## 1 k2

Since Ti and w, must be positive, and Pand kt are inherendy positive 2k, > k2 Weight of Fiber

Consider a ring lying between z and z + dz. A single fiber with inclination a to the latitudinal direction has a length da/sin a, where do is the width of the ring measured along a line of longitude. The number of fibers at inclination ai per unit length of a line of latitude is «j sin a,. The total length of fiber in the ring is therefore 4nr.

If the mass per unit length of fiber is Jt, the total mass of fiber covering the surface is

Minimum Weight

If each fiber supports its maximum tensile force Tj the minimum weight of fiber required to withstand the internal pressure P is simply related to the volume of the vessel Fby the following relationship

For example, for an ellipsoid of revolution in which Zis the semi-axis of revolution, and R is the semiaxis at right angles to Z, V = (4/3) nR2Z, and

Isotensoid Design

In isotensoid design every fiber is at the same tension t and if t is at the same time the maximum stress the fiber is permitted to carry, this also becomes the minimum weight design. For this to be true, the inequalities must become equalities in which r, = r.

Rewriting:

Adding these together u 4t/c, i kj

Any choice of cos2 a, and n, sin2 a¿, both functions of 2 which satisfy the equation, provides an isotensoid design of minimum mass M.

Geodesic-Isotensoid Design

On a surface of revolution, a geodesic satisfies the following equation rcosa = flcos p (2-153)

in which a is the inclination of the geodesic to the line of latitude that has a radial distance r from the axis, and (5 is the inclination of the geodesic to the line of latitude of radius R. Attention here is restricted to shells of revolution in which r decreases with increasing z2. an equator occurs at z = 0, all geodesies cross the equator, and all geodesies have an equation with R the radius at the equator.

If r>R cos ¡3, then cos a> 1 and a is imaginary. Therefore r = R cos (5 gives the extreme lines of latitude on the shell reached by the geodesic.

If N (¡3) sin ¡3df5 is the number of fibers per unit length of the equator with inclinations to it lying between ¡5 and ¡5 + d(3, it can be shown that for a sphere

The fiber distribution is independent of the angle (5. For a cone with half-vertex angle y

_ 3PRC0SP K 4r cos 7

For an ellipsoid of revolution

APfl2 p + (3/2) VCOS2y3 3 . ... fl, 1 -(-v) "jcos/n m

A/ (/3) = —= . —+ - -v cos —7—t—r:-^ 2-156

in which v + = (Z2 - R2)/R2 and provided that v > 0.

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