Walled Cylinder Arrestor ee FIG.5.2 DIFFERENT ARRESTOR DESIGNS - 160 - ip- On Arrestor Sl een Lowe | Basssy- name hh 8) FIG.5.3 PROBLEM PARAMETERS awl] <—— SBYUNSS3Yd YSAOSSOND SOLSSYYV JO NOILVNINYSLSQ WLNSWI8Sdx3 4JO OILVW3SHOS ¢G'OlS (9g) eanssesg Uol}DBDdosg edig nO Y ara OA © 1 evo4 (gd) e4nsseig JOAOSSOID JOJSOsIY v0, eisydsowyy edig ~ OL TT —~ ~ : 3 . an S4OjSeuay pejyorjiuy AjjOuseyxg 84NSSBIg +«——— 162 SSSNHOIHL YOLSSYYV HLIM SYNSS3SYd YSAOSSOYND 3O NOILVINVA G'G’Ols eansseig Suljyong t—4 = 22 Ol= gos = Z's al= o|~- 40}SOddy UO-dHS 163 YSLAW Ved SSBNMOIHL YOLSSYYV HLIM SYNSSSYd YSAOSSOND 3O NOILVIYVA 9°S'Sls eu i l ! i t l 1 (+).— o2 09 os Ov o¢ 02 Ol O ~ einsseisg Buijyong 5 | = = + = foareree ! 0'l or 0'0S 6 G*2l = g i = 4 \ Sg 40}SO44y UQ-dIS - 164 - S3SSS3Y1S QTSIA “LNSYSSSIO YOS YALSWVYVd SSANASIHL SA 3YNSS3Yd YSAOSSOND 2'¢'dl4 } w) ‘ O02 09 OS Ov O€ O02 Ol 0 _f, l i l | l i | me oe TTF x N ~ gansseig DUl|yONg = L | oS ° Td OD ~ | edld Og % 15) = Hi, ‘ ‘dl “= \ did soseusy ugo-dijs - 165 SYSLAWVYVd SSSYLS GISIA GNV SSANMOIHL YOLSSYYV HLIM SYNSS3Yd YSAOSSOND SO NOILVINVA 8S'9ls 4 °0 ¢ (a) ( 20 )~<— Oz O9 OS Ov O¢ 02 Ol 0 ! | l | | | YALSWVYEVd HLONS1 YOLSSYeV HLIM SYNSS3SYd WSAOSSOND JO NOILVIYVA6S'OlS + \ r O09 i OS 1 Ov | O¢ O2 Ol O ~ 166 40}SO44y UO-dIIS edid N ! \ | I~ @ einsseig Buijyong |. ¢ | 8d}q q Winn. 4b enon 4 d Ol o 0'0G = q: Seles oF - 167 SNOISNSWIGQ J3did LN3SY3SSIG YOS HLONSA) HYOLSSYYV HLIM SYNSSSYd YSAOSSOXD 4O NOILVIYVA ONS SIS (+)—<— 02 o9 os Op o¢ 02 Ol O “ig | | ! I | I l i 6 - + — Oo fo) 'g} + TO TOTO TTY OP ab e/a | L¢ a ‘+ , Sf ~~ “egg aya | wt, 4 Gaeree| j f — 4 \ Sg em ee cr ee re wee ee ee eee ee eee eee ee zeesva ("tg nad wo nN u Cael ~ \° Ao. tw 9) © N " =| o°| (973) IN3YS4sI0 YO3 NYALAWVYVd HLONS) YOLSSYYV SA 3YNSSAYd YSAOSSONO II'S'9Olg 4 20 09 os Ov o¢ 02 j i j l ~ 168 - Ho (€edig) ggg ¢ (2 8diq) g¢eg a (9 8did) gg! o %7 3 VIAWYNOS WO IdW3a AONZIOISS3S YOLSSYYV LSNIVOV G31101d SLINSSY WINAWIYadx3 el'S'9Ols (HMe)— 02 09 OG L i \ mae) + vO . > & ro La b& 9°0 3 a Q m -3s0 3 Q, @ ° peo) re} Qa ras -~170 - Air foie Region Of ” b- | = conservatism St "by 7 e Air A i a Water rs = T Lj Tt T T 7 t q . .2 4 6 8 1.0 —> "st. FIG.5.13 DYNAMIC VS STATIC ARRESTOR EFFICIENCY FOR AIR AND WATER AS PRESSURIZING MEDIA ~ 171i 7 (b) FIG. 5.14 BUCKLE ARREST (a) Buckle arrested by slip-on arrestor under quasi-static conditions. (b) Buckle arrested by slip-on arrestor under dynamic conditions. (c) FIG. 5.15 BUCKLE CROSSOVER (a) Buckle penetrating an arrestor in "U"' shaped mode under quasistatic conditions. (b) Buckle penetrating an arrestor under dynamic conditions. (c) Flip-flop mode of penetration common to dynamic and quasistatic cases. - 173 SNOISNSWIG YOLSSYYV IN38S35I0 YOs 3ZIS dVO HLIM 3YNSSSYd UYSAOSSOND 4O NOILVIYVA 91'S 914 y _— OOS OGL 000! 0022 O0ve 4\ 74 ze(t)(a7) SUOISUBWIQ JOJSedIY ed) ! A 4, _ oy @ZZA~ 2=- | edid tT ‘y _ yy 4 40}SO4ay UQ-dIIS e@ouour < S3ZIS dV9 IN3SYSSSI0 YOS SYBLSWNVYEVd OINLSWOSD YOLSSYYV SA 3YNSS3Yd YSAOSSOND JIG Sls hy 74 abaya) coog 00S2 0002 = 008!_—000I 00S O | 2 4 . r Ni bO rt t €0’ Yo rAey r lO" \ OG= a (0) ee or) eansseig Buljyong dy —-2175. - 1 faa Le a 4} H I FIG.SISDIFFERENT SPIRAL ARRESTOR ARRANGEMENTS AND WELDING METHODS ~ 1767 Concrete Coating ee ee LLLLIARL LALLA PLL. 1 ere AO A et wre Spiral Arrestor FIG.5.19 SPIRAL ARRESTOR USED UNDER CONGRETE COATING 177 - SQO0Y¥ YOLSSYYV IWHldS YOS SNOILORSSOND J WISSOd Oz's'914 uoijoessou5 uo1j9essolg POY 40jsadiy jDIIds jOuoBoxeyH = u0ljDesso1D WDEg nly 4DjnBudjoey = uodljo@ssouy ADjND419 edid x wi == ts ar -~ 178 7 b-b c-c yf t D J? CX FIG.5.21 PARAMETERS OF THE ARREST PROBLEM 179. FIG, 5.22 SPIRAL ARRESTOR STOPPING A PROPAGATING BUCKLE FIG, 5.23 BUCKLE PENETRATED ARRESTOR IN "U'"' MODE, - 180. HOLid YOLSSYYV AVWldS HLIM 3YNSSSYd YAAOSSOYD AO NOILVIYVA ves Sls q => . G v ¢ 2 \ 0 i I i j i ; i | an | ! | i te | ; \ 122 | =a \ °.0 | a Bulyono), susny | OZ =N os 4 | Gl°89-2== 0 | FE 4 d s2le=7y | Sd I" % | ee NS ee em ce es ee ee eS ce cen em me ST eansseig Buljyong - 181 - ai Buckling Pressure FD J T T ui | t O 5 lO 15 20 25 30 —m f\] FIG.5.25 VARIATION OF CROSSOVER PRESSURE WITH NUMBER OF TURNS IN SPIRAL t - 182 - Buckling Pressure N=14 2 =| fe) D 2393 t = =238.| % : t ! T tT T O 5 10 I5 20 us FIG.5.26 VARIATION OF CROSSOVER PRESSURE WITH SPIRAL ROD DIAMETER 3 - 183 AONZIDIssSS YOLSSYYNV SHL YOS NOISSSYdxX3 WOIMIdWS SHL LSNIVOV G3LL01d SLINSSY IWLNAWINSdXS 22G Sls NG) 5) = f 9 (A9U8191}}5 JOJSe4sy) & ~—— - 184 - SNOILVALIS O13l3 Ni G3SN SYOLS3NYV 4O SSIdWUXE I'S 37NeWL (00S) - (066) Cay) Ww esi felele) Os! ends s0jseuy (2g) (€02) (661) (18d) 40g 6S'¢ O'bv! Z’¢l ‘dg S9X-xX2@G G$9-x71-S Cb-X1-G | SPOID 404S8s4y 943819U09 8391909 nog (o'1) ($1) $z0'0 g¢0'0 821g dog (art) (0"1) (S2'1) (uy) of£0'0 S200 2c20'0 "MOIUL 40}SOquy (8'6) (22) (Sb) Cur Ww £ o¢s'| Ebvl'] y Sue sojseuay HOM edk} peynosy Aavey uo-dis peynoig peynoi9 Jojseqiy G9-xX1-G GS9OX-X27G 09-X1-S G9-K1-G O9-X1-S 8pod419 (S20) (218°0) ($2°0) ($20) (298°0) Cul) Ww 610°0 9020'0 6100 610'0 2c0'0 SSOUXDIYL Odig (2) (02) (9S) (2€) (9¢) (ur) wi €!lgO 80S'0 cob’ l ¢1i3'0 vi6O 48JOWDIG Bdig (OS>) (02 6!) (OS!) (02s) (1g) (33) W 4yydeg Zel OOS 9b 8S} $9! WNW XD, (00%) (+ 01€1) (409) (00%) (9ze) (44) W 4ugdeg AA +006 +8l rAPAl ool eboisay (s2n) (001) (82) (SZ2) ($822) ru) wy uybue7 102 ogi Sb OSE Soo ]OJO} Buljedid (98S ‘N) (9°S]"4831PEW (Oolxew j0°9) (D8S'N) (08S 'N) pjelg [e's] se1s404 Ky!d1g - DIsiuny | [g'¢} doo [2¢}]66144 | [1g] "}ue4g - 185 D t P. Po mm(in) mm (in) D/t bar(psi) | bar (psi) 25.48 0.508 4.4982 13 .034 (1,003) (0.020) 50.0 (65) (189) 38.18 0.889 49.2 7 241 19.93} (1.503) (0.035) , (105) (289) 31. 83 0.889 10.207 34.482 (1.253) (0.035) 38.8 (148) (500) 28.65 0.889 12.070 496 897 (1.128) (0.035) 32.2 (175) (680) 25.43 0.889 16.759 64.828 (1.001) (0.035) 28.6 (243) (940) 31.83 0.889 35.8 W.24] 33.655 (1.252) (0.035) ; (163) (488) 31.85 0.889 35.8 18.070 92.900 (1.254) (0.013) : (262) (1347) TABLE 5.2 CHARACTERISTICS OF PIPES USED IN ARRESTOR EXPERIMENTS ~"186 - meoum| “4 | bt | Lvt | Po/p, | Foo sep | Yu, | “st. | Moy oy “st |"sh% ain | 50 | 2.45] 12.5 | 1.454 | 1.55 60 |.246 | .275 11.8% ain |50 | 2.75 /12.5 |1.708 | 268 | 91 |.384| .912 137.5% wateR| 50 | 2.75 | 12.5 | 1.708 | 2.02 | 30 |.384 | .528 37.5 % air | so | 3.25/12.5 |] 2.23 | 2.81 | .95 | 681 | 937 37.6 % waTer| 50 | 3.25/125 | 2.23 | 268 | .36 |.68! | .916 34.5% AIR | 357| 2.86/ 5.71 | 1.557 | 1.84 | .65 | .265| .397 49.8 % WATER| 35.7| 2.86/ 5.71 | 1.557 | 1.68 36 | 265] .322 21.5% air | 357] 286 | 857| 2.25 | 2.81 Lt | 588] .853 45.1% WATER] 35.7) 2.86 | 8.57| 2.25 | 2.59 43 | 588| 750 27.6 % ain | 357; 2861] 1143| 2.55 | 3.12 x |.729| 1000] 37.2% WATER! 357| 2.86 | 1.43} 255 | 2.91 * | 729] .912 25.1% TABLE 5.3. COMPARISON OF DYNAMIC AND STATIC * Flip- Flop Mode ARRESTOR EFFICIENCY ~ 187 g uoibey [] Vv uolbey Fe] 1NOS8S 31AONE YOS AYLEWOSD G3ZIMVAad! 1'9'SI4