Patent Publication Number: US-2012043755-A1

Title: System and Method for Relieving Stress at Pipe Connections

Description:
RELATED APPLICATIONS 
     The present application claims the benefit of priority to U.S. Provisional Patent Application No. 61/375,751 entitled “Steel-HDPE Epoxy Bonded Bend Limiter” filed Aug. 20, 2010, the entire contents of which are hereby incorporated by reference. 
    
    
     FIELD 
     The present invention relates to flexible pipe structures used for the transport of liquids or gases, and more particularly to a system and method for managing stresses in pipe connections. 
     BACKGROUND 
     In many industrial, power plant, and shipbuilding situations pipes are rigidly connected to large structures including other pipelines through welds or bolted flanges. In such structural assemblies, loads applied to the pipe are concentrated near the structural attachment or termination end of the pipe. In addition, the strength of the pipe at the mechanical connection is often weaker than the parent pipeline. Therefore loads on the pipe often result in failures at the connection. 
     SUMMARY 
     The various embodiments provide structural systems and methods for relieving stress at pipeline connections including flanges. The various embodiments include positioning a rigid sleeve around or within the portion of the pipe close to attachment to the other structure and filling the volume between the pipe and the sleeve with a deformable material such as epoxy that adheres to both the sleeve and the pipe. The sleeve may be conical in shape, such as a frustum, or parabolic in shape. The sleeve may be positioned around the outside of the pipe such as to form a collar. Alternatively, the sleeve may be positioned within the inside of the pipe such as to form a narrowed portion. Under tensile, torsional or bending loads in the pipe the interaction of the pipe with the sleeve through the epoxy reduces the stress concentrations in the vicinity of the pipe attachment. The embodiments are particularly applicable to flexible pipes, including pipes made from high-density polyethylene (HDPE). The embodiments enable bolting a flexible pipe to a large structure or bolting two sections of flexible pipes together so that the entire assembly may be bent without the connection becoming the weak link, thereby reducing the chance of pipe failure at the connection. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Features of the present system and method are illustrated by way of example and not limited in the following figure(s). In the figures, like numerals indicate like elements. In some cases, elements in two figures which are similar or analogous, but represent somewhat different embodiments or different instances of the same element, may be represented by similar numbers with suffixes (for example,  110 ,  110   e ,  110   c ,  110 . 1 ,  110 . 2 , etc.). 
         FIG. 1A  is a cross sectional view of a pipe attachment assembly according to an embodiment. 
         FIGS. 1B and 1C  are diagrams illustrating structure and force references related to the pipe attachment assembly shown in  FIG. 1A . 
         FIGS. 2A and 2B  are longitudinal and lateral cross sectional views of a pipe attachment assembly according to an embodiment. 
         FIG. 3A  is a cross sectional view of a pipe attachment assembly according to another embodiment. 
         FIG. 3B  is a detail of a feature of the epoxy of the pipe attachment assembly illustrated in  FIG. 3A . 
         FIGS. 4A and 4B  are longitudinal and lateral cross sectional views of a pipe attachment assembly according to the embodiment shown in  FIG. 3A . 
         FIG. 5  is a cross sectional view of a pipe attachment assembly according to another embodiment. 
         FIGS. 6A and 6B  are longitudinal and lateral cross sectional views of a pipe attachment assembly according to an embodiment. 
         FIG. 7A  is an illustration of an embodiment assembly and  FIG. 7B  is a graph of a bonding material hardness along the length of the pipe shown in  FIG. 7A . 
         FIG. 8  is a cross sectional view of a pipe attachment assembly according to another embodiment. 
         FIG. 9  is a cross sectional view of a joint between two pipes with each pipe embodying a system for relieving pipe stresses at the connection according to an embodiment. 
         FIG. 10  is a cross sectional view of a pipe attachment assembly according to another embodiment. 
         FIG. 11A  is an illustration of an embodiment assembly and  FIG. 11B  illustrates elements useful for modeling of distribution of shear stresses in the embodiment illustrated in  FIG. 11A . 
         FIGS. 12A-12C  illustrates alternative configurations of a reinforcing member generally referred to herein as a frustum. 
         FIG. 13  is a process flow diagram of an exemplary method for constructing a system for distributing shear stresses in a pipe. 
     
    
    
     DETAILED DESCRIPTION 
     The various embodiments will be described in detail with reference to the accompanying drawings. Wherever possible, the same reference numbers will be used throughout the drawings to refer to the same or like parts. References made to particular examples and implementations are for illustrative purposes, and are not intended to limit the scope of the invention or the claims. Alternate aspects may be devised without departing from the scope of the invention. Additionally, well-known elements of the invention will not be described in detail or will be omitted so as not to obscure the relevant details of the invention. 
     The words “exemplary” and/or “example” are used herein to mean “serving as an example, instance, or illustration.” Any aspect described herein as “exemplary” and/or “example” is not necessarily to be construed as preferred or advantageous over other aspects. 
     Pipes are key structural elements in many structures, including natural gas pipe systems, water pipes, and other applications. In many applications, pipes are made from high-density polyethylene (“HDPE”) which has lower strength and elastic modulus than steel and aluminum pipes, meaning that HDPE pipes are more flexible than steel or aluminum. In many applications the pipes are subject to strong forces, such as bending forces, compression forces, stretching forces, or torsional forces. These forces can lead to large stresses in the pipe in the vicinity of joints or connections between the pipe and other structures or other sections of pipe. 
     Polyethylene thermoplastic pipes are used in some offshore seawater intake or outfall applications. When an HDPE pipe is used for offshore seawater intakes, hydrodynamic forces can result in large axial forces or moments applied to the pipes. As a result, such pipes may bend with a bend ratio (the bend radius divided by the diameter of the pipe) of roughly 20 to 30. Such a high bend places large loads on the mechanical flanges used to connect long lengths of fused pipe to larger structures (such as tanks and plenum) or to other lengths of pipe. This can create severe strains in the flange stub end of the pipe, which can result in opening a gap between the lengths of pipe and the connected structure or other pipe length. In extreme cases, the bending forces can cause a failure in the HDPE pipe or fusion joints, which can cause the entire pipeline to sink. Pipes have failed at mechanical joints in the past, and the potential for failure appears to be more severe with large diameter pipes. 
     To alleviate this problem, the inventors have explored numerous solutions in the past. One approach involved adding external snug-fitting stiffener sleeves to the HDPE for about 1 to 2 diameters on either side of the flange. Modeling of this approach showed it to be ineffective because there is no shear capability between the outer sleeve and the HDPE. The high stress in the bent pipe is still applied fully to the stub end, and it is greatly distorted. 
     Another approach evaluated involved beefing up the stub ends, by using a heavier stub end which reduces the stress in the area of increased pipe thickness. Physical tests show much less distortion which could reduce the potential for opening of the flange. However, this approach introduces quality control and availability issues. 
     Another approach used by the inventors has been to insert a press-fit inner sleeve inside the pipe adjacent to the connection and to utilize an external clamp with a roughened inner surface that very tightly squeezes both the HDPE pipe and the internal sleeve. This approach squeezes the HDPE pipe between two stiff inner and outer cylinders and develops a frictional shear capacity between the steel clamp and the HDPE pipe. However, because of the extreme differences in the elasticity of steel and HDPE, the shear distribution between the clamp and the HDPE is not uniform: it is excessively high at the clamp edge furthest from the flanged connection. The stresses are not reliably relieved at the connection under repetitive pipe loading and this approach is expensive to fabricate. 
     To address the stress concentration problem in a manner that is superior to previously considered approaches the various embodiments include a rigid sleeve positioned about a pipe with the volume between the pipe and the sleeve element filled with epoxy. The pipe with the rigid sleeve is also referred to herein as a “reinforced pipe”, a “pipe attachment assembly,” a “pipe with a stress reliever element,” and a “pipe with a stress reliever for substantially uniform distribution of stress.” The elements taught herein apart from the pipe proper, and in particular the sleeve and the epoxy, may be referred to a “pipe bend limiter,” a “bend limiter,” or a “pipe stress relief element,” or by substantially similar terms. 
     Epoxy is an adhesive polymer formed from reaction of a “resin” with a “hardener”. Epoxy has a wide range of applications, including as a general purpose adhesive. Epoxy is also referred to in this document as an “elastic potting material” (“potting” referring to a material which is pourable at least in initial use or application, and which has sufficient flow properties to fill relatively small voids, gaps, pockets, etc.); and is also referred to herein as an “elastomeric bonding material” (“elastomeric” referring to a material that is able to resume its original shape when a deforming force is removed). When used during a manufacturing process, an epoxy typically starts as relatively fluid though viscous, but then permanently hardens to a relatively more solid form, though a form still capable of bending, compressing and stretching. 
     In embodiments described herein, an epoxy is used to transfer a shear stress or force from a flexible polymer pipe to a rigid sleeve. In an embodiment, the epoxy is configured to transfer a pressure of approximately 250 pounds per square inch of pipe surface. There are epoxy and elastic potting materials that will adhere to HDPE and to the sleeve. 
     While the embodiments are described with reference to pipes made from HDPE; however, the embodiments may be applied to other types of flexible pipes (i.e., pipes with a relatively low elastic modulus). 
     As disclosed herein, a bend limiter is used to remove stress from a pipeline at a flange of the pipe. The bend limiter is made in part from a sheet of material, referred to as a sleeve, which is stiffer than the HDPE. The sleeve may be a metal, which may include for example and without limitation steel, titanium, or aluminum, or related alloys. The sleeve may also be a fiber glass. 
     Another element of the bend limiter is an epoxy which adheres to both the bend limiter and the HDPE pipe, and transfers forces from the HDPE pipe into the bend limiter via shear stress, thereby greatly lowering the stress in the HDPE at the stub end. 
     The various embodiments redistribute stress and strain in the vicinity of an HDPE pipe mechanical joint, and thus improve reliability during high loading conditions—particularly high bending. The embodiments reduce the high stress and strain that occur at stub ends in HDPE pipe attached to other structures (e.g., a tank or another pipe). An embodiment could be employed as a means for attaching to an HDPE pipe termination or mid-section for the purpose of adding wall anchors, pulling points, etc. 
       FIG. 1A  shows a longitudinal cross-sectional view of a pipe  105  encircled with a sleeve  110  and epoxy fill  115  which together form a pipe attachment assembly  100 .  FIG. 1B  illustrates a cylinder  105 . c  representing the pipe element only, illustrated in a three-dimensional view. Planar surface  102  illustrates a longitudinal plane bisecting the pipe  105 . c.    
     In the cross-sectional view shown in  FIG. 1A , the pipe  105  encompasses an inner space with a longitudinal axis  125 , the inner space typically being intended for use for the transport of liquids or gases. A radial direction  127  is also associated with the pipe  105 , as indicated in  FIG. 1A-1C  by dashed line  127  which is orthogonal to the long axis  125 . 
     In the illustrated embodiment, the pipe  105  terminates with a flange  120 . The end of the pipe  105  with the flange  120  is also referred to herein as the “stub end of the pipe,” or simply the “stub end.” 
     Starting at or near the flange  120  and extending for some length along the end of the pipe  105  is an epoxy or elastic potting material  115  which in an embodiment is bonded continuously to the surface of the pipe  105  and completely covers the circumferential length of pipe  105  from the flange  120  out to some length along the pipe which terminates at a designated endpoint  138  along the length of the pipe  105 . The epoxy  115  will have a high shear strength, meaning it can withstand stretching and bending forces without the epoxy tearing or cracking. For applications with large HDPE pipes  105 , the shear strength of the epoxy  115  may typically be on the order of 300 psi or greater. At the same time, the epoxy  115  has a low shear modulus (a measure of how stiff the epoxy is relative to torsion and twisting) and so will significantly distort, thus allowing the HDPE pipe  105  to move relative to a rigid sleeve  110  (described further immediately below). 
     In the pipe attachment assembly  100  of  FIG. 1 , the elastomeric potting material  115  is completely external to the pipe  105 . The width of the epoxy  115  varies along the length of the pipe  105 , being thinnest at or near the flange  120  and getting progressively thicker progressing along the length of the pipe away from the flange  120 , reaching a maximum at or substantially near to the endpoint  138 . In a pipe attachment assembly  100 , the width of the epoxy  115  (i.e., the gap between the pipe  105  and the sleeve  110  increases linearly from the flange  120  to the endpoint  138 ). As discussed in more detail below, in various embodiments, the gap between the sleeve  110  and the pipe  105  may be varied and/or the epoxy shear modulus may be varied in order to cause the shear to be uniform and within the shear limits of the bond between the pipe and the epoxy. It should be noted that in the pipe attachment assembly there will be a bond shear strength between the epoxy and the pipe or sleeve, and there is a shear strength within the epoxy itself, which when exceeded, the material itself shears off internal to the epoxy. In general, if the epoxy&#39;s bond strength is 300 psi, then the epoxy&#39;s shear strength will be 300 psi or higher. 
     Immediately external to the epoxy  115 , and bonded to the epoxy  115 , is a rigid sleeve  110 . The rigid sleeve  110  is made of a high elastic modulus material which is harder than the material of the pipe  105 . For example the rigid sleeve  110  may be comprised of a metal or fiberglass. In an exemplary embodiment the rigid sleeve  110  is composed of a metal such as steel, titanium, or aluminum. As the rigid sleeve  110  is continuously bonded to the epoxy  115 , the shape of the rigid sleeve  110  conforms to the shape of the outer surface of the epoxy  115 . Since the epoxy  115  varies in, width, the rigid sleeve  110  forms a frustum. A frustum shape is discussed further below with respect to  FIG. 12 . The rigid sleeve  102 , epoxy  115 , and sleeve flange  128  may be referred to together as a bend limiter  102  or, synonymously, as a pipe bend limiter  102 . 
     The narrow or smaller diameter end of the frustum of the rigid sleeve  110  substantially coincides with the flange  120  of the pipe  105 . The wide or larger diameter end of the sleeve  110  is longitudinally removed from the flange  120  of the pipe  105 , substantially coinciding with or being near the endpoint  138 . 
     This variation in width allows the present system and method to adjust for non-uniform relative movement between the HDPE pipe  105  and the sleeve  110 . When the pipe  105  is loaded by a moment, torsion, tension or compression, the relative movement between the pipe  105  and the sleeve  110  is greatest at point  138  and is least near the flange  120 . The variation in gap allows for a near-uniform shear in the epoxy  115  over the length of the sleeve  100  even when there is a non-uniform relative movement between the pipe  105  and the sleeve  110 . Therefore, with a near uniform shear strain in the potting material  115 , there is a near-uniform transfer of load from the pipe  105  into the sleeve  110  over the length of the epoxy  115 . 
     The illustrated pipe attachment assembly  100  includes a backup ring  128 , also referred to as a “sleeve flange,” which serves as a base at the narrow end of the sleeve  110 . The backup ring  128  is rigidly connected to the flange  120  (for example, by the pressure exerted on flange  120  as it is squeezed between backup ring  128  and a mating surface, not shown), ensuring that the sleeve  110  is rigidly connected at a joint which may be formed at the end of the pipe  105 . The bolt  130 , which may extend through the backup ring  128 , may be used to connect an end of the pipe attachment assembly  100  to a mating surface (not shown in  FIG. 1 ). The mating surface may be another pipe or may be a container or tank of some kind. Joints formed between the pipe attachment assembly  100  and other elements are discussed further below with respect to  FIGS. 9 and 10 . 
     While illustrated in the figure as penetrating only through the backup ring  128 , the bolt  130  may be configured to penetrate through the backup ring  128  and the flange  120 . Also, other types of attachment mechanisms may be used, including positioning the flange  120  within structural layers of the structure which it is attached, and using rivets, screws and/or adhesives instead of bolts. 
     Also shown in  FIG. 1C  are forces and moments  140  to which the pipe  105  may be subject. These forces  140  are illustrated in relation to the central longitudinal axis  125  of the pipe  105  and the mating surface plane  190 . Forces to which the pipe  105  may be subject include bending moments  150 . 1 ,  150 . 2  orthogonal to axis  125 , shear forces  155  orthogonal to axis  125 , axial forces  160  (compression and tension) along axis  125 , and torsional moments  157  about axis  125 . 
     Bending moments, tensile loads, and torsion moments result in stresses in the pipe  105 . These shearing stresses, in turn, can induce a gap at an end of the pipe  105  where the flange  120  is mated to a mating surface (not shown in the figure). These stresses can result in pipe failure, such as flange cracking or bursting. The various embodiments solve this problem by enabling movement of the pipe  105  relative to the sleeve  110  resulting in a shear stress within the epoxy layer  115 . The loads on the pipe  105  at or near the flange  120  are thus distributed via this shear stress in the epoxy  115  to the sleeve  110  over the full length of the epoxy  116 , thereby uniformly reducing the stresses along the length of the pipe  105 . In this way the risk of a gap formed between the flange  120  and a mating surface is reduced, and the risk of pipe failure at or near the flange  120  is reduced. 
     In an exemplary embodiment of the present system and method, the constant outer diameter of the pipe  105  may be somewhere between 12″ (0.3 meter) and 98.5″ (2.5 meter). In an embodiment, the thickness of the walls of the pipe  105  may be between about 0.5″ (1.3 cm) and 3″ (7.6 cm). In an embodiment, the length of the sleeve  110  from the base of the frustum to the top of the frustum may be 1 to 2 times the diameter of the pipe  105 , or about 5′ to 15′ (1.5 meters to 4.5 meters). In an embodiment, the thickness of the rigid sleeve  110  may be approximately ¼″ (0.64 cm). In an embodiment, the epoxy  115  may vary in width along the length of the pipe from approximately 1/10″ (0.25 cm) at or near the flange  120  to approximately 1-2″ (2.5 to 5 cm) wide at or near its terminus (at or near point  138 ). The dimensions listed above are merely to serve as an illustrative example, and other embodiments with differing dimensions fall within the scope of the claims. 
       FIGS. 2A and 2B  are longitudinal and lateral cross sectional views of a pipe attachment assembly  100  according to the embodiment shown in  FIG. 1A . Cross sectional view  210 . 1  and cross-sectional view  210 . 2  are both orthogonal to the long axis  125  of the pipe  105 . 
     Cross-sectional view  210 . 2  represents a lateral cross section through the pipe  105  at a location relatively closer to the flange  120 , and cross-sectional view  210 . 1  represents a lateral cross section through the pipe  105  at a point or a distance relatively further from the flange  120 . As can be seen, the cross sectional view  210 . 1  shows a wider portion of the epoxy  115  as compared to the narrower portion of the epoxy  115  shown in the cross-sectional view  210 . 2 . Consequently the, sleeve  110 , being exterior to the epoxy  115 , has a wider diameter in the cross-sectional view  210 . 1  and a narrower diameter in the cross-sectional view  210 . 2 . 
     The variation in width of the epoxy  115 , and the corresponding variation in the diameter of the sleeve  110  along the length of pipe  115 , which results in the sleeve  110  being a frustum with a narrow end close to the flange  120  and a wide end removed at some distance from the flange  120 , further results in a substantially more even and continuous distribution of shear stresses along the end of the pipe attachment assembly  100  which result from the primary forces  140  on the pipe  105 . 
     The shear stress distribution between the pipe  105  and the sleeve  110 , carried through the epoxy  115 , can be controlled by: varying the thickness of the epoxy  115  layer and thus the shape of the sleeve  110 , varying the shear modulus or stiffness of the epoxy  115 ; or by a combination of epoxy thickness variation and epoxy shear modulus variation. The objective of these variations is to have a near-uniform shear stress in the epoxy  115  and to not exceed the bond strength between the epoxy  105  and the sleeve  110 , or to not exceed the bond strength between the epoxy  115  and the pipe  105 . Providing a near-uniform shear stress in the epoxy  115  will result in uniformly distributed loads from the pipe  105  to the sleeve  110 . Embodiments illustrating these variations are discussed further below. 
       FIG. 3A  illustrates an exemplary embodiment of a pipe attachment assembly  300 . Many elements shown in  FIG. 3A  are the same or substantially similar to those described above with respect to the pipe attachment assembly  100 , and a detailed description will not be repeated here. However the pipe attachment assembly  300  has the epoxy  115  bonded to the interior surface of the pipe  105 . Further, the sleeve  110  is bonded to the interior surface of the epoxy  115 . The result is that the interior sleeve  110  now has a frustum-shape with the narrow end of the frustum removed at some distance from the flange  120 , substantially at or near point  138 , and the wide base of the frustum in close proximity to the flange  120 . A standard backup ring  315  may be employed to connect the pipe attachment assembly  300  to a mating surface (not shown). 
     As shown in  FIG. 3B , the epoxy  115  of the pipe attachment assembly  300  also has a minor frustum-shaped base  310  at the end farthest removed from the flange  120  with the slope of the frustum so inclined so as to facilitate smooth flow of fluids within the interior of the pipe  105 . 
       FIGS. 4A and 4B  are longitudinal and lateral cross sectional views of the pipe attachment assembly embodiment shown in  FIG. 3A . Cross sectional view  410 . 1  and cross-sectional view  410 . 2  are both orthogonal to the long axis  125  of the pipe  105 . 
     Cross-sectional view  410 . 2  represents a cross section through the pipe  105  at a location relatively closer to the flange  120 , and cross-sectional view  410 . 1  represents a cross-sectional view at a point or a distance relatively further from the flange  120 . As can be seen, the cross sectional view  410 . 1  shows a wider portion of the epoxy  115  as compared to the narrower portion of the epoxy  115  shown in the cross-sectional view  410 . 2 . Consequently, the sleeve  110 , being interior to the epoxy  115 , has a relatively narrower diameter in the cross-sectional view  410 . 1  and a relatively wider diameter in the cross-sectional view  410 . 2 . 
       FIG. 5  illustrates another exemplary embodiment of a pipe attachment assembly  500 . Many elements shown in  FIG. 5  are the same or substantially similar to those described above with respect to the pipe attachment assembly  100 ,  300  and a detailed description will not be repeated here. However the pipe with bend limiter  500  has the epoxy  115   e  with a nonlinear variation in width starting from the flange  120  and extending along the length of the pipe  105  towards the point  138 . 
     In one embodiment, the nonlinear variation may result in an exponential curvature of the surface of the epoxy  115   e  which is in contact with the sleeve  110   e . In turn, the frustum formed by the sleeve  110   e  has a curvature along the length of the long axis  125 , starting closer to the long axis  125  near flange  120  and curving away from the long axis  125  approaching the point  138 . In an exemplary embodiment, the frustum of the sleeve  110   e  may have an exponentially shaped curvature along the direction of long axis  125 . 
     To maximize the shear transfer loads (keeping an optimal high shear load along the entire epoxy-filled gap between the sleeve  110   e  and the HDPE pipe  105 ), it is an advantage to have the gap between the HDPE pipe  105  and the sleeve  110   e  be non-linear along the pipe axis  125 . An exponential shape may be ideal; a truncated shape approximating the exponential shape may be the most practical to construct. The nonlinear variation in the width of the epoxy  115   e , and the corresponding nonlinear variation in diameter of the sleeve  110   e  along the length of the pipe  115 , contributes to a more even and continuous distribution of shear stresses  140  through the epoxy  115  and between the pipe  105  and the sleeve  110 . 
     While  FIG. 5  illustrates the epoxy  115   e  and sleeve  110   e  being external to the pipe  105 , in an alternative embodiment the epoxy  115   e  and a curved frustum-shaped sleeve  110   e  may be positioned internal to the pipe  115 . 
       FIGS. 6A and 6B  are longitudinal and lateral cross sectional views of a pipe attachment assembly according to an embodiment. Many elements shown in  FIGS. 6A and 6B  are the same or substantially similar to those described above with respect to the pipe attachment assembly  100 ,  300 ,  500 , and a detailed description will not be repeated here. However, the pipe attachment assembly  600  illustrated in  FIG. 6A  includes a sleeve  110  with multiple sleeve sections  110   s . The sleeve sections  110   s  run parallel with the long axis  125 , but are separated from each other by longitudinal gaps  135  which run the length of the sleeve  110 . While four sleeve sections  110   s  are shown as separated by four gaps  135 , more or fewer sleeve sections and gaps may be employed as well. Also, the width of the gaps  135  shown is representational only, and gaps employed in practical application may be thinner, or wider relative to the sleeve sections  110   s.    
     Each sleeve section  110   s  may be attached to the backup ring  128  by a hinge  610 . Each hinge  610  may be a flexible element which securely connects a sleeve section  110   s  to the backup ring  128  while permitting the sleeve section  110   s  to pivot at a respective hinge  610 . Sleeve sections  110   s  thereby can expand or contract along a direction approximately orthogonal to the length of the sleeve section  110   s , that is, in an approximately radial direction of the pipe  105  as suggested by arrows  620  in the figure. This, in turn, permits the sleeve  110  as a whole, which is comprised of its sleeve sections  110   s , to expand or contract if the pipe  105  expands or contracts in a direction  127  orthogonal to axis  125 . The pipe  105  may, for example, expand or contract radially due to thermal expansion or contraction, compression or stretching directed along the axis  125 , or due to pressure of fluids within the pipe  105 , or for other reasons. 
     Each sleeve section  110   s  still takes axial shear without distortion and passes the load through its respective hinge  610  to the backup ring  128 . 
     The hinge  610  may be a barrel hinge, a piano hinge, a spring loaded element or elements, a flexible metallic strip, a flexible polymer, or other element or elements which permit flexing at the junction of the sleeve element  110   s  and the backup ring  128 . 
     While  FIGS. 6A and 6B  illustrate the epoxy  115 , the sleeve sections  110   s , and the hinges  610  as being external to the pipe  105 ; in an alternative embodiment the epoxy  115 , the sleeve sections  110   s , and the hinges  610  may be internal to the pipe  115 . 
       FIG. 7A  is an illustration of an embodiment pipe attachment assembly and  FIG. 7B  is a graph of epoxy  115  shear strength, or hardness, along the length of the pipe shown in  FIG. 7A . Many elements shown in  FIG. 7A  are the same or substantially similar to those described above with respect to the pipe attachment assembly  100 ,  300 ,  500 ,  600 , and a detailed description will not be repeated here. However, the pipe attachment assembly  700  may have an epoxy  115  with a shear modulus, or hardness, which varies along the axial length  125  of the pipe  105 . 
       FIG. 7B  is a plot  705  indicating several possible variations in the shear modulus or hardness of the epoxy  115  along the length of the pipe attachment assembly  700  shown in  FIG. 7A . In one exemplary embodiment, illustrated by plotline  710 , the shear modulus of the epoxy  115  is constant along the length of the pipe  105 . Note that while plotline  710  is indicative of a low constant shear modulus along the entire length of the epoxy  115 , in other embodiments the constant shear modulus may be a higher shear modulus. 
     In another exemplary embodiment, illustrated by plotline  720 , the shear modulus of the epoxy  115  is constant at a first constant value along a first segment of the epoxy  115 , and then is constant at a second and different constant value along a second segment of the epoxy  115 . While not shown, the epoxy  115  may have three or more segments, each segment having a different shear modulus from the others, but the shear modulus being constant within each segment. In an embodiment, the epoxy  115  will have a higher shear modulus closer to the backup ring  128  and/or the flange  120 , and a progressively lower shear modulus at distances further removed from the backup ring  128  and/or the flange  120 . 
     In another exemplary embodiment, illustrated by plotline  730 , the shear modulus of the epoxy  115  varies in a substantially linear fashion along the length of the pipe  105 . In an embodiment, the epoxy  115  will have a higher shear modulus closer to the backup ring  128  and/or the flange  120 , and a progressively lower shear modulus at distances further removed from the backup ring  128  and/or the flange  120 . 
     In another exemplary embodiment, illustrated by plotline  740 , the shear modulus of the epoxy  115  varies in a substantially exponential fashion along the length of the pipe  105 . In an embodiment, the epoxy  115  will have a higher shear modulus closer to the backup ring  128  and/or the flange  120 , and a progressively lower shear modulus at distances further removed from the backup ring  128  and/or the flange  120 . 
     Variations in the shear modulus of the epoxy  115  can be achieved by using different formulations of epoxy along the length, or by adding various degrees of hardening compounds along the length, as appropriate. Other means may be employed as well. 
     The variations in the shear modulus of the epoxy  115  may contribute to a more even and continuous distribution of shear stresses  140  along the end of the pipe attachment assembly  700 . While  FIG. 7A  illustrates the epoxy  115  and the sleeve  110  as being external to the pipe  105 , in an alternative embodiment the epoxy  115  and the sleeve  110  may be positioned internal to pipe  105 . 
       FIG. 8  illustrates another exemplary embodiment of a pipe attachment assembly  800 . Many elements shown in  FIG. 8  are the same or substantially similar to those described above with respect to the pipe attachment assemblies  100 ,  300 ,  500 ,  600 ,  700 , and a detailed description will not be repeated here. 
     The pipe attachment assembly  800  may include a sleeve  110  with two or more component parts; for example, a first component part  110   lp  and a second component part  110   ep . As shown in  FIG. 8 , the first part  110   lp  varies linearly in distance from the pipe  105  along the axial length  125  of the pipe  105 , while the second part  110   ep  varies nonlinearly in distance from the pipe  105  along the axial length  125  of the pipe  105 . Corresponding linear and nonlinear variations occur in the width of the epoxy  115  along the axial length  125  of the pipe  105 . Note that the nonlinear variation has been exaggerated in the figure for purposes of illustration only. 
     The pipe attachment assembly  800  may also include a length along the pipe  105  where an annular gap  810  exists. The annular gap  810  is a region where no epoxy  115  is used to bond a portion of the sleeve  110  with the pipe  105 , and where the sleeve  110  and the pipe  105  are not bonded. 
     Either or both elements, that is either of a sleeve  110  with two or more component parts  110   ep ,  110   lp , and/or an annular gap  810  in the epoxy  115 , may contribute to a more even and continuous distribution of stresses  140  through the epoxy  115  and between the pipe  105  and the sleeve  110 . 
     In general, the system and method of the various embodiments may vary the width of the gap between pipe  105  and the sleeve  110 , and/or may also vary the shear modulus of epoxy  115 , as described above. The system and method employ these variations so that the effect of either variation, or both in combination is to ensure that the shear stress between the pipe  105  and the epoxy  115  remains acceptable and substantially even. By varying the gap and/or varying the epoxy properties, the result is a bend limiter  102  which makes the shear substantially uniform and within the shear limits of the bond between the pipe  105  and the epoxy  115 . 
       FIG. 9  illustrates an exemplary mechanical joint  900  coupling a first pipe attachment assembly  100 . 1  with a second pipe attachment assembly  100 . 2 . Without embodiment bend limiting elements there is an increased risk that pipe bending, tension or torsion may introduce excessive stress in the pipe  105  at or near stub ends  120 , resulting in a pipe failure in this region or an undesired gap or breach at joint  900 , such as a separation between the flanges  120  of the pipe  105  associated with element  100 . 1  and the pipe  105  associated with element  100 . 2 . 
     With the bend limiting elements described in embodiments throughout this document, shear stresses on either or both of the first pipe attachment assembly  100 . 1  or the second pipe attachment assembly  100 . 2  are distributed in a substantially uniform fashion along the pipes  105  and also along the sleeves  110 . The distribution of shear stresses significantly reduces the risk of undesired gaps or breaches at the joint  900  or pipe failures at or near stub ends  120 . 
       FIG. 10  illustrates an exemplary mechanical joint  1000  coupling a pipe attachment assembly  100  with a mating surface  1005 , which may for example be an opening or portal into a compartment, container, well, or similar fluid bearing enclosure. Without the embodiment bend limiting elements there is an increased risk that pipe bending, tension or torsion may introduce excessive stress in the pipe  105  at or near stub ends  120  resulting in a pipe failure in this region or an undesired gap or breach at joint  900 , for example by inducing a separation between the flanges  120  of the pipe  105  and mating surface  1005 . 
     With the embodiment bend limiting elements positioned on a pipe  105  at an attachment, shear stresses on the pipe attachment assembly  1000  are distributed in a substantially uniform fashion along the pipe  105  and also along the sleeve  110 . The distribution of shear stresses significantly reduces the risk of undesired gaps or breaches at the joint  1000   900  or pipe failures at or near stub ends  120 . 
       FIG. 11A  is an illustration of an embodiment assembly and  FIG. 11B  illustrates elements useful for modeling of distribution of shear stresses in the embodiment illustrated in  FIG. 11A . 
     A pipe termination  100  loaded as shown in  FIG. 1C  with a downward shear load  150 . 2  or a counterclockwise moment  157  results in a moment applied to the pipe  105 . In such a situation, the upper portion of the pipe  105  is in tension and the lower portion of the pipe is in compression. A schematic of the pipe modeled is shown in  FIG. 11A . Preliminary analysis modeled an upper surface of the pipe as a pair of flat plates bonded by an epoxy and neglected the rest of the pipe. In that modeling, the upper plate is steel, the lower plate is HDPE, and the gap between is variable. 
     The system was modeled using finite element methods. Each element consisted of a one inch wide strip of HDPE  105 , a strip of epoxy  115 , and steel sleeve  110 . The parameters considered were the stresses and deflections on each end of the sleeve and the HDPE.  FIG. 11B  shows a free body diagram of a single element. The positive direction is to the left and points towards the free end of the system (directed from the flange  120  towards the point  138 ). The stresses s 1  and s 4  are unknown reactions-ultimately imposed by the fixed end (the flange end  120 ) while sh and sp are known stresses derived from the tension placed on the free end of the pipe (from the direction of point  138 ). The deflections d 1  through d 4  are absolute measurements of the change in position of each end of the pipe  105  and sleeve  110  due to elongation. The free end, represented by d 2  and d 3  are unknown while d 1  and d 4  are inferred from the rigid position of the fixed end. 
     Modeling the steel and epoxy as linearly elastic materials and the HDPE as a non-linear elastic material created relations between the eight parameters above. Four equations were derived and are presented below. 
     
       
         
           
             
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     The additional variables in the equations are defined as follows: Tp, Tg, and Th refer to the thickness of the pipe, epoxy, and steel respectively; Ep, and Eh refer to the tangent modulus of elasticity of the pipe and steel respectively; Eg refers to the shear modulus of the epoxy; and L refers to the length of the element. 
     The limiting factor in the design of the system is the maximum allowable shear stress in the epoxy. A preliminary value of 295 psi may be used as the ultimate shear strength of the epoxy based on literature provided by Reltek (“RELTEK LLC, 2345 Circadian Way, Santa Rosa, Calif. 95407”) about its BONDiT™ B-45 epoxy. The maximum allowable stress may be 150 psi to give a safety factor of 2. Two parameters affect the shear stress in the epoxy: the gap between the pipe and the sleeve; and the shear modulus of the epoxy. Several combinations may be considered in order to determine the most efficient configuration. An efficient configuration may be defined as one with a minimum gap and length. 
     In a first embodiment, the pipe attachment assembly may feature a linearly increasing gap between the pipe and the sleeve. In this embodiment, the steel sleeve represents a cone that would steadily increase the gap for the epoxy to fill. As described above, this gap setup may be configured to result in a steadily decreasing epoxy stress. 
     In a second embodiment, the pipe attachment assembly may feature two cones and a pipe. In this embodiment, the steel sleeve represents two cones rather than a single cone. This allows the gap to narrow faster near the free end yet still remain under 150 psi epoxy stress limit. A short segment at the fixed end with a constant gap further increased the efficiency of the system. This gap setup resulted in 2 sections of steadily decreasing epoxy stress with a sudden jump in stress at the interface of the two cones. The average stress in the epoxy was higher than with the linearly increasing gap. 
     In a third embodiment, the pipe attachment assembly may feature an exponential gap. In this embodiment, an exponentially increasing gap may be tailored to match the stress increase in the epoxy near the free end. This gap setup results in a nearly constant epoxy stress of 150 psi. 
     In a fourth embodiment, the pipe attachment assembly may feature a single epoxy used throughout the gap between the pipe and the sleeve. 
     In a fifth embodiment, the pipe attachment assembly may feature two epoxies used in the gap. Near the fixed end, where deflections are relatively small, a stiffer epoxy may be used. Near the free end, where deflections are larger, a more flexible epoxy may be used. Using two epoxies improved the efficiency of all gap setups. 
     The results of analyses of these embodiments are presented in the table below. 
     
       
         
           
               
               
               
               
               
             
               
                   
               
               
                   
                 Free End 
                 Fixed End 
                 Maximum 
                 System 
               
               
                   
                 Modulus 
                 Modulus 
                 Gap 
                 Length 
               
               
                 Gap Setup 
                 [psi] 
                 [psi] 
                 inch 
                 inch 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Linearly Increasing 
                 800 
                 800 
                 1.69 
                 30 
               
               
                 Linearly Increasing 
                 800 
                 1600 
                 1.50 
                 24 
               
               
                 2 Cones and a Pipe 
                 800 
                 800 
                 1.28 
                 21 
               
               
                 2 Cones and a Pipe 
                 800 
                 1600 
                 1.28 
                 20 
               
               
                 Exponential 
                 800 
                 800 
                 1.22 
                 20 
               
               
                 Exponential 
                 800 
                 1600 
                 1.20 
                 19 
               
               
                   
               
            
           
         
       
     
     An epoxy bonded bend limiter to relieve flange tension appears feasible. Nothing in the geometry or force balance indicates extraordinary challenges in the application of such a system. The gap between the sleeve and the pipe has the greatest effect on the size of the pipe attachment system. Using more than one epoxy can also help reduce the size of the pipe attachment system. 
     Elements herein are sometimes described in terms of a geometric shape known as a “frustum.”  FIGS. 12A-12C  illustrates alternative configurations of a reinforcing member generally referred to herein as a frustum. Element  1205  shown in  FIG. 12A  is a cone. Element  1205 .D is a planar surface bisecting cone  1205  and parallel to base  1205 .B of cone  1205 . Formed between base  1205 .B and planar surface  1205 .D is a lower portion  1205 .F of cone  1205 . Lower portion  1205 .F is a frustum. 
     A profile or vertical cross-sectional view of frustum  1205 .F is shown as element  1210  in  FIG. 12B . For purposes of this document, a cone with curved sides can also be used as a basis to define a frustum. As shown for example with element  1220  in FIG.  12 C, then, a frustum may have curved sides rather than straight sides, the curved sides extending between a wider horizontal base and a relatively narrower, parallel horizontal top of the frustum. 
     While the figures and the foregoing description describe embodiments in which the sleeve fully encircles an exterior or interior surface of the pipe, sleeve that partially encircle the pipe may be used in applications in which applied forces will be limited to particular directions. In some situations a pipe attachment may be subject to bending forces limited to a narrow angle, and not subject to bending over the remaining angles about the pipe center line. In such situations, the sleeve may be configured as a section of a frustum, with the sleeve positioned only within the angles of the pipe about which bending stresses are anticipated. Such embodiments will be configured substantially as shown in the figures and described above, with the exception that the sleeve and epoxy will not extend completely around or within the pipe. 
       FIG. 13  is a flow chart of an exemplary method  1300  for constructing a system for distributing shear stresses in a pipe. 
     The method begins by forming a frustum shaped sleeve that is configured to attach to a flange end of a pipe. The sleeve is comprised of a material which is stiffer than the pipe, and may for example be a metal, such as steel, titanium or aluminum, or fiberglass. 
     In decision step  1310 , a determination is made as to whether the sleeve is to be configured as an inner sleeve placed inside the pipe or as an outer sleeve to surround the pipe. 
     If an outer sleeve is to be used, then outer sleeve is attached to the flange or substantially near the flange at the end of the pipe in step  1315 , so that the narrow end of the frustum is attached to the bend limiter flange. 
     If the sleeve is to be configured as an inner sleeve, the inner sleeve is attached to the flange or substantially near the flange at the end of the pipe in step  1320 , so that the wide end of the sleeve of frustum is attached to the bend limiter flange. 
     In step  1325 , an epoxy of a desired shear modulus is poured into the gap which exists between the sleeve and the pipe. 
     In step  1330  the epoxy pouring process may be monitored to determine whether the gap has been completely filled to the desired length along the pipe. If so, the fabrication operation concludes at step  1345 . 
     If the gap has not been fully filled, a determination is made as to whether the shear modulus of the epoxy should be changed. If the shear modulus of the epoxy should not be changed, the epoxy pouring step  1325  continues. 
     If at step  1335  the shear modulus used for further filling of the gap is to be changed, a different epoxy with a different shear modulus may be selected in step  1340 , and poured into the remaining gap in step  1325 . 
     In an alternative embodiment, the epoxy may be attached to either the interior or the exterior of the pipe, using some kind of molding method or other manufacturing method, before the sleeve is in place area. After the epoxy is in place the sleeve may be attached (e.g., by epoxy bond) to the remaining exposed surface of the epoxy, as well as the sleeve being attached to the flange of the pipe. 
     While various embodiments of the present system and method have been described above, it should be understood that they have been presented by way of example, and not limitation. It will be apparent to persons skilled in the relevant art(s) that various changes in form and detail can be made therein without departing from the spirit and scope of the present system and method. Thus, the present system and method should not be limited by any of the above described exemplary embodiments, but should be defined only in accordance with the following claims and their equivalents. 
     In addition, it should be understood that the figures illustrated in the attachments, which highlight the structure, functionality and advantages of the present system and method, are presented for example purposes only. The architecture of the present system and method is sufficiently flexible and configurable, such that it may be implemented and utilized in ways other than that shown in the accompanying figures. 
     Further, the purpose of the foregoing Abstract is to enable the U.S. Patent and Trademark Office and the public generally, and especially the scientists, engineers and practitioners in the art who are not familiar with patent or legal terms or phraseology, to determine quickly from a cursory inspection the nature and essence of the technical disclosure of the application. The Abstract is not intended to be limiting as to the scope of the present system and method in any way.