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STUDY OF THE DOUBLE-CONE RING GASKET  by  Chao-lin Lou B . S c , National Taiwan University, 1963  A Thesis Submitted i n - P a r t i a l Fulfillment of the Requirements for the Degree of Master of Applied Science In the Department of Mechanical Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY:OF BRITISH COLUMBIA August, 1967  In p r e s e n t i n g t h i s  thesis  in p a r t i a l  f u l f i l m e n t of the requirements  f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia,  I agree  t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study.  I further  agree that p e r m i s s i o n f o r e x t e n s i v e  c o p y i n g of t h i s  t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h.i)s r e p r e s e n t a t i v e s . or p u b l i c a t i o n of t h i s w i t h o u t my w r i t t e n  Department of  permission.  The U n i v e r s i t y of B r i t i s h Vancouver 8, Canada r  1Q67  i s understood t h a t  copying  t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d  Mechanical  Date August, 2 5  It  Engineering  Columbia  ABSTRACT The double-cone ring gasket used as a static seal for high pressure operates on the principle of unsupported area by which the pressure applied to the gasket produces a large pressure at the seating surfaces thus maintaining a tight j o i n t . An analysis based on elastic behaviour and simplifying assumptions is developed for predicting the sealing forces and the strains in the gasket. An apparatus was built for testing the effectiveness of the :  gasket for sealing pressures to 10,000 p s i .  Gaskets^of several different  proportions were tested and-all were found to seal s a t i s f a c t o r i l y .  Strain  measurements made during the tests showed satisfactory agreement with predicted values for the assembly-condition, but there were discrepancies between the predicted and~observed values-for the-gaskets under pressure. These discrepancies -indicate that the simple assumptions used in the analysis are not sufficiently accurate. The general conclusions are -that the double-eone-ring gasket is satisfactory for high pressure s t a t i c seals and that the-proportions of the ring cross section- are not c r i t i c a l to the-effectiveness-of the seal.  ii  ACKNOWLEDGEMENT The author gratefully acknowledges Professor w, 0- Richmond for his invaluable guidance and encouragementthroughout this study.  Thanks  are extended to the staff in the Machine Shop of the Department of r  Mechanical"Engineering at the University of British Columbia, in particular to Mr. John Wiebe, for the assistance"in'preparing equipment and models used in the experimental part of this studyThis research project was financially supported by the National Research Council of Canada under Grant-In-Aid of Research Number A-1687<  iii  CONTENTS PAGE 1.  2.  3.  INTRODUCTION  1  1.1  Purpose of Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1  1.2  Outline of Thesis  .....1  THE DESIGN PRINCIPLE OF DOUBLE-CONE RING GASKET . . . . . . . . . . . . . . . .  3  2.1  Historical Review of High-Pressure Gasket Design . . . . . . . . . .  3  2.2  The Principle of Unsupported-Area . . . . . . . . . . . . . . . . . . . . . . . . .  6  2.3  Design Features of the Double-Cone Ring Gasket . . . . . . . . . . . .  6  THEORETICAL ANALYSIS  12  3.1  General  . . . . . . . . . . 12  3.2  Fundamental Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13  3.3  Derivation of the Governing Differential Equation . . . . . . . . . 14  3.4  Solution to the Governing Differential Equation . . . . . . . . . . . 16  3.5  Tangential Strains of Ring G a s k e t . . . . . . . . . . . . . . . . . . . . . . . . . . 21  3.6  Effects of Design Parameters-on"Ring Deformation and Seal ing Pressure-  4.  5.  6.  . . . . . . . . . . . . . . . 21  DESCRIPTION OF TEST MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1  Model Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27  4.2  Design. DetaiIs of Ring Models  4.3  Preparation of Test Model  . 28 w . . . . . . . . . . . . . . .  INSTRUMENTATION AND TEST PROCEDURE  29 33  5.1  Description of A p p a r a t u s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  5.2  Test Procedure . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . 39  EXPERIMENTAL RESULTS AND DISCUSSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.1  Results of S t r a i n - M e a s u r e m e n t s : . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42  6.2  Comparisons Between -Experimental"Data-and Theoretical Values......................o..............................  iv  46  PAGE 6 = 3 Observations Made During Tests . . . . . . . . . . . . . . . 6.4 7.  Design Criteria on Double-Cone Ring -Gaskets-  51 54  CONCLUSIONS AND RECOMMENDATIONS  58  7.1  Summary of Conclusions  58  7.2  Recommendations  59  REFERENCES  62  v  LIST OF ILLUSTRATIONS NUMBER  PAGE  1  Most Primitive Design of High-Pressure Gasket  ...........  3  2  Amagat's Type of Fully Enclosed Gasket . . . . . . . . . . . . . . . . . . . . .  4  3  High-Pressure Gasket Design U t i l i z i n g the Principle of Unsupported Area  5  4  A Double-Cone Ring Gasket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7  5  The Double-Cone Ring Gasket and Flanges Before Assembly . . . .  8  6  The Double-Cone Ring Gasket in the Assembled; State  8  7  A Double-Cone Ring-Gasket Assembly U t i l i z i n g Threaded Plug .  9  8  A Modified Double-Cone Ring Gasket Design-  9  A Longitudinal Sliver Isolated from a Ring Gasket . . . . . . . . . . 12  10  Simplified Boundary Conditions of a Ring Sliver . . . . . . . . . . . . 14  11  A n - I n f i n i t e s i m a l - B e a m - E l e m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14  12  Relationship Between K and Design Parameters, a and p , for  10  £  Double-Cone Ring Gaskets with e=12.5° . . . . . . . . . . . . . . . . . . . . . . 23 13  Relationship Between  and Design Parameters, a and p , for  Double-Cone Ring-Gaskets-with-e=12.5°  . . . . . . . . . . . . . . . . 24  14  Relationship Between n and Design Parameters, a and p , for Double-Cone Ring Gaskets witlr 0=12.5° . . . . . . . . . . . . . . . . . . 25  15  Relationship Between-n  c  and Design Parameters, a and p , r  for Double-Gone Ring Gaskets with 0=12.5° . . . . . . . . . . . . . . . . . . 26 16  Gage-Arrangements for (a) A-Wide-Ring Model, and (b) A Mid-Ring Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  17  Arrangement of the- Testing'Apparatus  18  Design Details of the Top Cover . . . . . . . . . . . . . . . . . . . . . .  19  Design Details of the Vessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  20  General Views of (a) The Top Cover, and (b) The Vessel . . . . . 37 vi  34 . . 35  NUMBER  PAGE  21  External Quarter-Bridge Circuit with Internal Dummy . . . . . . . .  39  22  Results of Strain-Measurements- Made""in Assembling WideRing Models ...... ................................  43  23  Results of Strain-Measurements'Made'in Assembling MidRing Models  44  24  Results of Strain-Measurements Made in Assembling NarrowRing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  45  25  p-e Curves for Wide-Ring Models  47  26  p-e Curves for Mid-Ring Models . . . . . . . . . . . . . . . . . . . . . . . . . . .  48  27  p-e Curves for Narrow-Ring Models  49  28  Typical'Stress-Strain Diagram for Mild Steel . . . . . . . . . . . . . .  51  29  A Wide-Ring Model After Tests  53  30  A Cross-Section of the Double-Cone Ring Gasket with Cylindrical Coordinates rez  s  g  s  vi i  ..  60  LIST OF TABLES NUMBER  PAGE  1  Physical Propertis of Model ^Materials'" .;..»•. •„•. . • . . . . . . . . . .  2  Design Parameters and Dimensions of Test-Models . . . . . . . . . . .  ;  vi i i  28 30  LIST OF SYMBOLS b:  Half-width of ring gasket, i n .  c :  Radial crush of ring gasket at edges,  c :  Total radial crush at edges of a ring gasket in the assembled state.  g  Q  C-j,  C^, C^:  Constants of integration.  D:  Flexural r i g i d i t y ,  Eh /12(l- ).  e:  Width of seating bevel, i n .  3  y  2  9  E:  Young's modulus, lb/in .  g:  I n i t i a l stand-off of flanges, i n .  G:  Shear modulus, G = E / 2 ( l + u ) .  h:  Thickness of ring w a l l , i n .  k: Pressure-contact r a t i o , k = e/h. K , K : Coefficients related to ({U) c p Eh x  M: M*: N:  Bending moment in ring w a l l ,  Ib-in/in.  Moment term, M* = Nhb sine/2Dr. 2  Sealing force normal per unit length of ring circumference, lb/in, 2  p:  Internal pressure applied in ring gasket, positive outward, lb/in .  P:  Pressure-force term, P = p(l-h/2r) - y(Nsine+ph)/r.  P*:  Particular integral,  r:  Mean radius of ring, i n .  R:  Maximum radius of flange bore, i n . 2  s : h  Hoop compression in ring w a l l , lb/in .  V:  Shear force in ring w a l l , lb/in.  w:  Total radial deflection of ring gasket, positive inward, i n .  w^:  Radial deflection of ring gasket due to bending, i n .  w:  Radial deflection of ring gasket due to shear, i n . ix  $  x:  Co-ordinate in axial direction of ring gasket, i n .  e : Tangential-strain at the centroid of the-axial - cross-section of ring gasket. 0  e : s  u: p, a: e:  Tangential strain on outer surface of ring  (x-0).  Poisson's r a t i o , Design parameters-of - ring, p •= r/h and a = b/h. Inclination angle of ring bevels.  n , rip: Coefficients related to tangential strain of ring gasket. c  w , 5 , A, y, TQ, r^j, r : 2  Operation parameters and notations defined for simplicity.  x  1 CHAPTER I INTRODUCTION 1.1  Purpose of Investigation It is obvious that the most vital•problem which must be faced in  every high-pressure application is the prevention of leaks.  This means in  every case some-form of sealing. To prevent the migration of high-pressure f l u i d across the j o i n t of a vessel-or-an assembly where no relative -motion occurs between the ;  joined parts, a great many forms of static seals have-been developed in the past years.  One of thesev the'double-cone ring gasket, is the subject  of the present study-.  This gasket is a pressure-energized ring seal used  in various fields of-high pressures.- Although'a-large amountof information about such ring-seals have-been published, much research work is yet to be 1  done.  Accordingly, to study- the behaviour of-the double-cone ring gasket r  as a> static seal with a view to determining the -optimum design properties and conditions-becomes the original-: object of this research project. ;  -. 1.2 - Outline of Thesis In the p r e s e n t t h e s i s a historical-review-of the design concepts of high-pressure-gaskets is reported,-and the-basic design principle of the double-cone ring gasket is discussed. A simplified- elastic analysis is- developed-, and equations for predicting the sealing- forces and the tangential-strains-in the doublecone ring gaskets are derived as the- r e s u l t s - o f - t h i s analysis. For the experimental part of the investigation, double-cone ring gasket models are made and tested.  The physical - properties of  2  model materials, the design details of the models, the testing apparatus :  and the test procedure are^separately described- in the thesis.  Then the  results of the- strain-measurements-made -in testing-the ring gasket models are presented-in graphic form and compared with the theoretically predicted values.  In addition, the observations - on- the - performance-characteristics  of the tested models are-also reported.  These observations lead to some  design c r i t e r i a for-the double-eone-ring gasket. Finally-,-a^-few statements-serving as-the general conclusions of the- present investigation are-mentioned-,-and suggestions-which may be ;  helpful -to-further- researehes -on the- behaviour-of - double-cone ring gasket :  are stated.  3 CHAPTER II THE-DESIGN PRINCIPLE OF DOUBLE-CONE RING GASKET 2.1  His tori cal ••Review--of~ High-Pressure-Gasket Design Three distinct-stages-in the evolution of high-pressure gasket  design may be distinguished- [1]*.  The- desigmeoneept developed in each of  these three stages can be discussed separately by way of introducing the problem of sealing the pipe connection in" a" high-pressure system. :  ;  A schematic- i-llustration of" the"most" primitive^ design is shown :  in Figure 1-. In this design, the' sealing-pressure exerted on the sealing :  surfaces between the-gasket-and the'rigid, members completely depends on the i n i t i a l bolt load applied in assembling this pipe connection. design has two shortcomings.  FIGURE 1 Most Primitive Design of-High-Pressure Gasket  ^Numbers in brackets identify references-at the end of thesis.  This  The f i r s t one is that during the assembly process-, in order to make the gasket proof against high pressure, the gasket must be- very- tightly compressed; as a consequence, the gasket-might flow sidewise, either into the interior of the pipe or to the outside;  Another shortcoming of this  design is that, when the hydrostatic pressure-in the-pipe-is very high, the gasket w i l l be blown out by i t . The second stage of e v o l u t i o n i s shown in Figure 2. :  In this  FIGURE 2 Amagat's•-Type of'. Fu 1 ly Enclosed Gasket improved design, a - f l a n g e i S t formed on the-end of -the pipe, and this :  :  pipe flange enters a recess -in such-a way-that the-gasket is entirely enclosed-by r i g i d metal walls. - Therefore, the gasket - can neither flow sidewise under- i n i t i a l - b o l t - load, nor'ean -it» be ^Town-out'by the hydro;  static pressure contained-. - The-.gaskets- of - such- a design are commonly u  known as the Amagat's- type-of~-fully-enclosed-gaskets-. -However, the upper pressure l i m i t at which an Amagat s gasket-can be effectively^ used is s t i l l ;  1  1  5  set by the intensity of' the i n i t i a l ' b o l t - l o a d imposed- in- assembly.  When  the hydrostatic pressure in the'pipe'has reached- an intensity equal to 1  that i n i t i a l l y exerted-on the sealing surfaces- by- the-screws, there is :  balance.  Any-further increase of hydrostatic-pressure-above this l i m i t 1  :  w i l l cause the gasket-to shrink away-from the-containing-wall and produce 1  :  a leak. Figure 3 shows the gasket-design-in-the-third-stage'of  evolution.  At f i r s t sight-, i t seems-that-this-design-appears-to-be-only- a- modification  FIGURE 3 - High-Pressure- Gasket-Design U t i l i z i n g - the Princip=le of• Unsupported Area ;  of the one shown in Figure 2-.  The location of gasket-has-been changed 1  from the front- to -the back side of -the- pi pe-f-Tange-.-- However, this :  :  apparently t r i v i a l change-has introduced-a-very-important change in the design concept-of high-pressure gasket-,- and has made the conventional, :  compressive gasket into one u t i l i z i n g the:principle of- unsupported area. This principle forms the basic design concept- of the double-cone ring  gasket and many other-pressure-energized''gaskets..  The next section w i l l  be devoted to describing this principle in d e t a i l . 1  2.2  The Principle- of Unsupported Area :  The principle- of unsupported area-can be- stated-as [1][2][3][5] 1  "If the geometrical design of a gasket assembly is such that- there exists a certain amount of unsupported area,- then the intensity of the sealing pressure on the sealing-surface-of -the gasket- can automatically be maintained at a fixed percentage higher-than- the pressure intensity in the contained f l u i d , and- leak-of -the f4uid cannot-occur as long as the j  retaining walls- of this gasket-assembly hold." In the design- shown in Figure* 3 , the unsupported area is the area of the cross-section of the pipe-.  This principle of unsupported  area is commonly credited to P.W. Bridgman, andhas been-widely employed 1  in many high-pressure gasket designs. - • A--survey- of- the - existing highpressure gaskets reveals that most of the-pressure-energized gaskets used- in- various fields- of-very- hi ghrpressures-ut-i-lize-.-this principle [8][9]. 2.3  Design- Features- of - the- Doub le-€one- Ring Gasket ;  The double-cone- ring-type-, of-igasket- was?-er4g4na'l ly developed in Germany-for highrpressure--reactors-. ^-AsH-mpMedr-byr^ts*- name, this gasket is essent-ia-1 -ly-.a^meta-H-ic-.Mng-.-go-int-made-~€t?omr.4forged'.-aluminiurn or mi 1 d steel-, cons-isting of- two cone'vtype'rseating-- beve-l-s- wh-teh-form part of the outer- surface.- • The-sehematie i 1 lustration-:of-.such-..a-ring joint is 1  shown in Figure 4.  :  Mean-radius of ring Half-width of ring Thickness of ring •Width of - seating bevel Ine-1ination'angle of seating bevel Pressure-contact ratio  P  = r/h  a = b/h I Design parameters of ring FIGURE 4 A Double-Cone Ring Gasket When used as a static seal, this ring gasket is placed between two flanges of specific design, as shown in Figure 5.  In order to create  the necessary i n i t i a l seal, so that the hydrostatic pressure subsequently applied to this assembly can act on the inner surface of ring only, the 1  flanges and the ring are so dimensioned thai:'the'seating'bevels of ring come in contact with the sealing surfaces of flanges when the flanges have a standoff g (see Figure- 5).  Tightening the studs swages the ring  inward until the flanges are face to face.  Under this' bolt-up condition,  no hydrostatic pressure exists and the f u l l bolt load is carried by the :  slightly coned ring. assembled state.  Figure 6- shows a double-cone' ring gasket in the  FIGURE 5 The Double-Gone Ring Gasket-and Flanges Before Assembly r  FIGURE 6 The Double-Gone R-ing= Gasket-=in-the-Assembled State  After the flanges are tightly bolted'and'the ring has been r  properly seated, hydrostatic pressure can then be applied.  A portion of  the applied pressure, which is acting on the inner surface of ring, ;  expands the ring into the sealing surfaces of flanges and forces i t to maintain a seal.  Since there-exists an unsupported'area in this gasket 1  design, which is equal to the difference between the inner surface of ring and the  projected area of ring bevels, the increase in the  intensity of sealing pressure w i l l be higher than the increase in the intensity of applied pressure.- Therefore, an effective seal can be retained no matter how high the applied pressure. Since the ring expansion mentioned in the/last paragraph, along with the hydrostatic pressure acting on the f l a t portions of the flanges, tends to separate the bolted flanges, this gasket assembly must be firmly held in place by some proper means.  In the original  design [6], a large threaded plug was used, as shown in Figure 7.  FIGURE 7 A Double-Gone Ring Gasket Assembly U t i l i z i n g Threaded Plug  10 This arrangement-is satisfactory from the strength-standpoint, but the 1  threads are subjected to- damage.-' It was then- decided' that large-diameter studs be adopted as the bolting devices. In some modern designs-, the-double-cone -type ring gasket is modified by cutting grooves on-the-seating bevels"to reduce the contact area.  An even more complicated modification has" also been-used in other 5  heavy-duty applications,- [5-][6-];- This modification-, which is schematically shown in Figure 8, consists of providing a mechanical-backing shoulder to  FIGURE 8 A Modified Double-Gone Ring'Gasket Design :  prevent the ring from- excessive' -inward-deflection-' i f i t ' becomes necessary :  to preloadi  .the studs tighter'than normal to overcome surface variation  in gasket or flanges.- A groove- is cut in this shoulder for i t s entire circumference and holes provided to- make certain that the applied hydrostatic pressure w i l l act-on the inner surface-of the ring. :  However, for  the sake of easy fabrication, these modifications were not applied to the ring models used in this investigation.  The ring models remained as  ain-bevel, non-backing type.  12 CHAPTER  III  THEORETICAL ANALYSIS 3J  General The elastic behaviour of the double-cone ring gasket used as  a static seal may be theoretically analyzed in the sense that a longitudinal s l i v e r 2b°rd0'h isolated from the ring gasket, as shown in Figure 9, can be treated as "a beam on elastic foundation".  \ \  i I  FIGURE 9 A Longitudinal-SIiver Isolated-from a Ring Gasket If rd0 is taken to be unity, then the ring siiver-becomes a beam of unit width, which rests on an^elastie foundation-consisting of the rest of the ring (covering an-angle-of 2-rr-d0).  This- beam-is acted upon by -  resultants of sealing pressure-N,-hydrostatic-pressure-p and hoop compressions s^o The hoop compressions, which are assumed'to be uniformly distributed along both sides of-the ring--s-1 iver,--are-not quite in opposite directions, but include-a small angle d0 between them.  Therefore, the  resultant of these hoop compressions-, S r - ^ b ' l r d0, -constitutes the  13 reaction force imposed on the "beam" by the "elastic-foundation", and is proportional to the deflection of the beam. 3.2  Fundamental Assumptions In developing the mathematical approach for-analyzing the ring  deformation under applied loads,-some simplifying^assumptions are to be made so that the resulting differential equation whielr governs the e l a s t i c line of a ring s l i v e r is simplified to such an extent that i t may be solved without too much mathematical  difficulty.  These-assumptions are  summarized as follows: 1) The material-from-whieh-the ring is made•is assumed to be homogeneousj-isotropic and-perfectly e l a s t i c . 2) The flanges between which the ring is assembled are assumed ;  to be absolutely r i g i d , and the studs-holding-these flanges in place are assumed to have  infinite  strength, or, in other  words, i t is assumed that no-flange deformation or stud stretch 1  could occur. 3) A ring s l i v e r is assumed to be a rectangular beam with uniform cross-sections, as-shown in Figure 10.  In addition, i t  is assumed that-the beam equation may be used for a relatively small length to depth r a t i o . 4) The resultant of the sealing; pressure acting on each ring bevel can be resolved into two orthogonal components (see Figure 10).  One of these components, Ncose, which is in the  radial direction of the ring, is' assumed to act at the extreme outer edge of ring, while the other component Nsine, also acting at the edge of the ring, is replaced by the s t a t i c a l l y equivalent  14 bending moment Nsine'h/2 and an axial force with a magnitude equal to Nsine.  Nsine'^-  Nsine^-  \ Nsine  Nsine— S Ncose  -J ^dx Ncose  FIGURE 10 Simplified- Boundary Conditions of a Ring Sliver 3.3  Derivation of the Governing Differential Equation If a set of rectangular co-ordinate-axes XYZ is chosen for the  beam (see Figure 10), an infinitesimal beam-element dx is acted upon by shear force V i bending moment M, hydrostatic pressure p and hoop compressions s^, as shown in Figure 11.  s^hdx-  S hdx h  FIGURE -11 An- Infinitesimal Beam Element  15 From the consideration-of equilibriunrof the forces-acting on this beam element, i t follows in the Z-direction  a? - 7 h PM - If) s  (3-')  +  The magnitude of the hoop compression is given-by the expression ^  =  E  ^„  l  M  M  n 9 ^ h l  ( 3 > 2 )  where E is Young's-modulus, y is Poisson's  ratio and w is beam deflection  in Z-direction consisting of the deflection attributable to bending, w^, and that attributable to shear, w , i . e . s w = w.b + w„s The bending moment in^the^beam is given by d w,  M = -D  —f-  (3.3)  dx  where D is the flexural  rigidity  D - 12(l-y — ) 2  The rate of change of the deflection due to shear [11] is given by dx  _ K5V . 3(1+u) . " hG Eh  , *'*>  v v  [  Combining Equations ( 3 . 1 ) t o (3.4), and replacing p(l - j^r) . *P, give y -(NsinQ + ph) c - J - by dfw_ M T T " " D dx  +  3(l+y) . r~  3(Uu) Eh  . " P  ( 3  Taking the derivative of (3.5) twice and using Equation (3.1) again, give  '  . 5 )  16  "  d4w  dx  4  1  dw . A  J  2  2  _  +  P  dx^  y  ,,  u  where A _ 3(l-y ) 2  hV  This differential equation expresses the-relation between the ring deformation and the applied loads, and is-coincident with that derived by S. Levy for short cylindrical shells [12] allowing for reverse in sign of w. 3.4  Solution to the Governing-Pifferenti al Equati on ;  Any solution to the governing differential-equation obtained in the preeeeding-section applies-only to a particular ring-gasket of mean radius r and thickness h.  To overcome this and reduce the number of  independent variables at the.same-time-, Equation (3;6)-is made dimensionless.  This is done by defining two dimensionless parameters C=£  (3.7a)  a) = *  (3.7b)  and  Combining Equation- (3.6) with Equations- (3^.7), and recalling that _ r " h  p  ,  _ b F  as defined in last chapter give-a dimensionless expression d o> 4  4  3(l+y)g du>+ 2  -  2  2  2  12(1-u )g 2  2  P  4  _  a) - -  12(l-y )g 2  E  a? p QK The solution of Equation (3.8) with y = 0.3 is  4  s  (r>  Q\  17 a) = C-| cosh A? cos y£ + C sinh XE sin  +  2  cosh XE, sin 5  + C sinh xs cos y£ - P*  Y  (3.9)  4  in which, P* is the particular integral P* = I  and, x and  (3.10)  P  are parameters defined for simplicity  Y  1.285  X =  — ,/p  = 1 . 2 8 5 ^jp  Y  and C p C^, C^,  + .5901 -  .5901  are constants of integration.  Since the deflection, w, is symmetrical with respect to the Z-axis C  3  = C  = 0  4  (3.11)  Other boundary conditions useful in evaluating the non-zero constants C-j and  are:  1) The end displacement of beam,-w , is a known quantity. 2) The bending moment at the ends of beam, NsinO'^-is known. These two end conditions can be expressed mathematically as w e :  ^=±l  =  (3.12)  F~  and 2 dw  3.9a  p  QE,  2  3.9a  2  p  _*  5.46a Nsin9 2  Eph  (3.13)  respecti vely. For s i m p l i c i t y , define c  e  = —  r  (3.14)  '  18 ana  M* = -  5.46a NsinQ Eph  (3.15)  Then the end conditions (3.12) and (3.13) become (3.16) and .2  d oj  3.9a^  —o—  3.9a + 2  n  ID - — 7 5 —  (3.17)  = M*  r*  Substituting Equation (3.9) into (3.16) and (3.17) and solving the resulting simultaneous equations, yield 1  2x (sinh A + cos )  1  2  Y  2  Y  + P*)[2AyCOSh A C O S Y  {(C  +  (A  2 2  - Y^  2  3.9a /p )sinh A s i n y ]  - M* sinh A siny}  (3.18)  and 2  2  2Ay(sinh . A + cos Y)  {(c + e  P*)[2Ay  sinh  A  siny -  (A  2  - y  2  -  3.9a /p )cosh x cosy] 2  2  + M* cosh x cosy}  (3.19)  At this point, i t i s recognized that to combine Equation (3.9) with Equations  (3.10),  (3.18) and (3.19) does not give the complete  solution to the deflection, because there i s s t i l l an unknown quantity, N, involved in P* and M*.  However, this can be c l a r i f i e d by noting  another boundary condition that dM dx  = NcosQ x=+b  (3.20a)  19  or, in alternate form 3 d  2 to  3,9a 2  10.92a  da)  J  dt  cose ,N_.  (3.20b)  5=+l After some cumbersome operations, the expression for the magnitude of sealing force per unit length of ring circumference, N, can be obtained by solving the following equation Eh  = K  •c  +K  (3.21) M  in which Kc and K are coefficients relating c r  K.  g  and |- to the quantity  [(2A r-, - r r ) c o s h x sinh x 2  Y  Y  2  0  - (2xy r + A r r ^ ) cos y siny] 2  Q  and K =  ( -0.8)[(2x YiyYr r )cosh x sinh 2  p  0  P  2  2  - (2x .r +xr r- )cos y siny] Y  2  Q  1  where F Q , r-j, r and Den. are defined for simplicity 2  TQ = X 1  ?2  2  - y  2  3.9a /p  -  2  2 2 X - 3y -  2 2 = 3X - y  -  2  2 2 3-.9u/p  2 2 /p  3.9a  and 2  Den. = 0.3 sine [2x yr-j -  Y^I^  +  18.2(a  2  /p)r Jcosh x sinh x  2  2  2  - 0.3 sin [2xy r + xr r-| 2  Q  - 18.2(a /p)r-|]cos y sin y  3 2 2 21.84(a /p)Xycose(sinh X + COS y)  (^)  20 Then, Equations (3.18), (3.19) and (3.10) reduce to c C 1  2x (sinh x + cos y) 2  Y  2  2 {[r  - 0.3(r  Q  - 18.2a /p)sine-K ]sinh x sin y  Q  c  + 2xy(l - 0.3 sine'K )cosh x cos y} 2 2 2xy(sinh m. X + COS y) p  +  {[(  - 0.8)r - 0.3(r  P  Q  + 2Xy[(p  0.8) -  -  - 1 8 . 2 a / ) s i n e ' K ] s i n h x sin 2  Q  0.3  P  p  sine *K ]coshxcos > p  Y  (3.22)  Y  c  c 2  e 2Xy(sinh x + cos y) 2  2  { 2 X y ( l - 0 . 3 sine°K )sinh x sin y 2 - [ r - 0 . 3 ( r - 18.2a /p)sine°K ]cosh x cos c  Q  *  Q  c  Y  )  eZI 2x (sinh x + cos y) Y  2  2  {2Xy[(p - 0.8) - 0.3 sine"K p ]sinh  - [ ( P - 0.8)r  - 0.3(r  Q  Q  X  sin y  - 18.2a /p)sine°K ]cosh x cos y} 2  p  (3.23) ana P* = - 0 . 3 s i n e " K - c + [(p - 0.8) - 0.3 sine'K ] ' | c  e  p  (3.24)  Substituting these values into Equation (3.9) gives the final solution to the differential equation (3.8).  This solution in turn i n d i -  cates the radial deflection of the ring. In particular, at the center of the ring s l i v e r concerned, i . e . at the origin of the co-ordinate system chosen, where x = 0,  or c = 0  21 the solution reduces to VO 3.5  =  w  0  =  C  l " * P  ( 3  °  2 5 )  Tangential Strains of Ring Gasket Since the double-cone ring gasket is axisymmetrical in geometry  and is subjected to axi symmetrical loading-, the tangential-strain at the centroid of any axial cross-section of the ring gasket is w «  0  - - - ° - -  •  0  <- > 3  26  the minus sign in the right-hand side denotes that-an inward deflection gives a compressive s t r a i n . Furthermore, the tangential strain at the point S (see Figure 10), which is on the outer surface of the ring gasket, is given as e  s  = c + M(Nsine + ph) "o Eh = e + 0.3K sine°c + K (1 + sine) £ o c e p E 0  v  (3.27)  Equation (3.27) is of practical interest-because in the experimental part of this investigation, the tangential strains at point S were actually measured by using strain gages. 3.6  Effects -of Design Parameters on- Ring-Deformation - and -Sealing Pressure The combination of Equation (3.26) with Equations (3.25), (3.22)  and (3.24) gives an alternate expression for the tangential-strain at the centroid of any axial cross-section of ring e = --i • + n • (£) o c e p E c  where  V  J  (3.28) \ /  22 n ^  o  r  2x (sinh {[r  Q  o-  + cos )  Y  - 0.3(r  Q  p - 18.2a-/p)s1ne*K ]sinh A sin y  + 2x (l - 0.3sine'K Jcosh A COS y] + 0.3K_ sin e P c Y  (3.29)  and -1 2Ay(sinh x + cos y) 2  {[(P  - 0.8)r  2  Q  --0.3(r  0  - 18.2a / )sine"K ]sinh A sin 2  P  p  Y  + 2Ay[(p - 0.8) - 0.3 sine'K lcoshxcos y) + (p - 0.8) - 0.3K are coefficients relating c respectively.  g  and  sin e  (3.30)  to the tangential strain of ring  By using a digital computer for rapid calculations, the  relationships between these two coefficients and the design parameters of ring, as well as the relationships between coefficients, K and K c  p  (see Equation (3.21)), and the design parameters, are shown in Figures 12 to 15.  From these figures, the effects of the design parameters, o  and a, on the ring deformation and the sealing force can be readily seen.  25  26  27 CHAPTER IV DESCRIPTION OF TEST MODELS 4,1  Model Material A l l of the ring models used in this investigation were fabricated  from SAE 1020 carbon steel.  As received, the model material appeared in the  form of round, seamless, cold-drawn tubing which was supplied by the Wilkinson Company, Limited.  The tubing was 6 1/4 i n . in outside diameter and  1/2 i n , thickness, without any heat-treatment when purchased. In fabricating the ring models, sections with suitable lengths were cut from the tubing.  Some of these tubing sections in as-received  condition were then turned on a lathe to the predetermined shapes, and the resultant rings were designated Models A l , B l , B2, Cl and C2 according to their dimensions, as outlined in next section.  As for Models A2, B3  and C3, the fabrication process was somewhat different from that just described.  The associated tubing sections were softened by annealing  before machining operations.  The annealing temperature was 1350° F.  Since the physical properties of the cold-drawn tubing depend upon the degree of cold work performed in producing each piece, tension tests were performed to secure reliable information about the proportional limits and the ultimate strengths of the model materials, which were of the most importance for analyzing the behaviour of the ring models. In performing the tension tests, test specimens were cut from the cold-drawn and the annealed tubing sections, and then loaded to failure in an Instron testing machine.  Table 1 shows the results of the  tension tests, along with the average Rockwell readings of the model materials as taken from the hardness tests.  28 TABLE 1 Physical Properties of Model Materials MATERIAL  PROPORTIONAL LIMIT  ULTIMATE STRENGTH  ELONGATION IN 2 IN.  ROCKWELL B.  Cold-drawn  52,000-psi.  87,000 p s i .  14%  94  Annealed  33,000 p s i .  57,000 p s i .  38%  70  4.2 Design Details of Ring Models In designing a ring model, suitable numerical values were f i r s t selected for the design parameters a and p, as well as the pressure-contact :  ratio k.  Then, the ratio of the radial component of end displacement of  the ring in the assempled state t o i t s mean r a d i u s , c , was predetermined ;  ;  by considering the required i n i t i a l sealing pressure.  Q  This ratio was  given by the following expression: c  Q  =^  (4.1)  in which g was the i n i t i a l stand-off of the flanges with the ring snugly located between them, and-e-was-the inclination'angle-of the-seating bevels of the ring, which, in this investigation^, was takenas 12.5 degress. ;  With al1 of these predetermined-values chosen, the thickness of the ring model was then ealeulated-fromthe-geome-trical relation R h = (1 - c )p + (1 -l/k)atane+ .5 Q  (4.2)  where R was the maximum radius of-the bore of 'flanges-, which was also a fixed quantity of 3 i n . in the-present'-investigation. In order to observe the effect on the performance of the ring gasket as the proportions between the ring dimensions were changed,  29 eight ring models were designed,••••••eaeh~w-ith different dimensions. r  According  to their a-values, these eight models were divided-into-three groups, namely, the wide-ring^ the mid-ring' and- the narrow-ring series.  Table 2  shows the design-parameters of the ring models, along-with their nominal dimensions. 4.3  Preparation of Test Model Because of t h e i r - s t a b i l i t y , - accuracy and relatively small s i z e s ,  the-bonded, variable-resistance-strain gages-had been-generally accepted as one of the-best strain-sensing devices for experimental stress analysis, and thus adopted in this investigation to measure the deformations of the ring models in working condition. The strain gages used were of the Budd C6-141-B type, with the following characteristics: Gage Factor:  2.05 ± 1/2%  Gage Resistance: Gage Length: Grid Form:  120 ± .2 ohms  1/4 i n . Etched MetalfiIm  In preparing for tests, the strain gages were-bonded to the ring models with-GA-2 polymeric cement which was-also-supplied by the Budd Company.  The bonding procedure was-performed following the recommendations  of the supplier. In order to measure both'axial and tangential strains of the ring, for each model of the-wide-ring series, four two-element rectangular gage-rosettes were - applied to - the-ring - and were- equally spaced around i t s outside circumference;  However,-for-other ring models, due to their  limited widths, only four single strain gages oriented in the tangential direction were applied to each ring. both kinds of- gage-arrangement.  The photograph in Figure 16 shows  TABLE 2 Design Parameters-and EHmensions".of Test Models  SERIES wide-ring  mid-ring  narrow-ring  DESIGN PARAMETERS  OUTSIDE DIA. IN.  INSIDE DIA. IN.  WIDTH IN.  TH I C K.NESS IN.  REMARKS  a  P  PRESSURECONTACT RATIO  Al  3  15  3  5.8590  5.-4810  -1,1340  .1890  cold-drawn  A2  3  15  6  5.8094  5.4346  1.1244  .1874  annealed  Bl  2  15  3  5.8900  • 5.5100  ,7600  .1900 -  cold-drawn  B2  2  15  3  5.9086  5.5274  .7624  ,1906  cold-drawn  B3  2  15  6  5,8776  5.4984  .7584  .1896  annealed  Cl  1  10  6  5,9325  5.3675  . 5650  .2825  cold-drawn  C2  1  12  6  5.9354  5.4606  .4748  .2374  cold-drawn  C3  1  12  6  5.9225  5.4487  ,4738-  .2369  annealed  MODEL NO.  FIGURE 16 Gage-Arrangements for (a)  A Wide-Ring Model, and (b)  A Mid-Ring Model  32 To provide the means of communicating-the signals from a strain gage to the strain-indicating device • during'the ring model tests, e l e c t r i cal lead was soldered to the gage subsequently to the curing of the bonding  1  material.  Usually, the lead used-in-this'investigation-was of the three-  wire type for the reason that thechanges in 'lead-wire' resistance due to 1  the temperature changes of a three-wire lead could be made self-compensating. c  33 CHAPTER V INSTRUMENTATION AND TEST PROCEDURE :  5.1  Description of Apparatus The apparatus used for testing the ring models is shown in the  photograph in Figure 17=  This apparatus consists of the following  components: Pressure Chamber SPS-245 steel was chosen for fabricating the pressure chamber because of i t s high strength and r i g i d i t y .  The chamber consisted of two  major parts; the top cover and the vessel.  Design details of these two  parts of chamber are shown in Figures 18-and 19, and their general views are photographically presented in Figure 20. For assembling the cover and the vessel together, twelve 3/4 i n . studs were provided.  These studs were firmly fastened on the vessel,, as  clearly shown in the photographic picture.  Of these twelve studs, three  were with a centre hole in- which- Budd-140-B strain- gages- were installed to measure the stud strains-during the tests. Hydraulic Pump The choice of the-pressure-source-in the- testing system was greatly influenced by the ease with-which-the pressure-applied to the pressure chamber could be controlled.  From t h i s p o i n t of view, an Enerpac :  P-85 hand pump manufactured by the-Blaekhawk IndustriaT Products Company ::  was selected.  This pump< was-a- self-eon-tainedf piston-type- hydraul ic pump  with a rated capacity of 10,000 p s i .  tn  FIGURE 18 Design Details of the Top Cover  .  :  FIGURE 19 Design Details-of the Vessel  co  CT,  /  B FIGURE 20 General Views of (a)  The Top Cover, and (b)  The Vessel  38  In order to form the union of the pump with the pressure chamber, a 3/8 i n . flexible hose with plug-in-type couplers-was also supplied by the pump manufacturer.  Through this hose, the hydraulic o i l under pressure  was transmitted from the pump to an inlet opening located at the bottom of the pressure chamber. Pressure Gauge The pressure gauge used in this investigation was supplied by the American Standard Advanced Technology Laboratories.  It was a Bourdon  spring-type gauge capable of indicating pressure up to 10,000 p s i . and with a readability of 50 p s i . In working, the pressure-sensing element of the gauge, which was essentially a tube flattened and bent to an arc of a c i r c l e , was exposed to the pressure in the pressure chamber through a bottom connection of the gauge, and then with a-suitable multiplying mechanism, the intensity of pressure was shown-by the position of a pointer on the gauge d i a l . Strain-Indicating Device The strain-indicating device used consisted mainly of a P-350 Strain Indicator which was manufactured by the Instrument-Division of the Budd Company.  This instrument was designed primarily for use in determining  static strainswhen used, with resistance-type strain gages-,, and with the following-characteristic features: Range:  ±50,000 y strains  Accuracy:  ±5 y strains or ±5%  Readability:  l p strain  Sensitivity:  40 - 2,000 y strains for f u l l scale one side meter deflection  External C i r c u i t s :  f u l l , half or quarter-circuit  39 Internal Dummies: Battery:  120 and 350 ohms  9 volts  Amplifier:  A-C transistorized.  This indicator was also equipped with a control for adjusting ;  i  c i r c u i t balance and a selector for switching the internal dummy halfbridge into external c i r c u i t for quarter- and half-bridge applications. In this investigation, only tTfe quarter-bridge external c i r c u i t , as shown in Figure 21, was employed.  Acti Gage  Internal 120-Ohm Dummy  Galvanometer  9-v Battery FIGURE 21  External Quarter-Bridge Circuit'with-Internal Dummy  Another major component of the strain-indicating device was a Budd 5B-1 Switch and Balance-Unit.  By-combining-this Switch and Balance  Unit with the-Strain Indicator just-described, a strain-measuring operation to read gage points up to a number of TO was possible. • 5.2  Test Procedure The procedure of testing- a ring model could be divided in two  distinct parts; one in which-the ring model was-assembled-into a proper  40 position- in the pressure chamber-,- and in the other the hydrostatic pressure 1  was applied to the ring.  1  These two parts of a ring-model test were  denoted as the- assembly-and the pressure test respectively, and w i l l be separately described in- the remainder of this section. Assembly The method- of assembling a ring- model'into the pressure chamber was as follows.  With the ring bevels and the sealing surfaces of the  pressure chamber coated with a thin layer of Rulon a n t i - s t i c k agent, the ring-model-was placed-into-the-pressure-chamber, and snugly rested in a position between the top cover- and the- vessel'of the chamber.  At this  point,- the gap between the-flange of the-cover and that of the vessel was measured and recorded.- T h i s - i n i t i a l value of gap, g, could be checked with the one predetermined- in the-design- of- the ring-model through the u t i l i z a t i o n of Equation (4.1).  Then-, after the-strain gages on the  ring model had been connected to the strain-indicating device, the top cover of the chamber was lowered-by tightening the nuts on the studs until i t s flange- came in contact with that of the vessel face to face and the assembly of the ring model was completed.  During this assembly process,  great care should be taken to-obtain an-even-tightening of nuts for maintaining an uniform gasket loading and preventing the ring from shifting out of place.  Usually, this was*done by conducting the assembly process  in- a suitable -number of steps.  At each step, the gap of the pressure  chamber was checked at four chosen-points and the strain measurement of the ring was taken. Pressure Test The purpose of a pressure test was to observe the sea-ling capacity of the ring model being tested,-and to measure the ring deformation due to  41 the applied hydrostatic pressure.-  In this test, the o i l under pressure  was transmitted from the hydraulic pump to the bolted chamber and acted on the assembled ring model.  The intensity of pressure-was increased from  0 to 10,000 p s i . in ten increments, and strain measurement of the ring was taken-at each increment.  In addition, the tensile strain readings of the  studs were also recorded in the test.  42 CHAPTER VI EXPERIMENTAL RESULTS AND' DISCUSSIONS 6.1  Results of Strain-Measurements The strain-measurements made in testing the ring models determined  the tangential strains on the outer surfaces of'the models tested. This was based on the consideration that these strains not only gave an indication of the ring deformations during tests, but also provided a means to check the validity of the theoretically developed formulas. During the tests, the strain-readings read out from the strain ;  indicator were merely apparent strains indicated by the strain gages placed in biaxial strain fields which differed from the b i a x i a l i t y of the strain f i e l d in which the gages were calibrated.  Although the correction  for transverse strain effects should be made to convert these apparent strains into true strains, i t was found that the accuracy of the s t r a i n measurements, the cross-sensitivity factor of the gages and the b i a x i a l i t y of the strain fields did not seem to j u s t i f y this correction.  Consequently,  the strain-readings taken from the experiments-were directly used for plotting the test curves without being corrected for transverse strain effects.  The errors involved were quite small and could be neglected. Figures 22, 23 and 24 show the results'of the strain-measurements  made in assembling the ring models.  The plotted strain values represent  the average tangential strains as measured- at four chosen points around the outer circumference of- each ring modelThese- measured strains are compared with the theoretical values, plotted-in dash Tines, directly computed from Equation (3.27) with p = 0.  43  4000  LEGEND <  SYMBOL © ©  3000  MODEL NO. Al A2  / /  -  /  /  e  re s-  +J to. a. CD  ©  3  2000  •  > If)  /  /  -/  /  /  /  /  /  <u Q.  E o to  •/ 1000 /  ©# j  f  0  /  0.001  0.002  0.003  0.004  0.005  FIGURE 22 Results of Strain-Measurements'Made in Assembling Wide-Ring Models c  0.006  5000 /  /  LEGEND SYMBOL MODEL NO. f) Bl d B2 m B3  «>  4 /  /  / *> A / / /  /  4 —r  /©  /  /(I /©  4 0  0,001  0.002  0.003  0.004  0.005  0.006  FIGURE 23 Results of Strain-Measurements Made in Assembling Mid-Ring Models  4000 p =  LEGEND SYMBOL MODEL NO.  —A /  10  Cl C2 C3  © ©  12  / / /®©  0.001  0.002 c  0.003  0.004  FIGURE 24 Results of Strain-Measurements" Made"in Assembling Narrow-Ring Models  46 In Figures 25, 26 and 27 are pressure-tangential strain curves for various ring models.- During pressure tests,-strain-measurement was taken-at-every 1,000 p s i . throughout the'pressure range up-to 10,000 p s i . for a l l ring models except No. A2 and No. C3.  The strain gages bonded on  these two models f a i l e d to perform normally when the hytrostatic pressures applied had intensities over 4,000 p s i . and 5,000 p s i . respectively.  The  strain-measuring operations for Model A2 and C3 were thus forced to stop at these pressures.  Again, for the-purpose of comparison, in addition to  the test curves, the computed curves are also shown in dash lines in the figures.  It should be noted that the corrections for the flange-movement  effects on ring deformations were made to the computed curves rather than to the test curves.  In every pressure t e s t , the average stretch of studs  at each pressure increment'was obtained, through c a l i b r a t i o n , from the tensile strains which were indicated by the strain gages installed in three of the studs.  This stud stretch represented- the associated flange  movement and was converted into the end displacement of ring by using a geometrical relation s i m i l a r t o that expressed by Equation (4.1). F i n a l l y , Equation (3.27) was employed to compute the theoretical strain values. 6.2  Comparisons Between^ Experimental Data-' and- Theoretical Values r  An-inspection-of--the-e--readings-taken-in-assembling the ring models showed that up-to points where yielding stresses were produced, they were in f a i r agreement-with the calculated values, and the differences were too small to be significant. On the other hand, from Figures 25, 26 and 27, it-was-seen that the strain values measured in'the pressure-tests were appreciably  50 different from that predicted by the theoretical calculations.  Attempts  were made to explain the discrepancies, and the following probable reasons are advanced. The most significant fact in explaining the unsatisfactory experimental results was that at the very beginning of every pressure test, the resultant of the sealing pressure exerted on a ring bevel, N, would act at points along the center c i r c l e of the bevel.  As the inten-  sity of applied pressure increased during the t e s t , the ring wall was graduately coned by the increasing bending due to pressure, and shifting of the sealing force from the center c i r c l e toward the inner edge of the bevel resulted. This violation of the assumed end conditions would have made the measured strain values smaller than that predicted by the analytical solutions, in which the sealing force was assumed to act at the extreme outer edge of the ring bevel.  Furthermore, i t was certain that  the effect of the violation of the assumed end conditions was larger for the narrow- and mid-ring modelsthan for models of the wide-ring series. Another suspicion was borne out to some extent, by noting that the results of the pressure tests on annealed models approximated to the theoretical values better than that on the cold-drawn models.  It was  possible that the axial grain-alignment formed in cold extrusion, i f not 1  relieved by annealing, would have had certain influence on the behaviours 1  of the ring models made of cold-drawn tubing. F i n a l l y , i t was noted that'in the assembled-state, a l l the ring models were being subjected to compressive stresses well beyond the proportional limits of the model materials, and were in the s t r a i n hardening regions [15].  Figure 28 shows the typical stress-strain  diagram for mild s t e e l , along with the unloading and reloading lines  51 originated in the strain-hardening region. Stress  Strain  FIGURE 28 Typical Stress-Strain Diagram for Mild Steel Although these unloading and reloading lines-can be approximated by straight lines that are parallel to the straight line in the region of proportionality, they are actually slightly curved lines forming narrow loops.  Hence, i t seemed to be understandable that the ring models would  not have had perfectly e l a s t i c distortions during the pressure tests. In spite of the reasons mentioned above, i t was s t i l l d i f f i c u l t to understand why in testing the cold-drawn models of the narrow-ring series, when the applied hydrostatic pressures were greater than 5,000 p s i , , the tangential strains of the ring'models decreased as the pressure increased.  No explanation had been found for the abnormal :  behaviour of these two models. 6.3  Observations Made During Tests A few observations in connection with the performance characteris-  tics of the ring models were made-during the tests. are summarized as follows:  These observations  52 Sealing Capacity When properly assembled, every ring model used in this i n v e s t i gation could create and maintain- a perfect^pressure-tight^seal r  throughout  the pressure range up to 10,000 p s i . " However,-the- uniform tightening of ;  studs in assemblies was found to be of utmost"importance- for the sealing r  capacities of the ring models.  D i f f i c u l t y in providing e f f i c i e n t i n i t i a l j  sealing was encountered in one"of the -pressure- tests when the ring model being tested had been shifted out of place in assembly. 1  Effect of Cycling Pressure Load A series of pressure tests"were conducted on ring models A l , B l , B2, B3 and Cl to cycle the hydrostatic pressures applied on the assembled ring models from zero to 10,000 p s i .  This was done for more than twenty  times for each of the ring models without leakage.  Furthermore, the fact  that no re-tightening of studs was ever found necessary during this series of tests gave conclusive evidence that a joint employing the double-cone ring gasket could be subjected to cycling pressure load without periodical f i e l d maintenance. Permanent Deformations After Tests More or l e s s , every tested" ring model was found permanently deformed.  The amount of permanent deformation-was approximately propor-  tional to the a-value of the ring model.  For-models of the wide-ring  series, the permanent deformations after t e s t s ; were as large as could be detected by the coned ring walls, as i l l u s t r a t e d in Figure 29. On the other hand, for models of the other two series, the permanent deformations were quite sma-1 T and could not be detected visually. Re-usabi1ity Three of the ring models, AT, Bl and-B2-, had been successively  54 subjected to assembly-processes^and pressure-tests-for more than four times, and no leakage was observed.  This effectively demonstrated the outstanding  re-usability of the double-cone ring gaskets-:" Additionally, i t was noted that the permanent- deformations in the ring walls-^-which- resulted from r  previous tests, had no practical effect on the sealing capacities of the ring models when re-used. Stud Elongations The excessive b o l t s t r a i n s often associated with the compression :  type high-pressure gaskets were not found in this-investigation. 10,000 p s i . , the total stud elongation averaged only 0.0015 i n .  At It was  found that, the stud-elongations in pressure tests-were directly proportional to the applied hydrostatic loads. 6.4- Design- Criteria- on-Double^Cone Ring Gaskets :  Experience-with this investigation^resulted in-a few c r i t e r i a on the design of the double-cone ring gaskets.  These- c r i t e r i a are reported  in this section. The focal point around which the problem of designing a doublecone ring gasket is to provide the i n i t i a l seal which is imperative to the 1  successful application of the gasket.  As the assembly process of the  gasket is completed, the intensity of the sealing-pressure exerted on the contact surfaces of ring bevels and flanges-must-be-high-enough to obtain r  a perfect metal-to-metal contact; for accomplishing the i n i t i a l seal provided by the plastic flow-of gasket-material into the tool marks and imperfections on the flange-facing:  Theoretical calculations using  Equation- (3.21) showed that-in this-'investigation-,-the-minimum value of the i n i t i a l sealing pressures-exerted^'on the-assembled-ring models was  55 in the order of 24,000 p s i . for Mode-lBl:  Hence, i t is estimated, on the  conservative side, that 25,000 psi .might" be taken-" as the required r  bearing load to ensure an e f f i c i e n t i n i t i a l ' s e a l - for the double-cone ring gaskets made of cold-drawn"SAE"1020 m i l d s t e e l . " As-for the annealed 1  gaskets, the required bearing loadcould be considerably reduced. The amount of unsupported-areaof a double-cone-ring gasket, and hence- the inherent a b i l i t y " o f the gasket to be pressure-energized in ;  ;  working, is control-led by i t s pressure-contact r a t i o , k; r  1  Theoretical l y ,  assuming no flange-movement-occurs and the ring not to y i e l d , a pressure1  contact ratio of one seems to be sufficient to ensure a good seal. However, this value is practically too small for two reasons.  The f i r s t  one is that the flange separation cannot be avoided-in every high-pressure j o i n t , which tends to relieve the bearingloads on the ring bevels. :  ;  The  second reason is due to the "fact-" that t h e p l a s t i c flowon the ring bevel, which is essential for an e f f i c i e n t seal, has a tendency to re-distribute the high 1ocalized sealing pressures to the adjacent- parts. :  r  Therefore,  higher pressure-contact ratios are necessary to be selected in designing the double-cone ring gaskets to compensate these effects. 1  In this i n v e s t i -  gation, two values of pressure-contact ratio, 3 and 6, were-tested, both 1  were found satisfactory. Of the design parameters-of the ring, the ratio of mean radius to thickness, p, is the-dominant"factor in"determining the thickness of ring wall (see Equation (4.2)).  Sincethe thickness-is affected by two  conflicting requirements; the sufficient strength and the ease with which the ring gasket' can- be"pressure-energized, the se-leetion-of p-values in designing the double-cone ring gaskets becomes- a compromise matter, P Values in the range- from 10'to"15 were used"in"this-investigation, and  56 the resultant ring gaskets were found- adequate for flanges with 3 i n . bore. It is apparent that the upper l i m i t of p-values can be extended when flanges with larger bore sizes are to be joined.  As for the ratio of  half-width to thickness, a , although values ranging from 1 to 3 were chosen in this investigation and found acceptable, i t seems that there w i l l be no reason to use a a-value higher than 1 as long as good sealing is the only consideration. Relatively tolerant in design is the inclination angle of ring bevel,0. A literature research [6][7][16][18] showed that angles of 12.5°, 23° and 30° from the verticalhave been used in practice for the doublecone ring gaskets or the similar ones.  Although smaller inclination  angles would impose closer manufacturing tolerances, 12.5° was adopted in this investigation for approximating the cross-sections of the longitudinal slivers of ring models to the rectangular shape assumed in the theoretical analysi s. The ring models used in this investigation were machined from cold-drawn and annealed tubing sections of SAE 1020 steel.  The results  of the pressure tests indicatedno differencethat would warrant any definite statement regarding the-effect of annealing on the pressureenergizing capacity of the double-cone ring gasket.  However, the effect  of annealing did appear in the degree'of surface finish of ring bevels r  that could be produced by the machining tools, and would make the plastic flow of gasket material easier to improve the sealing. Although the design considerations for the flanges joined by the double-cone ring gaskets is beyond the scope of this investigation. It is recommended that these flanges should be made of metals harder than the gasket material for avoiding permanent set on the contact surfaces,  and should butt face-to-face in assembly, as in this investigation, for the reason that the bolt load applied in assembly can be partly carried by the flange faces, and hence, the thickness of the gasket can be considerably reduced from the strength viewpoint.  58 CHAPTER VII CONCLUSIONS AND RECOMMENDATIONS 7.1  Summary of Conclusions Definite conclusions could not be readily drawn from study of a  limited scope such as reported in this thesis;  The principal value of  the present study was in providing a basis"for'further investigations of the pressure-energized type of ring-gaskets.  The results of the study  did, however, point out some important'merits"of the double-cone type ring gaskets, and they suggested-certain c r i t e r i a on design. The following statements b r i e f l y summarize the general results of this'investigation. 1)  The acceptability of the principle of unsupported-area in  high-pressure gasket design was confirmed. 2)  The advantage of using the double-cone ring gaskets as  static seals was j u s t i f i e d .  Although, in this investigation,  hydrostatic pressure up to only 10,000 p s i . had been sealed successfully, there was every indication-that the double-cone ring gaskets could be employed at s t i l l higher pressures. 3)  Experiments carried out with various-models revealed that  the optimum design propertiesof'the double-cone ring gasket were quite tolerant to changes in proportions. 4)  The focal point in designing the double-cone ring gasket  should be to provide an adequate i n i t i a l seal, since the sealing capacity of the gasket induced by pressure-energization  did  increase with the intensity of the contained pressure at an e f f i c i e n t rate.  5)  In the course of experimental'work, i t was noticed that the  satisfactory application of'a double-cone ring gasket could be secured only when extreme care was used-in assembling the gasket into proper position'in the pressure enclosure. 6)  The inherent sealing capacity and'the re-usability of a  double-cone-ring gasket were notaffected-by limited plastic :  deformation in ring walls. 7)  A mathematical treatment for the elastic behaviours of the  double-cone ring gaskets was examined, and the distortion of the ring gasket under applied loads was formulated. 8)  Strains on the outer circumferences of the ring gasket models  under working conditions were measured with variable-resistance strain gages.  In assembling'the'ring models, the measured  values agreed f a i r l y wel1 with that predicted by theoretical calculations.  However, great discrepancies were found between  the experimental data taken in pressure tests and the computed values due to plastic effect and violation of the assumed boundary conditions in the tests. 7.2  Recommendations It might be of value to investigate the elastic behaviour of the  double-cone ring gasket by using numerical methods.  Since a ring gasket  is an axisymmetrical solid generated by the"revolution of one of i t s cross-sections about the axis, as in Figure-30 where cylindrical coordinates r, e and z are used, and is subjected to a given axisymmetric i loading system, the stress components in such"a ring gasket are a l l independent of 6, and can be expressed in terms of two stress functions  60  0 FIGURE 30 A Cross-section of the"Double-cone Ring Gasket with Cylindrical Coordinates rez and ii^ in the forms derived by Southwell [19] 1 3*1 r (aT-  r  1 " "T *2 r  ^  +  [  H-y)^]  +  l r °z  where, i)^ and ^  1  and  r ar  =  zr  r az  are functions of r and z only, and must satisfy the  following simultaneous equations  r ar  ar  + —x-  az  2 a ^2 a/  -\ r  9  3  r  2 *2  az-  =o 2 ^ *1 az-  Then, on a square mesh drawn to cover the gasket cross-section shown in the figure, these governing equations can be converted into corresponding finite-difference equations and solved by relaxation process.  By  selecting the mesh size to a smallest possible l i m i t and using the high-order finite-difference approximations [20], solutions with great  61 accuracy could be expected.  However, in performing this numerical  analysis, some special points of the relaxation techniques concerned with the satisfaction of the appropriate boundary conditions must be noted 1  [19], and a large amount of labour has to be devoted. The numerical solutions for three types of thick axisymmetric plate of varying geometry had  been obtained by Kenney and Duncan [21],  and were compared with the parallel experimental stress analyses using the frozen-stress technique.  It might be similarly desirable to carry  out a three-dimensional photoelastic study of the stress distribution in the double-cone gasket for comparing with the result obtained by the numerical method described in last paragraph.  62 REFERENCES 1.  Bridgman, P. W., The Physics' of High Pressure, G. Bell and Sons, L t d . , London, 1931, p. 30-35 and p. 78-97.  2.  Bridgman, P. W., "The Technique of High Pressure Experimenting", American Academy Arts and Sciences Proceedings, Vol. 49, 1913, p. 628.  3.  Bridgman, P. W., "High Pressures and Five-Kinds of Ice", Journal of the Franklin Institute, Vol. 177, 1914, p. 315.  4.  Bridgman, P. W., "Breaking Tests Under Hydrostatic Pressure and Conditions of Rupture", Philosophical Magazine, Series 6, Vol. 24, 1912, p. 63.  5.  Bridgman, P. W., "General Survey of the Effects of Pressure on the Properties of Matter", Proceedings of the Physical Society, Vol. 41, 1929, p. 341.  6.  Freese, C. E., "Mechanical Design Problems Connected with Ammonia Synthesis", a paper presented at the ASME Petroleum Mechanical Engineering Conference, September, 1956, ASME Paper No. 56-PET-l.  7.  Freese, C. E., "Here are Ammonia Plant Design Tips", Petroleum Refiner, January, 1957, p. 193.  8.  Everett, M. H. and G i l l e t t e , H. G., "Static O-Ring Seals", Seals Book, a penton publication of the Machine Design, 1961, p. 100.  9.  Dunkle, H. H. and Gastineau, R. L., "Metallic Gaskets", Seals Book, a penton publication of the Machine Design, 1961, p. 103.  10.  Den Hartog, J . P., Advanced Strength of Materials, McGraw-Hill Book Company, Inc., New York, p. 164-165.  11.  Timoshenko, S . , "Strength of Materials, Part I", Third Edition, D. Van Nostrand Company, Inc., Toronto, 1955, p. 113-118 and p. 170-171.  12.  Levy, S . , "Shear and Bending of the Walls of Short Cylindrical Shells", a paper presented at the Winter Annual Meeting of the ASME, NovemberDecember, 1964, New York, ASME Paper No. 64-WA/MD-9.  13.  Dally, J . W. and Ri1 ley, W. F., Experimental• Stress Analysis, McGraw-Hill Book Company, New York, 1965, p. 366-476.  14.  Dove, R. C. and Adam, P. H., Experimental Stress Analysis and Motion Measurement, Charles E. Merri11 Books, Inc., Onio, 1964, p. 50-281.  63 15.  P h i l l i p s , A . , Introduction"to P l a s t i c i t y , the Ronald Press Company, New York, 1956, p. 3-4.  16.  Eichenberg, R., "Design of High-Pressure Integral and Welding Neck Flanges with Pressure_Energized Ring Joint Gaskets", Journal of Engineering for Industry, May, 1964, p. 199.  17.  Eichenberg, R., "High-Pressure WeiIhead Equipment - - Design Considerations for AWHEM 15 ,Q00-psi. Flanges' ».Mechanical? Engineering, Vol. 80, No. 3, p. 66. :  1  18.  Brooks, R. C , "High-Pressure Wellhead Equipment - - Wellhead Connections", Mechanical Engineering, Vol. 80, No. 3, p. 63.  19.  A l l e n , D. N. de G., Relaxation Methods, McGraw-Hill Book Company, Inc., New York, 1954,. p. 134-144.  20.  Wang, Chi-Teh, Applied E l a s t i c i t y , McGraw-Hill Book Company, Inc., New York, 1953, p. 122.  21.  Kenny, B. and Duncan, J . P., "An Assessment of Methods of Three Dimensional and Experimental Stress Analysis", departmental paper of the Department of Mechanical Engineering, University of British Columbia, December, 1966.