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PASS/NOZZLE-FEM 3.5. Program Manual | |
As well in the course of cylindrical shell at strength calculation of branch pipe junction zone, the relative nondimensional forces and moments are used, subject to geometrical parameters [8].
Nondimensional geometrical parameters:
| $$ \lambda = \frac{d+2s_1}{D+s+c} \sqrt{\frac{D+s+c}{s-c}}, $$ | (5.76) |
| $$ \eta = \frac{d+2s_1}{s_1-c_s}, $$ | (5.77) |
| $$ \rho = \frac{s-c}{s_1-c_s}. $$ | (5.78) |
Related nondimensional forces and moments in circumferential ($\theta$) direction:
| $n_{\theta F}$ | - | membrane force due to $F_R$ |
| $m_{\theta F}$ | - | bending moment due to $F_R$ |
| $n_{\theta MC}$ | - | membrane force due to $M_C$; |
| $m_{\theta MC}$ | - | bending moment due to $M_C$ |
| $n_{\theta ML}$ | - | membrane force due to $M_L$; |
| $m_{\theta ML}$ | - | bending moment due to $M_L$. |
Nondimensional forces and moments in axial (a) direction:
| $n_{RF}$ | - | membrane force due to $F_R$ |
| $m_{RF}$ | - | bending moment due to $F_R$ |
| $n_{RMC}$ | - | membrane force due to $M_C$; |
| $m_{RMC}$ | - | bending moment due to $M_C$ |
| $n_{RML}$ | - | membrane force due to $M_L$; |
| $m_{RML}$ | - | bending moment due to $M_L$. |
Circumferential membrane stresses of $F_R$ force:
| $$ \sigma_{m\theta}(F_R) = n_{\theta F} \frac{F_R}{(s-c)^2}, $$ | (5.79) |
where $n_{\theta F}$ is assigned as per [9].
As compared to membrane, circumferential bending stresses from $F_R$ force is considerably lower, therefore their values are neglected.
Longitudinal membrane stresses from $F_R$ force:
| $$ \sigma_{ma}(F_R) = \frac{F_R}{A}, $$ | (5.80) |
where $A=\frac{\displaystyle\pi \left((d+2s_1)^2-(d+2c_s)^2\right)}{\displaystyle 4}$ - design cross-sectional area of branch pipe.
Longitudinal bending stresses from $F_R$ force for all design points:
| $$ \sigma_{ba}(F_R) = \left(6m_{RF}-3n_{RF}\right) \frac{F_R}{(s_1-c_s)^2}, $$ | (5.81) |
where $m_{RF}$ and $n_{RF}$ is assigned as per [9].
Circumferential membrane stresses from $M_C$ moment:
| $$ \sigma_{m\theta}(M_C) = n_{\theta MC} \frac{M_C}{(s-c)^2(d+2s_1)}, $$ | (5.82) |
where $n_{\theta MC}$ is assigned as per [9].
As compared to membrane, circumferential bending stresses from $M_C$ moment
Longitudinal membrane stresses from $M_C$ moment
| $$ \sigma_{ma}(M_C) = \frac{M_C}{W_s}, $$ | (5.83) |
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.
Longitudinal bending stresses from $M_C$ moment:
| $$ \sigma_{ba}(M_C) = \left(6m_{RMC}-3n_{RMC}\right) \frac{M_C}{(s_1-c_s)^2(d+2s_1)}, $$ | (5.84) |
where $m_{RMC}$ and $n_{RMC}$ is assigned as per [9].
Circumferential membrane stresses from $M_L$ moment:
| $$ \sigma_{m\theta}(M_L) = n_{\theta ML} \frac{M_L}{(s-c)^2(d+2s_1)}, $$ | (5.85) |
where $n_{\theta ML}$ is assigned as per [9].
As compared to membrane, circumferential bending stresses from $M_L$ moment is considerably lower, therefore their values are neglected.
Longitudinal membrane stresses from $M_L$ moment:
| $$ \sigma_{ma}(M_L) = \frac{M_L}{W_s}. $$ | (5.86) |
Longitudinal bending stresses from $M_L$ moment:
| $$ \sigma_{ba}(M_L) = \left(6m_{RML}-3n_{RML}\right) \frac{M_L}{(s_1-c_s)^2(d+2s_1)}, $$ | (5.87) |
where $m_{RML}$ and $n_{RML}$ is assigned as per [9].
Due to the torsional moment shearing stresses are created in the junction between nozzle and shell:
| $$ \tau_{\theta x} = \frac{M_T}{2\pi r^2_0 (s_1-c_s)}. $$ | (5.88) |
Forces $F_C$ and $F_L$ cause shear membrane stresses in longitudinal (points 1-4) and circumferential (points 5-8) sections correspondingly:
| $$ \tau_{x\theta} = \frac{F_C}{\pi r_0 (s_1-c_s)}, $$ | (5.89) |
| $$ \tau_{\theta x} = \frac{F_L}{\pi r_0 (s_1-c_s)}. $$ | (5.90) |
Local membrane stresses from internal pressure are defined subject to a stress intensification factors $I_{\theta(x)p}$.
Circumferential stress from internal pressure in the longitudinal section (points 1-4) can be shown as follows:
| $$ \sigma_{\theta p} = p I_{\theta p} \frac{D+(s+s_2-c)}{2(s+s_2-c)}. $$ | (5.91) |
The same stress for cross-section (points 5-8):
| $$ \sigma_{\theta p} = p I_{xp} \frac{D+(s+s_2-c)}{4(s+s_2-c)}. $$ | (5.92) |
If design factors $I_{\theta(x)p}<1$, then for calculation of circumferential stresses in the longitudinal section (points 1-4) the following relation is used:
| $$ \sigma_{\theta p} = p \frac{1+I_{\theta p}}{2} \frac{D+(s+s_2-c)}{2(s+s_2-c)}. $$ | (5.93) |
For calculation of circumferential stresses in the cross-section section (points 5-8) the following relation is used:
| $$ \sigma_{\theta p} = p \frac{1+I_{xp}}{2} \frac{D+(s+s_2-c)}{4(s+s_2-c)}. $$ | (5.94) |
For calculation of longitudinal stresses from internal pressure the following relation is used:
| $$ \sigma_{ap} = p \frac{\pi(d+2c_s)^2}{4A}. $$ | (5.95) |
When $l_2 < 8(s+s_2-c)$, at calculation of stresses and stress intensification factor it is necessary to substitute $s$ for $s+s_2$ (pad thickness is neglected).
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 $F_R$, $F_C$, $F_L$ and moments $M_C$, $M_L$, $M_t$. 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 5.17.
In the presence of corrosive hydrogen sulphide environment, a supplementary calculation of tensile stresses on the shell inside surfaces of the nozzle is made (2, 4, 6, 8 points):
| $$ \sigma_{in} = \max{\left\{ \frac{1}{2} \left( \sigma_{\theta}+\sigma_a+\sqrt{(\sigma_{\theta}-\sigma_a)^2+4\tau^2_{\theta a}} \right); 0 \right\}}. $$ | (5.96) |
| Table 5.17. 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 297 | ||||||||
| Circumferential stresses, $\sigma_{\theta}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Membrane from $F_R$ | - | - | - | - | - | - | - | - |
| Membrane from $M_C$ | - | - | + | + | ||||
| Membrane from $M_L$ | - | - | + | + | ||||
| Circumferential stresses from pressure $\sigma_{\theta p}$ | + | + | + | + | + | + | + | + |
| Total circumferential membrane stresses $\sigma_{m\theta}$ | ||||||||
| Total circumferential stresses $\sigma_{\theta}$ | ||||||||
| Longitudinal stresses, $\sigma_a$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Membrane from $F_R$ | - | - | - | - | - | - | - | - |
| Bending from $F_R$ | - | + | - | + | - | + | - | + |
| Membrane from $M_C$ | - | - | + | + | ||||
| Bending from $M_C$ | - | + | + | - | ||||
| Membrane from $M_L$ | - | - | + | + | ||||
| Bending from $M_L$ | - | + | + | - | ||||
| Longitudinal stresses from pressure $\sigma_{ap}$ | + | + | + | + | + | + | + | + |
| Total longitudinal membrane stresses $\sigma_{ma}$ | ||||||||
| Total longitudinal stresses $\sigma_{a}$ | ||||||||
| Shearing stresses from $M_t$ | + | + | + | + | + | + | + | + |
| Shearing stresses from $F_C$ | + | + | - | - | ||||
| Shearing stresses from $F_L$ | - | - | + | + | ||||
| Total shearing stresses $\tau_{\theta a}$ | ||||||||
| Reduced total stresses $\sigma_{экв}$ | ||||||||
| Tensile stresses on the shell inside surface $\sigma_{in}$ | ||||||||
PASS/NOZZLE-FEM 3.5. Program Manual
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