Based on formula (13) the dependence of the maximum stress value from tapering ratio is given in Table 1. As it follows from Table 1. data, the thickening of the cable at the point of the Earth geosynchronous orbit leads to the decreasing of the maximum tension stress. E.g. for a continuous cable of a circular cross section the two times increase of the cable radius at the geosynchronous orbit point leads to the two times decrease of the maximum stress.

                                                                                                                                   Table 1

   The above-mentioned results determine the projects of the cable design when the "limit" length does not change, while the cable form permits decrease of maximum stress.
   We can reach the decreasing both the "limit" length and the maximum stress by appropriate choice of the cross section area function. Let us consider the cross section area functions as the quadratic polynomial functions of the following type

where a,b are arbitrary constants.
   Substituting (14) into (8), we obtain the stress function depending upon parameters a,b. Since the stress function expression is very cumbrous we do not present it here.
   By means of numerical analysis of this expression the cable forms, namely, parameters a,b are determined, providing the implementation of the following conditions

These numerical results for two considered cases are the following:
cable "limit" length L₀ = 18, (b = -1.04, a = 2.0), max_xs(x) = 0.61rg₀R₀,
cable "limit" length L₀ = 16.0, (b = -2.2, a = 4.23), max_xs(x) = 0.52rg₀R₀.
Let us now consider the cable with a "counterweight" of mass M₀ attached to its outward end. In this case, we have the following boundary condition

s(x) = rg0R0(x)

(x) = (L - x)[(1 + L)(1 + x)(2 + L + x) - 576] 576(1 + L)(1 + x) + M0L M й к л (1 + L) 288 - 1 (1 + L)2 щ ъ ы ,

(16)

where M₀ = rLR₀S₀ is the cable mass.
When the "counterweight" of mass M₀ is attached to the cable outward end, it is possible to consider a design of uniformly tensioned cable when mechanical stress is constant along the cable length. Such a design for a ponderable rod was considered first by S. Timoshenko [2].
Seeking in equation (1) solution s(x) = const we come to the following solution for the cable cross-section area function.

g(x) is the function defined by (4), S₀ is the cable cross-section area at the outward end, where the mass M₀ is attached. For a cable with "limit" length L₀ = 22.4 function S(x) reaches to its maximum value S^* at point x » 5.60.
The maximum value of the (17) can be written as

where s₀ = 0.76rg₀R₀ is the maximum stress of the of non tapered cable. From (18), (19) follows that decrease of the maximum stress leads to exponential increase of S_*. E.g. when b = 2, we have S_* = 7.39S₀. For a continuous cable of a circular cross section, it means that the cable radius increase 2.7 times.

                                                                                                                                          Table 2

   The "counterweight" decreases slightly the "limit" length and stress maximum values, when M >> M₀. In the contrary, the "counterweight" decreases essentially the "limit" length and stress maximum values, when the cable mass is compared with "counterweight" mass.
Let us now consider the elastic stability problem in the non tapered cable when its length is less than "limit" length, L < L₀. In this case the compression stresses arise localized near cable base and due to it, the cable may become unstable.
   The cable elastic stability equation and appropriate boundary conditions can be written as

d4W dx4 - l d dx й к л 0   (x) dW dx щ ъ ы = 0;

(20)

where the dimensionless function (x) is defined as in (6), EI is the bending rigidity of the cable material, W(x) transverse displacement of the elastic cable, l є rg₀R₀³S₀(EI)^-1. Using notation y(x) = [dW/dx], integrating equation (20) and taking into consideration the boundary conditions we come to the following equation and boundary conditions

d2y dx2 0 - l 0   (x)y = 0,

(22)

Equation (22) with conditions (23) is the self-adjoint boundary value problem determining eigenvalues l. Since the function (x) changes its sign in the interval x О [0,L], the eigenvalues can be both positive and negative ones. The minimum positive eigenvalue l₀ corresponds to the critical load behind which the cable becomes unstable.
Let function (x) change its sign from a negative value to a positive one at point x₀.
Then for minimum positive eigenvalue, l₀ the following inequality is valid

where function y₀(x) is any admissible function satisfying to the conditions y₀(0) = y₀(x₀) = 0.
Let us consider the cable with L = 22.0. For this cable x₀ = 0.04; and taking y₀(x) = x(x - x₀) we come to the inequality l₀ < 6326. Based on this inequality we can postulate that for a rod with circular cross section of radius r when r < r₀ the elastic rod under consideration is unstable, where

r0 = 0.025

Therefore, the solution of the self-adjoint boundary value problem (22-23) even in this case (L = 22.0) results in the instability of the cable for any values of elastic and geometrical parameters having practical and science forecasting meanings.

   Conclusions.
Based on the solution of the one dimension equation of the elasticity theory qualitative and quantitative results are obtained related to strength and stability problems of a space elevator cable/ribbon. It is assumed that the cable is subjected to action of the Earth gravity inward force, defined by Newton gravity law and the centrifugal outward force, due to the Earth's daily spinning. The numerical data related to the strength characteristics of new modern materials such as carbon nanotubs are omitted, since these data are in [1]. We confined ourselves to results related to the tapered cable design strength problems. Based on the solution of a one dimension elasticity equation solution of the tensile strength does not determine an absolute value of cable cross-section area. This cross section value parameter may be determined while considering the two dimensional problems or dynamics problems, especially elastic wave propagation along a tensioned cable.
On the other hand, there is no doubt that the future objects of the investigation should be constructions of elastic closed shells (pipes) type, which are of interest from an applied point of view. For such constructions, the new mechanical problems should be considered, which take into account circular, transversal stresses and displacements arising in the shells. Among these problems we can list the dynamic interaction of closed shells subject to external media, including electromagnetic, temperature and atmosphere fields actions.

   Institute of Mechanics of NAS RA