Improved design rules for pipe clamp connectors

International Journal of Pressure Vessels and Piping 81 (2004) 141–157 www.elsevier.com/locate/ijpvp

Improved design rules for pipe clamp connectors Cornelis J. Dekkera,*, Walther J. Stikvoortb a Continental Engineers BV, Zaandam, The Netherlands Consultant Mechanical Integrity Matters, Assen, The Netherlands

b

Abstract Pipe Clamp Connectors (or PCCs) are attractive substitutes for conventional flange connections, they are much lighter, require less space and are easier to tighten. However, the currently much used assessment method ‘Appendix 24’ from ASME VIII Div.1 for PCCs gives far too conservative results. Both hydrostatic testing to failure as well as FEM-analyses testify to this assertion, always much higher ‘failure’ pressures are reported than can be expected to correspond with ‘normal’ design pressures. To rectify this anomaly, we have developed an alternative assessment/design method for PCC-connections that utilizes better the inherent strength of PCCs and, consequently, allows much higher pressures. Note that photograph 1 (page 16) gives a nice view of an assembled PCC. Ideally, we would like to see this method implemented in the EN standards 13445 and/or EN 13480 resp. ISO 15649. q 2003 Elsevier Ltd. All rights reserved. Keywords: Pipe clamp connectors; Hydrostatic; Stress

1. Introduction PCCs consist of two hub parts which are held together by two 1808 clamps forced down over the slightly sloping contact areas of hub parts. Tightness is ensured by a self energized seal ring cramped in its place between the hubs. Photograph 1 of a PCC-model with cut-away sections gives a nicce view of an assembled PCC showing clearly the hub parts, the clamps with their bolts and the seal ring. As far as the hub parts of PCCs are concerned, the new method is based on limit load theory, and hence only primary stresses are considered. Gross distortion and subsequent failure is prevented by setting appropriate design limits. The limit load approach, as practised for flanges, converts all forces on the flange into a distributed torque ‘m’. Multiplying with the average radius of the flange ring gives the internal radial bending moment about the x – x axis in the hatched cross section.

* Corresponding author. Tel.: (þ 31) (0)75-6590187; fax: (þ 31) (0)756590199. E-mail addresses: [email protected] (C.J. Dekker); [email protected] (W.J. Stikvoort). 0308-0161/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2003.11.005

The ultimate load from the radial bending moment is limited to the flange’s yield stress times the smallest (plastic) section modulus of considered part, often the flange ring with tapered hub and the flange ring proper. For normal design loads a safety factor of 1.5 is to be introduced. However, assuming that the limit load is reached when the largest (absolute) stress is equal to the uni-axial yield stress is not satisfactory here, as longitudinal hub bending stresses will cause 3D stress situations in large parts of PCCs. In situations with principal stresses of opposite sign, plastic yielding may start when the highest (absolute) stress is still well below the actual yield stress! Hence, a ‘3D’ stress correction factor is essential here and will be introduced. As already mentioned, design methods based on limit loads deal only with primary stresses. The combination of primary and secondary stresses in PCCs could lead, in theory, to alternating yielding (i.e. ‘low cycle fatigue’ at gross structural discontinuities) each time the PCC-connection is

142

C.J. Dekker, W.J. Stikvoort / International Journal of Pressure Vessels and Piping 81 (2004) 141–157

Nomenclature A1 ; A2 ; A3 and A4 Aadjusted Aclamp Ahub Ar Aring Ashift

arm1 arm2 B Bc Ct Cw Dpipe Dpressure

E Ering eb Fadd:p

Faxial

Fbolting Fcirc Fgasket Fhor Fpressure Ftang faxial ; ffriction ; fnormal and fradial

cross sectional area’s in PCCs, see Fig. 13 adjusted area of hub (in discontinuity analysis) cross sectional area of the clamp, see Fig. 17 cross sectional area of hub proper ð¼ A1 þ A 2 þ A3 Þ auxiliary area within the clamp’s cross section cross sectional area of seal ring ‘shift’ area for neutral line of radial bending due to Fcirc in hub proper or extended hub lever arm for calculation of radial bending moment see at ‘arm1 ’ inside diameter of hub (and of connected pipe) distance of clamp bolts to C.L.of the PCC thickness (effective) of clamp width of clamp flexural rigidity of the pipe’s wall, 3 Etpipe =½12ð1 2 y 2 Þ largest diameter of seal ring recess exposed to pressure (defined by the very tips of their lips) modulus of elasticity of the PCCs hub modulus of elasticity of seal ring lever arm for moment in clamp due to Fbolting longitudinal pressure force on area between sealing diameter Øsealing and internal diameter of hub axial force on hubs resulting from Fbolting while accounting for translational hub-clamp friction tangential force in clamps due to bolts circumferential force in hub (proper or extended) reaction force from spacer part of seal ring on PCC horizontal force resulting from line load fhor longitudinal (end) force due to pressure tangential force in hub proper line contact loads of clamps on hubs

fb fhor ; fvert fhubface Iadjusted

Ibase

Iclamp Ihub lc lext: hub exp: lhub exposed lhub llip lpres:lip ltotal M; M0 ; ML and MH

M1 M2

Massembly;hub ; Moperating;hub Mass:;ext:hub ; Moper:;ext:hub Mcounter

Mradial bending Mtwist mtwisting ð…Þ

with respect to Øcontact : axial, friction, normal and radial line load, respectively, see Fig. 3 allowable (membrane) stress for clamp bolts line loads from seal ring lips on hub, see Fig. 7 vertical (radial) line load of seal ring on hub face hub’s moment of inertia (discontinuity analysis) adjusted with Icontribution seal and Icontribution clamp moment of inertia of the clamp’s cross section with respect to the baseline, see Fig. 17 clamp’s principal moment of inertia principal moment of inertia of hub proper effective length of clamp lips length of extended hub exposed to p, see Fig. 11 pressure exposed length of hub, i.e. ltotal 2 lpres:lip length of either the hub proper or the extended hub effective length of the seal ring lips, see Fig. 7 pressure loaded length of seal ring lips, see Fig. 8 total length of hub proper moment loads on edge per unit length in circumferential direction of pipe, of short cylinder (with edges ‘0’ and ‘L’) and hub, respectively contribution Faxial and Fhor to radial bending moment contribution of Fradial and Fvert: towards radial bending moment with respect to the considered neutral line radial bending moments on hub at assembly and in operating condition, respectively as before but for the extended hub countering moment due resistance of connecting part against rotation of hub (proper or extended) maximum radial bending moment in hub (proper or extended) total twisting moment acting on the hub twisting moment on hub per length


nb ODgasket p Q; Q0 ; QL and QH

Rclamp Rcog r rb rcontact rinside gasket S s tconnection

tcyl tlip tlipbase tpipe tpressure tweld end uhub W Wplastic wðM0 Þ; wðML Þ

wðQ0 Þ; wðQL Þ

w0 ; wL

wcyl:;p

unit due to load as specified between brackets. number of bolts per clamp lug (usually nb ¼ 2) outside diameter of the seal ring considered pressure (design/test) shear loads on edge per length unit in circumferential direction of pipe, of short cylinder (with edges ‘0’ and ‘L’) and hub, respectively outside radius of clamp centre of gravity radius of the hub’s cross section clamp corner radius of cross section bolt radius corresponding with min cross section radius corresponding with Øcontact ; i.e. 12 Øcontact inside radius of seal ring (gasket) static (or first) moment of area width of spacer ring (in Fig. 26) thickness of cylindrical part next to the hub proper (width cyl. part) or to the extended hub ðtweld end Þ thickness of cylindrical part next to the hub proper thickness of clamp lips at their (thin) end thickness of the clamp lips at their baseline pipe thickness connected at weld end of PCC thickness required for longitudinal pressure force thickness of weld end of PCC-hub radial displacement of hub proper’s centre of gravity nominal force per bolt in the clamps plastic section modulus of hub (proper or extended) edge load coefficients, to obtain radial edge displacements due to moment loadings on edge and on opposite edge, respectively. edge load coefficients, to obtain the radial edge displacement due to shear loadings on edge and on opposite edge, respectively edge displacements of short cylinder due to edge loads, edges are indicated with subscripts 0 and L coefficient for pressure displacement of cylinder with thickness tcyl and internal diameter B

wgap wpipe edge wpipe;M wpipe;Q wpipe;p

wspacer z ØA ØCi Øcontact Øring Øsealing factor

DFaxial Df vert

DM1 DM2

Du DWplastic a

bpipe d

dext dshift dclamp

143

gap width between inside tips of the clamp lips edge displacement of long pipe due to edge loads coefficient of M for the pipe’s edge displacement coefficient of Q for the pipe’s edge displacement coefficient for pressure displacement of pipe with thickness tpipe and internal diameter B width of spacer part of seal ring distance to neutral line for calculating Wplastic outside diameter of hub effective inside diameter of clamp for load transfer average diameter of contact between hub and clamp diameter of the centre of gravity circle of seal ring average diameter of sealing area at the seal ring lips factor adjusting plastic section moduli based on uni-axial stress conditions for the 3D stress effect friction correction on Faxial for translational sliding additional vertical (radial) line load from seal ring lips on hub due to pressure contribution of Fpressure and Fadd:p to the radial bending moment contribution of p and fhub face to the radial bending moment with respect to the considered neutral line radial compression of seal ring at assembly correction on plastic section modulus due to the neutral line’s shift. angle in use for defining integration interval auxiliary value for pipe flexibility distance centre of (elastic) rotation of hub proper to facing of hub (for spacer part of seal ring) distance neutral line in hub proper or extended hub to the hub’s facing reduction of lever arm eb with respect to tangential bending stresses in the clamp distance neutral line of clamp cross section to clamp’s baseline, see Fig. 17

144

1þ 11þ 1_

uðQ0 Þ uðQL Þ uðM0 Þ; uðML Þ

uhub u0 ; uL upipe edge upipe;M upipe;Q


global bolt load scatter value above nominal value positive scatter value for single bolt load global bolt load scatter value below nominal value edge load coefficient, to obtain the edge rotation due to shear load on edge edge load coefficient, to obtain the edge rotation due to shear load on opposite edge edge load coefficients, to obtain edge rotations due to moment loadings on edge and on opposite edge, respectively. rotation of hub proper’s centre of gravity edge rotations of short cylinder due to edge loads, edges are indicated with subscripts 0 and L edge rotation of long pipe due to edge loads coefficient of M for the pipe’s edge rotation coefficient of Q for the pipe’s edge rotation

pressurized or de-pressurized. Normally encountered geometries of PCC-hubs do not require check of alternating yielding, nevertheless the sum of primary and secondary stresses (or stress range), will be addressed later. As the actually found stress ranges are rather low, we do not considered it necessary for the new design method to include stress range assessment. It might be helpful to recall here the various stress categories and how one may deal with them if one bases the PCC-design (or approval) entirely on stress analysis: Primary stresses. Depending upon their character, (local) membrane or bending, the stress limit is either 2/3 of the yield stress or the yield stress (at design temperature). Limit load analysis deals adequately with primary stresses. Secondary stresses. No separate limit exists for secondary stresses, but the stress intensity limit for the sum of primary and secondary stresses is the sum of the yield stress at the highest and at the lowest temperature during the load cycle. If this stress limit is satisfied then elastic shakedown is considered to take place, preventing ongoing plastic deformation after the first few load cycles. Secondary stresses usually occur at gross structural discontinuities and can be found, although always together with primary stresses, by means of discontinuity analyses. Peak stresses. Contributing to fatigue cracks and/or brittle fractures in PCC-connections. In case of sufficiently

l

m

msealing s1 ; s2 ; s 3 slong:;uniform slong:;bending stang:; inside bending stang:; outside bending stang:; uniform syield y w wsealing x

axial distance of centre of (elastic) rotation of hub to position of Øcontact ; see Fig. 27 coefficient of friction between clamp and hub(not angle of friction!) angle of friction between seal ring and hub principal stresses uniform clamp stress in the direction of the PCCs central axis clamp bending stress in the direction of the PCCs central axis inside clamp bending stress in tangential direction outside clamp bending stress in tangential direction uniform clamp stress in tangential direction yield stress of considered material Poisson ratio angle of contact between clamp and hub angle of seal ring recess locally at lips of seal ring local angle for translational clamp sliding, see Fig. 5

ductile materials only of interest when the number of pressure cycles is exceptionally large. A threshold number of 7000 cycles is often used for piping systems, see e.g. ASME B31.3 ‘Process Piping’-par. 302.3.5. It is assumed that the number of cycles is below this threshold for the need of fatigue evaluation. 2. Forces in PCCs 2.1. Axial force introduced by the clamp bolts Two 1808 clamps are forced over the hubs by bolts. Load transfer takes place at annular contact areas, which are shaped like frustum surfaces of shallow right cones. The contact diameter is the average of the hub’s outside diameter ðØAÞ and the clamp’s effective inside diameter ðØCi Þ (Figs. 1 and 2). The clamp’s equilibrium equation in the absence of friction, considering that each clamp exerts two line loads: 2fradial ð¼ fnormal sin wÞ Bcontact ¼ 2Fbolting

ð1Þ

Despite lubrication some friction will resist the clamp’s (radial) sliding over the hub (Fig. 3). Using the coefficient of friction m; one has ffriction ¼ m fnormal

ð2Þ


145

Fig. 3. Effects of friction upon fradial and faxial .

Fig. 1. Equilibrium of bolt forces and radial clamp line loads.

The friction load is not directed radially as clamps cannot slide purely radially over the hub’s contact areas, because that would entail tangential shrinkage of clamps, and they are strained tangentially by increasing bolt forces during assembly. Due to the fairly large stiffnesses of clamps, a better assumption is that they move predominantly towards each other during the last phase of ‘bolting-up’. Hence, the friction direction coincides with intersections of the annular contact area and planes parallel to the CLs of both the PCC and of the bolts (Figs. 4 and 5). The angle x between frustum intersections and the translating direction (i.e. parallel with the CL of the bolts, their direction is drawn vertically here) is a function of the circumferential position, best described by means of angle a : tan x ¼ tan w sin a ðwith 08 # a # 1808 for the top clampÞ Establishing equilibrium in bolt direction, with fradial sina as vertical component of fnormal ; mfnormal cos x as vertical friction load component, and the bolt forces, one arrives at: 2

ðp 0

tan xða ¼ 08;1808Þ ¼ 0 and hence: cos x ¼ 1:0 tan xða ¼ 458;1358Þ ¼ 0:3297! cos x ¼ 0:9497: tan xða ¼ 908Þ ¼ 0:4663!cos x ¼ 0:9063: Replacing cos x by 1.0 results in a slight overestimation of the (negative) influence of friction on the axial force introduced in the PCC during assembly. Integrating this equilibrium equation, and by usage of faxial ¼ fradial =tan w ¼ cos w fnormal ; one finds: Faxial ¼ 2prcontact faxial ¼ Fbolting

ð3Þ

Assuming fnormal uniform along the circumference of the PCC, fradial is uniform too, as fradial ¼ sin w fnormal : With w ¼ 258 (all PCCs have hubs slopes of 258!) we note that:

Fig. 2. Vectorial decomposition of normal line load into a radial directed line load.

ð4Þ

With sin x < tan x ¼ tanw sin a for x # 258; integration of the horizontal friction component sin x ffriction over the range a from 0 to 908 gives a quarter of the additional force in horizontal (or axial) direction due to friction. ðp=2 mfnormal tan w sina rcontact da ð5Þ DFaxial ðfrictionÞ ¼ 4 0

Evaluating this term and including it in the resulting axial force in the PCC at ‘assembly’ renders: Faxial ¼

ðfradial sin a þ m fnormal cos xÞrcontact d a ¼ 2Fbolting

2p cos w 2sin w þ m p

2pcosw 4m sin w F 2 F 2sin w þ m p bolting 2sinw þ m p cos w bolting ð6Þ

In the case of forces which pull the PCC apart, become so large during the operating condition that some (minuscule) sliding takes place, then the direction of the friction

Fig. 4. Translational sliding parallel to the CL of the bolts.

146


Fig. 7. Line loads on the tips of sealing lips. Fig. 5. Segment (908) of contact area with intersections of planes parallel to the CL’s of both bolts and PCC.

reverses. Friction opposes no longer the bolt force, but helps the bolt force in building the axial force. Hence: Faxial ¼

2pcos w 4m sin w F þ F 2sinw 2 m p bolting ð2sin w 2 mpÞcos w bolting ð7Þ

2.2. Loads from the gasket Self energized seal rings have cross sections resembling a capital T upside-down. The vertical part serves as spacer ring for PCC-hubs and both endings of the horizontal T-bar (the ‘seal lips’) are in contact with the hub’s conical seal areas. The ring’s diameter at the seal lips is slightly oversized with respect to these conical seal areas. Hub parts have to be pushed against the spacer ring of the gasket’s upside-down T in order to force the seal lips into their ‘too narrow’ space. The diameter of the gasket ring decreases and circumferential (compressive) stresses buildup in the gasket. This kind of prestressing keeps the seal lips pressed against the hub’s sealing contact area (Fig. 6). When fully assembled, the spacer part of the seal ring is solidly fixed between the hubs. The seal lips are bridging the area between the spacer part and the tips of the seal lips in contact with the hubs. Any pressure load on the lips is transferred to the spacer part (and absorbed as friction by the hub’s facings) and to the tip’s contact zone. This load increases with increasing internal pressure, hence the term ‘self-energizing’ seal rings. If PCC-hubs would be pulled (almost) apart due to pressure in conjunction with any

Fig. 6. Enforced radial displacement of seal ring at assembly.

external load, then friction can no longer absorb the load on the spacer ring and consequently the seal tip loads increase and even better sealing is achieved. Assuming that the seal ring undergoes a (near) uniform radial displacement Du (derived from kinematic considerations), the resulting line load on the seal tips is easily established from the ring’s circumferential compressive stress. fvert ¼

2Du Aring Ering Bring Bsealing

ð8Þ

Herein Bring is the average diameter, Aring the cross sectional area and Ering the Young’s modulus of the seal ring. The sloping contact areas of the hubs and friction between these and the seal ring produce a horizontal line load as well: fhor ¼ tan ðwsealing þ msealing Þ fvert

ð9Þ

Integration over the whole circumference gives the horizontal force from the seal ring on the PCC hub at assembly (Fig. 7): Fhor ¼

2p Du Aring Ering tanðwsealing þ msealing Þ Bring

ð10Þ

In operation internal pressure creates additional loads on the seal lip tips and on the spacer part (friction-fixed against the flange facing!). Medium and, hence, pressure is supposed to be excluded from the seal ring contact areas. Dpressure ; the diameter at the beginning of the contact area, and lpres:lip ; the effective pressure exposed lip length (from flange facing to the position of Dpressure ) is introduced for the pressure generated loads like Dfvert and Fadd:pressure (Fig. 8) (formula for Fadd: pressure is given in the next paragraph).

Fig. 8. Additional loads on the seal ring due to pressure.


147

Additional pressure line load on the seal ring: Dfvert ¼

lpres:lip pressure 2

fhub face ¼

ðlpres:lip þ wspacer Þ pressure 2

ð11Þ ð12Þ

In normal operating cases the horizontal force on the hub from the seal ring is to be calculated with the sum of the vertical loads due to prestressing the seal ring ð¼ fvert Þ and due to the internal pressure ð¼ Dfvert Þ: The operating moments on the PCC-hubs cause some slight rotation. Both involved hubs rotate in opposite directions and hence the movement direction (or tendency) of the seal lip tips is opposite the ‘assembly’ movement, the friction direction reverses! fhor ¼ tanðwsealing 2 msealing Þ ðfvert þ Dfvert Þ

ð13Þ

Circumferential integration yields the horizontal force: ! 2p Du Aring Ering lpres:lip pressure Fhor ¼ þ pBsealing Bring 2 tanðwsealing 2 msealing Þ

ð14Þ

The radial distributed line loads fradial and fvert: cause distributed twisting moments around the centre of rotation lying on the neutral line of the hub’s cross section. mtwisting ðfradial Þ ¼ fradial ðdistanceneutral line to fradial Þ

ð19Þ

mtwisting ðfvert: Þ ¼ fvert: ðdistance neutral line to fvert Þ

ð20Þ

Contribution towards the radial bending moment in the flange ring of these twisting moments is:

2.3. Longitudinal equilibrium in PCCs Longitudinal (end) force due to the pressure is of course: p Fpressure ¼ B2 pressure ðB the PCCs inside diameterÞ 4 ð15Þ An additional longitudinal pressure force arises at the seal recess due to the difference between Dpressure and B: p Fadd:p ¼ ðD2pressure 2 B2 Þ pressure ð16Þ 4 Longitudinal forces acting on PCCs must be in equilibrium with each other, and that allows us to establish the force acting on the spacer part of the seal ring ð¼ Fgasket Þ: This gasket force, in normal operating cases usually rather small, is not distributed uniformly over the contact area between the flange facing and the seal ring’s spacer part. Due to minute (elastic) rotation it will concentrate near the tip of the spacer part, i.e. at the outside diameter of the seal ring. Faxial ¼ Fpressure þ Fadd:p þ Fhor þ Fgasket

Fig. 9. Forces (distributed loads) at ‘assembly’.

ð17Þ

M2 ¼ 2mtwisting ðfradial Þ 12 Bcontact 2mtwisting ðfvert: Þ 12 Bsealing ð21Þ Consequently, the total radial bending moment exerted on the hub proper during ‘assembly’ is: Massembly;hub ¼ M1 þM2 3.2. During normal operation in the hub proper The radial bending moment during operating condition is the ‘assembly’ bending moment (though M1 and M2 ; respectively, are to be calculated with the ‘operating’ forces F hor: and Fgasket and the ‘operating’ loads fradial and fvert þ Dfvert Þ but augmented with the pressure acting on the inside of the PCC together with the longitudinal pressures forces. Except for fhub face ; the shear load on the hub face (Fig. 10), the other pressure generated loads from the seal ring are already included in Fhor and Dfvert :

3. Radial bending moments in PCCs 3.1. At assembly in the hub proper Once the neutral plane is established for radial bending of the hub, the radial moment is derived quite directly (Fig. 9). The contribution of all longitudinal forces towards the radial bending moment in the flange ring (or hub proper): M1 ¼ ðFaxial arm1 þ Fhor: arm2 Þ=2p

ð18Þ

ð22Þ

Fig. 10. Additional ‘operating’ loads on the PCC-hub.

148


These additional contributions towards the radial bending moment on the hub proper are: DM1 ¼

Fpressure ðODgasket 2 B 2 tconnection Þ 4p

Fadd:p ðODgasket 2 ðBpressure þ BÞ=2Þ 4p DM2 ¼ 2mtwisting ðfhub face Þ rinside gasket þ mtwisting ðpÞ

ð23Þ

þ

1 2

B

Fig. 12. Hub proper of the PCC and idealized shape.

with : mtwisting ðfhub face Þ ¼ fhub face dext

Operational radial bending moment on the extended hub:

mtwisting ðpÞ ¼ p lhub exposed ðlpres:lip þ 1=2lhub exposed 2 dext Þ dext ¼ distance from neutral line to face of the hub ð24Þ

Moper:;ext: hub ¼ M1 ðoperatingÞ þ DM1 þ M2 ðoperatingÞ þ DM2 ð28Þ

The operational radial bending moment exerted on the hub proper is Moperating; hub ¼ M1 ðoperatingÞ þ DM1 þ M2 ðoperatingÞ þ DM2

ð25Þ

4.1. Plastic section moduli for radial bending moments

3.3. At assembly in the extended hub of the PCC The ‘M1’—part of the radial bending moment for the extended hub is exactly the same as for the hub proper. The ‘M2’—part due to the radial line loads fradial and fvert is almost the same too: it is to be established with respect to the extended hub’s neutral line on which the rotation centre is situated. 3.4. During normal operation on the extended hub The extended hub’s radial bending moment is calculated in a similar way as the radial bending moment on the hub proper (Fig. 11). Hence, the additional contributions towards the radial bending moment on the extended hub are: DM1 ¼

4. Plastic section moduli of PCCs

When the entire cross section of either the hub proper or the extended hub has become fully plastic then the maximum value for the radial bending moment is reached. Hub cross sections have complicated forms for calculating plastic section moduli. If the more or less triangular seal ring recess is replaced by an equally large rectangular area with its geometrical gravity centre closer to the hub face than that of the original area, then the plastic section moduli are slightly underestimated but they are much easier to calculate (Figs. 12 and 13). The rotation centres lie on neutral lines connected with radial bending of the hub proper (area’s A1 ; A2 and A3 Þ and of the extended hub (all four areas in Fig. 13), respectively. Neutral lines of plastic section moduli divide the considered section area exactly in half, with simple geometry these can

Fpressure ðO:D:gasket 2 B 2 tweld end Þ 4p

Fadd:p ðO:D:gasket 2 ðBpressure þ BÞ=2Þ 4p DM2 ¼ 2mtwisting ðfhub face Þ rinside gasket þ mtwisting ðpÞ 12 B þ

ð26Þ

with : mtwisting ðfhub face Þ ¼ fhub face dext mtwisting ðpÞ ¼ p lext:hub exp: ðlpres:lip þ 1=2lext:hub exp 2 dext Þ

dext ¼ distance from neutral line to the hub0 s face

Fig. 11. Additional ‘operating’ loads on the extended hub.

ð27Þ

Fig. 13. True and idealized cross sections of extended hub.


149

the ‘rotation’ neutral line and Ashift the area over which the shift took place: ð DWplastic ¼ 22 lyl dA ð31Þ Ashift

4.2. Design requirement for plastic section moduli

Fig. 14. Neutral line in extended hub in case A4 . A1 þ A2 þ A3 :

be found for the hub proper as well as for the extended hub (Fig. 14). Both plastic section moduli can be calculated as follows: ð Wplastic ¼ lzl dA ð29Þ A

with lzl the absolute distance of dA to the neutral line considered. However, due to radial line loads fradial ; fhubface ; ðfvert þ Dfvert Þ and pressure p; circumferential forces will occur in the hub. B Fcirc ¼ p ð2llip þ wspacer Þ rinsidegasket þ ðlhub 2 lpres lip Þ 2

As tangential rotation of the hub is resisted by the connected cylindrical part (i.e. A4 Þ; longitudinal bending stresses arise in A4 consequently. Note that the rotation of the extended hub is resisted by the pipe connected to the PCC-hub. The (theoretical) limit for these longitudinal bending stresses is syield ; the corresponding torque on hub proper (or extended hub) is readily transformed into a counteracting radial plastic moment: Mcounter ¼

ð30Þ with lhub the length of either the hub proper or the extended hub (Fig. 15). As a result the neutral line shifts to the left over an area of 21=2Fcirc =syield Corrections for the plastic section moduli due to the shifts are, lyl being the absolute distance of dA to

ð32Þ

Herein tconnection is the thickness of the cylindrical part for the hub proper or the pipe thickness for the extended hub, tpressure is the thickness required for longitudinal pressure stress: tpressure ¼

Bsealing B þ fvert 2 f radial contact 2 2

1 2 B þ tconnection 2 ðtconnection 2 tpressure Þ syield 4 2

pðB þ tconnection Þ 2syield

ð33Þ

The plastic section moduli and the counteracting moments both depend on the value of the yield stress. However, the resulting tangential stresses from the radial bending moment, longitudinal bending stresses from the counteracting moment and the radial pressure stress cause an in essence 3D stress situation. Hub yielding will occur at lower values

Fig. 15. Extended hub with circumferential stress patterns.

150


Fig. 16. Location of most critical 3D stress situation.

than the uni-axial yield stress, the Von Mises criterion will be applied to estimate how much lower (Fig. 16). Considering the stress situation at the inside of the hub proper where, due to the axi-symmetrical nature of the flange and its loading, the three principal stresses are as follows: (i) s1 —circumferential tensile (left of the neutral line!) stress due to radial bending moments, (ii) s2 —longitudinal bending stress as a plastic hinge develops here, compressive at the inside of the hub, and (iii) s3 —radial pressure stress, at the inside of the hub is s3 ¼ 2p: But with s1 . 0 . s3 ¼ 2p . s2 the onset of yielding is indeterminable with ðs1 2 s2 Þ2 þ ðs2 2 s3 Þ2 þ ðs3 2 s1 Þ2 ¼ 2 £ s2yield : Assuming s1 ¼ 2s2 then the Von Mises criterion results in: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ðp=syield Þ2 s1 ¼ 2 s2 ¼ ð34Þ £ syield 3 This square root expression can be seen as a correction for the uni-axial yield stress in the expressions as derived for the plastic section moduli to account for the 3D stress situation. Maximum radial bending moments with respect to the hub (both hub proper as well as extended hub) is reached if Mradial bending #{ðWplastic þDWplastic Þ syield þMcounter }

3Dfactor 1:5 ð35Þ

The design safety factor of 1.5 for primary stresses (limit load!) is incorporated in above formula, which shall be checked for both assembly as well as operating condition for the hub proper as well as for the extended hub.

Fig. 17. Lateral cross section of clamps.

The first (or static) moment of area with respect to the base line (Fig. 17):

tlipbase þ2tlip CA pr 2 4r þCt 2r 2l2c S ¼ t r þr ðCt 2rÞ2 þ 2 2 3p 3 ð37Þ The moment of inertia with respect to this base line: 1 2 p p Ibase ¼ Ct2 Ar þ rðCt 2rÞ3 þ r 4 þ r 2 ðCt 2rÞ2 3 3 8 2 þl3c

tlipbase þ3tlip 6

ð38Þ

The clamp’s principal axis of inertia runs at a distance of:

dclamp ¼

S Aclamp

ð39Þ

Hence, the moment of inertia with respect to this principal axis (neutral line!) is: Iclamp ¼ Ibase 2Aclamp d2clamp

ð40Þ

5.2. Longitudinal stresses in the clamps

5. Clamp rings

The axial force as introduced by the clamps on the sloping contact surface of the hubs, tries to pull apart, to bend open and to shear off the lips from the clamps (Fig. 18). The tensile stress in longitudinal direction (with respect to the CL of the PCC), assuming that both clamps cover the whole circumference of the hubs, is:

5.1. Cross section of clamps

slong:;uniform ¼

Cross sectional area of the clamp: 2

Aclamp ¼ Ar þ 1=2pr þ 2ðCt 2 rÞr þ lc ðtlip þ tlipbase Þ

ð36Þ

with tlipbase ¼ ðCw 2 wgap Þ=2 þ lc tan w and Ar ¼ ðCw 2 2rÞCt

Faxial pð2Rclamp 2 Ct ÞCt

ð41Þ

The offset of Faxial with respect to the mean diameter of the clamp’s lateral cross section causes a bending moment, the lever arm for Faxial is {ð2Rclamp 2 Ct Þ 2 ðA þ Ci Þ=2}=2: Consequently, the bending stress in the lateral cross section


151

Fig. 20. Lever arm reduction for tangential bending stresses. Fig. 18. Longitudinal clamp stresses.

ð42Þ

The maximum reduction for the offset ‘eb’ can be estimated by considering the bolt stresses. Let us limit the bolt stresses (membrane þ bending) to 1 12 times the usual allowable bolt stresses. Maximum possible shift ‘dshift ’ of the force away from the bolt’s CL is then the maximium reduction for ‘eb’. ( ) Fbolting 1 4dshift # 1 12 f b þ ð44Þ nb prb3 prb2

The bolts pulling together the clamps around the PCChubs cause both tensile as well as bending stresses in the clamps (Fig. 19). The uniform tangential stress is

with rb the radius of the bolt’s cross section for stress evaluation. Hence: ( ) rb 1 12 f b pr2b dshift ¼ 21 ð45Þ 4 Fbolting =nb

of the clamp (tensile at inside and compressive at outside):

slong:;bending ¼ ^

3{ð2Rclamp 2Ct Þ2ðAþCi Þ=2} Faxial pð2Rclamp 2Ct ÞCt2

5.3. Tangential stresses in the clamps

stang:;uniform ¼

Fbolting Aclamp

ð43Þ

The offset of the bolts with respect to the clamp’s centre of gravity (neutral plane) introduces a bending moment in the clamps. As the bolt force is a primary load, the tangential bending stresses are primary too. However, the curvature of the clamps cannot change really as they are wrapped around the hubs of the PCCs. Near the bolting lugs the side lips will bend slightly open in case of yielding and so the clamps are pressed further down over the hubs. The locally increased clamp curvature means that the bolting lugs dip slightly and the points of effort of the bolts shifts. The lever arm for the bending moment that generates these tangential bending stresses, decreases as a consequence (Fig. 20).

Moment’s lever arm (unreduced): eb ¼ Bc 2 Rclamp þ Ct 2 dclamp

ð46Þ

The tangential bending stress at the ‘base’-line of Fig.17 and at the outside of the clamp, respectively:

stang:;inside bending ¼ 2Fbolting ðeb 2 dshift Þ

stang:;outside bending ¼ þFbolting ðeb 2 dshift Þ

dclamp Iclamp

Ct 2 dclamp Iclamp

ð47Þ

ð48Þ

Equivalent clamp stress intensities can be derived from above membrane and bending stresses in the two directions. At the outside of the clamp longitudinal and tangential stresses have opposite signs and control most likely the clamp’s capacity. These primary SIs shall be limited to once the yield stress.

6. Design/assessment procedures of PCCs 6.1. Nominal bolt force and requirements to axial forces

Fig. 19. Tangential clamp stresses.

Assume a nominal assembly force and, depending the bolt tightening procedure, the actual bolt force will be either a little more or a little less than the intended nominal value. Use realistic estimates for the accuracy that can be achieved

152


when applying nominal bolt forces, e.g. the so-called ‘scatter’-values as provided in ‘Annex C’ of EN 1591-1 [1].

6.3. Checking of the strength of the PCC hubs

minimum Fbolting ¼ W nb ð1 2 12 Þ}

The maximum bolt force shall not be too large so that extensive yielding (plastic deformations) of the sloping contact area between clamp and hub and/or the gasket slope with the sealing ring would jeopardize the integrity and sealing performance of the PCC. Actual radial bending moments shall not be larger than those based on the plastic section modulus plus bending moment capacity of adjoining part with accounting for the 3D stress situation and a design margin of 1 12 : This shall be checked for both the hub proper as well as the extended hub in assembly and in operating condition.

nominal Fbolting ¼ W nb } maximum Fbolting ¼ W nb ð1 þ 1þ Þ}

ð49Þ

with W nominal force per bolt, nb number of bolts per lug and scatter values 1þ and 12 per par. 4.4.2 of EN 1591 [1]. With an appropriate estimate of the friction between hubs and clamps the axial forces that clamps may/can exercise on the hubs of the connection, can be determined. † Faxial ðassembly; min:Þ $ Fhor ðassemblyÞ Minimum axial assembly force is to be determined with minimum Fbolting and opposing(!) friction. † Faxial ðoper:; nom:Þ $ Fpressure þ Fadd:p þ Fhor ðoper:Þ Nominal axial operating force is to be determined with the nominal Fbolting and ‘zero’-friction. In operating condition (and testing) friction does not oppose Faxial but enlarges it. Reliable friction coefficients shall be used though, which would not be smaller due to smoother surfaces or better lubrication! In normal operating it is not safe to rely on friction. † Faxial ðoper:; max:Þ The maximum axial operating force is to be determined with the maximum Fbolting and ‘zero’-friction. It is clearly larger than the nominal axial force and so it is certainly capable of holding the connection together. The actual operating axial force will never be larger than this value and so this maximum axial force determines the maximum stress situation in the hubs, clamp and bolts in operating condition. † Faxial ðtestingÞ $ {Fpressure þ Fadd:p þ Fhor }testpressure The axial force in testing condition can be determined with maximum Fbolting and an appropriate value for beneficial(!) friction. In testing conditions, being ‘one-off’ conditions, one may rely on friction for keeping together the connection. In case the actual bolt force is less than the maximum bolt force, then the resulting axial force can be insufficient. Hubs will move apart and clamps will slide outwards over the hubs. The consequential bolt-elongation does increase the bolt force but never beyond the value of the maximum bolt force as then the corresponding axial force has grown sufficiently for the testing conditions.

6.4. Other strength aspects of PCCs Following other stress aspects shall be fulfilled too: (i)

(ii)

(iii)

(iv)

(v)

Primary stresses (membrane and bending) in the clamp shall be limited by the yield stress if the material is sufficiently ductile. Total primary S.I. due to tangential and axial stresses will proof to be decisive here. Accounting for the bolting lug dipping (lever arm reduction!) is allowed. The maximum bolt stress that may occur, i.e. based on the nominal bolt force times the ‘11þ ’ scatter factor (refer to Appendix C of EN 1591-1) [1]. Use bolt stress limits from a reliable flange assessment method. PED [2] restriction on general membrane stress for ferritic steel of 5/12 times the tensile strength shall be adhered to for pressure’s membrane stress in weld end and connecting pipe. Stresses (bending and shear) in the ‘cantilevered’ bolting lugs and the shear in the clamp’s lips are primary stresses, refer to par. 24-7 of Appendix 24 (ASME VIII, Div.1) [3] for suitable formulae. Contact stress of clamps on hubs 2fnormal ðA 2 Ci Þ shall not exceed the lower yield stress of either hub or clamp. Use Fnormal calculated with max. Fbolting for this stress check.

7. Discontinuity analysis of PCC hubs 7.1. Introduction

6.2. Calculation of all the forces in the PCC With the intended design pressure, all relevant dimensions plus the friction coefficient of the hub’s seal area and the seal ring all forces and line loads active in the PCC can be determined. For both the ‘assembly’ as well as the ‘operating’ conditions the maximum Fbolting shall be used. Now check whether above discussed three ‘axial force’requirements are met or not. If not then one has to select a larger nominal bolt force and/or a bolt tightening procedure with a smaller ‘minus’-scatter value and start all over again.

Secondary and primary stresses together may result in local yielding whenever PCCs are pressurized or depressurized. A discontinuity analysis was performed to investigate ‘low cycle fatigue’, especially a risk of the PCC weld end. The only relevant PCC load is the pressure cycle (0 ! full design pressure ! 0). Not relevant is the prestressing bolt load for the stress range as this load does not change during the cycle. The geometry of PCCs is too complicated and is simplified to a pipe, short thick cylindrical part and hub proper. The gross structural discountinuity at the weld end is emphasized


153

Fig. 21. Shape of PCC-hub and analysis-model with loads.

by the abrupt thickness change and so the stresses will be overestimated. The strength and stiffness contributions of seal ring and clamps will be accounted for later (Fig. 21). The hub proper is considered as a ring undergoing uniform rotation and radial translation. Knowing Rcog (‘centre of gravity’-radius of the hub’s cross section), Ihub (second moment of inertia about the radial axis) and Ahub (cross sectional area) then the centre of gravity’s displacements are:

uhub ¼

Mtwist Rcog 2pEIhub

ð50Þ

with Mtwist the hub’s total twisting moment, refer to Eq. (68) uhub ¼

Ftang Rcog EAhub

ð51Þ

The uniform radial displacement due to internal pressure is wcyl:; p p with the coefficient wcyl:; p ¼

ðB þ tcyl Þ2 ð1 2 12 nÞ 4Etcyl

7.1.1. Connected pipe at weld end Assuming a long enough pipe then the half-infinite solution for the edge displacement and rotation can be used. Refer to e.g. Timoshenko’s Plates and Shells (Chapter 15—General Theory of Cylindrical shells) [5] (Fig. 23). wpipe edge ¼

1 ðbpipe M þ QÞ ¼ wpipe;M M þ wpipe;Q Q 2bpipe 3 Dpipe ð57Þ

with ftang the tangential force in the hub, refer to Eq. (69) The cylindrical part is a short cylinder with edge loads, i.e. bending moments and shear loads both per unit length. With reference to Article 4-2 ‘Analysis of Cylindrical Shells’ from ASME VIII Div.2 [4], the edge displacements and rotations as function of the edge loads are (for edge load coefficient wðQ0 Þ and all other coefficient see referenced article) (Fig. 22): w0 ¼ wðQ0 Þ Q0 þ wðM0 Þ M0 þ wðQL Þ QL þ wðML Þ ML

ð52Þ

wL ¼ wðQL Þ Q0 þ wðML Þ M0 þ wðQ0 Þ QL þ wðM0 Þ ML

ð53Þ

2u0 ¼ uðQ0 Þ Q0 þ uðM0 Þ M0 þ uðQL Þ QL þ uðML Þ ML

ð54Þ

uL ¼ uðQL Þ Q0 þ uðML Þ M0 þ uðQ0 Þ QL þ ðM0 Þ ML

ð55Þ

ð56Þ

Fig. 22. Short cylinder with edges ‘0’ and ‘L’.

154


Fig. 24. Shear loads on pipe and ‘0’-edge of short cylinder. Fig. 23. Edge of pipe with edge loads.

1 upipe edge ¼ 2 ð2bpipe M þ QÞ ¼ upipe;M M þ upipe;Q Q 2bpipe Dpipe ð58Þ

As the unknown discontinuity loads (shear and moment load, respectively) Q; M; Q0 ; M0 ; QL ; ML ; QH and MH are loads per unit length along mean radii, the following relations exists (Figs. 24 and 25): Q¼2

with:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 12ð1 2 n2 Þ 4 bpipe ¼ t 2 ðB þ tpipe Þ2 tpipe

ð59Þ M¼

and 3 E tpipe Dpipe ¼ 12ð1 2 n2 Þ

ð60Þ

The pressure’s uniform displacement of the pipe is wpipe; p p with ðB þ tpipe Þ2 ð1 2 12 nÞ wpipe; p ¼ 4Etpipe

ODgasket 2 B 2 tcyl Fpressure 2

Dpressure þ B Fadd:p: þ ODgasket 2 2 2

s ltotal s þ pB ltotal þ 2d2 p 2 4 2 þ

Ftang

ð62Þ The other three continuity equations are:

upipe; M M þ upipe; Q Q ¼ 2ðuðQ0 ÞQ0 þ uðM0 ÞM0 þ uðQL ÞQL þ uðML ÞML Þ

ð63Þ

B þ tcyl s B p QH þ ltotal þ ¼ 2 2 d 2

For ltotal and d refer to Fig. 26.

ð68Þ

ð69Þ

The four continuity equations contain eight unknowns (Q; M; Q0 ; M0 ; QL ; ML ; QH and MH ) but above equilibrium relations eliminate four of the unknowns. One ends up with four unknowns in four linear equations which can be solved and then all the stresses can be found with appropriate formulae.

wðQL ÞQ0 þ wðML ÞM0 þ wðQ0 ÞQL þ wðM0 ÞML þ wcyl:; p p Ftang Rcog Mtwist Rcog þ £ ðltotal 2 dÞ ¼ EAhub 2pEIhub

uðQL ÞQ0 þ uðML ÞM0 þ uðQ0 ÞQL þ uðM0 ÞML ¼ 2

ð67Þ

Mtwist ¼ pðB þ tcyl Þ MH þ pðB þ tcyl Þðltotal 2 dÞ QH

Displacements and rotations at bordering elements must be necessarily equal and loads that bordering elements exercise on each other must conform to ‘action equals reaction’ (i.e. equilibrium). Total displacement of the pipe edge is equal to the displacement of the ‘0’-edge of the cylindrical part is:

¼ wðQ0 ÞQ0 þ wðM0 ÞM0 þ wðQL ÞQL þ wðML ÞML þ wcyl:; p p

B þ tcyl B2 ðtcyl 2 tpipe Þ p M0 2 8ðB þ tpipe Þ d B þ tpipe

ð66Þ

At the ‘short cylindrical-hub proper’ discontinuity one may simply establish that MH ¼ ML and QH ¼ 2QL : Now the total twisting moment on the hub and the total tangential force in the hub proper can be derived as applicable to this analysis (Fig. 26).

ð61Þ

wpipe; M M þ wpipe; Q Q þ wpipe; p p

B þ tcyl Q B þ tpipe 0

Mtwist Rcog 2pEIhub ð65Þ Fig. 25. Moments on pipe and ‘0’-edge of short cylinder.


155

Fig. 26. Hub proper with loads acting on it.

7.3. Exploring the effect of clamps and seal ring The seal ring undergoes the same radial translation as the hub facings, after all it is fixed between these facings. The same can be argued about the clamps, large normal loads between clamps and sloping contact surfaces of the hubs give rise to large enough friction forces to prevent any slippage between clamps and hubs when the hubs displace locally in radial direction (Fig. 27). The tangential force that causes the uniform radial translation of the hub, is absorbed simultaneously by the hub proper, the clamps and seal ring. So the area of the hub is to be adjusted by half of the cross sectional area’s of clamps and seal ring. Aadjusted ¼ Ahub þ ðAclamp þ Aring Þ=2

Aring 2

Icontribution clamp ¼ l2

Aclamp 2

ð71Þ ð72Þ

Hence, the second moment of inertia of the hub proper in the rotation formula is to be augmented with above contributions. Iadjusted ¼ Ihub þ Icontribution seal þ Icontribution clamp

8. Results of various PCC assessment methods For an industrial standard type 1000 PCC analyses were performed to determine the maximum allowable pressure.

ð70Þ

Any hub rotation forces additional uniform radial displacements on seal ring and clamp, e.g. hub rotation over w results in the seal ring moving over a distance of dw and the clamp moving over lw: The seal ring behaves in its entirety as if it were at a distance of d from the neutral line with the centre of rotation. Contributions of seal ring and, similarly, clamp towards the hub’s second moment of inertia are hence: Icontribution seal ¼ d2

It is proposed that one perform two discontinuity analyses, one with the normal values for Ahub and Ihub and another one with above adjusted values. Then one should use the larger of both found stress intensities for demonstrating that the sum of primary and secondary stresses stay within their allowable limits and that there is no danger for low cycle fatigue!

ð73Þ

(i)

Using a small friction coefficient of 0.1 the allowable pressure per Appendix 24 of ASME VIII Div.1 [3] was found to be 335 bar, a friction coefficient of 0.2 resulted in an allowable pressure of 347 bar. (ii) With ANSYS finite element analyses have been performed with linear-elastic but also with linearelastic perfectly plastic material behaviour. Sliding of the various PCC-parts (e.g. hub-clamp contact) was assumed to be frictionless. Based upon stress classification the maximum design pressure was reported to be about 680 bar, though the PED [2] restriction on general membrane stress lowered the pressure to 637 bar in the connected pipe. (iii) Using this new assessment method an allowable pressure of 580 bar was determined, though we realistically assumed the 191 kN nominal force per bolt was applied with a torque wrench

156


Fig. 27. Rotation forces additional radial translations.

(single bolt accuracy ^ 30%) and the friction between the PCC-parts to be as low as 5%. Increasing the force per bolt to 215 kN (^ 2%) and adapting a more realistic friction coefficient of 0.1 between hubs and

clamps, an allowable pressure of 637 bar could be found too. (iv) Discontinuity analysis for the same PCC showed that for 637 bar the sum of primary and secondary stresses reached not even 50% of their allowable limits and hence there is no danger for low cycle fatigue at all!.

9. Conclusion

Photograph 1. PCC-model with cut-away sections in both hub part and clamp section.

(1) The primary stresses control the maximum internal pressure in PCCs with proportions conforming to current industrial standards. Applying the limit load approach as discussed here, results in the allowable pressure being almost twice as high as with the ‘Appendix 24’ assessment method. (2) There is no need to take low cycle fatigue into consideration, and any stress range assessment of primary and secondary stresses is not at all required for current industrial standard PCCs. (3) Flanges are often the weakest link in piping systems, ‘normal’ PCCs (according to industrial standards) will be in most cases stronger than the connecting piping. This will facilitate piping design significantly: smaller piping layouts with all its associated savings! Hence, PCCs assessed with this


new design method contributes towards sustainability as they are anyway lighter in weight than conventional flanges.

Acknowledgements Authors are indebted to Galperti Engineering (Italy) who were forthcoming with dimensions not stated in standard product brochures on PCCs.

157

References [1] European Standard EN1591-1. Flanges and their joints—design rules for gaskets circular flange connections—part 1: calculation method. CEN European Committee for Standardization, April 2001. [2] Pressure Equipment Directive 97/23/EC of the European Parliament and of the Council of 29 May 1997. [3] Appendix 24-ASME VIII, Div.1. American Society of Mechanical Engineers. [4] ASME VIII, Div. Z. American Society of Mechanical Engineers. [5] Timoshenko SP, Woinowsky-Krieger S. Theory of plates and shells. New York: McGraw-Hill; 1959.