Design the composite shell of a cylindrical pressure vessel with an inner diameter of 200 mm and a cylindrical length of 800 mm. Internal pressure is 20 MPa. The end domes are hemispherical in shape and reinforced by polar end fittings that have equal composite pole opening diameters of 40 mm (Figure 14.19).
Use unidirectional carbon/epoxy with the following material data:

Question

[latex]E_1=130 \ GPa,E_2=6.5 \ GPa,G_{12}=5.0 \ GPa, u _{12}=0.25,X^T= 2000 \ MPa[/latex]

Accepted Answer

Let us provide meridional stiffness to the pressure vessel entirely by helical windings, that is, we design the shell using helical and hoop plies only. With equal

TABLE 14.7
Required Thickness along the Meridional Contour (Example 14.8)

Location	s (mm)	r (mm)	[latex]\beta(^\circ)[/latex]	[latex]\alpha(^\circ)[/latex]	Thickness Required
Location	s (mm)	r (mm)	[latex]\beta(^\circ)[/latex]	[latex]\alpha(^\circ)[/latex]	[latex]t_h (mm)[/latex]	[latex]t_c (mm)[/latex]
A	4.6	24.5	75.821	54.736	3	0
B	21.2	40.1	66.344	29.897	1.33	1.67
C	37.7	54.7	56.866	21.463	1.16	1.85
D	54.2	67.7	47.388	17.182	1.10	1.90
E	70.8	78.9	37.911	14.684	1.07	1.93
F	87.3	87.9	28.433	13.146	1.06	1.95
G	103.9	94.6	18.955	12.208	1.05	1.95
H	120.4	98.6	9.478	11.699	1.04	1.96
I	136.9	100	0	11.537	1.04	1.96

composite pole openings, geodesic winding is feasible, in which the cylinder angle of winding is given by

[latex]\alpha_{cyl}=\sin^{-1}\left\lgroup\frac{40}{200} ight group=11.537^\circ [/latex]

Then, taking a factor of safety of 2.0, using Equations 14.68 and 14.69, the required helical and hoop thicknesses at various points along the meridional contour are determined and tabulated in Table 14.7. Note that point A is the point of inflection, where the helical angle is given by [latex]\alpha_i= an^{-1}\sqrt{2} [/latex] and the required hoop thickness is zero.

[latex]2(X^T_ht_h\cos^2\alpha+X^T_mt_m)\cos \beta =pr \quad\quad\quad\quad\quad (14.68) \ (X_h^Tt_h\sin^2 \alpha+X_c^Tt_c)\cos \beta=pr \quad\quad\quad\quad\quad (14.69)[/latex]
Let us provide in the cylindrical portion, three helical plies of 0.4 mm thickness each and four hoop plies of 0.5 mm each as per the following ply sequence:

[latex][90^\circ / \pm \alpha /90^\circ / \pm \alpha / 90^\circ / \pm \alpha / 90^\circ ][/latex]

Helical plies cover the entire pressure vessel—from one pole opening to the other. The helical thickness provided increases from the equator toward the pole opening and the same is determined using Equation 14.70 and tabulated in Table 14.8.

[latex]rt_h\cos\alpha=K \quad\quad\quad\quad\quad (14.70)[/latex]

TABLE 14.8
Thickness Provided along the Meridional Contour (Example 14.8)

Location	s (mm)	r (mm)	[latex]\beta(^\circ)[/latex]	[latex]\alpha(^\circ)[/latex]	Thickness Required
Location	s (mm)	r (mm)	[latex]\beta(^\circ)[/latex]	[latex]\alpha(^\circ)[/latex]	[latex]t_h (mm)[/latex]	[latex]t_c (mm)[/latex]
A	4.6	24.5	75.821	54.736	8.31	2
B	21.2	40.1	66.344	29.897	3.38	2
C	37.7	54.7	56.866	21.463	2.13	2
D	54.2	67.7	47.388	17.182	1.82	2
E	70.8	78.9	37.911	14.684	1.54	2
F	87.3	87.9	28.433	13.146	1.37	2
G	103.9	94.6	18.955	12.208	1.27	2
H	120.4	98.6	9.478	11.699	1.22	2
I	136.9	100	0	11.537	1.2	2

Hoop winding, on the other hand, is not feasible in the domes and the need for circumferential reinforcement is met by local lay-ups (doilies). It is seen from the first table that the requirement of circumferential reinforcement gradually decreases as we move from the equator toward the pole and it vanishes at the inflection point and beyond. Note that the above calculations are based on the netting model, which does not consider interaction of the shell with polar bosses and other elements. Also, during doily lay-up, owing to its very geometrical shape of the doily developments, it is not feasible to orient the local reinforcement accurately everywhere. To accommodate such uncertainties, we provide 2-mm-thick circumferential reinforcements (i.e., same as at the cylindrical region) everywhere in the dome. The dome plies can be modified by finite element analysis.

Question 14.8: Design the composite shell of a cylindrical pressure vessel ...

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