International Journal of Pressure Vessels and Piping 188 (2020) 104247 Contents lists available at ScienceDirect International Journal of Pressure Vessels and Piping journal homepage: http://www.elsevier.com/locate/ijpvp Analytical modeling of FRP hubbed flanges Ali Khazraiyan Vafadar, Abdel-Hakim Bouzid *, Anh Dung Ngô Ecole de Technologie Superieure, 1100 Notre-Dame Ouest, Montreal (Quebec), Canada, H3C 1K3 A R T I C L E I N F O A B S T R A C T Keywords: Analytical modelling Composite flanges FRP flanges Structural integrity Experimentation Stress analysis In spite of the increasing use of FRP composites in bolted flange joints and the good knowledge of these structures and their material behavior the procedure used for their design is that of metallic flanges. There is a major concern to appropriately address the anisotropic behavior of composite materials in FRP flange design. There­ fore, it is necessary to revisit this procedure by implementing a more suitable model to analyse the stresses and strains of FRP bolted flange joints. This paper presents an analytical model that treats FRP bolted flange joints integrity and tightness based on anisotropy and a flexibility analysis of all joint elements including the gasket, bolts and flanges. The theories that treats all parts of the flange are described in details. Finally, the developed analytical model is validated with experimental testing and numerical modeling of different size FRP flanges. 1. Introduction bolted joint can only be expected if all the system components work together in harmony. To understand the behavior of fiber reinforced plastic bolted flange joints, one of the early work was conducted by Sun in his thesis project [4–6]. The author focused on the integrity of FRP flanges by proposing an analytical study validated numerically using finite element modeling (FEM). But it turns out that the analytical part was not reliable due to some errors in his analytical flange ring model. The numerical model was based on anisotropy material as the laminated shell theory was not yet incorporated in the available software to allow a better handling of the problem in 1995. Blach [7] used this model to propose a method for a stress analysis of fiber reinforced plastic flanged connections with full face gaskets. They used a few assumptions to simplify their proposed method. Although they used a rigorous mathematical analysis to describe all the complexity of the laminate materials, the physical con­ straints related to the presence of the hub and the resulting two junctions with the ring and shell was not considered in the analysis. Kurz and Roos [8] studied the behavior of GRPF bolted flange joints, analytically and experimentally. In their study, they divided the flange into several parts each considered as a beam clamped at one end and subjected to a bending moment at the other end. Consequently, their model only il­ lustrates the effective portion of the flange subjected to a virtual radial stress neglecting the stress in the tangential direction. The analytical model that Estrada and Parsons proposed in their study, considers the circumferential stresses and the rotation of the FRP stub flange. The results obtained from FE models and experimental tests show a good Fiber reinforced plastic composites have recently experienced a spectacular development in the areas of pressure vessels and piping. They are used in applications ranging from water and gas services to nuclear and petrochemical industries. Their special properties such as weight, chemical resistance, resistance to fatigue, low maintenance cost, and even aesthetics have let these composites take precedence over traditional materials [1,2]. Because of the ability to resist corrosive environment, they are increasingly used in the chemical and petro­ chemical process plants. They are a good alternative to nickel (Ni) molybdenum (Mo) chromium (Cr) alloys at low temperatures [3]. Fluid handling and transportation require safe bolted flange assemblies and pressurized equipment. Indeed, the main purpose of a bolted flange joint is to ensure the containment fluid and thus protect the immediate environment against contamination from leakage of harmful fluids or fluid escapes that are nauseating, toxic, and dangerous. Leakage may result in a reduction of the efficiency of the installation and cause haz­ ards to the workers or the public. In addition, the none-compliance to the strict regulations on fugitive emission, the cost in the loss of revenue and repair can be substantial. Therefore, the proper functioning of bolted flange joints in a particular process is an important responsibility of the user and the designer, as they must ensure compliance with the laws, regulations and standards. The performance of a bolted flange joint depends on its various components that it is composed of namely the bolt gasket and the flanges. The satisfactory performance of the * Corresponding author. E-mail addresses: Ali.khazraiyanvafadar.1@ens.etsmtl.ca (A.K. Vafadar), hakim.bouzid@etsmtl.ca (A.-H. Bouzid), AnhDung.Ngo@etsmtl.ca (A.D. Ngô). https://doi.org/10.1016/j.ijpvp.2020.104247 Received 21 June 2019; Received in revised form 21 August 2020; Accepted 28 October 2020 Available online 19 November 2020 0308-0161/© 2020 Elsevier Ltd. All rights reserved. A.K. Vafadar et al. International Journal of Pressure Vessels and Piping 188 (2020) 104247 agreement with the analytical results [9]. Campbell proposed a new technique to fabricate FRP flanges by adopting a one-piece integral flange [10]. His design was based on the metal flange counterpart given by ASME B16.5. In spite of the fact that a simple design method of FRP flanges with full face gaskets is suggested in Ref. [11] the flange, bolts and gasket flexibility as well as the elastic interaction between them are neglected [12,13]. It is worth noting that in the ASME code designs of FRP flanges and metal flanges, the effect of temperature on the redis­ tribution of load and stresses is not addressed [14]. An overview of all researches done on FRP bolted flanged joints, shows that most of the developed models are based on metallic flange design considerations. In order to achieve accuracy in evaluating the stresses in FRP flanges, it is necessary to adopt reliable design models that describe the true material behavior and in particular the anisotropic properties of the composite flanges [15]. Finally, it is worth noting that very few papers discuss the leak rate performance of FRP bolted flange joints [16]. This study concentrates mainly on the development of an analytical model for fiber reinforced plastic bolted flanged joints based on lami­ nation theory. The latter is applied to composite materials to account for the anisotropic properties such as those encountered in FRP flanges. The developed analytical model is based on the force and moment equilib­ rium, the compatibility of the radial displacements and the rotations of the flange different elements at their junctions as well as the axial compatibility necessary to solve the statically undeterminate problem to give the stresses at the inside and outside surfaces of the flange. The proposed analytical model is validated by comparison with experi­ mental data and finite element modeling conducted on NPS 3 and 12 class 150 FRP hubbed flanges. 2. Analytical model In developing the analytical model of FRP bolted flange joints, one of the objectives was that the approach follows the ASME code design philosophy [3]. Therefore, the methodology is based on the two important factors: structural integrity and leakage tightness. Although the paper focus on structural integrity, the leakage verification can easily be achieved by conducting experimental leak tests such as those specified by ASTM F2836 or EN13555 that link the gasket load and pressure with leakage. The developed model takes into account the flexibility of the gasket, the bolts and the flanges as well as the elastic interaction between them. It covers the two operating conditions: Fig. 1. Analytical model for the FRP flange with the hub (a) bolt-up (b) pressurization. following assumptions are made: • The tightening torque during phase i, is applied to all bolts of the flange equally and simultaneously. • The axial force resulting from tightening all the bolts is considered to act as a ring load on a circle of constant radius of the flange known as the bolt circle radius. This assumption makes it possible to consider the flange as an axisymmetric case to simplify the analysis. • A plane of symmetry is considered at the midplane of the joint. This limits the model to the case of identical pair of flanges. • Initial tightening or phase i: in this first stage the bolt-up of the flange is conducted to properly seat the gasket. • Pressurization of final phase f: an internal pressure is applied to the inside of the vessel or pipe and consequently the flange. The inside pressure acts on the flange wall in the radial direction but also cre­ ates a hydrostatic end effect, which is represented by an axial stress acting on the flat surfaces at the end of the cylindrical shell. The analytical model of the FRP bolted flange joint with the hub is illustrated in Fig. 1. In order to model the bolted flange joint analyti­ cally, the flange is divided into three distinct parts which are connected: these are the flange ring or plate, the hub and the shell. Fig. 1a shows the flange in phase n = i or bolt-up and Fig. 1b shows the flange in phase n = f or pressurization. Point 1 shown in Fig. 1 is the junction between the flange ring and the hub and point 2 is the junction between the hub and the shell. The flange ring is treated as a thin composite laminated plate sub­ jected to bending in the tangential direction and as a thick laminate cylinder subjected to lateral pressure. The shell is treated as long lami­ nate cylinder and the hub as short laminate cylinder subjected to pres­ sure and edge loads for which the beam on elastic foundation theory is applied. To simplify the analytical model of the FRP bolted flange joint the 2.1. Flange ring model The theory of plates based on the Kirchhoff-Love hypothesis char­ acterize well the behavior of the composite flange ring. It is considered to be a circular plate subjected to a moment in the tangential direction and ring loads in axial direction on its free inner edge and balanced by a simple support on its outer edge. In addition, the theory of thick-walled cylinder is used to obtain the radial displacement of the ring produced by pressure and the radial edge load. Applying equilibrium of forces in x and r directions and moment about θ to the plate element of volume rdθ by dr by t shown in Fig. 2 and using the governing equations of the classic laminate plate theory the following is obtained. 2 A.K. Vafadar et al. International Journal of Pressure Vessels and Piping 188 (2020) 104247 Nr = du0 dψ u0 ψ Ap,rr + Bp,rr + Ap,rθ + Bp,rθ dr dr r r (14) and therefore Nr = du0 u0 Ap,rr + Ap,rθ dr r (15) similarly ∫t/2 du0 u0 Ap,rθ + Ap,θθ dr r (16) σ r xdx = dψ ψ Dp,rr + Dp,rθ dr r (17) σθ xdx = dψ ψ Dp,rθ + Dp,θθ dr r (18) σθ dx = Nθ = − t/2 ∫t/2 Mr = − t/2 Fig. 2. Plate element model. ∫t/2 Mθ = − t/2 d(rNr ) =0 dr Nθ − (1) (2) d (rQrx ) = 0 dr ( ) 1 dw ψ + 0 Ap,rx 2 dr (19) − t/2 Noting that the FRP plate stiffnesses are given by (3) ( with the following boundary conditions: At r = ri : ∫t/2 ) Ap,rr , Ap,rθ , Aθθ = (20) (Qrr , Qrθ , Qθθ )dx − t/2 Nr = − pt + Qn1 (4) Qn t + 1 2 (5) Mr = M1n τr dx = Qrx = d (rMr ) = 0 dr Mθ + rQrx − ∫t/2 ( (Qrr , Qrθ , Qθθ )xdx = (0, 0, 0) (21) − t/2 (6) Qrx = − F ∫t/2 ) Bp,rr , Bp,rθ , Bp,θθ = ( ) Dp,rr , Dp,rθ , Dp,θθ = and at = r0 : ∫t/2 (Qrr , Qrθ , Qθθ )x2 dx (22) − t/2 (7) Nr = 0 ∫t/2 (23) Mr = 0 (8) Ap,rx = w0 = 0 (9) Now, substituting Eqs. (15) and (16) into the equilibrium equation (1) gives the differential equation governing the radial displacement of the flange ring at its mid-thickness. According to the strain displacement field, we have: εr = du0 dψ +x dr dr (10) εθ = u0 ψ +x r r (11) ( εrx = 1 dw ψ+ 0 2 dr r2 (12) ∫t/2 − t/2 + Qrθ ∫t/2 σr dx = (Qrr εr + Qrθ εθ )dx = − t/2 r (24) d2 ψ dψ Dp,θθ ψ ri F + − =− Dp,rr dr2 dr Dp,rr r is: ) ∫t/2 [ ( du0 dψ Qrr +z dr dr (25) After application of the boundary conditions, the solution of Eq. (24) ( ) ( ) pt − Qn1 (r0 )m2 − 1 pt − Qn1 (r0 )m1 − 1 √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ) rm1 − ( √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ) rm2 uf0 = ( 4 Ap,rθ + Ap,θθ Ap,rr Δ 4 Ap,rθ − Ap,θθ Ap,rr Δ − t/2 ] ) ] ∫t/2 [ ( (u (u ψ) du0 dψ ψ) 0 0 Qrr +z dx = +z + Qrθ +z dx r r dr dr r r d2 u0 du0 Ap,θθ +r u0 = 0 − dr2 dr Ap,rr and substituting Eqs. (17)–(19) into the equilibrium equation (2) and noting that at r = ri , rQrx = − ri F, gives the differential equation gov­ erning the rotation of the flange ring at its mid-thickness. ) u0 and w0 are the radial and transverse displacement of the ring mid­ plane. The membrane force is given by Nr = Qrx dx − t/2 where: m1 = (13) √̅̅̅̅̅̅̅ Ap,θθ Ap,rr , m2 = − m1 Δ = (2ro )m1 − 1 (2ri )m2 − 1 − (2ro )m2 − 1 (2ri )m1 − − t/2 In terms of plate stiffness the membrane force is (26) 1 (27) In the case of equivalent elastic properties of the ring in the radial 3 A.K. Vafadar et al. International Journal of Pressure Vessels and Piping 188 (2020) 104247 and tangential directions, we have Ap,θθ = Ap,rr and m1 = − m2 = 1 and therefore Δ= ro2 − ri2 2ro2 − ri2 (28) Therefore, the radial displacement of the ring is given by: ( ) ( ) pt − Qn1 ri2 pt − Qn1 ro2 ri2 1 )( ) )( )× ( u0 (r) = ( × r − r Ap,rθ + Ap,rr ro2 − ri2 4 Ap,rθ − Ap,rr ro2 − ri2 (29) The rotation about a tangential axis of the ring is obtained from the solution of the differential equation (25) noting that the transformed lamellae stiffness Dp,rr and Dp,θθ are equal ( ( ) ) dw Fr r Cn C n r (30) ψ(r) = = 2 ln − 1 − 2− 1 dr 8πDp,rr ro 2 r The transverse displacement w of the flange ring in the transverse direction subjected to a concentrated force F and edge loads Mn1 and Qn1 at junction 1, can be obtained by integrating Eq. (30): ( ( ) ) ( ) Fr2 r r C n r2 ln − 1 − C2n ln − 1 + C3n w(r) = (31) ro ro 8πDp,rr 4 where F is the gasket force applied on the flange ring. The constants Cn1 , Cn2 and Cn3 are determined from the following boundary conditions: at r = ri , w=0 at r = ri , Mr = M1n − at r = ro , Fig. 3. Shell element model. (32) tp n Q 2 1 Mr = 0 (33) (34) ⎫ ⎡ ⎧ A11 Nx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ A12 Nθ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎬ ⎢ ⎨ Nxθ A16 =⎢ ⎢ M ⎪ ⎪ x ⎪ ⎢ B11 ⎪ ⎪ ⎪ ⎪ M ⎪ ⎪ ⎣ B12 ⎪ ⎪ ⎭ ⎩ θ⎪ Mxθ B16 where n is equal to i for bolt-up and f for pressurization. Appendix A gives the detail constants used for the flange ring. 2.2. Hub and cylinder model The cylinder part of the FRP flange is modeled as an axisymmetric laminated shell subjected to internal pressure and edge loads. It is considered as a long cylinder of semi-infinite length i.e. the length is √̅̅̅̅̅̅ greater than π/βs = 2.45 ri ts . The thin shell theory will apply in this case. At the finite end, the shell is subjected to the discontinuity shear force Q and edge moment M. The long thin cylinder equations for the displacements, rotation, bending moments, and shearing forces in terms of conditions at any locations x are given by the theory of beams on elastic foundation. The hub can also be considered as a thin cylinder subjected to axisymmetric loading. However, the length of the hub is considered short and less than π/β. In this case, both ends of the hub are subjected to the discontinuity shear force Q and moment M in addition to the hy­ drostatic end force Nx . Applying the equilibrium of forces in the axial x and radial r directions and moment about the tangential direction θ to the shell or hub element of volume rdθ by dx by t shown in Fig. 3, the following governing equations are derived: dNx =0 dx Q− dMx =0 dx dQ Nθ − =− p dx r A12 A22 A26 B12 B22 B26 A16 A26 A66 B16 B26 B66 B11 B12 B16 D11 D12 D16 B12 B22 B26 D12 D22 D26 ⎧ 0 dw ⎫ εx = ⎪ ⎪ ⎪ dx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u ⎪ ⎪ 0 ⎪ ε = ⎪ ⎪ ⎤⎪ θ ⎪ ⎪ ⎪ r ⎪ B16 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B26 ⎥ ⎪ ⎪ ⎪ dv ⎥⎪ 0 ⎨ γ = ⎬ ⎥ B66 ⎥ xθ dx ⎥ D16 ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ D26 ⎦⎪ d2 u ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ κ = − ⎪ ⎪ x 2 ⎪ D66 ⎪ dx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ κ = 0 ⎪ ⎪ θ ⎪ ⎪ ⎩ ⎭ (38) κ0xθ = 0 where: Aij , Bij and Dij are the stiffness coefficients. According to the sym­ metrical laminate shell A16 = A26 = D16 = D26 = 0 and [B] = 0. A subscript s is added to these stiffness coefficients when referring to the cylinder and h to the hub. The stress resultants related to the shell displacement are: Nx = A11 Nθ = dw u + A12 dx r ) ( u A2 A12 A22 − 12 + Nx r A11 A11 (39) (40) (35) Mx = − D11 d2 u dx2 (41) (36) Mθ = − D12 d2 u D12 =− Mx dx2 D11 (42) (37) Mxθ = − D16 Mx D11 (43) Substituting Eq. (39) and Eq. (40) into Eq. (36) and Eq. (37) and combining the last two gives the following: ( ) d4 u u A2 A12 Nx D11 4 + 2 A22 − 12 + =p (44) dx r A11 A11 r The relation between the stress resultants and strains are: 4 A.K. Vafadar et al. International Journal of Pressure Vessels and Piping 188 (2020) 104247 The displacement of the shell after integration is: β x − β x u = e [C1 cos(β x) + C2 sin(β x)] + e [C3 cos(β x) + C4 sin(β x)] requirement. The final bolt load after pressurization should then be verified for leakage after calculation. Consequently, during bolt-up the system of equations is reduced to eight unknowns and the axial compatibility equation is not required. The radial displacements, rotations, radial shear force and moment of the hub are equal to those of the flange ring at junction 1 and to those of the shell at junction 2: At junction 1 (45) The rotation ψ, Moment M and shear forces Q are given as follows: ψ= du dx Mx = − D11 (46) d2 u dx2 (47) un1,p = un1,h (49) (48) ψ n1,p = ψ n1,h (50) The integral constants C1 , C2 , C3 and C4 can be determined from the following boundary conditions for both bolt-up and pressurization, for the hub: At x = 0, then Mx = Mn1 and. Q = Qn1 At x = lh , then Mx = Mn2 and Q = Qn2 . Appendix B gives the detailed expressions for the radial displacement, rotation, moment and shear force for the hub. for the shell: Since the cylinder is considered as a long and infinite shell, the displacement, rotation, moment and shear force of the shell are zero when x tends to infinity. This gives C1 = 0 and C2 = 0. Also, at x = 0, then, M = Mn2 and Q = Qn2 . Appendix C gives the detailed expressions for the radial displacement, rotation, moment and shear force for the shell. Qn1,h = Qn1,p (51) n n M1,h = M1,p (52) 2 Q= dMx d w = − D11 2 dx dx At junction 2 un2,h (53) = un2,s ψ n2,h = ψ n2,s (54) Qn2,s = Qn2,h (55) n n M2,s = M2,h (56) Fig. 4 illustrates the compatibility of displacement based on the axial distance traveled by the nut during the tightening process. It is to be noted that this distance is the sum of the displacement of the flange due to rotation, the elongation of the bolt and the compression of the gasket all in the axial direction [2]. Considering the axial compatibility: 2.3. Compatibility equations The equations of compatibility requiring continuity of rotation and radial displacement at the two junctions can be used with the equations of equilibrium of radial shear forces and moments to form a system of equations with eight unknowns in the seating condition and nine un­ 3 ∑ f knowns for the pressurization condition because the final bolt force FB is not known. For the operating condition this load can be obtained from the geometric compatibility consideration in the axial direction. The distance traveled by the nut during tightening remains constant and is therefore equal in both the initial bolt-up and pressurization states. Before applying the internal pressure, the bolt initial force FiB , which is the minimum required bolt load that mus satisfy the seating i=1 wii = 3 ∑ wfi (57) i=1 or wib + wig + wip = wfb + wfg + wfp In terms of stiffness and angular rotation Eq. (58) becomes: Fig. 4. Axial compatibility: a) hand tightening, b) bolt-up), c) pressurization. 5 (58) A.K. Vafadar et al. International Journal of Pressure Vessels and Piping 188 (2020) 104247 FBf FGf Fi Fi + + (rB − rG )ψ f1 = B + G + (rB − rG )ψ i1 KB KG KB KG ⎡ (59) ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ [V] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢( ⎣ 1 Considering the axial equilibrium. Initial pre-tightening state (i) (60) FiB = FiG Final pressurization state (f) (61) FfB = FfG + pAp Substitution of Eqs. (60) and (61) into Eq. (59) gives ( ) ( ) ) ( 1 1 1 1 AP FBf + + p = F i + 2(rB − rG ) ψ I1 − ψ f1 + KB KG KB KG B KG (62) After assembling the above equations, a system of 9 unknowns is obtained. The equations are put into a matrix form to solve for the unknowns. 0 1 Ah,12 2 p 4 4βh Ah,1 Dh,11 Ah,11 − − 0 1 As,12 2 − p 4 4βs As,11 Ds,11 As,11 − 0 − (L21 − L22 )tp p ) L27 ( − (rG − ri ) rG2 + ri2 p 4rm ) 1 AP F i + 2(rB − rG )ψ I1 + + p KB KG B KG (63) [U] = [C]\[V] 1 Ah,12 2 − p 4 4βh Ah,1 Dh,11 Ah,11 − [U] is a vector composed of the 9 unknown elements: the two edges forces and two edge moments, the radial displacements and the rotations at both junctions and the bolt force. ⎡ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (65) ⎤ − B11 ⎢ 2β3 D ⎢ h h,11 ⎢ ⎢ − B ⎢ 12 ⎢ 2 ⎢ 2βh Dh,11 ⎢ ⎢ ⎢ − G11 ⎢ 3 ⎢ 2βh Dh,11 ⎢ ⎢ ⎢ G12 ⎢ 2 ⎢ 2β Dh,11 ⎢ h ⎢ ⎢ 1 [C] = ⎢ ⎢ 2β3 D ⎢ s s,11 ⎢ ⎢ − 1 ⎢ ⎢ 2 ⎢ 2βs Ds,11 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎣ 0 B12 2β2h Dh,11 G11 2β3h Dh,11 G12 2β2h Dh,11 − 1 0 0 0 B22 2βh Dh,11 G12 2β2h Dh,11 G22 2βh Dh,11 0 − 1 0 0 G12 2β2h Dh,11 B11 2β3h Dh,1 B12 2β2h Dh,11 0 0 − 1 0 − G22 2βh Dh,11 − B12 2β2h Dh,11 − B22 2βh Dh,11 0 0 0 − 1 1 2β2s Ds,11 0 0 − 1 0 0 0 − 1 βs Ds,11 0 0 0 − 1 0 0 0 − (L21 − L22 ) 0 0 0 − 1 tf 2 0 tf L26 2 − L26 0 0 0 − 1 0 0 0 0 0 0 2(rB − rG ) [ ]T [U] = Qn2 , M2n , Qn1 , M1n , un2 , ψ n2,h , un1 , ψ n1 , FBn 0 ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ rB − rG ⎥ L27 ⎥ ⎥ 2 π rm ⎥ ⎥ 1 1 ⎦ KB + (66) KG To obtain the final bolt force FBf which depends on the flange pa­ rameters as well as the inside pressure, first the unknown vector [U] has to be solved for the initial bolt-up condition with a known initial bolt force FBi . The solution gives in particular the rotation of the flange at the initial tightening condition that is used in the last row in [U] in the final (64) [C] is a square matrix and [V] is a vector defined as follows: condition to solve for the bolt force FBf . 2.4. Stress calculation To calculate the stresses and displacements in the hub and shell, the solutions of the two vectors [Ui ] and [Uf ] relative to the initial and final conditions n = i and n = f are first obtained. The longitudinal and tangential stresses at the inside and outside surfaces of the hub are given 6 A.K. Vafadar et al. International Journal of Pressure Vessels and Piping 188 (2020) 104247 σ nh,l (x) = n Nh,l (x) 6Mhn (x) ± th2 th (69) σ nh,t (x) = n Nh,l t(x) 6Dh,12 Mhn (x) ± th th2 Dh,11 (70) The longitudinal and tangential stresses at the inside and outside surfaces of the shell for the initial and final conditions n = i and n = f are given by: i Ns,l (x) = 0 and ( As,11 − ⎡ n ⎢us (x) n Ns,t (x) = ⎣ rs f Ns,l (x) = + p rs 2 ) As,12 2 4β4s As,11 Ds,11 (71) ⎤ ( ) A2s,12 p⎥ As,12 n + N (x) ⎦ As,22 − rs As,11 As,11 s,l (72) σ ns,l (x) = n Ns,l (x) 6Msn (x) ± ts2 ts (73) σ ns,t (x) = n Ns,t (x) 6Ds,12 Msn (x) ± ts Ds,11 ts2 (74) 3. FEM modeling Two finite element models representing the NPS 3 and 12 class 150 FRP bolted flanged joints were developed under the finite element software ANSYS® R16.2 in order to validate the proposed analytical model. The 3D solid models are divided into three parts; the flange, the gasket and the bolt. The model of the bolted flange assembly includes only one flange and the gasket with half of its thickness due to the symmetry of bolted flange joint with respect to a plane that passes through the mid gasket thickness (Fig. 5). The FRP flanges are made out of laminates of E-glass fibers for reinforcement and vinyl-ester as the matrix. The three major parts (ring, hub and shell) have different thickness and arrangement of plies but are all composed of mat and woven roving laminates. It is worth noting that the laminates are hand Fig. 5. Finite element model for the NPS 12 FRP flange. in terms of the longitudinal and tangential membrane forces Nl and Nt : i Nh,l (x) = 0 and p rh 2 ( ⎡ n ⎢uh (x) n Nh,t (x) = ⎣ f Nh,l (x) = rh ) Ah,11 − + Ah,12 2 4β4h Ah,11 Dh,11 (67) ⎤ ( ) A2h,12 p⎥ Ah,12 n + N (x) ⎦ Ah,22 − rh Ah,11 Ah,11 h,l (68) Fig. 6. NPS 3 class 150 FRP Flange a) Experimental Test Rig. b) strain rosettes bounded to the hub. 7 A.K. Vafadar et al. International Journal of Pressure Vessels and Piping 188 (2020) 104247 Fig. 7. Gasket load-compression data curve. Fig. 10. Longitudinal stress distribution of NPS 12 class 150 flange after pressurization. Fig. 8. Longitudinal stress distribution of NPS 12 class 150 flange after bolt-up. Fig. 11. Tangential stress distribution of NPS 12 class 150 flange after pressurization. fundamental building blocks of a good simulation to reproduce the real behavior of the bolted flange joint is to employ the correct element type of the contact surface in the FE model. The contact element used in all deformable contact surfaces is CONTA174 element that has 8 nodes and is used in conjunction with 3D target interface surface element TARGE170. The flanged connection is assumed to be sufficiently far from the end enclosures of the vessel, such that the bending and shear between the shell and cap are only limited to their junction and does not affect the length of the shell under consideration. Because of the axisymmetric analysis, the boundary conditions applied in these finite element models are the symmetry about a plane passing through gasket mid-thickness. The nodes that belong to this plane are constrained from any move­ ment in the axial direction. The bolt symmetrical plane is free to move in the radial direction while the rotation about a plane perpendicular to the flange axial direction is fixed. Since the flange is symmetrical in nature, the cyclic symmetry is applied to the two sides of the cut section. A standard friction model is used to simulate the contact and controls sliding. The coefficients of static friction of 0.7 for rough surface and 0.15 for smoother surface are used. The pressure applied to the internal surface of the NPS 3 and 12 flanges is about 1 MPa (150 psi). The lateral pressure is applied to the flange internal wall while the hydrostatic end load is applied to the end of the pipe. By applying a pre-stress equivalent Fig. 9. Tangential stress distribution of NPS 12 class 150 flange after Bolt-up. layered-up with midplane symmetry. Fig. 5 shows the mesh of the model of the NPS 12 flange with the hub. In Ansys workbench, all parts of the flange are modeled with SOLID185 elements while the gasket is meshed with INTER195 elements. Both element types have 8 nodes. One of the 8 A.K. Vafadar et al. International Journal of Pressure Vessels and Piping 188 (2020) 104247 Fig. 12. Radial displacement in NPS 12 class 150 flange. Fig. 14. Tangential stress distribution of NPS 3 class 150 flange after bolt-up. Fig. 13. Longitudinal stress distribution of NPS 3 class 150 flange after bolt-up. Fig. 15. Longitudinal stress distribution of NPS 3 class 150 flange after pressurization. to a load of 14600 and 10,000 N on the bolts of the two flanges respectively, the initial gasket seating is achieved. In the ANSYS the torque load is applied through the pre-existing command bolt pretension. In this model, the gasket deforms filling the irregularities on the flange face, insuring full contact over its entire surface. The mechanical behavior of the gasket is given by the nonlinear curves of gasket contact stress versus axial displacement shown in Fig. 6. hub locations on the flange as shown in Fig. 7b. The test was conducted at room temperature (22 ◦ C) and all measuring sensors were calibrated at this temperature. During the test, the bolt force, the gasket displacement, the strains at selected locations of the flange surface, the temperature and internal pressure were continuously monitored through a data acquisition and control unit connected to a computer. A special software working under LabView was written to monitor and control the test parameters. 4. Experimental set-up 5. Results and discussion The test rig used in the experimental investigation of FRP bolted flanged joints is shown in Fig. 7a. The test rig is made of an ANSI B16.5 NPS 3 Class 150 FRP full face flange used with a metallic flange of the same type and size and a 1/16 Teflon full face gasket. The FRP flange is made of glass mat and woven roving impregnated with vinyl ester resin. An initial bolt load of 14600 N was applied gradually by a torque wrench to each instrumented bolt using a tightening procedure based on the criss-cross pattern conducted with three load step levels. The load is measured through a high temperature extensometer device made of mainly a ceramic rod and tube that transmit the displacement to a straingaged beam located far from the heated flanges. The extension of the displacement difference between the drilled bolt and the central rod welded to its end gives the load applied with a good precision. A gas pressure of 1 MPa was applied to the bolted joint. Three full bridge strain rosettes were bonded with adhesive to the outside surface at the selected For validation of the analytical model the results are presented and compared to the numerical results obtained from FE modeling and experimentation. The longitudinal and tangential stress distributions at the inside and outside surfaces of the flange at bolt-up are illustrated in Figs. 8 and 9 for the NPS 12 and 13 and 14 for NPS 3. These figures show clearly that the analytical and FE stress distributions at the inside and outside flange surfaces during bolt-up have a similar trend. The distri­ butions of the longitudinal and tangential stresses in both cases are in a good agreement and the difference is less than 4% far from the two junctions where stress concentration is expected. These results match well with the experimental stresses deduced from the measured strain gages on the NPS 3 class 150 flange. As anticipated during bolt-up, due to the lack of the pressure inside of the FRP bolted flange joints, the 9 A.K. Vafadar et al. International Journal of Pressure Vessels and Piping 188 (2020) 104247 stresses. Although at the junctions, the stress at the inner and outer surfaces show a higher difference, the distributions are consistent. In the case of the NPS 12, there is a location where a drastic change of longitudinal stress occurs. This is less visible in the case of the NPS 3 flange. Figs. 13–16 show a relatively good agreement between the analytical and FE models and the experimental data during pressurization except at the junctions. Nevertheless, while the results of the tangential stresses at the inner and outer flange surface are shown to have the same trend, those at the hub show a discrepancy because the hub is considered as a cylinder with an equivalent thickness rather than tapered. This simpli­ fication is acceptable in the design of such a complex structure. In addition, at the junction of the hub and the shell where stress concentration exists, the stresses at this location increases drastically by more than 92%. This phenomenon also occurs near the hub to the flange ring junction. It is to be noted that since the small tapered portion of the hub of the NPS 12 flange at the junction with the shell that is not accounted for in the analytical model, the results would be affected. Nonetheless, considering the fact that the analytical model is developed to analyse general membranes stresses and not peak stresses generated by stress concentration, the difference is not significant and is acceptable for design purposes. The radial displacement results obtained from the numerical and analytical models of the two FRP bolted flange joints at bolt-up and pressurization are compared in Figs. 12 and 17. As shown in these fig­ ures, the general distributions have a remarkably similar trend. The difference observed between the analytical and numerical results is once again higher at the junctions but compares well elsewhere in both cases. Finally, the radial displacement of the shell is shown to be much higher than the flange ring and the hub during pressurization because these two structures are much stiffer. The results of the radial displacements confirm the robustness of the analytical developed model. Fig. 16. Tangential stress distribution of NPS 3 class 150 flange after pressurization. 6. Conclusion This study presents a stress analysis methodology that handles the anisotropy behavior of the complex structures of FRP bolted flange joints. The modelling of the FRP flange composed of the three elements ; the ring, the hub and the shell, is done using the different theories of laminate materials. The analytical developed model was tested and validated using FEM and experimental tests conducted on two different size flanges. It follows the general ASME code design philosophy but accounts for the anisotropy multilayer modelling as well as flexibility. From the results obtained numerically and experimentally, it could be stated the proposed analytical model for FRP flanges, has proved to be efficient, accurate and reliable in predicting the longitudinal and tangential stress distributions at inside and outside flange surfaces including the radial displacement of the flange. However, at the junc­ tions of the hub with the ring and shell, the predictions are less accurate due to stress concentrations and laminate interconnections between the elements. The tangential stresses at the hub inside surface are those that show the highest difference. However, in terms of radial displacement, the importance of considering anisotropy is demonstrated. The results of this work have led to the conclusion that the presented model has great potentials: it provides a new formulation that could be used as an alternative to the ASME code section X standard while allowing accu­ racy, reliability and thorough stress analysis of FRP bolted flange joints. Additional experimental tests on FRP materials with different properties in the two longitudinal and tangential directions are curently being conducted to confirm these findings. Fig. 17. Radial displacement in NPS 3 class 150 flange. stresses in the shell far away from the hub and shell junction are equal to zero. Figs. 10 and 11 and 15 and 16 present the longitudinal and tangential stress distributions of the NPS 12 and 3 flanges respectively obtained after application of an internal pressure of 1 MPa. A similar stress dis­ tribution trend along the hub and cylinder length is observed in both the analytical and FE models. However, the FE numerical results show higher stress values at the outside flange surface near the junctions of the hub and the shell and the hub and ring flange. On comparing the analytical and numerical results in the shell far away from the junctions, the highest difference observed in the case of the NPS 12 flange is around 22% and 18% for the longitudinal stress distribution at the outside and inside diameters respectively. While this difference may seem acceptable, for the case of the NPS 3 flange, the tangential stress at the inside diameter of the hub with both methods are completely the opposite, especially near the junction with the flange. It is difficult to confirm with the experimental measurement since the bounding of strain gages at the inside diameter is not an easy task because of the complexity of introducing the wiring inside the vessel. Nevertheless, such differences are anticipated because these junctions are smoother and the laminates are not interconnected in the FE models. Moreover, the presence of the high local stresses in the composite flanges affects considerably the stress distribution particularly at the outside surfaces of the hub and the shell. Since the material properties vary through the thickness of the composite flanges, there is an even distribution of Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 10 A.K. Vafadar et al. International Journal of Pressure Vessels and Piping 188 (2020) 104247 Nomenclature Ab Ag Ap Ai,mn As Bi,mn C1 to C4 Di,mn db E F K l M n N p Q Qrx r rb t u w x Poisson ratio Hub constant [mm− 1] Shell constant [mm− 1] Strain Tangential direction Normal stress [MPa] Shear stress [MPa] Rotation of the plate [rad] Bolt area [mm2] Full gasket contact area [mm2] Pressure area [mm2] In-plane laminate modulus [MPa.mm] Section cut area of the shell [mm2] Coupling laminate modulus [MPa.mm2] Constants of integration Flexural laminate modulus [MPa.mm3] Diameter of the bolt [mm] Young Modulus [MPa] Force [N] Uniaxial stiffness [N/mm] Length [mm] Discontinuity moment [N] Number of bolts Membrane force [N/mm] Internal pressure [MPa] Discontinuity or shear force [N] Shear force in the plane rx [N] Radial direction or radius [mm] Bolt circle [mm] Thickness [mm] Radial displacement [mm] Axial displacement [mm] Axial direction Subscript 0 1 2 b, B f g, G h i l r s t o p x refers refers refers refers refers refers refers refers refers refers refers refers refers refers refers ν βh βs ε θ σ τ ψ to mid plane to junction between ring and hub to junction between hub and shell to bolt to flange to gasket to hub to inside to longitudinal direction to radial to shell to tangential direction to outside to plate or ring to axial Superscript i Refers to initial tightening of the bolt condition f Refers to final condition (pressurization) n Refers to working condition, n=i,f Appendix a. Flange ring Details The constants of the flange ring axial displacement and rotation used in the analysis are obtained for the two conditions of bolt-up n=i and pressurization n=f. The bolt force F and edge loads Q and M are also related to those two conditions: 11 A.K. Vafadar et al. 1 ( )( 4πDp,rr ro2 − ri2 Dp,rr C1n = − C2n = International Journal of Pressure Vessels and Piping 188 (2020) 104247 4πDp,rr ( ( ) ] ( )( ) ( ) ( r t ) ) Dp,12 − Dp,rr ro2 − ri2 F + 2 Dp,rθ + Dp,rr ri2 log i − 8πri2 Dp,rr M1n − f Qn1 ro 2 + Dp,rθ [ ( ) ] [ ( ) ro2 ri2 tf n ) ( ri n )( ) π D − + D M − D F log 4 Q p,rr p,rθ p,rr 1 1 2 ro ro2 − ri2 Dp,rr − Dp,12 (A1) (A2) [( )( ( )( ( ) (r ) ) − ri2 t ) ) D2p,rθ − 3D2p,rr + 2Dp,rθ Dp,rr ro2 − ri2 F − 8πri2 D2p,rr − Dp,12 Dp,rr M1n − f Qn1 − 2ro2 D2p,rθ − D2p,rr log i F − 4ro2 ( ) 2 ro 16πDp,rr ro2 − ri2 D2p,rr − D2p,rθ ( ) ) ( ] ( ) ( ) ri ri ( n tf n ) D2p,rθ + D2p,rr + 2Dp,rθ Dp,rr log 2 (A3) F − 16π ro2 D2p,rr − Dp,rθ Dp,rr log M1 − Q1 ro ro 2 C3n = ( In order to alleviate the system of equations the following is used. To simplify the derivations the following coefficients are defined: L21 = ( L22 = ( − r3 )( i ) r2o − r2i Ap,12 + Ap,11 (A4) − r r2 )( i o ) Ap,12 − Ap,11 (A5) r2o − r2i ( ) ri − 2ri log L23 = (A6) 8π Dp,11 ( L24 = − ri ro ( ) )( ) ) ( r2o ri + r3i Dp,12 − Dp,11 + 2r3i log rroi Dp,11 + Dp,12 ( )( ) 8πDp,11 r2o − r2i Dp,11 + Dp,12 (A7) ( ) ) ( − r2o ri log rroi Dp,11 + Dp,12 ( )( ) L25 = 4Dp,11 π r2o − r2i Dp,11 − Dp,12 (A8) (A9) L26 = L21 + L22 (A10) L27 = L23 + L24 + L25 Appendix B. HUB DETAILS Hub laminate:the equations of this appendix are used to calculate the displacement and stresses in the hub: ]1/4 [ Ah,22 Ah,11 − A2h,12 βh = 4rh2 Dh,11 Ah,11 1 F11 (βh x) = [cosh(βh x)sin(βh x) − sinh(βh x)cos(βh x)] 2 (B1) F12 (βh x) = sin(βh x)sinh(βh x) (B2) 1 F13 (βh x) = [cosh(βh x)sin(βh x) + sinh(βh x)cos(βh x)] 2 (B3) F14 (βh x) = cosh(βh x)cos(βh x) (B4) / 1 B11 = [sinh(2βh lh ) − sin(2βh lh )] [sinh2 (βh x) − sin2 (βh x)] 2 (B5) / 1 B12 = [cosh(2βh lh ) − cos(2βh lh )] [sinh2 (βh x) − sin2 (βh x)] 2 (B6) B22 = [sinh(2βh lh ) + sin(2βh lh )] / [sinh2 (βh x) − sin2 (βh x)] (B7) G11 = − cosh(βh lh )sin(βh lh ) − sinh(βh lh )cos(βh lh ) sinh2 (βh x) − sin2 (βh x) (B8) (B9) G12 = [2 ​ sinh(βh lh )sin(2βh lh )] / [sinh2 (βh x) − sin2 (βh x)] 12 International Journal of Pressure Vessels and Piping 188 (2020) 104247 A.K. Vafadar et al. G22 = − 2 cosh(βh lh )sin(βh lh ) + sinh(βh lh )cos(βh lh ) sinh2 (βh x) − sin2 (βh x) (B10) Substituting the above equations, the lamina stresses and displacement in the hub can be determined from the following relations: unh (x) = Qn1 Mn ψn F11 (βh x) + 2 1 F12 (βh x) + 1 F13 (βh x) + un1 F14 (βh x) βh 2β3h Dh,11 2βh Dh,11 [ ψ nh (x) = βh Qn1 Mn F12 (βh x) + 2 1 F13 (βh x) + 3 2βh Dh,11 βh Dh,11 ψ n1 βh (B11) ] F14 (βh x) − 2un1 F11 (βh x) (B12) ] Qn1 M1n ψ n1 n F (β x) + F (β x) − F (β x) − u F (β x) 13 14 11 12 h h h h 1 βh 2β3h Dh,11 2β2h Dh,11 [ Mhn (x) = 2β2h Dh,11 [ Qn1 F14 (βh x) − 2β3h Dh,11 Qn1 (x) = 2β3h Dh,11 M1n F11 (βh x) − β2h Dh,11 ψ n1 βh (B13) ] (B14) F12 (βh x) − 2un1 F13 (βh x) Appendix C. SHELL DETAILS Shell laminate: the equations of this appendix are used to calculate the displacement and stresses in the shell: ]1/4 [ As,22 As,11 − A2s,12 βs = 4rs2 Ds,11 As,11 (C1) f1 (βs x) = e− βs x [cos(βs x)] (C2) f2 (βs x) = e− βs x [cos(βs x) − sin(βs x)] (C3) f3 (βs x) = e− βs x [cos(βs x) + sin(βs x)] (C4) f4 (βs x) = e− βs x [sin(βs x)] (C5) substituting the above equations, the lamina stresses and displacement in the shell can be determined from the following relations: uns (x) = Qn2 Mn f1 (βs x) + 2 2 f2 (βs x) + 3 2βs Ds,11 2βs Ds,11 [ ψ ns (x) = βs − Qn2 f3 (βs x) − 2β3s Ds,11 As,11 − 12As,12 p 4β4s As,11 Ds,11 + ] M2n f1 (βs x) β2s Ds,11 (C7) [ Qn2 Mn f4 (βs x) + 2 2 f3 (βs x) 3 2βs Ds,11 2βs Ds,11 Msn (x) = 2β2s Ds,11 [ Qns (x) = 2β3s Ds,11 (C6) Qn2 f2 (βs x) − 2β3s Ds,11 M2n f4 (βs x) β2s Ds,11 ] (C8) ] (C9) References [9] H. Estrada, I.D. Parsons, Strength and leakage finite element analysis of a GFRP flange joint, Int. J. Pres. Ves. Pip. 76 (1999) 543–550. [10] J.M. Campbell, FRP flange for process pipe and tanks,” Proc. FRP Symposium, National Association of Corrosion Engineers, Niagara Frontier section, USA, 1990. [11] A.E. Blach, FRP bolted flanged connections subjected to longitudinal bending moments, in: Computer Aided Design in Composite Material Technology International Conference, 1996, pp. 11–20. [12] A. Bouzid, A. Chaaban, Flanged joint analysis: a simplified method based on elastic interaction, CSME Transactions 17 (2) (1993) 181–196. [13] A. Bouzid, H. Galai, A new approach to model bolted flange joints with full face gaskets, ASME Journal of Pressure Vessel Technology 133 (2) (2011), 021402-6. [14] A. Bouzid, A. Nechache, An analytical solution for evaluating gasket stress change in bolted flange connections subjected to high temperature loading, ASME Journal of Pressure Vessel Technology 127 (4) (2005) 414–427. [15] A.K. Vafadar, A. Bouzid, A.D. Ngo, Effect of Material Anisotropy on the Structural Integrity of Composite Bolted Flanged Joints, Waikoloa, Hawaii, USA, 2017. [16] J. Czerwinski, V.K. Garikipati, C.N. Jones, B. Pires, J.P. Ludman, D.H. Mccauley, B. Linnemann, A collaborative approach to achieving a gas-tight seal using expanded polytetrafluorethylene (EPTFE) gaskets in a fiberglass-reinforced plastic (FRP) lap-joint flanges, ASME Pressure Vessels and Piping Division, PVP2012 2 (2012) 239–246, 2012. [1] M. Aljuboury, M.J. Rizvi, S. Grove, R. Cullen, Bolted fibre-reinforced polymer flange joints for pipelines: a review of current practice and future challenges, Proc IMechE Part L: J Materials: Design and Applications 233 (8) (2019) 1698–1717. [2] A.E. Blach, S.V. Hoa, Bolted flange connections for glass fiber reinforced plastic pipes and pressure vessels, in: Proceedings 11th International Conference on Fluid Sealing, Cannes, France, 1987, pp. 445–457. [3] ASME, ASME Boiler and Pressure Vessel Code Section X, Fiberglass reinforced plastic pressure vessels, New York, 2015. [4] L. Sun, Bolted Flanged Connections Made of Fiber Reinforced Plastic Materials, Ph. D. Dissertation, Concordia University, Montreal, Canada, 1995. [5] A.E. Blach, L. Sun, Fiber reinforced plastic bolted flanged connections, in: Proceedings 2nd International Symposium on Fluid Sealing, La Baule, France, September, 1990, pp. 445–457. [6] A.E. Blach, S.V. Hoa, Effects of pull-back on stresses in FRP flanges, Exp. Tech. 12 (11) (1988) 12s–16s. [7] A.E. Blach, Fiber reinforced bolted flanged connections discussion of ASME code rules, in: Proceedings of the International Conference on Pressure Vessel Technology, ICPVT, Design and Analysis, vol. 2, 1996, pp. 301–306. [8] H. Kurz, E. Roos, Design of floating type bolted flange connections with GRP flanges, Int. J. Pres. Ves. Pip. 89 (2012) 1–8. 13