# Flat face flanges with metal-to-metal contact outside the bolt circle for class 1 flange assembly (identical flange pairs)

## Values for calculation

$T$ $\mathrm{°C}$
$S_f$ $\mathrm{MPa}$
$S_n$ $\mathrm{MPa}$
$S_b$ $\mathrm{MPa}$
$P$ $\mathrm{MPa}$
$A$ $\mathrm{mm}$
$B$ $\mathrm{mm}$
$C$ $\mathrm{mm}$
$n$
$d_b$ $\mathrm{mm}$
$D$ $\mathrm{mm}$
$G$ $\mathrm{mm}$
$h_C$ $\mathrm{mm}$
$t$ $\mathrm{mm}$
$g_0$ $\mathrm{mm}$
$g_1$ $\mathrm{mm}$
$h$ $\mathrm{mm}$
$A_b$ $\mathrm{mm^2}$
$H_G$ $\mathrm{N}$
$t_s$ $\mathrm{mm}$
$r_E$

## Calculation

### Factor $h_0$

$$h_0=\sqrt{B\cdot g_0}$$

### Factor $A$

$$A=\left(g_1/g_0\right)-1$$

### Factor $C$

$$C=43.68\cdot\left(h/h_0\right)^4$$

### Factor $C_1$

$$C_1=1/3+A/12$$

### Factor $C_2$

$$C_2=5/42+17\cdot A/336$$

### Factor $C_3$

$$C_3=1/210+A/360$$

### Factor $C_4$

$$C_4=11/360+59\cdot A/5040+\left(1+3\cdot A\right)/C$$

### Factor $C_5$

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

### Factor $C_6$

$$C_6=1/120+17\cdot A/5040 +1/C$$

### Factor $C_7$

$$C_7=215/2772+51\cdot A/1232+\left(60/7+225\cdot A/14+75\cdot A^2/7+5\cdot A^3/2\right)/C$$

### Factor $C_8$

$$C_8=31/6930+128\cdot A/45045+\left(6/7+15\cdot A/7+12\cdot A^2/7+5\cdot A^3/11\right)/C$$

### Factor $C_9$

$$C_9=533/30240+653\cdot A/73920+\left(1/2+33\cdot A/14+39\cdot A^2/28+25\cdot A^3/84\right)/C$$

### Factor $C_{10}$

$$C_{10}=29/3780+3\cdot A/704-\left(1/2+33\cdot A/14+81\cdot A^2/28+13\cdot A^3/12\right)/C$$

### Factor $C_{11}$

$$C_{11}=31/6048+1763\cdot A/665280+\left(1/2+6\cdot A/7+15\cdot A^2/28+5\cdot A^3/42\right)/C$$

### Factor $C_{12}$

$$C_{12}=1/2925+71\cdot A/300300+\left(8/35+18\cdot A/35+156\cdot A^2/385+6\cdot A^3/55\right)/C$$

### Factor $C_{13}$

$$C_{13}=761/831600+937\cdot A/1663200+\left(1/35+6\cdot A/35+11\cdot A^2/70+3\cdot A^3/70\right)/C$$

### Factor $C_{14}$

$$C_{14}=197/415800+103\cdot A/332640-\left(1/35+6\cdot A/35+17\cdot A^2/70+A^3/10\right)/C$$

### Factor $C_{15}$

$$C_{15}=233/831600+97\cdot A/554400+\left(1/35+3\cdot A/35+A^2/14+2\cdot A^3/105\right)/C$$

### 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 $C_{17}$

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

### Factor $C_{18}$

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

### Factor $C_{19}$

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

### Factor $C_{20}$

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

### Factor $C_{21}$

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

### Factor $C_{22}$

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

### Factor $C_{23}$

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

### Factor $C_{24}$

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

### Factor $C_{25}$

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

### Factor $C_{26}$

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

### Factor $C_{27}$

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

### Factor $C_{28}$

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

### Factor $C_{29}$

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

### Factor $C_{30}$

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

### Factor $C_{31}$

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

### Factor $C_{32}$

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

### Factor $C_{33}$

$$C_{33}=0.5\cdot C_{26}\cdot C_{32}+C_{28}\cdot C_{31}\cdot C_{29}-\left(0.5\cdot C_{30}\cdot C_{28}+C_{32}\cdot C_{27}\cdot C_{29}\right)$$

### Factor $C_{34}$

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

### Factor $C_{35}$

$$C_{35}=-C_{18}\cdot\left(C/4\right)^{3/4}$$

### Factor $C_{36}$

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

### Factor $C_{37}$

$$C_{37}=\left[0.5\cdot C_{26}\cdot C_{35}+C_{34}\cdot C_{31}\cdot C_{29}-\left(0.5\cdot C_{30}\cdot C_{34}+C_{35}\cdot C_{27}\cdot C_{29}\right)\right]/C_{33}$$

### 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 $E_4$

$$E_4=1/4+C_{37}/12+C_{36}/4-E_3/5-3\cdot E_2/2-E_1$$

### Factor $E_5$

$$E_5=E_1\cdot\left(1/2+A/6\right)+E_2\cdot\left(1/4+11\cdot A/84\right)+E_3\cdot\left(1/70+A/105\right)$$

### Factor $E_6$

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

### Factor $F$ for integral type flanges

$$F=-\cfrac{E_6}{\left(\cfrac{C}{2.73}\right)^{\cfrac{1}{4}}\cdot\cfrac{\left(1+A\right)^3}{C}}$$

### Factor $F_L$ for loose type flanges

$$F_L=-\cfrac{C_{18}\cdot\left(\cfrac{1}{2}+\cfrac{A}{6}\right)+C_{21}\cdot\left(\cfrac{1}{4}+\cfrac{11\cdot A}{84}\right)+C_{24}\cdot\left(\cfrac{1}{70}+\cfrac{A}{105}\right)-\left(\cfrac{1}{40}+\cfrac{A}{72}\right)}{\left(\cfrac{C}{2.73}\right)^{\cfrac{1}{4}}\cdot\cfrac{\left(1+A\right)^3}{C}}$$

### Factor $V$ for integral type flanges

$$V=\cfrac{E_4}{\left(\cfrac{2.73}{C}\right)^{\cfrac{1}{4}}\cdot\left(1+A\right)^3}$$

### Factor $V_L$ for loose type flanges

$$V_L=\cfrac{\cfrac{1}{4}-\cfrac{C_{24}}{5}-\cfrac{3\cdot C_{21}}{2}-C_{18}}{\left(\cfrac{2.73}{C}\right)^{\cfrac{1}{4}}\cdot\left(1+A\right)^3}$$

### Factor $F'$

$\text{if }\ \text{flange }$$\text{category}= \text{Category 1}$$F'=g_0^2\cdot\left(h_0+F\cdot t\right)/V$$\text{else if }\ \text{flange }$$\text{category}= \text{Category 2}$
$$F'=g_0^2\cdot\left(h_0+F_L\cdot t\right)/V_L$$
$\text{else}$
$$F'=0$$

$\text{if }\ \text{flange }$$\text{category}= \text{Category 1}$$f=C_{36}/\left(1+A\right)$$\text{else}$$f=1$$### Diameter B_1 \text{if }\ \text{flange }$$\text{category}= \text{Category 3}$
$$B_1=B$$
$\text{else if }\ \text{flange }$$\text{category}= \text{Category 1}\wedge f\geq 1$$B_1=B+g_0$$\text{else}$$B_1=B+g_1$$### Radial distance from bolt circle to point of intersection of hub and back of flange$$R=\cfrac{C-B}{2}-g_1$$### Radial distance from the bolt circle, to the circle on which H_D acts \text{if }\ \text{flange }$$\text{category}= \text{Category 1}$
$$h_D=R+0.5\cdot g_1$$
$\text{else}$
$$h_D=\cfrac{C-B}{2}$$

### Radial distance from bolt circle to outer edge of flange or spacer

$$h_{Cmax}=\cfrac{A-C}{2}$$

$$h_C< h_{Cmax}$$

### Shape factor for full face metal‐to‐metal contact flanges

$$β=\cfrac{C+B_1}{2\cdot B_1}$$

### Shape factor

$$a=\cfrac{A+C}{2\cdot B_1}$$

### Bolt hole aspect ratio used in calculating bolthole flexibility factor $r_B$

$$\overline{AR}=\cfrac{n\cdot D}{π\cdot C}$$

### Factor $r_B$

$$r_B=\cfrac{1}{n}\cdot\left(\cfrac{4}{\sqrt{1-\overline{AR}^2}}\cdot\tan^{-1}\left(\sqrt{\cfrac{1+\overline{AR}}{1-\overline{AR}}}\right)-π-2\cdot \overline{AR}\right)$$

### Factor $J_S$

$$J_S=\cfrac{1}{B_1}\cdot\left[\cfrac{2\cdot h_D}{β}+\cfrac{h_C}{a}\right]+π\cdot r_B$$

### Factor $J_P$

$$J_P=\cfrac{1}{B_1}\cdot\left[\cfrac{h_D}{β}+\cfrac{h_C}{a}\right]+π\cdot r_B$$

$$h_G=\cfrac{C-G}{2}$$

$\text{if }\ \text{flange }$$\text{category}= \text{Category 1}$$h_T=\cfrac{R+g_1+h_G}{2}$$\text{else}$$h_T=\cfrac{h_D+h_G}{2}$$### Total hydrostatic end force$$H=\cfrac{π}{4}\cdot G^2\cdot P$$### Hydrostatic end force on area inside of flange$$H_D=\cfrac{π}{4}\cdot B^2\cdot P$$### Difference between total hydrostatic end force and the hydrostatic end force on area inside of flange$$H_T=H-H_D$$### Moment due to H_D , H_T , H_G$$M_P=H_D\cdot h_D+H_T\cdot h_T+H_G\cdot h_G$$### Flange Moment Due to Flange‐Hub Interaction$$M_S=-\cfrac{J_P\cdot F'\cdot M_P}{t^3+J_S\cdot F'}$$### Slope of Flange at Inside Diameter Times E$$Eθ_B=\cfrac{5.46}{π\cdot t^3}\cdot\left(J_S\cdot M_S+J_P\cdot M_P\right)$$### Contact force between flanges at h_C$$H_C=\left(M_P+M_S\right)/h_C$$### Bolt load at operating conditions$$W_{m1}=H+H_G+H_C$$### Operating bolt stress$$σ_b=W_{m1}/A_bσ_b\le S_b$$### Strain length of bolt$$l=2\cdot t+t_s+1/2\cdot d_b$$### Design prestress in bolts$$S_i=σ_b-\cfrac{1.159\cdot h_C^2\cdot\left(M_P+M_S\right)}{a\cdot t^3\cdot l\cdot r_E\cdot B_1}$$### Radial flange stress at bolt circle$$S_R=\cfrac{6\cdot\left(M_P+M_S\right)}{t^2\cdot\left(π\cdot C-n\cdot D\right)}S_R\le S_f$$### Radial flange stress at inside diameter \text{if }\ \text{flange }$$\text{category}= \text{Category 1}$
$$S_R=-\left(\cfrac{2\cdot F\cdot t}{h_0+F\cdot t}+6\right)\cdot\cfrac{M_S}{π\cdot B_1\cdot t^2}$$
$\text{else if }\ \text{flange }$$\text{category}= \text{Category 2}$$S_R=-\left(\cfrac{2\cdot F_L\cdot t}{h_0+F_L\cdot t}+6\right)\cdot\cfrac{M_S}{π\cdot B_1\cdot t^2}$$\text{else}$$S_R=0S_R\le S_f$$### Ratio of outside diameter of flange to inside diameter of flange$$K=A/B$$### Factor Z$$Z=\cfrac{K^2+1}{K^2-1}$$### Tangential flange stress at inside diameter \text{if }\ \text{flange }$$\text{category}= \text{Category 1}$
$$S_T=\cfrac{t\cdot Eθ_B}{B_1}+\left(\cfrac{2\cdot F\cdot t\cdot Z}{h_0+F\cdot t}-1.8\right)\cdot\cfrac{M_S}{π\cdot B_1\cdot t^2}$$
$\text{else if }\ \text{flange }$$\text{category}= \text{Category 2}$$S_T=\cfrac{t\cdot Eθ_B}{B_1}+\left(\cfrac{2\cdot F_L\cdot t\cdot Z}{h_0+F_L\cdot t}-1.8\right)\cdot\cfrac{M_S}{π\cdot B_1\cdot t^2}$$\text{else}$$S_T=\cfrac{t\cdot Eθ_B}{B_1}S_T\le S_f$$### Longitudinal hub stress \text{if }\ \text{flange }$$\text{category}= \text{Category 1}$
$$S_H=\cfrac{h_0\cdot Eθ_B\cdot f}{0.91\cdot\left(g_1/g_0\right)^2\cdot B_1\cdot V}$$
$\text{else if }\ \text{flange }$$\text{category}= \text{Category 2}$$S_H=\cfrac{h_0\cdot Eθ_B}{0.91\cdot\left(g_1/g_0\right)^2\cdot B_1\cdot V_L}$$\text{else}$$S_H=0$$\text{if }\ \text{product }$$\text{form }$$\text{flange}= \text{Castings}$$S_H\le S_f$$\text{else}$$S_H\le 1.5\cdot S_f$$\text{if }\ \text{product }$$\text{form }$$\text{nozzle }$$\text{neck }$$\text{vessel }$$\text{or }$$\text{pipe }$$\text{wall}= \text{Castings}$
$$S_H\le S_n$$
$\text{else}$
$$S_H\le 1.5\cdot S_n$$

## Requirements

$$\left(S_H+S_R\right)/2 \le S_f$$$$\left(S_H+S_T\right)/2 \le S_f$$