Question 8.3: Calculate the stress at points A, B, and C of the vessel sho...

From Figures 8.6 and 8.7 and Eq. (5.1), the membrane and secondary stresses, in psi, at points A and B are shown as follows:

\sigma_{\theta}=\frac{P r}{t}                         (5.1)

The equivalent stresses, in psi, at points A and B are obtained from Eq. (8.3) and are shown as follows:

S_{e}=0.707\text { [ }\left(S_{1}-S_{2}\right)^{2}+\left(S_{2}-S_{3}\right)^{2} +\left[\left(S_{3}-S_{1}\right)^{2}\right]^{1 / 2}                       (8.3)

The maximum equivalent membrane stresses, P_{m}, are within the limit S as shown in Table 8.2.

The maximum equivalent primary plus secondary stresses (P_{m} + Q) are within the limit 3S as shown in Table 8.2.

Point C. The discontinuity forces at point C are shown in Figure 8.8. From this figure,

= 7734 in.-lb∕in.

Also,

M_{ h }=M_{ s }+N_{x} e=M_{ s }+7734 ,

r_{ s }=60+\frac{2.0625}{2}=61.0313 \text { in.} ,

r_{ h }=60+\frac{1.0313}{2}=60.5157 \text { in. } ,

\beta=\sqrt[4]{\frac{3\left(1-\mu^{2}\right)}{r_{ s }^{2} t_{ s }^{2}}}=0.1146 ,

\lambda=\sqrt[4]{3\left(1-\mu^{2}\right)\left(\frac{r_{ h }}{t_{ h }}\right)^{2}}=9.8465 ,

D_{ s }=\frac{E_{0} t_{ s }^{3}}{12\left(1-\mu^{2}\right)}=0.8035 E_{0}.

The first compatibility equation is given by

deflection of shell = deflection of head

or

w_{ p }+w_{N_{x} e}+w_{M_{ s }}-\left.w_{ Q }\right|_{\text {shell }}=w_{ p }+w_{ Q } +\left.w_{M_{h}}\right|_{\text {head }} ,                   (1)

where for the shell,

w_{ p }=\frac{P R^{2}}{E_{0} t_{ s }}(1-\mu / 2)=\frac{741820}{E_{0}} ,

w_{N_{x} e}=\frac{N_{x} e}{2 \beta^{2} D_{ s }}=\frac{366454}{E_{0}} ,

w_{M_{ s }}=\frac{47.3822 M_{ s }}{E_{0}} ,

w_{ Q }=\frac{Q}{2 \beta^{3} D_{ s }}=\frac{413.4569 Q}{E_{0}} ,

and for the head,

w_{ p }=\frac{P R^{2}}{2 E_{0} t_{ h }}(1-\mu)=\frac{621430}{E_{0}} ,

w_{ Q }=\frac{2 r Q_{ h } \lambda}{E_{0} t_{ h }}=\frac{1155.5665 Q}{E_{0}} ,

w_{M_{ h }}=\frac{2 M_{ h } \lambda^{2}}{E_{0} t_{ h }}=\frac{188.022 M_{ s }}{E_{0}}+\frac{1,454,162}{E_{0}} .

Substituting these values into Eq. (1) gives

M_{ s }+10.8719 Q=-6878                          (2)

The second compatibility equation is obtained from

rotation in shell = rotation in head

or

\theta_{N_{x} e}+\theta_{M_{ s }}-\left.\theta_{ Q }\right|_{\text {shell }}=-\theta_{M_{ b }}-\left.\theta_{ Q }\right|_{\text {head }}                         (3)

where for the shell,

\theta_{N_{x} e}=\frac{N_{x} e}{\beta D_{ s }}=\frac{83,991}{E_{0}} ,

\theta_{M_{ s }}=\frac{10.86 M_{ s }}{E_{0}} ,

\theta_{ Q }=\frac{Q}{2 \beta^{2} D}=\frac{47.3822 Q}{E_{0}} ,

and for the head,

\theta_{M_{ h }}=\frac{4 \lambda^{3} M_{ h }}{E_{0} t_{ h } r_{ h }}=\frac{61.1861 M_{ s }}{E_{0}}+\frac{473,213}{E_{0}} ,

\theta_{ Q }=\frac{2 \lambda^{2} Q}{E_{0} t_{ h }}=\frac{188.022 Q}{E_{0}} .

Substituting these values into Eq. (3) gives

M_{ s }+1.9521 Q=-7733.99                           (4)

Solving Eqs. (2) and (4) gives

Q = 95.97 lb∕in.,

M_{ s } = −7921.3 in.-lb∕in.,

M_{h} = −187.3 in.-lb∕in.,

total    w=\frac{693266}{E} .

The actual forces are shown in Figure 8.9. We have

hoop stress at point  C=\frac{E_{0} t_{ s } w}{r_{ s }}=23,430 psi ,

axial stress at point  C=\frac{P R}{2 t_{ s }}=7275 psi ,

axial bending stress at point  C=\frac{6 M}{t_{ s }^{2}}

= 264 psi,

circumferential bending stress at point C = (0.3)(264) = 80 psi.

These stresses are divided into two categories in accordance with Table 8.2:

1) Local membrane stress ( P_{L}):

\sigma_{\theta} = 23,430 psi,

\sigma_{t} 7275 psi,

\sigma_{ r } = −250 psi,

maximum stress difference = 20,950 psi.

From Tables 8.1 and 8.2, the maximum allowable local membrane stress is equal to

1.5 S = 22,500 > 20,950 psi acceptable

2) Local membrane plus secondary stress ( P_{L} + Q):

\sigma_{\theta} = 23,430 − 80 = 23,350 psi,

\sigma_{t} = 7275 − 264 = 7011 psi,

\sigma_{ r } = −250 − 250 = −500 psi,

maximum stress difference = 21,120 psi.

From Tables 8.1 and 8.2, the maximum allowable local membrane plus secondary stress is equal to

3 S = 45,000 psi > 21,120 acceptable

Table 8.1 Classification of stresses.

Table 8.2 Stress categories and limits of equivalent stress

Stress Category	Primary			Secondary Membrane plus Bending	Peak
	General Membrane	Local Membrane	Bending
Description (For examples, see Table 8.1)	Average primary stress across solid section. Excludes discontinuities and concentrations. Produced only by mechanical loads	Average stress across any solid section. Considers discontinuities but not concentrations. Produced only by mechanical loads.	Component of primary stress proportional to distance from centroid of solid section. Excludes discontinuities and concentrations. Produced only by mechanical loads.	Self-equilibrating stress necessary to satisfy continuity of structure. Occurs at structural discontinuities. Can be caused by mechanical load or by differential thermal expansion. Excludes local stress concentrations.	1.   Increment added to primary or secondary stress by a concentration (notch). Certain thermal stresses which may cause fatigue but not distortion of vessel shape.
Symbol	P_{m}	P_{L}	P_{b}	Q	F
S_{ PL }=1.5 S_{;} S_{ PS }=3 S ; S_{ a }= alternating stress used in fatigue analysis. Source: ASME.