Question 16.3: A vertical vessel has the dimensions shown in Figure 16.4. T...
A vertical vessel has the dimensions shown in Figure 16.4. The skirt weighs 12,000 lbs or 100 lbs/in., and the weight of the shell, heads, and fluid is 240,000 lbs or 1000 lbs/in. Moment of inertial of the shell is 2.074 × 10^{6}in.^{4}, and moment of inertial of the skirt is 0.518 × 10^{6} in.^{4}. Modulus of elasticity of all components is 30 × 10^{6} psi. Determine the fundamental natural frequency and fundamental period of the vessel.
Mass of the shell=1000/386.4=2.588 lbs-s ^{2}/in.^{2}
Mass of the skirt=100/386.4=0.259 lbs-s ^{2}/in.^{2}
EI of shell= 6.222 × 10^{13} lbs-in.^{2}
EI of skirt= 1.554 × 10^{13} lbs-in.^{2}
After a number of trials, the deflection of the beam is assumed as shown in line (1) of Figure 16.7. The mass of the beam multiplied by this deflection and multiplied by the quantity \rho^{2} is assumed as an applied distributed inertia load as shown in line (2), where \rho is the natural frequency. The equivalent concentrated forces due to the distributed load at points 2–4 are calculated from Eqs. (1)–(3) and are shown in line (3). Lines (4) and (5a) show the shear and moment diagrams. In line (5b), the moment is divided by the quantity EI.
Line (6) shows the forces at points 1–3 of the equivalent M/EI load. Lines (7) and (8) show the rotation and deflection diagrams. Point 4 of the deflection diagram is adjusted to match the deflection assumed in line (1) as shown in line (9), and all other points in line (8) are adjusted accordingly as shown in line (9). There is no deviation between the calculated and original deflection curves. Thus, the analysis is complete.
The natural frequency for the beam is obtained by equating the maximum deflections in lines (1) and (8).
10 K=1947.6 K \rho^{2} / 10^{6}or
\rho = 71.7 rad∕s
f = 11.4 cps.
and T =0.088 s.
