# Reverse narrow face flange for external pressure

## Values for calculation

$T$ $\mathrm{°C}$
$T_{test}$ $\mathrm{°C}$
$T_{assembly}$ $\mathrm{°C}$
$P_e$ $\mathrm{MPa}$
$P_{e_{test}}$ $\mathrm{MPa}$
$R_{p0.2/T}$ $\mathrm{MPa}$
$R_{p0.2/T_{test}}$ $\mathrm{MPa}$
$R_{p1.0/T}$ $\mathrm{MPa}$
$R_{p1.0/T_{test}}$ $\mathrm{MPa}$
$R_{m/20}$ $\mathrm{MPa}$
$R_{m/T}$ $\mathrm{MPa}$
$R_{m/T_{test}}$ $\mathrm{MPa}$
$R_{p0.2/bolt/T}$ $\mathrm{MPa}$
$R_{p0.2/bolt/T_{test}}$ $\mathrm{MPa}$
$R_{p0.2/bolt/T_{assembly}}$ $\mathrm{MPa}$
$R_{m/20/bolt}$ $\mathrm{MPa}$
$R_{m/bolt/T}$ $\mathrm{MPa}$
$R_{m/bolt/T_{test}}$ $\mathrm{MPa}$
$R_{m/bolt/T_{assembly}}$ $\mathrm{MPa}$
$A$ $\mathrm{mm}$
$C$ $\mathrm{mm}$
$B$ $\mathrm{mm}$
$D_i$ $\mathrm{mm}$
$G$ $\mathrm{mm}$
$e$ $\mathrm{mm}$
$d_b$ $\mathrm{mm}$
$n$
$g_0$ $\mathrm{mm}$
$g_1$ $\mathrm{mm}$
$h$ $\mathrm{mm}$
$b$ $\mathrm{mm}$
$m$
$y$ $\mathrm{MPa}$
$ν$
$A_B$ $\mathrm{mm^2}$

## Calculation

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

$\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Cast steels}$
$$f_d=\min\left(\cfrac{R_{p0.2/T}}{1.9}, \cfrac{R_{m/20}}{3}\right)$$
$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge\text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}\geq 35$$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]$$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}< 35$
$$f_d=\cfrac{R_{p1.0/T}}{1.5}$$
$\text{else}$
$$f_d=\min\left(\cfrac{R_{p0.2/T}}{1.5}, \cfrac{R_{m/20}}{2.4}\right)$$

### Maximum allowed value of the nominal design stress for testing load cases

$\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Cast steels}$
$$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.33}$$
$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge\text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}\geq 35$$f_{test}=\max\left(\cfrac{R_{p1.0/T_{test}}}{1.05}, \cfrac{R_{m/T_{test}}}{2}\right)$$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}< 35$
$$f_{test}=\cfrac{R_{p1.0/T_{test}}}{1.05}$$
$\text{else}$
$$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.05}$$

### Bolt nominal design stress for normal operating load cases

$\text{if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{bolt}= \text{Austenitic steels}$
$$f_B=\cfrac{R_{m/bolt/T}}{4}$$
$\text{else}$
$$f_B=\min\left(\cfrac{R_{p0.2/bolt/T}}{3}, \cfrac{R_{m/20/bolt}}{4}\right)$$

### Bolt nominal design stress for testing load cases

$\text{if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{bolt}= \text{Austenitic steels}$
$$f_{B_{test}}=\cfrac{R_{m/bolt/T_{test}}}{4}\cdot 1.5$$
$\text{else}$
$$f_{B_{test}}=\min\left(\cfrac{R_{p0.2/bolt/T_{test}}}{3}, \cfrac{R_{m/20/bolt}}{4}\right)\cdot 1.5$$

### Bolt nominal design stress at assembly temperature

$\text{if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{bolt}= \text{Austenitic steels}$
$$f_{B,A}=\cfrac{R_{m/bolt/T_{assembly}}}{4}$$
$\text{else}$
$$f_{B,A}=\min\left(\cfrac{R_{p0.2/bolt/T_{assembly}}}{3}, \cfrac{R_{m/20/bolt}}{4}\right)$$

### 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

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

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

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

### Net pressure load on the flange faces

$$H_T=H_D-H$$

### Net pressure load on the flange faces for testing load cases

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

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

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

### 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

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

### 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

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

### Total moment acting upon flange for testing load cases

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

### 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

$$K=B/A$$

### 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

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

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

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

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

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

### 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

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