HOOP STRESS IN A CYLINDER

In this section we will investigate how well MOSES computes the hoop stresses in a thin-walled cylindrical shell due to external pressure. Here we consider a cylinder with a radius $r=$ 31.5 ft, wall thickness $t=$ 1 in, and total length $l=$ 300 ft. The cylinder was submerged 295 ft with no trim or heel.

Since MOSES employs a finite element method, the computed answer will depend on how accurately we model the geometry of the cylinder. Thus, results for four models were obtained: two models with their nodes placed on the perimeter of the cylinder and two models with their nodes placed at a distance from the center of the cylinder in a manner that preserves the surface area of the cyclinder. For each scheme of placing the nodes, models were created that utilised 8 and 16 plates evenly distributed around the circumference of the cylinder.

Plots of the nodes are shown below in Figures 6–8. Notice that, in order to accurately preserve the surface area of the cylinder, a different distance $d$ is needed for each selected number of plates $n$:

$$ \begin{aligned} d = \sqrt{\frac{2\pi r^2}{n \sin\!\left(\frac{2\pi}{n}\right)}} \end{aligned} \tag*{(11)} $$

For the cylinder of radius $r=$ 31.5 ft that we have, this corresponds to a distance $d=$ 33.2 ft for $n=$ 8 plates and $d=$ 31.91 ft for $n=$ 16 plates. Figures 6 through 8 show the geometric differences in detail.

Figure 6: Top View of Simple Models

Figure 7: Top View of Detail Models

Figure 8: Side View of Detail Models

The "standard results" are given by:

$$ \begin{aligned} \sigma_r &= 0 \\ \sigma_\theta &= \frac{Pr}{t} \\ \sigma_z &= \frac{Pr}{2t} \\ P &= \rho g h \end{aligned} \tag*{(12)} $$

Here, $\sigma_r$ is the radial stress, $\sigma_\theta$ is the tangential stress, $\sigma_z$ is the longitudinal stress, $r$ is the inner radius, $t$ is the shell thickness, $P$ is the pressure, $\rho$ is the density of water, $g$ is the gravitational constant, and $h$ is the water depth of the point being measured.

Membrane Stress of a Thin-Walled Cylindrical Shell

It is interesting that the results for the equivalent area models are substantially better than the ones from the perimeter models. Also, the results are surprisingly close for such coarse models.