computation of unloading stiffness of nonlinear gaskets using dual ...

COMPUTATION OF UNLOADING STIFFNESS OF NONLINEAR GASKETS USING DUAL KRIGING INTERPOLATION Henri Champliaud1, Abdel-Hakim Bouzid2 École de technologie supérieure Mechanical Engineering Department 1100, Notre-Dame West Montreal (Qc), H3C 1K3, Canada 1 email : [email protected], tel1: (514) 396 8597 email2: [email protected], tel2: (514) 396-8563 Abstract: The leakage behaviour of bolted joint is very much dictated by the gasket contact stress. In particular, the non-uniform distribution of this stress in the radial direction caused by the flange rotational flexibility has a major influence on the leak tightness of some gasket types. One of the difficulties in developing an analytical solution is the modeling of the non-linear gasket material behaviour. In the past, a model was developed to handle this nonlinearity using dual kriging interpolation of the gasket material data curves. However, during the unloading process the model does not account for the radial variation of the gasket stiffness. In this paper, we will present a method based on dual kriging interpolation of the material data curves combined with finite difference computations to handle the flexibility of the gasket with reasonable accuracy. Together with the flange rotational flexibility, this technique implemented in the ‘‘SuperFlange’’ program is supported and validated by numerical FEA conducted on different size and type flanges used with different gasket materials.

Keywords: gasket, kriging, nonlinear displacement, interpolation, flange, leakage, finite difference Nomenclature θf,rad θf,rad a(X), ai aj A,in Ag,in² Ap,in2 b(X), B,in C,in d,in Do,in Ef,psi G(X) G,in Gi,in Go,in Gm,in h,in hD,in hG,in HB,lb HD,lb

flange rotation Poisson’s ratio of flange average behavior or drift of X coefficient i of a function pi coefficient j of a function K(|X-Xj|) flange outside diameter initial gasket area partition gasket contact area error term or fluctuation of X flange inside diameter bolt circle diameter bolt nominal diameter flange centroidal diameter flange modulus of elasticity Observation phenomena of var. X effective gasket diameter inside gasket diameter outside gasket diameter diameter at gasket average contact stress hub length lever arm from hydrostatic end force to bolt circle lever arm from gasket reaction force to bolt circle bolt load hydrostatic end force on area inside of flange

HG,lb k kp,

gasket load column number factor (equal to 1 for a joint with blind cover and 2 for identical flanges) K ratio of outside diameter to inside diameter of flange K(|X-Xj|) element of fluctuation or generalized covariance Kb,lb/in bolt uniaxial stiffness Kg,lb/in gasket uniaxial stiffness KfM,in.lb flange uniaxial stiffness due to a moment KfP,psi flange uniaxial stiffness due to a pressure Mf,in.lb/in total equivalent flange moment n row number nb number of bolts p,psi internal pressure pi element of drift function r radius position s normalized value for Sg Sg,psi gasket stress at radius r Sgm,psi mean gasket stress t normalized value for ug tg,in gasket thickness tf,in flange thickness u,in axial displacement ug,in gasket axial displacement X phenomena (Sg,ug) Y factor involving K W initial bolt load Subscript b g f

refers to bolt refers to gasket refers to flange

Superscript i f

initial seating condition final operating condition

1. Introduction Leakage is the main cause of failure of bolted flange joints. The lack of ability of current flange designs to evaluate the gasket contact stress and its variation both in the circumferential and the radial directions is the main cause to blame. One of the major

difficulties in developing an analytical solution is the modeling of the gasket behaviour. Its highly nonlinear non-elastic mechanical behaviour makes the analytical solution more complicated to elaborate. The importance of a good evaluation of the gasket contact stress is very important when predicting leakage. The gasket stress is not uniform because of flange rotation. Although, the mandatory ASME [1] code flange design procedure limits the rotation to a maximum value of 0.3 degrees, the resulting maximum gasket stress at the gasket outside diameter could reach twice the average gasket stress [2]. This effect can have a major impact on the tightness performance of the gasket. While this high local stress is beneficial in terms of leak path capillary size reduction with certain gasket styles it is detrimental to those soft gaskets that are prone to crushing [3]. The accurate prediction of gasket stress requires a good estimation of the gasket compression or displacement in the flange axial direction. Generally, for critical applications the distribution of contact stress is obtained from sophisticated finite element analysis [4, 5]. Several researchers studied the analytical distribution of the gasket contact stress by considering the gasket as a linear spring with an elastic linear behaviour [6, 7]. The nonlinear behaviour during loading and the change in stiffness across the width are not accounted for. As a result the evaluated contact stress in not representative of the real behaviour. The use of the nonlinear behaviour of the gasket was first introduced in the late seventies and has been used since in several applications [8, 9, 10]. Most of the analytical methods developed involve the sectioning of the gasket into several parallel spring elements of varying stiffness. In [2] an analytical approach was presented and models the nonlinear seating behaviour successfully. However, the operating behaviour was less successful

This paper presents a method of evaluating gasket contact stress using nonlinear gasket modeling and takes into account the radial stiffness variation of gasket during unloading. The method is based on dual kriging interpolation of gasket data having complex nonlinear behaviour. Finite element analysis is used to validate the approach through different size diameter flanges combined with different gasket types. 2. Theoretical Modelling 2.1 Dual Kriging Methodology Kriging is a generalization of the interpolation least square method used initially by Krige [11] in mining exploitation. This method was later used to define the complete surface of complex bodies based on few sample coordinate points [12, 13]. Biomechanics is an area where this technique is exploited for modeling hands, bones and human tissues. In our paper the gasket mechanical behaviour is described by the two functions Sg(Xi) for gasket stress and ug(Xi) for gasket displacement which are based on dual kriging interpolation in such a way that they fit all sample gasket data points Xi obtained from a compression test. Each function is decomposed into the sum of two terms, in the form a(Xi)+b(Xi), where a(Xi) represents the average behaviour called the drift and b(Xi) is an error term called the fluctuation. Xi is a gasket sample data point pairs formed by Sg and ug.

20000 Kringing interpolation by row Kringing interpolation by column Interpolated unloading curve Compression test data points

16000

Gasket stress, Sg

because the model could not accommodate the real gasket unloading curves which were oversimplified by linear lines of the same slope. In [10] while the stress during unloading was obtained with the kriging technique, the equivalent gasket unloading stiffness used in the interaction analysis was not accurately defined.

12000 i

i

( ug ,Sg )

8000 o

( ug ,0 )

4000

f

f

( ug ,Sg )

0 0

0,05

0,1

0,15

0,2

Gasket displacement, ug

Figure 1: Kriging interpolation of the gasket nonlinear data The result is a parametric grid that allows the determination of the gasket loading or unloading stress when the displacement is known and vice versa (Fig. 1). The observations G of a phenomenon (equal to Sg or ug in this case) at point X along a row or a column can be interpolated using the basic dual kriging model written in the form: M

N

j=1

i =1

G (X) = ∑ a jp j (X) + ∑ b i K ( X − X i )

(1)

By using the normalized values s and t for Sg and ug along rows and columns, two parametric equations for each parameter are obtained by dual kriging such that the gasket stress Sg is given by for a row k M

N

j=1

i =1

Sg k (s) = ∑ a s i p j (s) + ∑ b si K ( s − s i )

(2)

for a column n M

N

j=1

i =1

Sg n ( t ) = ∑ a t j p j ( t ) + ∑ b t i K ( t − t i )

(3)

The coefficients as, bs at, and bt are obtained by solving the equations Sgnk = Sgk(sn) = Sgn(tk) at each data point.

Finally it can be shown that:

Sg (s, t ) = [Sg k (s)]T [Sg n ( t )]

(4)

And a similar function for ug(s,t);

u g (s, t ) = [u g k (s)]T [ u g n ( t )]

(5)

These functions are then used to obtain one parameter of Sg or ug on the basis of the other.

θf

2.2 Flange Model The axial equilibrium consideration of the joint is obtained for both the initial seating state i, and final operating state f, such that (see Fig. 2 for symbols identification):

H iB = H iG = W

(6)

Figure 2: Bolted flange model

and from the operating axial equilibrium

H fB = H fG + H D

(7)

Where HB and HG are the bolt and gasket loads and HD is the hydrostatic end thrust such that: HD = p Ap

(8)

The problem is statically indeterminate. To solve Eq. (5) for the two unknowns H fB and H fG , the geometric compatibility equation is required. This is achieved by considering the axial displacement of the joint. The sum of the axial displacements of the gasket compression, the flange rotation and the bolt elongation must be the same in both the initial bolt-up and final operating state [14]¸ 4

4

∑ue = ∑ue 1

i

f

(9)

1

In terms of the individual axial displacement of the joint and for identical flanges, Eq. (9) becomes: u ib + u ig + 2 u if = u fb + u fg + 2 u ff

(10)

In terms of the individual element equivalent stiffness, the displacement of the gasket and bolt and the rotation of the flange are: ub =

HB M H p ; u g = G ; θfM = f ; θ fp = (11) K fM Kb K fp Kg

The axial displacement between the gasket reaction location G and the bolt circle C caused by flange rotation is given by: u f = h G (θ fM + θ fp )

(12)

Using the geometric axial displacement compatibility, it can be shown that the gasket operating load HG is therefore:

HG

⎛ H Dh D HD p + 2h G ⎜ + ⎜ πD K Kb K 0 fM fp ⎝ =W− 1 1 1 + + 2 h G2 Kb Kg π D 0 K fM

⎞ ⎟ ⎟ ⎠ (13)

2.3 Nonlinear Modeling of Contact Stress

The proposed analytical approach is based on the assumption that the contact stress at any radial position on the gasket is a function of the flange-to-gasket contact surface axial displacement. The nonlinear relationship between stress and deflection is given by

dual kriging of load deflection data described previously. Some assumptions are made to simplify the model. The flange-to-gasket contact surface remains plane after application of load. This implies that the deflection of the gasket, ug varies linearly with the radius. Consequently this linear variation across the gasket width depends on the average stress, Sgm and the rotation of the flange θf. In addition, there is no radial expansion of the gasket due to Poisson effect resulting from axial compression and therefore the gasket area, Ag is assumed to remain constant after loading. This is justified in cases of raised face and full face flanges that use gaskets cut to fit exactly the contact face.

radial displacement using similar equations to Eq. (8).

The gasket axial displacement, ug may be obtained at any radial position r, according to the linear relation:

Figure 3: Gasket deformation and contact stress model

ug (r) = u gm + k p (

Gm - r) θf 2

(14)

where kp is a constant equal to 2 for a joint with a pair of symmetrical flanges and equal to 1 for a joint with an assumed rigid cover plate. The average displacement of the gasket, ugm occurs at some diameter location, Gm and its value can be obtained by Eq. (5) based on s=sim and t=0. sim is found from Eq. (4) knowing the average initial gasket stress. The diameter location, Gm of ugm is, however, different from the gasket reaction location G. In practice, during gasket seating, the initial average stress Sigm applied to the gasket is straightforward to obtain. It is calculated from the bolt-up load and the gasket area. However, the operating gasket stress, Sfgm is more difficult to obtain since the interaction between all joint members is considered. The details of this derivation are spelled out elsewhere [14]. The radial distribution of gasket stress for a particular operating condition is obtained by considering the

½ufgi ½ufgm

½uigi

½ufg(r)

½ufgo

½uigo

i

½u gm

½uig(r)

θf f

Gasket Deformation (half thickness)

θf i

Flange operating position

Gi

Go

Flange seating position

Gfm

Gasket Stress Distribution

r

Sgo

Gim

Sfg(r) Sigm

Sfgm

Seating

Sgi Operating

dr

Sig(r)

Gasket width

2.4 Gasket Unloading Equivalent Stiffness

The gasket unloading stiffness is used when evaluating the contact stresses during application of pressure and subsequent loads. The gasket is divided into several sections in the radial direction. The equivalent rigidity can be calculated on the basis of the individual stiffness of each section. Referring to Fig. 3, the elementary gasket force is dF = 2π r dr Sg

(15)

Expressing the stress variation Sg in terms of the slope gives

dF = 2π r dr

ΔSg ΔDg

ug

(16)

The gasket force is therefore F = 2π ∫

ΔSg ΔDg

u g r dr = K g u gm

(17)

Substituting for ug and neglecting the rotation term gives ΔS K g = 2π ∫ g r dr (18) ΔDg

where the ratio term in (18) is numerically evaluated using finite difference.

The proposed method to evaluate the contact stress distribution of non-linear gasket materials using the dual kriging technique was validated numerically using FEA on four flange geometries used in conjunction with three different gasket styles. All flange joints were carbon steel flanges used in pairs and are, a 24 in. and 52 in. raised face (RF) flanges and a 10 in. and 24 in. full face (FF) flanges. A compressed asbestos fiber sheet CAF, a spiral wound SW and a corrugated metal sheet CMS gaskets were used with these flanges (see Table 1). B

A

C

tf

g0

g1

h

d

nb

10

16

14

1.25

⅜

⅜

0

1⅛

16

24

32

29.5

2

0

1¼

24

23¼ 29.5 277⁄16 1⅞ 58 51 56¼ 5⅝ ⅜

11⁄16 11⁄16 ⅜

5⁄8

1¼

⅞

24

⅜

13⁄16

1¼

1

76

Table 1: Flanges dimensions in inches

Gasket stress, (psi)

3. Numerical Fe Modeling

40000

30000

CMS gasket SW gasket

20000

10000

0 0

0.01

0.02

0.03

0.05

0.06

Figure 5: Gasket deformation and contact stress model The general-purpose non-linear numerical finite element program Ansys [15] was used to model the flange and the bolt with 3D 8 node solid elements as shown in Fig. 4. A circular equivalent head was used to model the hexagonal head of bolt. Special interface nonlinear gasket elements were used to model the gasket nonlinear mechanical behaviour. The new version of the software allows the introduction of the loading and unloading curves obtained from the loadcompression test as shown in Fig. 5. In all models, because of symmetry of the geometry and loading only one portion of the flange including one bolt was modeled. The initial bolt-up load was applied by imposing an equivalent axial displacement to all bolt nodes that belong to the section lying on the mid-gasket symmetrical plane. An equivalent displacement to 24.5 ksi bolt-up stress and 200 psi was applied to the 10 in. FF flange and 23 ksi bolt-up stress and 400 psi to the 24 in. FF flange. In the case of both the 24 in. and 52 in. of RF heat exchanger flanges, a 40 ksi bolt-up stress was used before applying a pressure of 400 psi. 4. Results and Analysis

Figure 4 3-D finite element model

0.04

Gasket displacement, (in)

0.02

0.02 Bolt-up to 40 Ksi

0.018

Pressurization to 400 psi

Gasket displacement, in.

Gasket displacement, in

0.014 0.012 0.01 0.008 0.006

Filled markers: CAF gasket unfilled markers: CMS gasket

0.004

Dashed lines: FE model (Ansys) solid lines: Analytical dual kriging method

0.002

Bolt-up to 40 Ksi Pressurization to 400 psi

0.018

0.016

0.016

0.014 0.012

0.01

Filled markers: CAF gasket unfilled markers: CMS gasket

0.008

0

Dashed lines: FE model (Ansys) solid lines: Dual kriging method

0.006

0

0.2

0.4

0.6

0.8

1

0

0.2

Gasket width ratio

0.4

0.6

0.8

1

Gasket width ratio

Figure 6: Gasket compression in 24 in. raised face flange

Figure 8: Gasket compression in 52 in. raised face flange

Small gasket displacements ranging from 1 to 20 mils (25 to 500 μmm) are shown in Figs. 6,8, 10 and 12. These figures show the compressions from which the contact stresses are estimated. The gasket contact stress distributions are shown for various flanges and gasket combinations in Figs. 7,9,11 and 13. In general, the results of the analytical model based on dual kriging interpolation compare well with those of the numerical FEA model. The stresses are lower at the gasket ID and higher at the gasket OD.

This is caused by flange rotation which may produce a zero stress at the gasket ID and twice as high tress as the average stress at the gasket OD. Figs. 6 to 9 presents the results for the 24 in. and 52 in. raised face flanges having a small gasket width whereas Figs. 10 to 13 show the result for the 10 and 24 full face flanges having a gasket over their whole contact surface area. It can be seen that for both seating and operating conditions, the analytical model predict similar gasket behavior to the numerical model in that the non-linear radial distributions of gasket contact stress is confirmed.

40000

45000 Pressurization to 400 psi Bolt-up to 40 Ksi

35000

Bolt-up to 40 Ksi

40000

30000

Filled markers: CAF gasket unfilled markers: CMS gasket Dashed lines: FE model (Ansys) solid lines: Analytical dual kriging method

30000

25000

Gasket stress, psi

Gasket stress, psi

Pressurization to 400 psi

35000

20000 15000

Filled markers: CAF gasket unfilled markers: CMS gasket

Dashed lines: FE model (Ansys) solid lines: Dual kriging method

25000 20000 15000

10000

10000

5000

5000 0

0 0

0.2

0.4

0.6

0.8

Gasket width ratio

Figure 7: Gasket contact stress in 24 in. raised face flange

1

0

0.2

0.4

0.6

0.8

Gasket width ratio

Figure 9: Gasket contact stress in 52 in. raised face flange

1

0.035

0.01

0.008

0.035

Bolt-up to 24.7 ksi

0.009

Bolt-up to 23 ksi

Pressurization to 400 psi

0.007

0.03

Pressurization to 200 psi

0.03

0.02

0.005 0.015

0.004 Filled markers: PTFE gasket unfilled markers: CAF gasket

0.003

0.01 0.005

0 0.4

0.6

0.8

0.015 0.003

Filled markers: PTFE gasket unfilled markers: CAF gasket

0.01 0.005

0.001

0 0.2

0.02 0.004

Dashed lines: FE model (Ansys) solid lines: Analytical dual kriging method

0.001

0

0.005

0.002

Dashed lines: FE model (Ansys) solid lines: Analytical dual kriging method

0.002

0.025

0

0

1

0

0.2

Gasket width ratio

0.4

0.6

0.8

1

Gasket width ratio

Figure 10: Gasket compression in 10 in. full face flange

Figure 12: Gasket compression in 24 in. full face flange

The results for the full face flanges are less accurate, because of higher rotation sensibility. In fact the flange warps as a result of bending in the radial direction. The rotation is more pronounced in the vicinity of the connection with the shell than at the bolt as may be appreciated from the non-linear displacements of Fig. 10 and 12 as compared to the raised face flanges.

Gasket displacements and contact stress values compare well because the gasket behavior is well defined by the dual kriging interpolation method, and the local rigidity of the gasket required for the operating condition is accurately computed by finite difference. 5000 Bolt-up to 23

Gasket stress, psi

The general performance of the analytical model is acceptable for the two types of flanges namely the raised face flanges and the full face flanges.

4000

Pressurization to 200 psi Filled markers: PTFE gasket unfilled markers: CAF gasket

3000

Dashed lines: FE model (Ansys) solid lines: Analytical dual kriging method

2000

8000 7000

1000

Bolt-up to 24.7 ksi Pressurization to 200 psi

Gasket stress, psi

6000 5000

Filled markers: PTFE gasket unfilled markers: CAF gasket

4000

Dashed lines: FE model (Ansys) solid lines: Analytical dual kriging method

0 0

0.2

0.4

0.6

0.8

1

Gasket width ratio

Figure 13: Gasket contact stress in 24 in. full face flange

3000 2000

5. Conclusion

1000 0 0

0.2

0.4 0.6 Gasket width ratio

0.8

Figure 11: Gasket contact stress in 10 in. full face flange

1

The dual kriging method is a powerful tool to analytically model the complex gasket contact stress problem due to nonlinear mechanical behaviour of the gasket. The gasket contact stresses relies on the ability of the model to calculate the displacements

PTFE compression, in

0.006

0.006 CAF compression, in

0.025

0.007

PTFE compression, in

CAF compression, in

0.008

based on flange rotation, and the gasket stiffness evaluation during the unloading state. The validity of the model used was checked against the more accurate nonlinear FE method. The distribution of the gasket contact stress is found to be non-uniform across the gasket width with a relatively high difference between the inner and outer peripheries depending on the gasket type and the flange rotational flexibility. Gasket stress at the inner periphery was found to be as low as zero during operation in some cases while at the outer periphery it is two times higher than the average stress. References

[1]

[2]

[3]

[4]

[5]

ASME Boiler and Pressure Vessel Code, 2001, Section VIII, Division 2, APPENDIX 2, Rules for Bolted Flange Connections with Ring Type Gaskets. Bouzid, A. and Derenne, M., (2002), Analytical Modeling of the Contact Stress with Nonlinear Gaskets, ASME Journal of Pressure Vessel Technology, 124, p. 47-53. Bouzid, A., Derenne, M. and Birembaut, Y., (1999), Safe Load Limits for Gaskets Related to Tightness and Mechanical Integrity, 5th Progress report, PVRC project 97-06, Committee on Bolted Flange Connections. Sawa, T., Higurashi, N. and Akagawa, H., (1991), A Stress Analysis of Pipe Flanged Connections, ASME Journal of Pressure Vessel Technology, 113, p. 497-503. Boneh, B., Ayrault, D. and Boillot, L., (1986), Amélioration d'une méthode d'analyse des brides circulaires boulonnées, International Symp. on Fluid Sealing, Application to Bolted Flanged Connections, Nantes, p. 107-119.

[6] Singh, K.P. and Soler, A.I., (1984), Mechanical Design of Heat Exchangers and Pressure Vessel Components, Arcturus Publishers, New Jersey. [7] Soler, A.I., (1980), Analysis of Bolted Flanged Joint with Non-linear Gasket, ASME Journal of Pressure Vessel Technology, 102, p. 249-256. [8] Nagy, A., (1997), Time Dependent Characteristics of Gaskets at Flange Joints, International Journal of Pressure Vessels and Piping, 72, p. 219-229. [9] Champliaud, H. and Lê, N.V, (2000), Finite Element Analysis of Crowning Sealing Caps, 2000 Ansys Conference, Pittsburgh, USA. [10] Bouzid, A. and Champliaud, H., (2004), Contact Stress Evaluation of Non-Linear Gaskets Using Dual Kriging Interpolation, ASME Journal of Pressure Vessel Technology, 126, p. 445-450. [11] Krige, D.G., (1951), A Statistical Approach to Some Basic Mine Evaluation Problems on the Witwaterstand, Journal of Chemical, Mettal. Min. Soc. S. Afr., 52, p. 119-139. [12] Matheron, G., (1973), The Intrinsic Random Functions and their , Adv. Appli. Prob., 5, Applications p. 439-468. [13] Champliaud, H., Duchaine, F. and Lê, V.N., (2002), Structured 3D Solid Mesh of Complex Thin Parts Using Dual Kriging Interpolation, 29th Intern. Conference on Computers and Industrial Engineering, Montreal, Canada, p. 564-568. [14] Bouzid, A. and Chaaban, A., (1993), Flanged Joint Analysis: A Simplified Method based on Elastic Interaction, 17(2), Transaction of CSME, p. 181-196. [15] ANSYS, 2004, ANSYS inc. Standard Manual, Version 8.1