Narrow face flange - smooth bore for external pressure

Values for calculation

Calculation

Maximum value of the nominal design stress for normal operating load cases

- Cast steels$$f_d=\min\left(\cfrac{R_{p0.2/T}}{1.9}, \cfrac{R_{m/20}}{3}\right)$$- Austenitic steels, A ≥ 35%$$f_d=\max\left[\cfrac{R_{p1.0/T}}{1.5}, \min\left(\cfrac{R_{p1.0/T}}{1.2}, \cfrac{R_{m/T}}{3}\right)\right]$$- Austenitic steels, 30% ≤ A < 35%$$f_d=\cfrac{R_{p1.0/T}}{1.5}$$- Steels other than austenitic, A ≤ 30%$$f_d=\min\left(\cfrac{R_{p0.2/T}}{1.5}, \cfrac{R_{m/20}}{2.4}\right)$$

Maximum value of the nominal design stress for testing load cases

- Cast steels$$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.33}$$- Austenitic steels, A ≥ 35%$$f_{test}=\max\left(\cfrac{R_{p1.0/T_{test}}}{1.05}, \cfrac{R_{m/T_{test}}}{2}\right)$$- Austenitic steels, 30% ≤ A < 35%$$f_{test}=\cfrac{R_{p1.0/T_{test}}}{1.05}$$- Steels other than austenitic, A ≤ 30%$$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.05}$$

Bolt nominal design stress for normal operating load cases

- Austenitic steels$$f_B=\cfrac{R_{m/bolt/T}}{4}$$- Steels other than austenitic$$f_B=\min\left(\cfrac{R_{p0.2/bolt/T}}{3}, \cfrac{R_{m/20/bolt}}{4}\right)$$

Bolt nominal design stress for testing load cases

- Austenitic steels$$f_{B_{test}}=\cfrac{R_{m/bolt/T_{test}}}{4}\cdot 1.5$$- Steels other than austenitic$$f_{B_{test}}=\min\left(\cfrac{R_{p0.2/bolt/T_{test}}}{3}, \cfrac{R_{m/20/bolt}}{4}\right)\cdot 1.5$$

Bolt nominal design stress at assembly temperature

- Austenitic steels$$f_{B,A}=\cfrac{R_{m/bolt/T_{assembly}}}{4}$$- Steels other than austenitic$$f_{B,A}=\min\left(\cfrac{R_{p0.2/bolt/T_{assembly}}}{3}, \cfrac{R_{m/20/bolt}}{4}\right)$$

Total hydrostatic end force

$$H=\cfrac{π}{4}\cdot\left(G^2\cdot P_e\right)$$

Total hydrostatic end force for testing load cases

$$H_{test}=\cfrac{π}{4}\cdot\left(G^2\cdot P_{e_{test}}\right)$$

Compression load on gasket to ensure tight joint

$$H_G=2\cdot π\cdot G\cdot b\cdot m\cdot P_e$$

Compression load on gasket to ensure tight joint for testing load cases

$$H_{G_{test}}=2\cdot π\cdot G\cdot b\cdot m\cdot P_{e_{test}}$$

Hydrostatic end force applied via shell to flange

$$H_D=\cfrac{π}{4}\cdot\left(B^2\cdot P_e\right)$$

Hydrostatic end force applied via shell to flange for testing load cases

$$H_{D_{test}}=\cfrac{π}{4}\cdot\left(B^2\cdot P_{e_{test}}\right)$$

Hydrostatic end force due to pressure on flange face

$$H_T=H-H_D$$

Hydrostatic end force due to pressure on flange face for testing load cases

$$H_{T_{test}}=H_{test}-H_{D_{test}}$$

Radial distance from bolt circle to circle on which $ H_D $ acts

$$h_D=\left(C-B-g_1\right)/2$$

Radial distance from gasket load reaction to bolt circle

$$h_G=\left(C-G\right)/2$$

Radial distance from bolt circle to circle on which $ H_T $ acts

$$h_T=\left(2\cdot C-B-G\right)/4$$

Minimum required bolt load for assembly condition

$$W_A=π\cdot b\cdot G\cdot y$$

Minimum required bolt load for operating condition

$$W_{op}=0$$

Minimum required bolt load for testing load cases

$$W_{test}=0$$

Total required cross-sectional area of bolts

$$A_{B,min}=\max\left(\cfrac{W_A}{f_{B,A}}, \cfrac{W_{op}}{f_B}, \cfrac{W_{test}}{f_{B_{test}}}\right)$$$$A_B\geq A_{B,min}$$

Design bolt load for assembly condition

$$W=0.5\cdot\left(A_{B,min}+A_B\right)\cdot f_{B,A}$$

Total moment acting upon flange for assembly condition

$$M_{T-A}=W\cdot h_G$$

Total moment acting upon flange for operating condition

$$M_{T-op}=H_D\cdot\left(h_D-h_G\right)+H_T\cdot\left(h_T-h_G\right)$$

Total moment acting upon flange for testing load cases

$$M_{T-test}=H_{D_{test}}\cdot\left(h_D-h_G\right)+H_{T_{test}}\cdot\left(h_T-h_G\right)$$

Distance between centre lines of adjacent bolts

$$δ_b=C\cdot\sin{\cfrac{π}{n}}$$

Bolt pitch correction factor

$$C_F=\max\left(\sqrt{\cfrac{δ_b}{2\cdot d_b+\cfrac{6\cdot e}{m+0.5}}}, 1\right)$$

Ratio of the flange diameters

$$K=A/B$$

Length parameter

$$l_0=\sqrt{B\cdot g_0}$$

Factor $ β_T $

$$β_T=\cfrac{K^2\cdot\left(1+8.55246\cdot\log_{10}{\left(K\right)}\right)-1}{\left(1.0472+1.9448\cdot K^2\right)\cdot\left(K-1\right)}$$

Factor $ β_U $

$$β_U=\cfrac{K^2\cdot\left(1+8.55246\cdot\log_{10}{\left(K\right)}\right)-1}{1.36136\cdot\left(K^2-1\right)\cdot\left(K-1\right)}$$

Factor $ β_Y $

$$β_Y=\cfrac{1}{K-1}\cdot\left(0.66845+5.7169\cdot\cfrac{K^2\cdot\log_{10}{\left(K\right)}}{K^2-1}\right)$$

Moment exerted on the flange per unit of length for assembly condition

$$M_{A}=M_{T-A}\cdot\cfrac{C_F}{B}$$

Moment exerted on the flange per unit of length for operating condition

$$M_{op}=M_{T-op}\cdot\cfrac{C_F}{B}$$

Moment exerted on the flange per unit of length for testing load cases

$$M_{test}=M_{T-test}\cdot\cfrac{C_F}{B}$$

Factor $ A $

$$A=\cfrac{g_1}{g_0}-1$$

Factor $ C $

$$C=48\cdot\left(1-ν^2\right)\cdot\left(\cfrac{h}{l_0}\right)^4$$

Factor $ C_1 $

$$C_1=\cfrac{1}{3}+\cfrac{A}{12}$$

Factor $ C_2 $

$$C_2=\cfrac{5}{42}+\cfrac{17\cdot A}{336}$$

Factor $ C_3 $

$$C_3=\cfrac{1}{210}+\cfrac{A}{360}$$

Factor $ C_4 $

$$C_4=\cfrac{11}{360}+\cfrac{59\cdot A}{5040}+\cfrac{1+3\cdot A}{C}$$

Factor $ C_5 $

$$C_5=\cfrac{1}{90}+\cfrac{5\cdot A}{1008}-\cfrac{\left(1+A\right)^3}{C}$$

Factor $ C_6 $

$$C_6=\cfrac{1}{120}+\cfrac{17\cdot A}{5040}+\cfrac{1}{C}$$

Factor $ C_7 $

$$C_7=\cfrac{215}{2772}+\cfrac{51\cdot A}{1232}+\left(\cfrac{120+225\cdot A+150\cdot A^2+35\cdot A^3}{14}\right)\cdot\cfrac{1}{C}$$

Factor $ C_8 $

$$C_8=\cfrac{31}{6930}+\cfrac{128\cdot A}{45045}+\left(\cfrac{66+165\cdot A+132\cdot A^2+35\cdot A^3}{77}\right)\cdot\cfrac{1}{C}$$

Factor $ C_9 $

$$C_9=\cfrac{533}{30240}+\cfrac{653\cdot A}{73920}+\left(\cfrac{42+198\cdot A+117\cdot A^2+25\cdot A^3}{84}\right)\cdot\cfrac{1}{C}$$

Factor $ C_{10} $

$$C_{10}=\cfrac{29}{3780}+\cfrac{3\cdot A}{704}-\left(\cfrac{42+198\cdot A+243\cdot A^2+91\cdot A^3}{84}\right)\cdot\cfrac{1}{C}$$

Factor $ C_{11} $

$$C_{11}=\cfrac{31}{6048}+\cfrac{1763\cdot A}{665280}+\left(\cfrac{42+72\cdot A+45\cdot A^2+10\cdot A^3}{84}\right)\cdot\cfrac{1}{C}$$

Factor $ C_{12} $

$$C_{12}=\cfrac{1}{2925}+\cfrac{71\cdot A}{300300}+\left(\cfrac{88+198\cdot A+156\cdot A^2+42\cdot A^3}{385}\right)\cdot\cfrac{1}{C}$$

Factor $ C_{13} $

$$C_{13}=\cfrac{761}{831600}+\cfrac{937\cdot A}{1663200}+\left(\cfrac{2+12\cdot A+11\cdot A^2+3\cdot A^3}{70}\right)\cdot\cfrac{1}{C}$$

Factor $ C_{14} $

$$C_{14}=\cfrac{197}{415800}+\cfrac{103\cdot A}{332640}-\left(\cfrac{2+12\cdot A+17\cdot A^2+7\cdot A^3}{70}\right)\cdot\cfrac{1}{C}$$

Factor $ C_{15} $

$$C_{15}=\cfrac{233}{831600}+\cfrac{97\cdot A}{554400}+\left(\cfrac{6+18\cdot A+15\cdot A^2+4\cdot A^3}{210}\right)\cdot\cfrac{1}{C}$$

Factor $ C_{16} $

$$C_{16}=C_1\cdot C_7\cdot C_{12}+C_2\cdot C_8\cdot C_3+C_3\cdot C_8\cdot C_2-\left(C_3^2\cdot C_7+C_8^2\cdot C_1+C_2^2\cdot C_{12}\right)$$

Factor $ C_{17} $

$$C_{17}=\left[C_4\cdot C_7\cdot C_{12}+C_2\cdot C_8\cdot C_{13}+C_3\cdot C_8\cdot C_9-\left(C_{13}\cdot C_7\cdot C_3+C_8^2\cdot C_4+C_{12}\cdot C_2\cdot C_9\right)\right]\cdot\cfrac{1}{C_{16}}$$

Factor $ C_{18} $

$$C_{18}=\left[C_5\cdot C_7\cdot C_{12}+C_2\cdot C_8\cdot C_{14}+C_3\cdot C_8\cdot C_{10}-\left(C_{14}\cdot C_7\cdot C_3+C_8^2\cdot C_5+C_{12}\cdot C_2\cdot C_{10}\right)\right]\cdot\cfrac{1}{C_{16}}$$

Factor $ C_{19} $

$$C_{19}=\left[C_6\cdot C_7\cdot C_{12}+C_2\cdot C_8\cdot C_{15}+C_3\cdot C_8\cdot C_{11}-\left(C_{15}\cdot C_7\cdot C_3+C_8^2\cdot C_6+C_{12}\cdot C_2\cdot C_{11}\right)\right]\cdot\cfrac{1}{C_{16}}$$

Factor $ C_{20} $

$$C_{20}=\left[C_1\cdot C_9\cdot C_{12}+C_4\cdot C_8\cdot C_3+C_3\cdot C_{13}\cdot C_2-\left(C_3^2\cdot C_9+C_{13}\cdot C_8\cdot C_1+C_{12}\cdot C_4\cdot C_2\right)\right]\cdot\cfrac{1}{C_{16}}$$

Factor $ C_{21} $

$$C_{21}=\left[C_1\cdot C_{10}\cdot C_{12}+C_5\cdot C_8\cdot C_3+C_3\cdot C_{14}\cdot C_2-\left(C_3^2\cdot C_{10}+C_{14}\cdot C_8\cdot C_1+C_{12}\cdot C_5\cdot C_2\right)\right]\cdot\cfrac{1}{C_{16}}$$

Factor $ C_{22} $

$$C_{22}=\left[C_1\cdot C_{11}\cdot C_{12}+C_6\cdot C_8\cdot C_3+C_3\cdot C_{15}\cdot C_2-\left(C_3^2\cdot C_{11}+C_{15}\cdot C_8\cdot C_1+C_{12}\cdot C_6\cdot C_2\right)\right]\cdot\cfrac{1}{C_{16}}$$

Factor $ C_{23} $

$$C_{23}=\left[C_1\cdot C_7\cdot C_{13}+C_2\cdot C_9\cdot C_3+C_4\cdot C_8\cdot C_2-\left(C_3\cdot C_7\cdot C_4+C_8\cdot C_9\cdot C_1+C_2^2\cdot C_{13}\right)\right]\cdot\cfrac{1}{C_{16}}$$

Factor $ C_{24} $

$$C_{24}=\left[C_1\cdot C_7\cdot C_{14}+C_2\cdot C_{10}\cdot C_3+C_5\cdot C_8\cdot C_2-\left(C_3\cdot C_7\cdot C_5+C_8\cdot C_{10}\cdot C_1+C_2^2\cdot C_{14}\right)\right]\cdot\cfrac{1}{C_{16}}$$

Factor $ C_{25} $

$$C_{25}=\left[C_1\cdot C_7\cdot C_{15}+C_2\cdot C_{11}\cdot C_3+C_6\cdot C_8\cdot C_2-\left(C_3\cdot C_7\cdot C_6+C_8\cdot C_{11}\cdot C_1+C_2^2\cdot C_{15}\right)\right]\cdot\cfrac{1}{C_{16}}$$

Factor $ C_{26} $

$$C_{26}=-\left(\cfrac{C}{4}\right)^{1/4}$$

Factor $ C_{27} $

$$C_{27}=C_{20}-C_{17}-\cfrac{5}{12}+C_{17}\cdot C_{26}$$

Factor $ C_{28} $

$$C_{28}=C_{22}-C_{19}-\cfrac{1}{12}+C_{19}\cdot C_{26}$$

Factor $ C_{29} $

$$C_{29}=-\left(\cfrac{C}{4}\right)^{1/2}$$

Factor $ C_{30} $

$$C_{30}=-\left(\cfrac{C}{4}\right)^{3/4}$$

Factor $ C_{31} $

$$C_{31}=\cfrac{3\cdot A}{2}-C_{17}\cdot C_{30}$$

Factor $ C_{32} $

$$C_{32}=\cfrac{1}{2}-C_{19}\cdot C_{30}$$

Factor $ C_{33} $

$$C_{33}=\cfrac{C_{26}\cdot C_{32}}{2}+C_{28}\cdot C_{31}\cdot C_{29}-\left(\cfrac{C_{30}\cdot C_{28}}{2}+C_{32}\cdot C_{27}\cdot C_{29}\right)$$

Factor $ C_{34} $

$$C_{34}=\cfrac{1}{12}+C_{18}-C_{21}-C_{18}\cdot C_{26}$$

Factor $ C_{35} $

$$C_{35}=C_{18}\cdot C_{30}$$

Factor $ C_{36} $

$$C_{36}=\left(C_{28}\cdot C_{35}\cdot C_{29}-C_{32}\cdot C_{34}\cdot C_{29}\right)\cdot\cfrac{1}{C_{33}}$$

Factor $ C_{37} $

$$C_{37}=\left(\cfrac{C_{26}\cdot C_{35}}{2}+C_{34}\cdot C_{31}\cdot C_{29}-\cfrac{C_{30}\cdot C_{34}}{2}-C_{35}\cdot C_{27}\cdot C_{29}\right)\cdot\cfrac{1}{C_{33}}$$

Factor $ E_1 $

$$E_1=C_{17}\cdot C_{36}+C_{18}+C_{19}\cdot C_{37}$$

Factor $ E_2 $

$$E_2=C_{20}\cdot C_{36}+C_{21}+C_{22}\cdot C_{37}$$

Factor $ E_3 $

$$E_3=C_{23}\cdot C_{36}+C_{24}+C_{25}\cdot C_{37}$$

Factor $ E_4 $

$$E_4=\cfrac{3+C_{37}+3\cdot C_{36}}{12}-\cfrac{2\cdot E_3+15\cdot E_2+10\cdot E_1}{10}$$

Factor $ E_5 $

$$E_5=E_1\cdot\left(\cfrac{3+A}{6}\right)+E_2\cdot\left(\cfrac{21+11\cdot A}{84}\right)+E_3\cdot\left(\cfrac{3+2\cdot A}{210}\right)$$

Factor $ E_6 $

$$E_6=E_5-C_{36}\cdot\left(\cfrac{7}{120}+\cfrac{A}{36}+\cfrac{3\cdot A}{C}\right)-\cfrac{1}{40}-\cfrac{A}{72}-C_{37}\cdot\left(\cfrac{1}{60}+\cfrac{A}{120}+\cfrac{1}{C}\right)$$

Factor $ β_F $

$$β_F=\cfrac{-E_6}{\left[\cfrac{C}{3\cdot\left(1-ν^2\right)}\right]^{1/4}\cdot\cfrac{\left(1+A\right)^3}{C}}$$

Factor $ β_V $

$$β_V=\cfrac{E_4}{\left[\cfrac{3\cdot\left(1-ν^2\right)}{C}\right]^{1/4}\cdot\left(1+A\right)^3}$$

Hub stress correction factor

$$φ=\cfrac{C_{36}}{1+A}$$

Factor $ λ $

$$λ=\left(\cfrac{e\cdot β_F+l_0}{β_T\cdot l_0}+\cfrac{e^3\cdot β_V}{β_U\cdot l_0\cdot g_0^2}\right)$$

Longitudinal stress in hub for assembly condition

$$σ_{H_{A}}=\cfrac{φ\cdot M_{A}}{λ\cdot g_1^2}$$

Longitudinal stress in hub for operating condition

$$σ_{H_{op}}=\cfrac{φ\cdot M_{op}}{λ\cdot g_1^2}$$

Longitudinal stress in hub for testing load cases

$$σ_{H_{test}}=\cfrac{φ\cdot M_{test}}{λ\cdot g_1^2}$$

Radial stress in flange for assembly condition

$$σ_{r_{A}}=\cfrac{\left(1.333\cdot e\cdot β_F+l_0\right)\cdot M_{A}}{λ\cdot e^2\cdot l_0}$$

Radial stress in flange for operating condition

$$σ_{r_{op}}=\cfrac{\left(1.333\cdot e\cdot β_F+l_0\right)\cdot M_{op}}{λ\cdot e^2\cdot l_0}$$

Radial stress in flange for testing load cases

$$σ_{r_{test}}=\cfrac{\left(1.333\cdot e\cdot β_F+l_0\right)\cdot M_{test}}{λ\cdot e^2\cdot l_0}$$

Tangential stress in flange for assembly condition

$$σ_{θ_{A}}=\cfrac{β_Y\cdot M_{A}}{e^2}-σ_{r_{A}}\cdot\cfrac{K^2+1}{K^2-1}$$

Tangential stress in flange for operating condition

$$σ_{θ_{op}}=\cfrac{β_Y\cdot M_{op}}{e^2}-σ_{r_{op}}\cdot\cfrac{K^2+1}{K^2-1}$$

Tangential stress in flange for testing load cases

$$σ_{θ_{test}}=\cfrac{β_Y\cdot M_{test}}{e^2}-σ_{r_{test}}\cdot\cfrac{K^2+1}{K^2-1}$$

Stress factor

$\text{if }\ B\le 1000$$$k=1$$$\text{else if }\ B\geq 2000$$$k=1.333$$$\text{else}$$$k=\cfrac{2}{3}\cdot\left(1+\cfrac{B}{2000}\right)$$

Requirements

$$g_1\le h+g_0$$$$k\cdot σ_{H_{A}}\le 1.5\cdot f_d$$$$k\cdot σ_{r_{A}}\le f_d$$$$k\cdot σ_{θ_{A}}\le f_d$$$$0.5\cdot k\cdot\left(σ_{H_{A}}+σ_{r_{A}}\right)\le f_d$$$$0.5\cdot k\cdot\left(σ_{H_{A}}+σ_{θ_{A}}\right)\le f_d$$$$k\cdot σ_{H_{op}}\le 1.5\cdot f_d$$$$k\cdot σ_{r_{op}}\le f_d$$$$k\cdot σ_{θ_{op}}\le f_d$$$$0.5\cdot k\cdot\left(σ_{H_{op}}+σ_{r_{op}}\right)\le f_d$$$$0.5\cdot k\cdot\left(σ_{H_{op}}+σ_{θ_{op}}\right)\le f_d$$$$k\cdot σ_{H_{test}}\le 1.5\cdot f_{test}$$$$k\cdot σ_{r_{test}}\le f_{test}$$$$k\cdot σ_{θ_{test}}\le f_{test}$$$$0.5\cdot k\cdot\left(σ_{H_{test}}+σ_{r_{test}}\right)\le f_{test}$$$$0.5\cdot k\cdot\left(σ_{H_{test}}+σ_{θ_{test}}\right)\le f_{test}$$