Bolted Flange Joint: Method

Bolted Flange Joint: Method, Conventions and Limits

What the tool does

The tool checks a bolted flanged joint with a ring-type gasket by the method of Mandatory Appendix 2 of ASME Boiler and Pressure Vessel Code Section VIII Division 1, the "Taylor Forge" method that Waters, Rossheim, Wesstrom and Williams published in 1937 and that the Code has carried since 1943. The Code text and its tables are copyrighted and are cited only; what the tool uses is the mathematics of the method, which is public in the 1937 paper, in the public-domain reports of Oak Ridge National Laboratory that implemented it (ORNL-5035 and ORNL/Sub/2913-3, 1976) and in every pressure-vessel textbook. Three flange types are covered: the integral flange (a weld neck whose tapered hub is welded to the pipe or shell, Code Figure 2-4 sketches 6, 6a and 7, and optional flanges calculated as integral), the loose flange with a hub (a lap-joint flange whose hub is considered, sketches 1 and 1a) and the loose ring flange without a hub (or a hubbed flange the designer calculates without its hub). The Code allows a flange that conforms to ASME B16.5 or B16.47 to be used at its pressure–temperature rating without this calculation (UG-44); the check below is for flanges designed to Appendix 2 and for the engineer who wants to see how a standard flange fares under it.

Gasket and bolt loads

The gasket contact width N = (G_o − G_i)/2 gives the basic seating width b₀ by the facing rule of Table 2-5.2: N/2 for a flat gasket on a flat or raised face, w/8 for a ring joint of width w. The effective seating width is b = b₀ when b₀ ≤ 1/4 in and b = 0.5√b₀ (inches; 2.52√b₀ in millimetres) when it is wider, the Code's allowance for the fact that a wide gasket seats on its outer part as the flange rotates; G, the diameter of the gasket load reaction, is the mean gasket diameter in the first case and G_o − 2b in the second. Two bolt loads are required. In operation the bolts must hold the hydrostatic end force H = πG²P/4 and keep a residual compression H_p = 2bπGmP on the gasket, m times the pressure on the effective area: W_m1 = H + H_p. At assembly they must seat the gasket to its seating stress y on that area: W_m2 = πbGy. The required bolt area A_m is the larger of W_m1/S_b and W_m2/S_a, with S_b and S_a the bolt allowables at design and atmospheric temperature; the actual area A_b is the root area of the bolts, (π/4)(d − 1.299038p)² for a 60° thread, taken from the thread forms of the Thread Strength library (NBS Handbook H28). The gasket factors m and y are entered; the tool's list holds single values of the kind tabulated in the Code's Table 2-5.1 as illustrative defaults, and the gasket maker's values should be used. A self-energizing gasket (an O-ring) takes m = y = 0, so that the bolts carry the end force only.

For the gasket-seating design of the flange the Code uses W = (A_m + A_b)S_a/2, the mean of the required and actual areas at the allowable: the bolts that are actually there will be tightened harder than the minimum, and the flange must stand it. The bolt spacing B_s = πC/n is compared with the maximum 2a + 6t/(m + 0.5) of equation (3), and when it exceeds 2a + t the moments may be multiplied by B_sc = √(B_s/(2a + t)) (equation (7)), which the Code requires in lethal service or when the user specifies it; the tool computes the factor and applies it on request.

External loads

A bending moment M_e and an axial tension F_e in the attached pipe are converted to the equivalent pressure of Kellogg's Design of Piping Systems, P_eq = 16M_e/(πG³) + 4F_e/(πG²), the uniform pressure that produces the same force on the gasket reaction circle at its most loaded point, and the operating loads are computed with P + P_eq. This is the usual piping practice (ASME III NB-3647.1 uses the same form); it is conservative for the bolts because the moment loads only one side of the joint.

Flange moments

Each force acts about the bolt circle with the arm of the Code's Table 2-6. The end force H_D = πB²P/4 on the area inside the flange acts through the hub, at R + g₁/2 from the bolt circle for an integral flange (R = (C − B)/2 − g₁) and at (C − B)/2 for a loose one; the face force H_T = H − H_D on the annulus between the bore and G acts at (R + g₁ + h_G)/2 or (h_D + h_G)/2; the gasket force H_G = W_m1 − H at h_G = (C − G)/2. The operating moment is M_o = H_Dh_D + H_Th_T + H_Gh_G; the gasket-seating moment is Wh_G, since at assembly the whole bolt load is reacted by the gasket.

The Waters analysis and the flange factors

Waters and his co-authors modelled the flange ring as an annular plate twisted by the moment of its loads and the hub as a conical shell of linearly varying thickness clamped to the ring, with the pipe beyond it a long cylinder; the ring and hub share the rotation θ at their junction and the hub's radial displacement there is taken as zero. The ring alone gives the factors involving K = A/B that the Code prints in Figure 2-7.1: T = [K²(1 + 8.55246 log₁₀K) − 1]/[(1.04720 + 1.9448K²)(K − 1)], U = [K²(1 + 8.55246 log₁₀K) − 1]/[1.36136(K² − 1)(K − 1)], Y = [0.66845 + 5.71690K² log₁₀K/(K² − 1)]/(K − 1) and Z = (K² + 1)/(K² − 1), all with Poisson's ratio 0.3. Y is exactly the tangential stress factor of Timoshenko's annular plate with free edges and a uniform shear W/(2πr) through it (the test script reproduces it from the plate equation to seven digits); as K → 1 it tends to 6/[π(K − 1)], the thin-ring twist formula. The hub gives three factors of A = g₁/g₀ − 1 and h/h₀, h₀ = √(Bg₀): F = h₀Q₁/(2M₁) is the ratio of the hub's shear to its moment at the ring junction, which with the ring thickness t makes the term te in L; V = (1 − ν²)Eg₀²h₀θ/(6BM₁) is the hub's rotational flexibility, which with U makes d = (U/V)h₀g₀² and the term t³/d; and f is the ratio of the longitudinal bending stress at the small end of the hub to that at the large end, used only when it exceeds 1. For a hub of uniform thickness the semi-infinite-shell solution gives F = 0.908920 and V = 0.550103, the values printed on the Code's figures. The Code's Figures 2-7.2 to 2-7.6 were drawn from Waters' power-series solution of the tapered shell, and the same solution is printed in formula form as Table 2-7.1 (the 45 expressions C₁ … C₃₇, E₁ … E₆); the tool evaluates those formulas. A direct numerical solution of the tapered-hub boundary-value problem with scipy (filed with the sources) agrees with the formulas to five digits for g₁/g₀ ≤ 2 and to within 1 to 3 % in F and V, and up to 8 % in f, at g₁/g₀ = 5: the series was truncated for the charts, and the Code's values are the ones the Code requires. For a loose hubbed flange the small end of the hub is free instead of continuing into a pipe, which gives F_L and V_L, and f = 1.

Stresses and limits

With e = F/h₀, d = (U/V)h₀g₀² and L = (te + 1)/T + t³/d, the Code's stresses are the longitudinal hub stress S_H = fM_o/(Lg₁²B), the radial flange stress S_R = (4/3 te + 1)M_o/(Lt²B) and the tangential flange stress S_T = YM_o/(t²B) − ZS_R, each at the operating moment with the allowable S_f at design temperature and at the gasket-seating moment with S_f at atmospheric temperature. The Code prints the radial coefficient as 1.33; the derivation gives 4/3, which the ORNL program also uses and which reproduces the program's printed example to every digit, so the tool uses 4/3 (the difference is 0.1 % in S_R). The limits of 2-8 are S_H ≤ 1.5S_f (reduced to the smaller of 1.5S_f and 2.5S_n for a hub welded to the pipe, sketch 6, or 1.5S_n for an optional flange or sketch 7, with S_n the pipe allowable), S_R ≤ S_f, S_T ≤ S_f, (S_H + S_R)/2 ≤ S_f and (S_H + S_T)/2 ≤ S_f; a loose ring flange has only S_T = YM_o/(t²B) ≤ S_f. The margins are allowable/stress − 1 for each check, and the margin boxes show the governing check of each condition, named "ASME VIII-1 App. 2".

Rigidity and rotation

Since the 1998 addenda the Code also limits the flange rotation through the rigidity index of 2-14: J = 52.14VM_o/(LEg₀²K_Ih₀) ≤ 1 for integral and loose hubbed flanges and J = 109.4M_o/(Et³K_I ln K) ≤ 1 for loose ring flanges, with K_I = 0.3 for integral and optional flanges and 0.2 for loose ones, evaluated at both conditions with the modulus at the corresponding temperature. The coefficients show where the index comes from: 109.4/57.30 = 6/π, so the ring form is the thin-ring twist θ = 6M_o/(πEt³ ln K) in degrees divided by K_I, and 52.14/57.30 = 1 − ν², so the hub form is the ring rotation of the Waters model, θ = (1 − ν²)VM_o/(LEg₀²h₀), divided by K_I. The tool therefore reports the rotation as K_IJ degrees, an estimate from the Code's own elastic model with rigid-bolt and linear-gasket assumptions; measured rotations of real joints also depend on the gasket's stiffness.

Bolt torque and assembly

The torque table gives, for the operating minimum W_m1, the seating design load W and an entered target assembly stress, the force per bolt, the bolt stress on the root area, the nut-factor torque T = KFd and the gasket stress on the effective area πbG. The Code's loads are minimums; the assembly stress is set by the joint-assembly procedure, for which ASME PCC-1 (cited and named only) gives the bolt-tightening patterns, the incremental rounds and target stresses. A standard flange with a modern spiral-wound gasket often shows a bolt area short for seating under Appendix 2, as the default case does: B16.5 ratings were set with the lower seating stresses of the gaskets of the 1920s (ORNL/Sub/2913-3 used m = 2.75 and y = 3,700 lb/in² for its B16.5 study), and such joints are assembled to bolt stresses well above the Code allowable, which PCC-1 addresses. The torque–tension relation, friction scatter and the joint diagram are in the Bolt Preload and Torque tool.

Checks and sources

The library is checked by test-scripts/test_bolted_flange_joint.php against the printed Example 1 of ORNL-5035 (the 60 in API 605 300 lb tapered-hub flange at 720 lb/in², Table A.2: b₀, W_m1, W_m2, the bolt stresses, M_o, the seating moment and all ten flange stresses), the bolt loads of Table 3 of ORNL/Sub/2913-3 for B16.5 flanges, the limiting values of the Code's figures, the exact annular-plate solution for Y, a scipy solution of Waters' tapered hub for F, V, f, F_L and V_L, a commercial program's printed case, and twenty seeded flanges run through an independent numpy implementation of the whole method. The sources, page renders and the check scripts are filed in References/Bolted-Flange/.

Conventions and limits

Dimensions follow Figure 2-4 of the Code: A the flange outside diameter, B the bore, t the ring thickness, g₀ and g₁ the hub thickness at its small and large ends, h the hub length, C the bolt circle. The tool works in inches and pounds internally (the gasket rule is written in inches) and converts to and from SI through the site dictionary. Not covered: external pressure (2-11), reverse flanges (2-13), flat-face flanges with metal-to-metal contact outside the bolt circle (Appendix Y, the Part B rules), split and non-circular flanges, nubbin, serrated and tongue-and-groove facings, the gasket's own stiffness and relaxation, thermal gradients between bolts and flange, bolt bending, fatigue of the bolts, and the leakage-based methods of PCC-1 Appendix O and EN 1591.