Denoting by ^ the angle between any direction at P and the principal direction at P corresponding to "1" (Fig. 6), the normal curvature kn in the given direction at P is defined by Euler's formula, as kn = k| cos' <4> + k2 sin"

P gik^idUfc

where hik = r1 x r2 • rik = - ri • Nk, with rik = 32r/3ui3uk and N = 9N/9uk. The quadratic differential form hikduiduk inEq. (19) is the second fundamental form of the surface and the symmetric second-order tensor formed on the hikcomponents represents the second fundamental tensor of the surface, with determinant h = det(h) = hjjh22 — (hj2)2.

At the a certain point, i.e. the umbilical point, one has that g = Àh, and the curvature is the same for all

Actually by considering the product, K = ki k2 = h/g of the principal curvatures, which represents the Gaussian curvature of the surface at P, one may check if the surface is developable, since, in this case, the Gaussian curvature is zero at ali points of the surface.

2.2 Shells of revolution

With reference to shells of revolution (Baratta & Corbi 2007b), one may refer to the generic element ABCD identified by the cut of the surface along vertical planes passing through two couples of adjacent parallels and meridians, as shown in Fig. 7. The generic point on the shell may be located by its Cartesian coordinates x1,x2 in the reference system (O x1,x2,x3), whilst the surface equation is x3= z(x1,x2).

After denoting by r the vector detecting the position of the generic point of the shell, whose projection on the x1x2 plane is denoted by r, one can move from

Figure 7. Shell of revolution: Cartesian and polar co-ordinates.

Cartesian coordinates x1 ,x2 to polar coordinates 9, r in the horizontal plane, leaving <z> as the ordinate axis, as shown in Figures 7 and 8, where the position of the edge A of the element ABCD of the shell is identified. Moreover a point on the meridian is located by the angle 9.

Therefore the following relations exist between polar and Cartesian coordinates at the generic point

Figure 8. Normal section of a shell of revolution and principal curvatures.

Figure 9. Generic shell element of dimension dA.

where the variable z depends on the shape of the meridian line of the shell, and it is thus dependent on r, i.e. z = z(r).

After calculating the first order derivatives of r, one can evaluate the first fundamental form of the surface gik dxi dxk (with dxi, dxk the variables), which involves the coefficients gik = ri • rk and, with reference to the generic surface element, calculate the length of the arch curves along the parallel ds9 and meridian dsr directions respectively, and its area dA (Fig. 9) as follows

Figure 9. Generic shell element of dimension dA.

specific case r1 = rr, r2 = r9, after some algebraic operations and further developments, one may obtain, according to the Meusnier theorem (Mitrinovic & Ulcar 1969, Ugural 1999), the principal curvatures k1 and k2 solving the equation |h — k g|= 0 with respect to the curvature k, as where p1 and p2 are the principal radii of curvature.

Since the normal vector N of the surface at point P has the direction of the vector rr x r9, one has

By the second fundamental form of the surface (Ugural 1999) hik dxi dxk (with dxi, dxk the variables), involving the coefficients hik = r1 XJ|' r'k, where in the whence, after some algebraic operations, one gets

In the case of shell of revolution the local principal radii of curvatures on the surface can be immediately identified. With reference to Figure 8, at some point on the shell the meridian has a radius of curvature p1, which is the radius of curvature of the small arch length ds.

The patch of the shell cut out by two meridians and two parallel circles has, however, a second radius of curvature p2; the normals at adjacent points on two meridians intersect on the axis of the shell, and p2 is the length of the normal from the point on the shell to the axis.

Thus the shell surface is described by the four parameters 9, 9, p1, and p2, but these quantities are not all independent and, for example, with reference to Figure 8, r = p2 sin 9.

Figure 9 shows the small element of the shell and it is seen that the dimensions of the element are which can be shown to be in agreement to what previously found in Eq. (24).

By comparison with Eq. (24) one has

After some further developments, one gets d(p =

m'Fz^) , (l + z'2)-=—--- dr = -f--{— z dr=—>-^

Figure 10. Local equilibrium of the generic shell element. whence

which yield the same result as in Eq.(25).

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