Shell Membrane Stresses in Material Coordinate System
Stresses in the local material coordinate system, particularly in the fiber direction, are desirable to determine the failure mode of a composite laminate shell. The local stresses, in terms of the global strains, are
The shear term, , is zero because the shell geometry, loading and material properties are all axisymmetric. Using Eqs. 13 and 16, the local stresses are
The stresses and are determined from dividing the membrane stress resultants, and (Eqs. 4 and 5) by the thickness, t, (Eq. 12). The final stress in the fiber direction becomes
The shell stresses, and , cannot be directly rotated to the fiber position like a homogeneous orthotropic material. Even though the angle ply laminate is symmetric and the fibers in both layers will be stressed equally, it is still a laminate and therefore only the strains can be rotated. Furthermore, unlike the commonly used netting theory, these local stress resultants do not assume zero stress in the transverse and shear direction.
Table 1. Material properties for IM6/2020 graphite-epoxy composite
The membrane stresses for three different domes are shown in Figs. 5-7, along with the finite element stresses. Figure 5 is a general ellipsoid shell where the major axis is a = 1 and semi-major axis is b = 0.6. This ellipsoid shape closely matches the dome contour of a netting dome. Figure 6 is for a sphere and Fig. 7 is for a netting dome. All domes have an initial radius of 1 m and a thickness of 1 mm at the outside edge. The thickness increases to about 15 mm near the dome opening. The lamina material properties used are shown in Table 1 for Hercules IM-6 fibers and Hercules 2020 epoxy resin. These material properties are also used in all subsequent examples. The pressure is P = 1 MPa for all domes.
The membrane laminate stresses in the fiber direction reduces considerably near polar opening for the ellipsoid and spheroid, but stays fairly constant for the netting dome. This is primarily caused by the rapid increase in the shell thickness. The transverse stress and shear stress also decrease as the fiber approaches the polar opening. The maximum fiber direction stress for all three shells are similar and are below the maximum allowable of approximately 1.7 GPa (Gramoll, 1990b). However, the maximum stress for both the transverse and shear stress is high for the ellipsoid and spheroid, but low for the netting dome. Due to the high shear and transverse stress, the graphite/epoxy composite would fail both the ellipsoid and spheroid, which is why they should not be used in commercial composite pressure vessels.
Displacements of an Ellipsoidal Composite Shell
The meridian and hoop strains of an arbitrary shell element are given by Flugge (1973) as
where v and w are the displacements tangent and normal to the meridian, respectively. Inverting Eq. 13, gives
Combining Eqs. 19 and 20 gives a single, first order partial differential equation for the tangential displacement, , as
The solution to Eq. 21 is
Where C1 is the constant of integration that is determined from the boundary condition, = 0 at = . After substituting for t, S and N, Eq. 22 becomes
For a spheroid, this simplifies to
where is obtained from Eq. 8 and the C terms are given by Eq. 14. Due to the complexity of the integral, (the angle and C terms are complex functions of ) a closed form solution is not possible even with computer based symbolic solvers. In this paper a simpler displacement variable, the radial displacement, , will be used for investigating displacements of an elliptical shell. The radial displacement of an axisymmetric shell is
After making various substitutions, Eq. 25 becomes
and for the spheroid
The radial displacement, using Eq. 26, for the previously mentioned graphite-epoxy filament wound ellipsoid and spheroid are shown in Figs. 8 and 9, respectively. The radial displacements for the netting dome, based on finite elements, are shown in Fig. 10. Also, for comparison purposes, the radial displacement for steel case (E = 200 GPa, = 0.33) with a uniform thickness of 1 mm (same as the composite cases at = 90o) is shown in Fig. 11. By comparing a steel case to a filament wound composite, their respective differences are readily evident.
The steel case is slightly stiffer than the fibers in the composite case, but the composite case is generally thicker since the thickness increases as the polar opening is approached. On the other hand, the composite case is below failure load for fiber, but above for shear load. The steel case is slightly above ultimate failure stress for high strength steel. But overall, the two cases are similar in strength. However, the displacements for the composite case are substantially higher. This emphasizes the difference between a filament wound composite shell structure and an isotropic shell structure of comparative strength (note that the ellipsoid is not the optimum shape for either a composite shell or isotropic shell). The large deflections cause further bending, which limits the use of bending theory for composite shells.
The netting dome has substantially less radial deflection than the ellipsoid and spheroid. This is because the fibers are aligned in the direction of the resultant membrane load. This minimizes the transverse stress and thus the transverse deflection, which is the main component in the hoop deflection. The spheroid has a large positive deflection whereas the ellipsoid has an initial negative deflection. For an ellipsoid shell with a/b ratio less than about 0.7, the hoop direction will be in the compression and could lead to buckling.
© 1997 Kurt Gramoll, The University of Oklahoma, All Rights Reserved.