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The modelling of bolted flange joints used with disc springs and tube spacers to reduce relaxation
International Journal of Pressure Vessels and Piping 87 (2010) 730e736
Contents lists available at ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
The modelling of bolted flange joints used with disc springs and tube spacers to reduce relaxation Abdel-Hakim Bouzid a, *, Akli Nechache b a b
Ecole de Technologie Superieure, 1100 Notre-Dame Ouest, Montreal, Quebec, Canada H3C 1K3 SNC-Lavalin, 18, rue Mustapha Khalef, 16306 Ben Aknoun, Algérie
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 August 2009 Received in revised form 16 September 2010 Accepted 28 September 2010
Bolted flange joints are prone to leakage when exposed to high temperature. In several cases, the root cause is relaxation that takes place as a result of material creep of the gasket, the bolt and the flange. One way to overcome this problem is to make the joint less stiff by introducing disc springs or the use of longer bolts with spacers. Although widely used, these two methods have no reliable analytical model that could be used to evaluate the exact number of washers or length of the bolts required to reduce relaxation to a minimum acceptable level. This paper describes an analytical model based on the flexibility and deflection interactions of the joint different elements including the axial stiffness of the flange and bolts, used to evaluate relaxation. The developed analytical flange model can accommodate either disc springs or longer bolts with spacer tubes to reduce the bolt load loss to a maximum acceptable value. This model is validated by comparison with the more accurate FEA findings. Calculation examples on a bolted flanged joint are presented to illustrate the suggested analytical calculation procedure. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Bolted flange joints Relaxation Disc springs Tube washers Analytical modelling Joint stiffness
1. Introduction Bolted gasketed flange joints are widely used in the chemical, petrochemical and nuclear power plants. A lot of research has been conducted in this area to tackle the several issues related to the actual ASME flange design code procedure [1] that is based on the early work of Waters et al. [2] and is known as Taylor Forge method [3]. Over the years, under the auspices of the Pressure Vessels Research Council (PVRC), a fairly large amount of research related to high temperature leakage in bolted joints was conducted as a result of Payne’s survey in 1985 [4]. The principal cause of leakage encountered in the power industries is related to high temperature operating bolted joints. It was found that the most frequently observed cause of leaks was due to loose bolts. This is essentially due to the creeperelaxation phenomenon that takes effect at high temperature although other effects such as gasket degradation and thermal expansion difference can be present. The load redistributions as a result of thermal transients and steady state were tackled both analytically and numerically in Refs. [5e8].
* Corresponding author. Tel.: þ1 514 396 8563; fax: þ1 514 396 8530. E-mail addresses: [email protected] (A.-H. Bouzid), akli.nechache@ snclavalin.com (A. Nechache). 0308-0161/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2010.09.001
In high temperature application, material creep has a significant effect on the relaxation of a joint. Gasket and bolt are the joint members that give the largest bolt load relaxation ratio over time. Bazergui [9] has observed significant creep taking place at room temperature over a short period of time with some gaskets. Few researchers treated the analytical modeling of relaxation in bolted joints. The most recent studies are of Bouzid et al. [10] who presented an analytical approach to treat relaxation and estimate the effect of gasket creep on the bolt and gasket load loss. Recently, Nechache et al. [11e13] extended the approach by considering the bolt, the flange ring, the hub and the cylinder creep. It was found that more than 50% of bolt load loss after 10,000 h of operating time was frequent in high temperature joints. It was concluded that the gasket and bolt creep are the parameters that affect more the loss of gasket seating stress and bolt preload and their effect is severe with time and increasing temperature. To maintain bolt load over time, disc springs also known as Belleville springs and tube spacers are often utilized. Their use makes the joint more flexible; a design parameter that can reduce the bolt load relaxation due to creep to within acceptable values. The dimensions and quantity used for a specific application are based on experience and approximate calculations. Engineers have no accurate calculation procedure to follow when disc springs are required. This paper describes an analytical procedure based on
A.-H. Bouzid, A. Nechache / International Journal of Pressure Vessels and Piping 87 (2010) 730e736
Nomenclature
a d n D qf Ab As A B C Di De E Fb Fg go g1 G h ho hG K Kb KfM KfP Kg Kj Kw Ks lb
washer constant washer outside to inside diameter (mm) Poisson’s ratio change in parameter flange rotation (rad) total bolt area (mm2) total spacer tube area (mm2) outer diameter of flange (mm) inner diameter of flange (mm) bolt circle diameter (mm) internal diameter of washer (mm) external diameter of washer (mm) Young’s modulus (MPa) bolt force (N) gasket force (N) thickness of cylinder (mm) thickness of hub big end (mm) gasket reaction force diameter (mm) unloaded height of washer truncated cone (mm) hub length (mm) flange lever arm (mm) flange inside to outside diameter ratio bolt uniaxial stiffness (N/mm) flange rotational stiffness due to moment (N mm) flange rotational stiffness due to pressure (N/mm2) gasket uniaxial stiffness (N/mm) global joint uniaxial stiffness (N/mm) total washer uniaxial stiffness (N/mm) total spacer uniaxial stiffness (N/mm) bolt effective length (mm)
ls L Mf nb p tw t V VL w
spacer tube height (mm) ASME code factor flange moment per unit circumference (mm.N/mm) number of bolts internal pressure (MPa) thickness of washer (mm) thickness of flange (mm) ASME code factor for integral type flanges ASME code factor for loose-type flanges axial deflection of the joint element (mm)
Superscript c refers to creep f refers to final pressurization state r refers to relaxation state Subscript b c w s f g
refers refers refers refers refers refers
to bolt to creep to washer to spacer tube to flange to gasket
Acronyms ASME American Society of Mechanical Engineers BPVC Boiler and Pressure Vessel Code CAF Compressed Fiber CMS Corrugated Metal Sheet FEM Finite Element Method HOBT Hot Blow-Out Test
a flexibility model which leads to an accurate design of the dimensions of bolt spacers and number of the disc springs required in a specific case based on the maximum tolerated relaxation. Calculation examples on various cases of a symmetrical flanged joint are presented to illustrate the suggested analytical approach. The results obtained analytically are compared to those provided by FE modeling. 2. Analytical model 2.1. Bolted joint with spring washers Fig. 1 shows the analytical model used to estimate the required spring washers or spacers to reduce relaxation to a specified maximum value. It includes the elastic rigidity corresponding to the flanges, KfM the bolts, Kb and the spring washers, Kw or the spacers Ks. These rigidities are in series with those of the bolts and the flange as shown in Fig. 1. 2.2. Bolt stiffness The stiffness of the bolts is given by
Kb ¼
Eb Ab ðlb þ la Þ
(1)
where lb is the effective length of the bolt without washers or spacers and la is the additional length with the spacers.
Fig. 1. Bolted flange analytical model.
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732
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2.3. Spacer stiffness The stiffness of the spacers is given by
Ks ¼
Es As 2ls
(2)
2.4. Flange stiffness The axial stiffness of a flange depends on its type [14]. Using the notation of the ASME flange design code, for an integral flange,
KfM ¼
LEf go2 hG V 1 n2f
(3)
For a loose-type flange with hub,
KfM ¼
LEf go2 hG VL 1 n2f
(4)
For a loose-type flange without hub,
KfM ¼
Fig. 2. Belleville springs load deflection relationship.
p 6
Ef t 3 ln K
(5) Ks ¼
nb Es As 2ls
(9)
2.5. Disc springs The paper by Almen and Laszlo [15] forms the basis for the calculations of spring washers used in standards such as DIN 2092. It has been modified in the last few years to include disc springs with contact flats. To calculate the stiffness of a disc spring, one defines the following parameters [16], namely; the diameter ratio of the spring washer:
d¼
De Di
(6)
wfb þ wfw þ 2wff þ wfg ¼ wrb þ wrw þ 2wrf þ wrg
and the coefficient a given by the following equation:
a ¼
1
2
d1 d
(7)
p dþ1 2 d 1 ln d
h ww tw tw
h ww tw 2tw
þ1
(8)
With Nw ¼ nb/2nw for the spring washers placed in series and Nw ¼ nbnw/2 for the spring washers placed in parallel, where ww represents the deflection and tw the thickness of the washers and nb and nw are the numbers of bolts and washers respectively. Fig. 2 shows the relationship between the axial force and the axial deflection for two combinations of spring washers placed in series. The stiffness is defined as the tangent to the curve and can vary with load depending on the size, number and configuration. On tightening the bolts, the disc springs transmit the entire load to the gasket while getting compressed down. When a stacking is used, the design is generally made to take into account a maximum spring washer deflection which represents approximately 75% compression of its initial height.
(10)
Knowing that the gasket total axial deflection wrg is the combination of the mechanical and the creep components, wfg and Dwcg this can be written as follows:
wrg ¼ wfg þ Dwcg
The force applied to the spring washer is given as:
4Ew t 4 ww w Fw ¼ Nw 1 n2w aD2e tw
Knowing that the more the joint is flexible the less the bolt load relaxation is, the axial bolted joint stiffness is made less rigid by the presence of the disc springs. The global axial joint stiffness is calculated by considering the combination in series of the bolts, the flange and the washer rigidities. This can be deduced from the compatibility equation of displacement which considers that the axial distance traveled by the nut remains unchanged as detailed in Refs. [11,14] and presented by the following relationship:
(11)
Substituting Eq. (11) into Eq. (10) gives:
wfb þ wfw þ 2wff ¼ wrb þ wrw þ 2wrf þ Dwcg
(12)
Using the force deflection relationship and applying axial equilibrium as in Ref. [17], Eq. (12) can be rearranged to give:
Fbf Ff Mf Fr Fr Mr p þ b þ 2hG f þ 2hG ¼ b þ b þ 2hG f Kb Kw KfM KfM Kfp Kb Kw p þ 2hG þ Dwcg Kfp
ð13Þ
By expressing the variation of the bending moment acting on the flange as the bolt load multiplied by the radial distance between bolt circle and the reaction point of the gasket, the following Eq. (14) is obtained:
Fbf Fbr hG
2.6. Spacers (sleeves)
Mff Mfr ¼
The total stiffness of the spacers used in pairs for each bolt is calculated as follows:
Substituting Eq. (14) into Eq. (13) and after simplifications and rearrangements gives:
(14)
A.-H. Bouzid, A. Nechache / International Journal of Pressure Vessels and Piping 87 (2010) 730e736
DFb ¼ Fbf Fbr ¼
1 2h2 1 1 þ þ G Kb Kw KfM
!Dwcg
733
(15)
The variation of the bolt load DFb depends on the combination of the rigidities of the flange, the bolts and the spring washers. It is important to note that, the relaxation of bolted flanged joint does not depend on the gasket stiffness [10] but rather of its creep characteristic as well as the other joint member rigidities. During relaxation, the change in bolt load is equal to the change in gasket load and therefore Eq. (15) can be written in the form of
DFg ¼ DFb ¼ Kj Dwcg
(16)
The right term involves the global joint axial stiffness Kj and is given by the following equation:
2h2G 1 1 1 ¼ þ þ Kj KfM Kb Kw
(17) Fig. 3. Relaxation with two different gasket styles in an NPS 3 class 150 slip-on joint.
2.7. Bolted joint with spacers If spacers are used instead of washers, the bolts will be longer and therefore produce a joint less stiff. The new combined stiffness of the system boltespacer may be introduced in Eqs. (15) and (17) in which the bolt stiffness Kb is calculated with a bigger length and Kw is replaced by Ks. The length of the spacers can be obtained by substation of the rigidities such that:
2h2G 1 ¼ þ Kj KfM
1 1 þ Eb Ab E s As ðlb þ 2ls Þ 2ls
circumferential direction, it is possible to model only an angular portion of the joint bounded by two longitudinal planes (Fig. 4) that pass through one bolt and between two adjacent bolts and include half of the bolt and spring washers. The loadings are applied in three steps. The first step consists of applying an initial bolt-up. This is achieved by imposing to the bolt mid-plane nodes, an equivalent axial deflection to produce the target initial bolt-up stress of 275 MPa for the 52-inch flange and 175 MPa for the NPS 4 class 600 flange. An internal pressure that
(18)
or
1 2h2G l b Kj KfM Eb Ab ls ¼ 2 2 þ Eb Ab Es As
(19)
3. Experimental and finite element validation The concept of the global stiffness has been discussed in several papers [10,18]. In addition, tests with several gaskets having different rigidities conducted on a modified HOBT fixture confirmed that the relaxation load is proportional to gasket deflection (gasket thickness loss) and the slope is a constant representing the global stiffness. The test bench on which this experimental finding was observed consists of a pair of NPS 3 class 150 lb WN flanges and is equipped with special transducer to measure both the bolt load and gasket deflection. In Fig. 3, samples of results from two gasket styles of different thicknesses and stiffness show two parallel lines when relaxation takes place as the temperature is increased. To further validate the analytical model, two three-dimensional numerical FE models were built and run on ANSYS [19]. These are two bolted joints; the first one being an NPS 4 cl. 600 welding-neck and a 52-inch heat exchanger flanges used in pairs. The dimensions of the 52-inch flange are in mm (inches): A ¼ 1483 (583/8), B ¼ 1295 (51), C ¼ 1429 (561/4), t ¼ 143 (55/8), go ¼ 16 (5/8), g1 ¼19 (3/4), ho ¼ 32 (11/4). It has 76 bolts of 25.4 mm (1 inch) diameter and is used with a 12.5 mm (1/2 inch) wide gasket of 1349 mm (531/8 inch) outside diameter. Because of symmetry with respect to plane that passes through the gasket mid thickness and the bolts and also the repeated loading acting at an equidistance in the
Fig. 4. 3D FE model with 5 disc springs.
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depends on the case studied is applied in the second step such that a radial pressure is applied to the shell and the flange while an equivalent longitudinal stress is applied to the shell end to simulate the hydrostatic end thrust. Creep of the gasket, the flanges and the bolts is considered in the third step. From these simulations, the relaxations of the bolts and the gasket loads over time are evaluated. The materials selected to run the analysis on these bolted joints are: ASTM A-105 or equivalent for the flange for which E ¼ 210 GPa and n ¼ 0.3 ASTM A-193 B7 for the bolt for which E ¼ 200 GPa and n ¼ 0.3 Two types of gaskets were used in conjunction with these two bolted joints; CMS corrugated metal sheet and CAF Compressed Asbestos Sheet. The non-linear behaviour and creep constants of these gaskets are given in Refs. [9,11]. Due to the limited gasket creep data to short term and for a better assessment of the method, the gasket creep case was run for up to 10,000 s. However, the bolt creep case was run for 10,000 h. It is worth noting that, in general, bolted joints relax extensively during the first few hours of service due to the excessive short term creep of the gasket. However, in the long term, the contribution of the bolt creep becomes significant especially at high temperature.
4. Discussion 4.1. Model validation Figs. 5 and 6 show the bolt stress relaxation due to gasket creep over time of the NPS 4 class 600 and the 52 HE joints. The results show over 36% of load loss in the first case and 51% of load loss in the second case respectively just after 10,000 s of the operating time. The use of five-disc springs on each side of the two flanges per bolt placed in series enabled to maintain 95% of the load. It is to be noted that the results obtained from the analytical model are in good agreement with those obtained by the finite element model. The creep of the flange components made of metal has a similar effect on the bolt load drop. As an example, to illustrate the creep effect of the flange and the bolts, another analysis was run on the 52-inch HE flange with the gasket creep not taking place. Creep constants of the flange and the bolts are given in Ref. [11]. The
Fig. 5. Effect of disc springs in a 52 inch HE flange.
Fig. 6. Effect of disc springs in an NPS 4 class 600 joint.
relaxation due to creep of the bolts and the flanges including the cylinder, the hub and the flange ring is quantified to approximately 45% of the initial bolt stress in this case (Fig. 7). The use of the disc springs decreases relaxation considerably. In effect, the introduction of three spring washers reduced relaxation by 30% whereas five-disc spring placed in series reduces relaxation by 50%. This confirms that the more the bolted joint is flexible, the less the load relaxation is. Once again, it is observed that the relaxation curves obtained from the analytical model are in good agreement with those produced by finite elements. Fig. 8 shows the gasket thickness loss vs time in the 52-inch flange for two different gaskets types. As expected, in both cases the creeperelaxation behaviour is non-linear. In addition, the loss of gasket thickness is smaller with the more flexible joint case (i.e. with the presence of disc springs). It is to be noted that the comparison between the analytical model and the FEA is within acceptable limits. The plots of the bolt load relaxation vs the gasket creep axial deflection are presented in Fig. 9. It is clearly observed
Fig. 7. Effect of the number of disc springs in a 52-inch HE flange.
A.-H. Bouzid, A. Nechache / International Journal of Pressure Vessels and Piping 87 (2010) 730e736
735
that their stiffness combined with that of the longer bolts would give the same global axial joint stiffness as that when disc springs are used. To illustrate the technique of spring washer or spacer selection, a numerical example is presented. The calculations are carried out for the case of the 52-inch heat exchanger joint used with a pair of symmetrical flanges. 4.1.1. Joint without disc springs Kb ¼ 20.07 106 N/mm is calculated using Eq. (1). Using Eq. (3), then KfM/h2G ¼ 28.2 106 N/mm. Using Eq. (17), Kj ¼ 8.28 106 N/mm. From the slope of Fig. 9, Kj ¼ 8.691 106 N/mm which gives a 5% difference. 4.1.2. Joint with disc springs The different rigidities are Kb ¼ 16.49 106 N/mm. Kw ¼ 1.39 106 N/mm. KfM/h2G ¼ 28.2 106 N/mm.
Fig. 8. Gasket thickness loss (relaxation) vs time with and without disc springs in a 52-inch HE joint.
that with both the finite element and the analytical model, the variation of the bolt load is directly proportional to the gasket thickness loss. This is consistent with the experimental observation conducted on the NPS 3-inch flange as shown in Fig. 3. The slopes of these lines represent the global axial joint stiffness. It is to be noted that the stiffness is significantly smaller in the case of the joint with five-spring washers and its reduction is about 87%. In addition, it is noticed that for the same relaxation period, the gasket thickness loss is slightly larger in the case of the joint with disc springs. This is due to the fact that the gasket creeps more because its decreasing load is comparatively bigger at all times. Nevertheless, the load relaxes a lot less in this case than in the case of a joint without spring washers. Another means of reducing the global axial joint assembly stiffness is to increase the length of the bolts using spacer tubes. The spacer dimensions are selected so 8000 Unfilled markers: With disc springs Filled markers : Without disc springs 7500
Using Eq. (17), Kj ¼ 1.18 106 N/mm. From the slope of Fig. 9 Kj ¼ 1.1 106 N/mm leading to an 8% difference. Another analysis could be focused on the load variation for the same gasket axial deflection in the case of the joint assembly with and without spring washers. The results obtained are presented hereafter. Case without disc springs
DFb ¼ (7.22 4.76) 106 ¼ 2.46 106 N is determined from Fig. 9. Case with disc springs
DFb ¼ (7.37 6.99) 106 ¼ 0.38 106 N By calculation, one can compare the reduction of the global axial joint assembly stiffness and the reduction of the bolt load variation. These are 87% and 85% respectively. These differences are due to the lever arm hG which varies according to the position of gasket load reaction location that is taken in the middle of the gasket width for the calculated value. The gasket load reaction location point depends on the radial distribution of the gasket compressive stress which changes during creep and consequently produces a small change in the flange axial stiffness.
7000
Bolt load, kN
4.2. Calculation of required disc springs and tube spacers 6500
To reduce the load relaxation by two third in the case of 52-inch flanges, one must determine the numbers of disc springs or the spacer dimensions which can be placed in series with the bolts. An illustrative calculation example is presented hereafter.
6000
5500
5000
CAF gasket creep CMS gasket creep
4500
4000 0.10
Dotted lines : Analytical model Solid lines : FE model
0.20
0.30
0.40
0.50
0.60
0.70
0.80
Gasket displacement, mm Fig. 9. Bolt load vs gasket thickness loss (relaxation) with and without disc springs in a 52-inch HE joint.
4.2.1. Joint with disc springs Since the variation of the bolt and gasket load relaxation is directly proportional to the global axial joint stiffness, to reduce relaxation by 2/3, the global axial joint stiffness should be reduced by the same amount. Knowing that without springs washers Kj ¼ 8.27 106 N/mm and therefore 8.27 106 2/3 ¼ 2.76 106 N/mm and, Kb ¼ 17.7 106 N/mm. Kf/h2G ¼ 28.2 106 N/mm. Kw ¼ 4.29 106 N/mm, which corresponds to the use of 6 springs washer of type 1 (heavy duty) per bolt placed in series.
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Each spring washer has a stiffness of 0.34 106 N/mm and can carry more than 118 kN at 75% of its maximum deflection or dish. 4.2.2. Joint with tube spacers When spacers are used, the calculation is carried out with Eq. (10) which determines the length of the spacer tubes in order to obtain the same global axial joint stiffness 2/3 smaller as with six disc springs placed in series. With Eb ¼ Es ¼ 205 GPa, Kj ¼ 2.76 106 N/mm and Kf/h2G ¼ 28.2 106 N/mm, the length of the spacer tube is 330 mm. For cases where space is limited, the disc springs might be suitable. However, one may be cautious as to the problems associated with disc springs and in particular corrosion cracking [20], nevertheless, they may be more suitable compared to the spacers which require a very long length.
5. Conclusion This paper illustrates the influence of the global axial joint assembly stiffness on the bolt and gasket load relaxation. It was shown that its value depends on the stiffness of the flanges, the bolts and the disc springs or the spacer tubes. The more rigid the joint is the more relaxation takes place. The developed model and concept were validated and verified using FEA. One of the practical means of reducing relaxation is the use of disc springs. An accurate method to size spring washers and their numbers based on a maximum tolerated relaxation has been developed and validated. An illustrative example on a 52-inch heat exchanger joint showed that spring washers can be more practical than spacer tubes although the latter could be used in some applications.
References [1] ASME Boiler and Pressure Vessel Code. Section VIII, Division 2, Appendix 2. Rules for Bolted Flange Connections with Ring Type Gaskets; 2001.
[2] Waters EO, Rossheim DB, Wesstrom DB, Williams FSG. Formulas for stresses in bolted flanged connections. Transactions of the ASME 1937;59: 161e9. [3] Waters EO, Rossheim DB, Wesstrom DB, Williams FSG. Development of general formulas for bolted flanges. Chicago, Illinois: Taylor Forge and Pipe Works; 1949. [4] Payne JR. PVRC Flanged Joint User’s Survey. Bulletin no. 306. Welding Research Council; 1985. [5] Bouzid A, Nechache A. Thermally induced deflections in bolted flanged connections. ASME Journal of Pressure Vessel Technology 2005;127:394e401. [6] Bouzid A, Nechache A. An analytical solution for evaluating gasket stress change in bolted flange connections subjected to high temperature loading. ASME Journal of Pressure Vessel Technology 2005;127:414e27. [7] Brown W, Derenne M, Bouzid A. Determination of gasket stress levels during thermal transients. Proceedings of the 2002 ASME-PVP conference, PVP-Vol. 433, Analysis of Bolted Joints, Paper No PVP2002-1078, Vancouver, Canada; 2002. p. 21e28. [8] Brown W. The effect of thermal transients on flange sealing. Ph.D. thesis. Ecole Polytechnique of Montreal; 2001. [9] Bazergui A. Short term creep and relaxation behavior of gaskets. Welding Research Council Bulletin no. 294. Welding Research Council; 2001. p. 9e22. [10] Bouzid A, Chaaban A. An accurate method for evaluating relaxation in bolted flanged connections. ASME Journal of Pressure Vessel Technology 1997; 119:10e7. [11] Nechache A, Bouzid A. Creep analysis of bolted flange joints. International Journal of Pressure Vessel and Piping 2007;84(no. 3):185e94. [12] Nechache A, Bouzid A. The effect of cylinder and hub creep on the load relaxation in bolted flanged joints. ASME Journal of Pressure Vessel Technology 2008;130(3). 031211 (9 pages). [13] Nechache A, Bouzid A. On the use of plate theory to evaluate the load relaxation in bolted flanged joints subjected to creep. International Journal of Pressure Vessel and Piping 2008;84(7):486e97. [14] Bouzid A, Beghoul H. The design of flanges based on flexibility and tightness, Proceedings of the 2003 ASME-PVP conference. Analysis of Bolted Joints, Paper No PVP2003-1870, Cleveland, Ohio; 2003. p. 31e38. [15] Almen JO, Laszlo A. The uniform-section disc spring. Transactions of ASME 1936;58:305e14. [16] Bickford J, Nassar S, editors. Handbook of bolts and bolted joints. Marcel Dekker Inc; 1998 [chapter 14: Belleville Springs by Davet GP]. [17] Bouzid A, Chaaban A. Flanged joints analysis: a simplified method based on elastic interaction. CSME Transactions 1993;17(No 2):181e96. [18] Bouzid A, Chaaban A, Bazergui A. The effect of creep relaxation on the leakage tightness of bolted flanged joints. ASME Journal of Pressure Vessel Technology 1995;117:71e8. [19] ANSYS. Standard manual, version 7.1. ANSYS Inc; 2001. [20] Atxaga G, Pelayo A, Irisarri AM. Failure analysis of a set of stainless steel disc springs. Engineering Failure Analysis 2006;13:226e34.
Contents lists available at ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
The modelling of bolted flange joints used with disc springs and tube spacers to reduce relaxation Abdel-Hakim Bouzid a, *, Akli Nechache b a b
Ecole de Technologie Superieure, 1100 Notre-Dame Ouest, Montreal, Quebec, Canada H3C 1K3 SNC-Lavalin, 18, rue Mustapha Khalef, 16306 Ben Aknoun, Algérie
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 August 2009 Received in revised form 16 September 2010 Accepted 28 September 2010
Bolted flange joints are prone to leakage when exposed to high temperature. In several cases, the root cause is relaxation that takes place as a result of material creep of the gasket, the bolt and the flange. One way to overcome this problem is to make the joint less stiff by introducing disc springs or the use of longer bolts with spacers. Although widely used, these two methods have no reliable analytical model that could be used to evaluate the exact number of washers or length of the bolts required to reduce relaxation to a minimum acceptable level. This paper describes an analytical model based on the flexibility and deflection interactions of the joint different elements including the axial stiffness of the flange and bolts, used to evaluate relaxation. The developed analytical flange model can accommodate either disc springs or longer bolts with spacer tubes to reduce the bolt load loss to a maximum acceptable value. This model is validated by comparison with the more accurate FEA findings. Calculation examples on a bolted flanged joint are presented to illustrate the suggested analytical calculation procedure. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Bolted flange joints Relaxation Disc springs Tube washers Analytical modelling Joint stiffness
1. Introduction Bolted gasketed flange joints are widely used in the chemical, petrochemical and nuclear power plants. A lot of research has been conducted in this area to tackle the several issues related to the actual ASME flange design code procedure [1] that is based on the early work of Waters et al. [2] and is known as Taylor Forge method [3]. Over the years, under the auspices of the Pressure Vessels Research Council (PVRC), a fairly large amount of research related to high temperature leakage in bolted joints was conducted as a result of Payne’s survey in 1985 [4]. The principal cause of leakage encountered in the power industries is related to high temperature operating bolted joints. It was found that the most frequently observed cause of leaks was due to loose bolts. This is essentially due to the creeperelaxation phenomenon that takes effect at high temperature although other effects such as gasket degradation and thermal expansion difference can be present. The load redistributions as a result of thermal transients and steady state were tackled both analytically and numerically in Refs. [5e8].
* Corresponding author. Tel.: þ1 514 396 8563; fax: þ1 514 396 8530. E-mail addresses: [email protected] (A.-H. Bouzid), akli.nechache@ snclavalin.com (A. Nechache). 0308-0161/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2010.09.001
In high temperature application, material creep has a significant effect on the relaxation of a joint. Gasket and bolt are the joint members that give the largest bolt load relaxation ratio over time. Bazergui [9] has observed significant creep taking place at room temperature over a short period of time with some gaskets. Few researchers treated the analytical modeling of relaxation in bolted joints. The most recent studies are of Bouzid et al. [10] who presented an analytical approach to treat relaxation and estimate the effect of gasket creep on the bolt and gasket load loss. Recently, Nechache et al. [11e13] extended the approach by considering the bolt, the flange ring, the hub and the cylinder creep. It was found that more than 50% of bolt load loss after 10,000 h of operating time was frequent in high temperature joints. It was concluded that the gasket and bolt creep are the parameters that affect more the loss of gasket seating stress and bolt preload and their effect is severe with time and increasing temperature. To maintain bolt load over time, disc springs also known as Belleville springs and tube spacers are often utilized. Their use makes the joint more flexible; a design parameter that can reduce the bolt load relaxation due to creep to within acceptable values. The dimensions and quantity used for a specific application are based on experience and approximate calculations. Engineers have no accurate calculation procedure to follow when disc springs are required. This paper describes an analytical procedure based on
A.-H. Bouzid, A. Nechache / International Journal of Pressure Vessels and Piping 87 (2010) 730e736
Nomenclature
a d n D qf Ab As A B C Di De E Fb Fg go g1 G h ho hG K Kb KfM KfP Kg Kj Kw Ks lb
washer constant washer outside to inside diameter (mm) Poisson’s ratio change in parameter flange rotation (rad) total bolt area (mm2) total spacer tube area (mm2) outer diameter of flange (mm) inner diameter of flange (mm) bolt circle diameter (mm) internal diameter of washer (mm) external diameter of washer (mm) Young’s modulus (MPa) bolt force (N) gasket force (N) thickness of cylinder (mm) thickness of hub big end (mm) gasket reaction force diameter (mm) unloaded height of washer truncated cone (mm) hub length (mm) flange lever arm (mm) flange inside to outside diameter ratio bolt uniaxial stiffness (N/mm) flange rotational stiffness due to moment (N mm) flange rotational stiffness due to pressure (N/mm2) gasket uniaxial stiffness (N/mm) global joint uniaxial stiffness (N/mm) total washer uniaxial stiffness (N/mm) total spacer uniaxial stiffness (N/mm) bolt effective length (mm)
ls L Mf nb p tw t V VL w
spacer tube height (mm) ASME code factor flange moment per unit circumference (mm.N/mm) number of bolts internal pressure (MPa) thickness of washer (mm) thickness of flange (mm) ASME code factor for integral type flanges ASME code factor for loose-type flanges axial deflection of the joint element (mm)
Superscript c refers to creep f refers to final pressurization state r refers to relaxation state Subscript b c w s f g
refers refers refers refers refers refers
to bolt to creep to washer to spacer tube to flange to gasket
Acronyms ASME American Society of Mechanical Engineers BPVC Boiler and Pressure Vessel Code CAF Compressed Fiber CMS Corrugated Metal Sheet FEM Finite Element Method HOBT Hot Blow-Out Test
a flexibility model which leads to an accurate design of the dimensions of bolt spacers and number of the disc springs required in a specific case based on the maximum tolerated relaxation. Calculation examples on various cases of a symmetrical flanged joint are presented to illustrate the suggested analytical approach. The results obtained analytically are compared to those provided by FE modeling. 2. Analytical model 2.1. Bolted joint with spring washers Fig. 1 shows the analytical model used to estimate the required spring washers or spacers to reduce relaxation to a specified maximum value. It includes the elastic rigidity corresponding to the flanges, KfM the bolts, Kb and the spring washers, Kw or the spacers Ks. These rigidities are in series with those of the bolts and the flange as shown in Fig. 1. 2.2. Bolt stiffness The stiffness of the bolts is given by
Kb ¼
Eb Ab ðlb þ la Þ
(1)
where lb is the effective length of the bolt without washers or spacers and la is the additional length with the spacers.
Fig. 1. Bolted flange analytical model.
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2.3. Spacer stiffness The stiffness of the spacers is given by
Ks ¼
Es As 2ls
(2)
2.4. Flange stiffness The axial stiffness of a flange depends on its type [14]. Using the notation of the ASME flange design code, for an integral flange,
KfM ¼
LEf go2 hG V 1 n2f
(3)
For a loose-type flange with hub,
KfM ¼
LEf go2 hG VL 1 n2f
(4)
For a loose-type flange without hub,
KfM ¼
Fig. 2. Belleville springs load deflection relationship.
p 6
Ef t 3 ln K
(5) Ks ¼
nb Es As 2ls
(9)
2.5. Disc springs The paper by Almen and Laszlo [15] forms the basis for the calculations of spring washers used in standards such as DIN 2092. It has been modified in the last few years to include disc springs with contact flats. To calculate the stiffness of a disc spring, one defines the following parameters [16], namely; the diameter ratio of the spring washer:
d¼
De Di
(6)
wfb þ wfw þ 2wff þ wfg ¼ wrb þ wrw þ 2wrf þ wrg
and the coefficient a given by the following equation:
a ¼
1
2
d1 d
(7)
p dþ1 2 d 1 ln d
h ww tw tw
h ww tw 2tw
þ1
(8)
With Nw ¼ nb/2nw for the spring washers placed in series and Nw ¼ nbnw/2 for the spring washers placed in parallel, where ww represents the deflection and tw the thickness of the washers and nb and nw are the numbers of bolts and washers respectively. Fig. 2 shows the relationship between the axial force and the axial deflection for two combinations of spring washers placed in series. The stiffness is defined as the tangent to the curve and can vary with load depending on the size, number and configuration. On tightening the bolts, the disc springs transmit the entire load to the gasket while getting compressed down. When a stacking is used, the design is generally made to take into account a maximum spring washer deflection which represents approximately 75% compression of its initial height.
(10)
Knowing that the gasket total axial deflection wrg is the combination of the mechanical and the creep components, wfg and Dwcg this can be written as follows:
wrg ¼ wfg þ Dwcg
The force applied to the spring washer is given as:
4Ew t 4 ww w Fw ¼ Nw 1 n2w aD2e tw
Knowing that the more the joint is flexible the less the bolt load relaxation is, the axial bolted joint stiffness is made less rigid by the presence of the disc springs. The global axial joint stiffness is calculated by considering the combination in series of the bolts, the flange and the washer rigidities. This can be deduced from the compatibility equation of displacement which considers that the axial distance traveled by the nut remains unchanged as detailed in Refs. [11,14] and presented by the following relationship:
(11)
Substituting Eq. (11) into Eq. (10) gives:
wfb þ wfw þ 2wff ¼ wrb þ wrw þ 2wrf þ Dwcg
(12)
Using the force deflection relationship and applying axial equilibrium as in Ref. [17], Eq. (12) can be rearranged to give:
Fbf Ff Mf Fr Fr Mr p þ b þ 2hG f þ 2hG ¼ b þ b þ 2hG f Kb Kw KfM KfM Kfp Kb Kw p þ 2hG þ Dwcg Kfp
ð13Þ
By expressing the variation of the bending moment acting on the flange as the bolt load multiplied by the radial distance between bolt circle and the reaction point of the gasket, the following Eq. (14) is obtained:
Fbf Fbr hG
2.6. Spacers (sleeves)
Mff Mfr ¼
The total stiffness of the spacers used in pairs for each bolt is calculated as follows:
Substituting Eq. (14) into Eq. (13) and after simplifications and rearrangements gives:
(14)
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DFb ¼ Fbf Fbr ¼
1 2h2 1 1 þ þ G Kb Kw KfM
!Dwcg
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(15)
The variation of the bolt load DFb depends on the combination of the rigidities of the flange, the bolts and the spring washers. It is important to note that, the relaxation of bolted flanged joint does not depend on the gasket stiffness [10] but rather of its creep characteristic as well as the other joint member rigidities. During relaxation, the change in bolt load is equal to the change in gasket load and therefore Eq. (15) can be written in the form of
DFg ¼ DFb ¼ Kj Dwcg
(16)
The right term involves the global joint axial stiffness Kj and is given by the following equation:
2h2G 1 1 1 ¼ þ þ Kj KfM Kb Kw
(17) Fig. 3. Relaxation with two different gasket styles in an NPS 3 class 150 slip-on joint.
2.7. Bolted joint with spacers If spacers are used instead of washers, the bolts will be longer and therefore produce a joint less stiff. The new combined stiffness of the system boltespacer may be introduced in Eqs. (15) and (17) in which the bolt stiffness Kb is calculated with a bigger length and Kw is replaced by Ks. The length of the spacers can be obtained by substation of the rigidities such that:
2h2G 1 ¼ þ Kj KfM
1 1 þ Eb Ab E s As ðlb þ 2ls Þ 2ls
circumferential direction, it is possible to model only an angular portion of the joint bounded by two longitudinal planes (Fig. 4) that pass through one bolt and between two adjacent bolts and include half of the bolt and spring washers. The loadings are applied in three steps. The first step consists of applying an initial bolt-up. This is achieved by imposing to the bolt mid-plane nodes, an equivalent axial deflection to produce the target initial bolt-up stress of 275 MPa for the 52-inch flange and 175 MPa for the NPS 4 class 600 flange. An internal pressure that
(18)
or
1 2h2G l b Kj KfM Eb Ab ls ¼ 2 2 þ Eb Ab Es As
(19)
3. Experimental and finite element validation The concept of the global stiffness has been discussed in several papers [10,18]. In addition, tests with several gaskets having different rigidities conducted on a modified HOBT fixture confirmed that the relaxation load is proportional to gasket deflection (gasket thickness loss) and the slope is a constant representing the global stiffness. The test bench on which this experimental finding was observed consists of a pair of NPS 3 class 150 lb WN flanges and is equipped with special transducer to measure both the bolt load and gasket deflection. In Fig. 3, samples of results from two gasket styles of different thicknesses and stiffness show two parallel lines when relaxation takes place as the temperature is increased. To further validate the analytical model, two three-dimensional numerical FE models were built and run on ANSYS [19]. These are two bolted joints; the first one being an NPS 4 cl. 600 welding-neck and a 52-inch heat exchanger flanges used in pairs. The dimensions of the 52-inch flange are in mm (inches): A ¼ 1483 (583/8), B ¼ 1295 (51), C ¼ 1429 (561/4), t ¼ 143 (55/8), go ¼ 16 (5/8), g1 ¼19 (3/4), ho ¼ 32 (11/4). It has 76 bolts of 25.4 mm (1 inch) diameter and is used with a 12.5 mm (1/2 inch) wide gasket of 1349 mm (531/8 inch) outside diameter. Because of symmetry with respect to plane that passes through the gasket mid thickness and the bolts and also the repeated loading acting at an equidistance in the
Fig. 4. 3D FE model with 5 disc springs.
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depends on the case studied is applied in the second step such that a radial pressure is applied to the shell and the flange while an equivalent longitudinal stress is applied to the shell end to simulate the hydrostatic end thrust. Creep of the gasket, the flanges and the bolts is considered in the third step. From these simulations, the relaxations of the bolts and the gasket loads over time are evaluated. The materials selected to run the analysis on these bolted joints are: ASTM A-105 or equivalent for the flange for which E ¼ 210 GPa and n ¼ 0.3 ASTM A-193 B7 for the bolt for which E ¼ 200 GPa and n ¼ 0.3 Two types of gaskets were used in conjunction with these two bolted joints; CMS corrugated metal sheet and CAF Compressed Asbestos Sheet. The non-linear behaviour and creep constants of these gaskets are given in Refs. [9,11]. Due to the limited gasket creep data to short term and for a better assessment of the method, the gasket creep case was run for up to 10,000 s. However, the bolt creep case was run for 10,000 h. It is worth noting that, in general, bolted joints relax extensively during the first few hours of service due to the excessive short term creep of the gasket. However, in the long term, the contribution of the bolt creep becomes significant especially at high temperature.
4. Discussion 4.1. Model validation Figs. 5 and 6 show the bolt stress relaxation due to gasket creep over time of the NPS 4 class 600 and the 52 HE joints. The results show over 36% of load loss in the first case and 51% of load loss in the second case respectively just after 10,000 s of the operating time. The use of five-disc springs on each side of the two flanges per bolt placed in series enabled to maintain 95% of the load. It is to be noted that the results obtained from the analytical model are in good agreement with those obtained by the finite element model. The creep of the flange components made of metal has a similar effect on the bolt load drop. As an example, to illustrate the creep effect of the flange and the bolts, another analysis was run on the 52-inch HE flange with the gasket creep not taking place. Creep constants of the flange and the bolts are given in Ref. [11]. The
Fig. 5. Effect of disc springs in a 52 inch HE flange.
Fig. 6. Effect of disc springs in an NPS 4 class 600 joint.
relaxation due to creep of the bolts and the flanges including the cylinder, the hub and the flange ring is quantified to approximately 45% of the initial bolt stress in this case (Fig. 7). The use of the disc springs decreases relaxation considerably. In effect, the introduction of three spring washers reduced relaxation by 30% whereas five-disc spring placed in series reduces relaxation by 50%. This confirms that the more the bolted joint is flexible, the less the load relaxation is. Once again, it is observed that the relaxation curves obtained from the analytical model are in good agreement with those produced by finite elements. Fig. 8 shows the gasket thickness loss vs time in the 52-inch flange for two different gaskets types. As expected, in both cases the creeperelaxation behaviour is non-linear. In addition, the loss of gasket thickness is smaller with the more flexible joint case (i.e. with the presence of disc springs). It is to be noted that the comparison between the analytical model and the FEA is within acceptable limits. The plots of the bolt load relaxation vs the gasket creep axial deflection are presented in Fig. 9. It is clearly observed
Fig. 7. Effect of the number of disc springs in a 52-inch HE flange.
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that their stiffness combined with that of the longer bolts would give the same global axial joint stiffness as that when disc springs are used. To illustrate the technique of spring washer or spacer selection, a numerical example is presented. The calculations are carried out for the case of the 52-inch heat exchanger joint used with a pair of symmetrical flanges. 4.1.1. Joint without disc springs Kb ¼ 20.07 106 N/mm is calculated using Eq. (1). Using Eq. (3), then KfM/h2G ¼ 28.2 106 N/mm. Using Eq. (17), Kj ¼ 8.28 106 N/mm. From the slope of Fig. 9, Kj ¼ 8.691 106 N/mm which gives a 5% difference. 4.1.2. Joint with disc springs The different rigidities are Kb ¼ 16.49 106 N/mm. Kw ¼ 1.39 106 N/mm. KfM/h2G ¼ 28.2 106 N/mm.
Fig. 8. Gasket thickness loss (relaxation) vs time with and without disc springs in a 52-inch HE joint.
that with both the finite element and the analytical model, the variation of the bolt load is directly proportional to the gasket thickness loss. This is consistent with the experimental observation conducted on the NPS 3-inch flange as shown in Fig. 3. The slopes of these lines represent the global axial joint stiffness. It is to be noted that the stiffness is significantly smaller in the case of the joint with five-spring washers and its reduction is about 87%. In addition, it is noticed that for the same relaxation period, the gasket thickness loss is slightly larger in the case of the joint with disc springs. This is due to the fact that the gasket creeps more because its decreasing load is comparatively bigger at all times. Nevertheless, the load relaxes a lot less in this case than in the case of a joint without spring washers. Another means of reducing the global axial joint assembly stiffness is to increase the length of the bolts using spacer tubes. The spacer dimensions are selected so 8000 Unfilled markers: With disc springs Filled markers : Without disc springs 7500
Using Eq. (17), Kj ¼ 1.18 106 N/mm. From the slope of Fig. 9 Kj ¼ 1.1 106 N/mm leading to an 8% difference. Another analysis could be focused on the load variation for the same gasket axial deflection in the case of the joint assembly with and without spring washers. The results obtained are presented hereafter. Case without disc springs
DFb ¼ (7.22 4.76) 106 ¼ 2.46 106 N is determined from Fig. 9. Case with disc springs
DFb ¼ (7.37 6.99) 106 ¼ 0.38 106 N By calculation, one can compare the reduction of the global axial joint assembly stiffness and the reduction of the bolt load variation. These are 87% and 85% respectively. These differences are due to the lever arm hG which varies according to the position of gasket load reaction location that is taken in the middle of the gasket width for the calculated value. The gasket load reaction location point depends on the radial distribution of the gasket compressive stress which changes during creep and consequently produces a small change in the flange axial stiffness.
7000
Bolt load, kN
4.2. Calculation of required disc springs and tube spacers 6500
To reduce the load relaxation by two third in the case of 52-inch flanges, one must determine the numbers of disc springs or the spacer dimensions which can be placed in series with the bolts. An illustrative calculation example is presented hereafter.
6000
5500
5000
CAF gasket creep CMS gasket creep
4500
4000 0.10
Dotted lines : Analytical model Solid lines : FE model
0.20
0.30
0.40
0.50
0.60
0.70
0.80
Gasket displacement, mm Fig. 9. Bolt load vs gasket thickness loss (relaxation) with and without disc springs in a 52-inch HE joint.
4.2.1. Joint with disc springs Since the variation of the bolt and gasket load relaxation is directly proportional to the global axial joint stiffness, to reduce relaxation by 2/3, the global axial joint stiffness should be reduced by the same amount. Knowing that without springs washers Kj ¼ 8.27 106 N/mm and therefore 8.27 106 2/3 ¼ 2.76 106 N/mm and, Kb ¼ 17.7 106 N/mm. Kf/h2G ¼ 28.2 106 N/mm. Kw ¼ 4.29 106 N/mm, which corresponds to the use of 6 springs washer of type 1 (heavy duty) per bolt placed in series.
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Each spring washer has a stiffness of 0.34 106 N/mm and can carry more than 118 kN at 75% of its maximum deflection or dish. 4.2.2. Joint with tube spacers When spacers are used, the calculation is carried out with Eq. (10) which determines the length of the spacer tubes in order to obtain the same global axial joint stiffness 2/3 smaller as with six disc springs placed in series. With Eb ¼ Es ¼ 205 GPa, Kj ¼ 2.76 106 N/mm and Kf/h2G ¼ 28.2 106 N/mm, the length of the spacer tube is 330 mm. For cases where space is limited, the disc springs might be suitable. However, one may be cautious as to the problems associated with disc springs and in particular corrosion cracking [20], nevertheless, they may be more suitable compared to the spacers which require a very long length.
5. Conclusion This paper illustrates the influence of the global axial joint assembly stiffness on the bolt and gasket load relaxation. It was shown that its value depends on the stiffness of the flanges, the bolts and the disc springs or the spacer tubes. The more rigid the joint is the more relaxation takes place. The developed model and concept were validated and verified using FEA. One of the practical means of reducing relaxation is the use of disc springs. An accurate method to size spring washers and their numbers based on a maximum tolerated relaxation has been developed and validated. An illustrative example on a 52-inch heat exchanger joint showed that spring washers can be more practical than spacer tubes although the latter could be used in some applications.
References [1] ASME Boiler and Pressure Vessel Code. Section VIII, Division 2, Appendix 2. Rules for Bolted Flange Connections with Ring Type Gaskets; 2001.
[2] Waters EO, Rossheim DB, Wesstrom DB, Williams FSG. Formulas for stresses in bolted flanged connections. Transactions of the ASME 1937;59: 161e9. [3] Waters EO, Rossheim DB, Wesstrom DB, Williams FSG. Development of general formulas for bolted flanges. Chicago, Illinois: Taylor Forge and Pipe Works; 1949. [4] Payne JR. PVRC Flanged Joint User’s Survey. Bulletin no. 306. Welding Research Council; 1985. [5] Bouzid A, Nechache A. Thermally induced deflections in bolted flanged connections. ASME Journal of Pressure Vessel Technology 2005;127:394e401. [6] Bouzid A, Nechache A. An analytical solution for evaluating gasket stress change in bolted flange connections subjected to high temperature loading. ASME Journal of Pressure Vessel Technology 2005;127:414e27. [7] Brown W, Derenne M, Bouzid A. Determination of gasket stress levels during thermal transients. Proceedings of the 2002 ASME-PVP conference, PVP-Vol. 433, Analysis of Bolted Joints, Paper No PVP2002-1078, Vancouver, Canada; 2002. p. 21e28. [8] Brown W. The effect of thermal transients on flange sealing. Ph.D. thesis. Ecole Polytechnique of Montreal; 2001. [9] Bazergui A. Short term creep and relaxation behavior of gaskets. Welding Research Council Bulletin no. 294. Welding Research Council; 2001. p. 9e22. [10] Bouzid A, Chaaban A. An accurate method for evaluating relaxation in bolted flanged connections. ASME Journal of Pressure Vessel Technology 1997; 119:10e7. [11] Nechache A, Bouzid A. Creep analysis of bolted flange joints. International Journal of Pressure Vessel and Piping 2007;84(no. 3):185e94. [12] Nechache A, Bouzid A. The effect of cylinder and hub creep on the load relaxation in bolted flanged joints. ASME Journal of Pressure Vessel Technology 2008;130(3). 031211 (9 pages). [13] Nechache A, Bouzid A. On the use of plate theory to evaluate the load relaxation in bolted flanged joints subjected to creep. International Journal of Pressure Vessel and Piping 2008;84(7):486e97. [14] Bouzid A, Beghoul H. The design of flanges based on flexibility and tightness, Proceedings of the 2003 ASME-PVP conference. Analysis of Bolted Joints, Paper No PVP2003-1870, Cleveland, Ohio; 2003. p. 31e38. [15] Almen JO, Laszlo A. The uniform-section disc spring. Transactions of ASME 1936;58:305e14. [16] Bickford J, Nassar S, editors. Handbook of bolts and bolted joints. Marcel Dekker Inc; 1998 [chapter 14: Belleville Springs by Davet GP]. [17] Bouzid A, Chaaban A. Flanged joints analysis: a simplified method based on elastic interaction. CSME Transactions 1993;17(No 2):181e96. [18] Bouzid A, Chaaban A, Bazergui A. The effect of creep relaxation on the leakage tightness of bolted flanged joints. ASME Journal of Pressure Vessel Technology 1995;117:71e8. [19] ANSYS. Standard manual, version 7.1. ANSYS Inc; 2001. [20] Atxaga G, Pelayo A, Irisarri AM. Failure analysis of a set of stainless steel disc springs. Engineering Failure Analysis 2006;13:226e34.