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PASS/NOZZLE-FEM 3.5. Program Manual | |
In the course of branch pipe calculation relative nondimensional forces and moments are used, which have been obtained as the result of spherical shell calculation.
Circumferential membrane stresses of $F_R$ force:
| $$ \sigma_{m\theta}(F_R) = n_{\theta F} \frac{F_R}{(s-c)^2}, $$ | (5.97) |
where $n_{\theta F}$ is assigned as per [9].
Circumferential bending stresses in the branch pipe from $F_R$ force in comparison with the membrane ones are considerably lower, so they are neglected.
Axial membrane stresses of $F_R$ force:
| $$ \sigma_{ma}(F_R) = \frac{F_R}{A}, $$ | (5.98) |
where $A=\frac{\displaystyle\pi \left((d+2s_1)^2-(d+2c_s)^2\right)}{\displaystyle 4}$ - design cross-sectional area of the nozzle.
Axial bending stresses of $F_R$ force for all design points:
| $$ \sigma_{ba}(F_R) = \left(6m_{RF}-3n_{RF}\right) \frac{F_R}{(s_1-c_s)^2}, $$ | (5.99) |
where $m_{RF}$ and $n_{RF}$ is assigned as per [9].
Circumferential membrane stresses in design points 1-4 from the moment $M_2$ and points 5-8 from the moment $M_2$:
| $$ \sigma_{m\theta}(M) = n_{\theta M} \frac{M}{(s-c)^2(d+2s_1)}, $$ | (5.100) |
where $n_{\theta M}$ is assigned as per [9].
Circumferential bending stresses of $M_{1(2)}$ moment in comparison with the membrane ones are considerably lower, so they are neglected.
Axial membrane stresses in design points 1-4 from the moment $M_1$ and points 5-8 from the moment $M_2$:
| $$ \sigma_{ma}(M) = \frac{M}{W_s}, $$ | (5.101) |
where $W_s=\frac{\displaystyle\pi\left((d+2s_1)^4-(d+2c_s)^4\right)}{\displaystyle 32(d+2s_1)}$ - design resisting moment of the nozzle cross section bending.
Axial bending stresses in design points 1-4 from the moment $M_1$ and points 5-8 from the moment $M_2$:
| $$ \sigma_{ba}(M) = \left(6m_{RM}-3n_{RM}\right) \frac{M}{(s_1-c_s)^2(d+2s_1)}, $$ | (5.102) |
where $m_{RM}$ and $n_{RM}$ is assigned as per [9].
Due to the torsional moment shearing stresses are created in the junction between nozzle and shell:
| $$ \tau_{\theta x} = \frac{M_T}{2\pi r^2_0 (s_1-c_s)}. $$ | (5.103) |
Forces $F_1$ in points 5-8 and $F_2$ in points 1-4 create shear membrane stresses:
| $$ \tau_{x\theta} = \frac{F_{1(2)}}{\pi r_0 (s-c)}. $$ | (5.104) |
As in the case of the spherical shell, local membrane stresses from internal pressure are calculated depending on stress intensification factor (SIF), obtained for cross section of the shell $I_{\theta p} = I_{xp}$.
Circumferential stress from internal pressure in all design points:
| $$ \sigma_{\theta p} = p I_{\theta p} \frac{D+(s+s_2-c)}{4(s+s_2-c)}. $$ | (5.105) |
If a design ratio is $I_{\theta p} < 1$, then for calculation of circumferential stresses in all branch connection design points the stress intensification factor (SIF) is to be substituted by $0.5(1+I_{\theta p})$, which shall be included in the (5.105) equation.
Axial stress from internal pressure in all design points is calculated by equation (%ref(equ_wrc_22)%).
In general, all the external loads applied to the nozzle can be distributed by three directions, i.e. can be shown as simultaneously acting forces $F_R$, $F_1$, $F_2$ and moments $M_1$, $M_2$, $M_t$. After calculation of stresses from effective forces and pressure, total stresses in design points (1-8) are calculated with taking into consideration of signs (table 5.18).
In the presence of corrosive hydrogen sulphide environment, a supplementary calculation of tensile stresses on the nozzle inside surfaces is made (2, 4, 6, 8 points):
| $$ \sigma_{1in} = \max{\left\{ \frac{1}{2} \left( \sigma_{\theta}+\sigma_a+\sqrt{(\sigma_{\theta}-\sigma_a)^2+4\tau^2_{\theta a}} \right); 0 \right\}}. $$ | (5.106) |
| Table 5.18. Local stresses (taking into account the signs) of the nozzle in branch connection design points loaded by the internal pressure and external loads as per WRC 107(297) | ||||||||
| Circumferential stresses, $\sigma_{\theta}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Membrane from $F_R$ | - | - | - | - | - | - | - | - |
| Membrane from $M_1$ | - | - | + | + | ||||
| Membrane from $M_2$ | - | - | + | + | ||||
| Circumferential stresses from pressure $\sigma_{\theta p}$ | + | + | + | + | + | + | + | + |
| Total circumferential membrane stresses $\sigma_{m\theta}$ | ||||||||
| Total circumferential stresses $\sigma_{\theta}$ | ||||||||
| Axial stresses, $\sigma_a$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Membrane from $F_R$ | - | - | - | - | - | - | - | - |
| Bending from $F_R$ | - | + | - | + | - | + | - | + |
| Membrane from $M_1$ | - | - | + | + | ||||
| Bending from $M_1$ | - | + | + | - | ||||
| Membrane from $M_2$ | - | - | + | + | ||||
| Bending from $M_2$ | - | + | + | - | ||||
| Axial stresses from pressure $\sigma_{ap}$ | + | + | + | + | + | + | + | + |
| Total membrane axial stresses $\sigma_{ma}$ | ||||||||
| Total axial stresses $\sigma_{a}$ | ||||||||
| Shearing stresses from $M_t$ | + | + | + | + | + | + | + | + |
| Shearing stresses from $F_1$ | - | - | + | + | ||||
| Shearing stresses from $F_2$ | + | + | - | - | ||||
| Total shearing stresses $\tau_{\theta a}$ | ||||||||
| Reduced total stresses $\sigma_{eqv}$ | ||||||||
| Tensile stresses on the shell inside surface $\sigma_{1in}$ | ||||||||
PASS/NOZZLE-FEM 3.5. Program Manual
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