PRESTRESSED COOLING TOWER

Abstract: Structure of flexible tensile members which requires only two arrays of said members (4, 6) for obtaining lateral stiffness of said structure, the ends of each member (4, 6) being attached to contour elements (8, 18, 30 or 32) and the members of each array having opposite curvatures for forming an axisymmetric geodesic network which is a section of a surface of revolution. The members of both arrays (4, 6) are prestressed and said network is geometrically arranged to be torque-balanced within said contour elements.

Description: DESCRIPTION PRESTRESSED COOLING TOWER Background of the Invention The present invention relates to structures composed of tensile members and more particularly to such a structure having lateral stiffness and being torque-balanced using only two arrays of flexible tensile members. Such structures may form a continuous envelope, thus making them suitable for a variety of applications, such as for cooling towers, suspended roofs, and other shell-type structures. Presently, such structures constructed from three sets of tensioned structural members or a continuous membrane are known for natural draft cooling towers. One such ' structure " is known as the Schmehausen cable ' net tower described in German Patents-Nos.2,243,222 and 2,255,793 (corresponding to U. S. Patent No. 3^ *^5,106). Such a tower is constructed from three intersecting sets of cables which form a structural cage which is subsequently clad from the inside with corrugated aluminum sheeting. One " set of cables is in the meridional direction and the other two cable sets are set at equal but opposite angles from the meridian, and the intersection of such sets of cables is secured. Another variation of such a three-set " tensioned structural member natural draft cooling tower is shown in German Patent No. 2,15^,530, while in U. S. Patent No. 4,010,580 the envelope of the tensioned cooling tower is constructed from a contin¬ uous membrane. All such structures can be difficult to erect (espe¬ cially for cooling towers several hundred feet in height) and can be costly because of the number of fixed intersec¬ tions of members required or the difficulty in uniformly tensioning a continuous membrane. Also, maintenance of such structures can be expensive. Advantages of the present invention include the necessi¬ ty for using only two- sets of structural members for forming the structure, which sets or arrays of structural members are not connected at their intersection. Also, such a s ture possesses lateral stiffness, is torque-balanced, and " uniform tension throughout each member; thus, providing timum strength of the structure. Further, erection and tenance of a cooling tower employing such novel structure comparatively easier to accomplish and less expensive th prior three-set or membrane structures. These and other advantages will become readily apparent to those skilled the art from the disclosure of the invention detailed here Broad Statement of the Invention The present invention is a structure of flexible ten sile members which requires only two arrays of said membe for obtaining lateral stiffness or ostensible rigidity of said structure. The ends of each member are attached to contour elements and the members of each array have oppos curvatures for forming an axisymmetric geodesic networkw is at least a section of a surface of revolution. The fi array forms an angle α with respect to the meridian of sa surface of revolution, where is- greater than -90° but l than 0° from said meridian. The second array forms an ang β with respect to the meridian of said surface, of revolut where β is greater than 0° but less than 90° from said me ian. The absolute value of the angle β is greater than t absolute value of the angle in the structure. The memb of both arrays are prestressed and said network is torque balanced within said contour elements. The members of th two arrays are not secured or fastened at any of their points of intersection. A process for erecting such structure wherein said n work forms a surface of revolution having a negative Gaussian curvature (a hyperbolic surface of revolution) comprises attaching ends of said members of said first ar to a rigid upper ring member and positioning said array-a tached ring member about the upper end of a central suppo (compression) ' column. A first portion of the ends of sai OMPI members of said first array (for example, every other member) is attached to a lower rigid ring member having points of attachment for said first array and said second array, and concommitantly therewith the other portion of the ends of said members of said 'first array are temporarily attached to the points of attachment o said second array on said lower ring member. Both portions of said members of the first array are prestressed to induce a sparse, slightly prestressed network of said members. The portion of said members temporarily attached to said .second array attachment points then are replaced one-by-one with the second array while the other ends of said second array are attached to said upper ring and the second array tensioned, and the re¬ placed other portion of said members of said first array are attached and tensioned to their final point of attachment on the lower ring member. Optionally, a final tension adjust¬ ment can be made for all individual members of both arrays. Brief Description of the Drawings Pig. 1 is a natural draft cooling tower constructed from the novel structure.. Pig. 2A is a cross section elevation view of the tower of Pig. 1, and Pig. 2B is an alternative design for such tow¬ er. Pigs. 3-6 show mathematical and geometric arrangements of the members of the structure and of the structure itself and will be described in detail in connection with a de¬ scription of the mathematical theory behind the novel struc¬ ture given ' below. Detailed Description of the Drawings The suspended natural draft cooling tower of Pig. 1 has a thin-wall envelope formed by the two arrays of tensile members and 6. The upper ends of the members are connect- ed to upper ring member 8 and the lower ends of the members are connected to transition joint 10. Upper ring element 8 is suspended by cables 12 from compression column 14 in OMPI -it- conventional fashion. Anchoring cables l β firmly secure transition joint 10 by attachment to tension foundation an chors 18. .The precise geometrical arrangment of members 4 and β will be described later. Pig. 2A shows the tower of Fig. 1 in cross-sectional elevation along a meridian of surface (or envelope) 2. Fi 2B shows an alternative to the design of the tower in Fig. 2A wherein members 4 and 6 are connected to lower ring mem ber 30 which is secured to the ground ' b anchoring cables and foundation support 32, and by means of lower suspensio cables 3 . Such an arrangement' excludes the ground as a link in the self-balanced system of prestressing-forces of the structure and results in a much smaller foundation for • the tower. Where ground space 'is limited or unsuitable fo large foundations, the design depicted in Fig. 2B may be desirable. It should be appreciated that either or both o * the two ring members may be adjustably attached to the com pression column to (uniformly) adjust the tension in the arrays by movement of the rings up-and-down the column. . Detailed Description of the Invention The novel structure of the present invention is Uniqu in its lateral stiffness in a torque-balanced network whic requires only two arrays of flexible tensile members. The lateral stiffness or rigidity results from the opposed cur atures of the two arrays of members. The tensile members each array have opposed curvatures. Such- opposed curvatur permits prestressing and results from the unique geometric arrangement of each array. The arrays form an axisymmetri geodesic network which is an areal section of a surface of revolution. A network composing a surface of revolution (bounded by the contour .elements) is suitable for forming .cooling tower as described above.. A network composed of a areal section of such surface is suitable for forming a su pended roof as well as other shell-type structures as the skilled artisan will appreciate. Regardless of the size o the section of the surface of revolution which the network OM forms, each array can be mathematically described relative to the surface of revolution of the network, and such de¬ scription will be used in this application. As mentioned above, because of the unique geometrical arrangement of the network of members, prestressing of the members causes the two arrays of members to exert mutual lateral pressure against each other. Such mutual pressure is the source of the lateral 'stiffness or ostensible rigid- ity of the network. In order for the structure tobe torque- balanced, the circumferential components of the initial forces of prestressing in the two arrays should cancel each other. Such cancellation means that the resultant force of the arrays acts in the meridional direction and the network as a whole, taken as a free body, is ' torque-balanced. It shouldbe emphasized that only two arrays of tensile members are required for achieving the novel structure. While additional arrays of tensile members can be added to the structure, they are not required for achieving the lat¬ eral stiffness and torque-balancing of the structure. The first array of tensile members forms an angle with respect to the meridian of said surface of revolution, where α is greater than -90° but less than 0° .from said mer- idian, and preferably between abou -β0° and -5° • The second array forms an angle β with respect to the meridian of said surface of revolution, where β is greater than 0° but less than 90° from said meridian, and preferably between about 7° and 80°. The absolute value of angle β always is greater than the absolute value of angle α (i.e. jβ > ). That angles and β do not have the same value is necessary in order to have opposite curvatures for the two arrays. Thus, the two arrays are not symmetric with respect to the merid¬ ian. For a further understanding of the novel structure of the present invention, the following brief mathematical discussion of the network is given. The mathematical re¬ lationships and equations developed below have been developed ,^ "] R£Λ OMPI fromthe general theories relating to flexible member stru tures which can be found in the following references: "Theory of Instantaneously Rigid Nets," E. N. Kuznetsov, Translation of the Soviet Journal of Applied Mathematics and Mechanics, P M Volume 29, No. 3, Pergamon Press Ltd. (1965); "Introduction to the Theory of Cable Systems," E. N. Kuznetsov, Stroiijdat, Moscow, (1968) and "Design of Shells out of Bands," E. N. Kuznetsov, IASS Pacific Symposium - Part II on Tension Struc¬ tures, Tokyo, (1971), the disclosures of which are expressly incorporated herei reference. According to the theories propounded in the foregoing articles, a flexible member network as a mechan cal system is characterized by its static vector, S. , whi is a function of geodesic and normal curvatures and some other parameters of the network. The cornerstone theorem for such a system may be stated as follows: In order for a cable network consisting of two arrays of tensile -members to allow prestressing, it is necessary and sufficient that its static vector be gradient: : S i = -& 1 i = ^_ 2 ' C This equation means that the two components of the static vector are partial derivatives of some scalar function, S called the static potential, where x, "~ ancT p ~ are any two coordinates on the surface of the network. The initial forces induced by prestressing in the two arrays of membe can be expressed by the following equations: TJ = Cσ expS, T* = -Cσ α expS " (2 In equation 2, C is an arbitrary constant, σ Ct and σ Ω p are t normal curvatures of the two respective arrays of members and the forces Tα* and Tfβ refer to unit width strip^s ds Q P = and ds = 1, respectively, i.e. to unit increments of the complementary linear elements. .Note, that for the forces the two arrays of members to be positive for the required tension therein, the normal curvatures must be of opposit sign, which means that the surface of the network is of negative Gaussian curvature. The structure, of the present invention is defined as at least an areal section of a surface of revolution.. A surface of revolution can be defined by its lines of principal curv¬ ature, meridians and parallel circles. The angles and β, formed by the two respective < arrays of members with the meridian (see ' Figs.3 and ) depend only on the axial coordinate, z, directed along- the axis of revolution. ' Since and ξ. are the same for all members of ' tfreir respective array, the network possesses axial symmetry. For such an axisymmetric geodesic network, the static vector in the system can be expressed by the following equation: ( ,u...-v.cosω)-λ, fi (,v__.-u.cosω~ 1)?ϊ where ω=β- is the net angle, the prime denotes a derivative with.-respect to z λ and λ β are , he Tchebyshev curvatures of the net as expressed by: λ α = γβ'cos + + jBg 1 Sin , (4) and u. , ui, v. and vi are the directional unit vectors of the member lines of the array. • The Lame' parameters, A and B, for the coordinate lines (z,φ) of the principal curvature of the surface can be ex¬ pressed as: -A = VT+r', B = r, (5 where r = r(z) is the radius of revolution and φ is the polar angle. The static potential, S, is a function solely of z. • Because of axial symmetry, both components S-, and S also are functions of z only. Therefore, the derivative of the static potential, 3S/3φ, must equal 'zero.. Satisfying this latter O PI condition uniquely separates the networks of the present invention from all other axisymmetric geodesic.networks.B virtue of the Clairaut theorem, rsin = b_, rsinβ =. c (6 where constants b and c define the two respective arrays geodesic lines on a surface of revolution. Thus, the fol lowing closed form equation characterizing the networks o « the present invention can be derived: where C, is an arbitrary constant. Of all axisymmetric geodesic networks, only those . sa isfying Equation (7) will permit prestressing of the memb When prestressed, the two arrays of members in the netwo will exert mutual lateral pressure against each other bec of their opposed curvatures. Such mutual pressure is the source of the lateral stiffness or effective rigidity of t network of the novel structure of the present invention. Since there is a constant in Equation (7), the network ca additionally be designed so that the circumferential comp nents of the two initial forces of the arrays cancel each other (see Fig. 4). In this situation the resultant forc acts in the meridional direction only and the network as whole, taken as a free body, would be torque-balanced, in which case the following equation holds true: Tα*ds 0 βsin + T*βdsαsinβ = 0 . (8 Since the angle β>0 and both of the forces in Equati (8) are assumed to be positive, it follows that the angle must be negative for such a torque-balanced network. It also follows that for such a network, the arbitrary const in Equation (7) becomes uniquely specified as: C = slnα = b ( o 1 sinβ c ' . y Substituting Euler's Formula in the foregoing relationshi the static potential of the network can be expressed as follows: S " ln α 0 s-ln n (e÷c,) + ln0 (1 " where C is an arbitrary constant reflecting the intensity of prestressing, and the initial forces in the two arrays of members can be expressed as: 's cSln os 5lnB - -° Bln co B slna <----• 5 The forces expressed in Equation (11) relate to.unit incre¬ ments of linear elements, dsp σ and dsα, respectively. More physically meaningful for the design and construc¬ tion of the novel structure of the present invention are the forces, T and T 0 , per unit increments du = 1 and dv = 1 of 10 coordinates of the member lines " (see Fig. 3 and Fig. 4). Each of these unit increment strips contains a certain con¬ stant number of members. To convert, to forces T and T R , the initial prestressing forces must be multiplied by the Lame ' parameter B n and A of the first quadratic form of the net- . 15 work. Using the equations of the u-lines and the v-lines on the surface of the network (see Fig. 3), forces T and T g can be expressed as follows: τ = B T* = ? co ?g , Csin ( β -α)sin β = = ( 2) α n sm (β- ) cos β = A n β = "" Crsln = ~Cb ( 1 3) 20 where b<0 because α<0, so that the forces are of the same sign. Equations (12) and (13) show that the forces are con¬ stant alongthe members. The ratio of these forces can be expressed as follows: T . T α c • °1 * U4 ■ • 25 The relationship expressed in Equation (14) represents a pivotal link in the statical-geometric interrelation and is characteristic of the whole class of torque-balanced axisym¬ metric geodesic networks of the present invention ' . ' In order to ascertain the configuration of the surface 30 ofthe network and other geometric " attributes of the net¬ works of the present invention, the following two Equations are required: ~ tan -tanβ = {ln[rsin(α+β)]} ' (15) C 2 sin θ = r sin ( + β ) = x ( l where G is the angle between the normal to the surface an the axis of revolution (see Pig. 5) and C 2 is a geometric parameter. Thus, the following Equation also holds: 5 §§ =-r * = cotan θ . (1 The first principal radius of curvature of a surface revolution is given by the 'following formula: R 1-i dΠθΪ (1 Combining Equations (β), ' «(Ϊ6) and (18), yields the f 10 lowing important Equation: where, p 2 = (c-b) 2 , q 2 = (c+b) 2 . (2 Equation (19), which expressed R-. as an explicit function 15 angle θ, is the "natural" equation of the meridian of the surface of the networks of the novel structure o-f the present invention. Parameters b, c, and * C p can be evaluated using Equa¬ tions (6) and (lβ) as soon as values, of the radius r and angles and β are selected for any desired value of the 20 .angle θ. Then, values r, z, α and β can be determined as ■respective functions of θ for all θ's of interest utilizin Equations (6), (lβ) and (17). However, a more practical a ternative is to. assume only one of the angles, say the ang g : 3, and the ratio ~r- according to Equation (14). The most α 25 convenient location for the assignment of the initial data is the equator of the surface where θ = I p T-, because its pla is the plane of symmetry of the surface and the origin of i the axial coordinate z. Consequently, all the necessary computations can be made covering only the part of the range of interest for z>0, because the geometry above the equator for the network is a mirror image of the geometry below the equator (see Pig. β). Regardless of the set of initial data selected, a pre¬ stressed torque-balanced axisymmetric geodesic network be¬ comes uniquely determined by specifying the values of three arbitrary parameters, for example, a, α and β . Obviously parameter (a) determines only the physical size of the net- work and thus simply is a scale factor. The remaining two parameters, α and β , govern the configuration of the net¬ work and provide a wide variety of forms of the network from which to choose. Thus, the geometrical arrangement of the two arrays described above produces a condition where the members of one array have a curvature opposite to the curvature of the other array. As a result of these opposed curvatures, when the members are prestressed between the contour elements, they exert mutual lateral pressure. The geometry and pre- stressing provide lateral stiffness of the network (which can be an envelope) and equilibrium (torque-balance) at the contour elements. In the description of the invention and particularly in Pigs. 3 and 4, the assignment of the angles α and β is mere- ly for convenience and it must be recognized that such an¬ gles (and the geometrical arrangement of their corresponding arrays) may be reversed. In order to more fully appreciate the flexibility which the novel structure of this invention provides in the number of networks which can be constructed, several numerical values for the parameters, β and were used in order to ' calculate the other variables discussed above for several different networks. In the following table, the angles are expressed in degrees while the length units of z and r a not specified, so that the actual size of the network depends on an arbitrary scale factor. Figs. 3, 5 and β should be referred to also in.studying the following tabl TABLE I 100 200 300 4 - = 0.4 r: 100 103 M12 126 ' α α: -9.0 -8.7 -7-9 -7.1 - 3 0 = 23° β: 23.0 22.2 20.2 17 . .8 1 θ: 90.0 '" 86.3 83-3 81.1 7 = 0.5 r: 100 110 136 169 α: -16. ' 7 -15 i " -12 ' .2 ' -9.6 - 3 0 = 35° β: 35.0 31-5 24.8 19-3 1 θ: 90.0 79. -J 73*.2 70.1 68 = 0.5 r: 100 107 126 152 α: -14.5 -13.4 * -11.4 -9.3 -7 3 0 = 30° 3: 30.0 27-7 23-4 18.9 1 θ: 90.0 82.1 77.0 74.1 72 T - 0.7 r: 100 110 137 172 2 α: -20.5 -18.5 -14.7 -11.5 -9 3 0 = 30° 3: ' 30.0 27.0 21.2 16.5 13 θ: 90.0 79.0 72.6 69.6 68 = 0.9 r: 100 113 147 191 2 α: -26.7 -23.3 -17.6 -13-3 -10 3 0 = 30° β: 30.0 26.1 19.6 14.8 11 θ: 90.0 75.7 68.5 65.5 64 From the foregoing discussion, it is r- -: dily apparen that structures having a variety of network.*- can be con- structed according to the precepts of the present inventi It must be recognized, however, that strict adherence to OMP values developed from the mathematical equations given in this application is limited by present day engineering and construction techniques. Thus, for design and construction for structures of the present invention, adherence to the values which can be calculated is recommended, and construc¬ tion within an engineering tolerance is quite acceptable. Cooling towers constructed according to present inven¬ tion can cover the entire * range of currently existing and future demand in terms of height. For example, present natural draft towers generally range from about 300 to β OO feet in height. Of course, towers of greater height will have increased utility. Thus, construction of such towers according to the present invention may mean erecting struc¬ tural members . which are several hundred feet in length. A unique way for erecting a cooling tower substantially like that tower shown in Fig. 1, can be practiced as follows. One array of members comprises bands which can be overlapped for forming a shell if desired. The other array comprises cables. The cables are attached to the upper spacer ring (ring member 8 in Fig. 1) which -ring is suspended from the column (column 14) and lifted to its design position. The lower ends of all odd-numbered, for example, cables are anchored, one by one in their design position on the lower rins (transition joint member 10 in Fig. 1) while the even-n mber.ed cables are anchored temporarily in the posi¬ tions of the bands. The cables of one set (say the odd- numbered cables) are prestressed which induces tension in the second set of cables. As a result, a sparse, slightly prestressed cable network is formed which is capable of resisting wind load. Then, the even-numbered cables are replaced one-by-one by the bands and the lower ends of the even-numbered cables are shifted to their proper design position and fixed, while simultaneously mounting the bands, by anchoring and tension- ing them in their design position. If required, a final tension adjust eat can be made individually for the bands and cables within an assigned tolerance. An alternative method of erecting the tower is to attach all cables and bands in the proper positions on the upper spacer ring and the lower transition joint, and lift the spacer ring to its design position. Of course, other methods for erecting the novel tower may be developed and are included within the teaching of this disclosure. For the alternative design given in Fig. 2B, the erec¬ tion procedure is similar to the procedure described above. In this case, the lower ring has to be attached to the spars cable network and lifted by the network into its design po- sition. If the ring is to be made of reinforced concrete, for example, a thin-sheet annular casing can be lifted and then concrete poured into it. The casing should be complet with devices for anchoring the bands and the ' cables. The weight of the reinforced concrete spacer ring is a positive factor since it provides a certain part of the prestressing load required and, therefore, lightens the lower suspension system. The process of lifting the upper spacer, ring (and a lower spacer ring if there is one) with all of the cable attached can be completed in a relatively short time becaus of the absence of any simultaneous assembly of the cables and bands. Also, the cable and bands are not -attached to each other at their intersectionwhich also simplifies the erection process. For a further illustration of the presen invention, the following design example is given, but it should not be construed as a limitation of the present inven tion. DESIGN EXAMPLE To further illustrate a natural draft cooling tower constructed from the novel structure of the present inven¬ tion, this design example details the typical structural components that may be required for a 544 foot high tower (from ground to top of network) which employs steel bands and cables as members of the two arrays. ' The bands are o- verlapped to form a shell. The tower is like that shown in O PI Pig. 2B. All steel members employ steel of a strength typi¬ cal for the particular structural shape. Referring to Fig. 2B^ the vertical height between ring 8 and the top of column 14 is l4θ feet and the vertical distance between ring 30 and the ground is 112 feet. The network (or shell formed from the two arrays) is 43 feet in height, has a top diameter of 321 feet, and a bottom diameter of 406 feet. The 'diameter is a rv?.niraum at the equator of the surface. This minimum diameter is 270 feet and is located 158 feet from the top of the netvork. The following table displays the details of the structural com¬ ponents of such a tower. OMPI m -16- o oo TABLE II t Total Prestressin g Number of Spacing (ft) Weight U Force OH Structural Member Members Req'd. Cross Section Length (ft) Top/Bottom (x 1000 lbs) (x 1000 lbs) 'Tower Bands (steel) 236 6 ft x 1/lβ in 457 4.3/5.4 1,648 7 /Band Tower Cables (steel) 236 1-5/16 in Dlam. 617 4.3/5-4 527 14/Cable Upper Spacer Ring (steel) 1 162 in 2 1000 N/A 557 3,898 (Coπpre Lower Spacer Ring 1 11.8 ft 2 1275 N/A 2,342 3,396 (Coπpre (reinforced concrete) Upper Suspension Cables 5 4 in Dia . 210 1.4/18.7 381 611/Cable (steel) Lower Suspension Cables 54 3-5/8.In Diam. 250 23.6/2.0 372. 585/Cable (steel) O Foundation Ties 54 31/8 in Diam. 120 23.6/30.4 150 N/A (steel rods) ' ' • ? Central Column 700 N/A 11,650 20,735 (top) (reinforced concrete) 31,651 (Botto vo o O CM o o CO o Obvlously, it will be appreciated that the tabulated re¬ sults are merely examplary and are not a limitation of the invention. Also, such results are for design and illustra¬ tion purposes 'only and are not necessarily to be relied μpon as the full engineering for construction of such a tower. Several advantages are realized with the tower of this design example which are not readily apparent. The modulus of elasticity is different for each array (i.e. for the array of bands and for the array of cables). This difference is beneficial in that the overlapped bands which form the shell have a higher modulus of elasticity and a greater cross-sec¬ tional area than the cables. The higher modulus of elastic¬ ity and greater cross-sectional area ofthe bands both posi¬ tively contribute to increased stiffness of the tower which results in better (less) lateral deflection of the tower under wind or other loads. Thus, the tower of this design example is designed to withstand a 100 mph wind load. Of . course, different material for each array can be used to provide favorable differences in " the modulus of elasticity of the two arrays as the skilled artisan will appreciate. Suitable materials for the arrays include metal such as, for example, steel, aluminum, various alloys, etc: fiberglass and other fiber reinforced resins (plastics): and like materials. ' . " Another unique feature of * the tower of this design example (and the towers of Figs. 1 and 2) is the combination of an interior compression column and the network which has a net force (or forces) acting in the meridional direction (i.e. the network is being stretched eridionally or ver- tically with respect to the column). The compression of the central column clearly is advantageous "for such self-balanced network and compression column combination.

Claims: CLAIMS 1. A structure of tensile members requiring only two arrays of said members for obtaining ostensible rigidity of said structure, the ends of each member being attached to contour elements, the members of each array having opposite curvatures for forming an axisymmetric geodesic network whi is a section of a surface of, revolution, the first array fo ing an angle α with respect to the meridian of said surface revolution, where is greater than -90° but less than 0° from said meridian, the second array forming ah angle β wit respect to the meridian of said surface of revolution, wher 3 is greater than 0° but less than 90° from said meridian, where |βl>fα|, said members of both arrays being prestresse and said network being torque-balanced within said contour elements. 2. The structure of claim 1 wherein α is between abou -5° and -60° and β is between about 7° and 80°. 3- The structure of claim 1 wherein said network i.s a surface of revolution and said contour elements are an upper rigid ring member and a lower transition member, the planes of said members being normal ' to the axis of revoluti there being a compression column interior of said network ' ■ and along said axis of revolution, the lower end of " said -column being fixed to the ground, said upper ring member being attached to said compression column about its upper end and said lower transition member being attached to " the groun . 4. The structure of claim 1 wherein said network is a surface of revolution and said contour elements are an upper and a lower rigid ring member whose plane is normal to the axis of revolution, there being a compression column interior of said network and along said axis of revolution, "the lower end of said column being fixed to the ground, said upper ring member being attached to said compression column about its upper end .and said lower ring member being attache Ao said column about its lower end. - XTB.E 5. The structure of claim 1 wherein for prestressing said members, the following equations for said members are satisfied: where and σ 0 are the normal curvatures of the α p first and.second array, respectively, and for said torque-balancing, t C, = 5in« = ≥ , 1 smβ c ' where b and c are the parameters of " the geodesic lines for said first and second array, respectively, on said surface of revolution. 6. The structure of claim 5 wherein for said torque- balancing, the following equation also is satisfied, T lH = b - r T ~ c ~ l* α where T and T fi are the forces per unit increments du = 1 and dv = 1 of the coordinates of the members for said first, array and said second array, respectively, where' u and v are the vectors of the members of said first array and said second array respectively.. 7 - The structure of claim 6 wherein the first princi¬ pal radius of curvature (R, ) of said surface of revolution is given by the following equation: where, p 2 = (c-b) 2 , q 2 = (c+b) 2 , x = C 2 sin θ = r sin (α+β), z is a distance measured along said. axis, r sin α = b . r sin β •= c, θ is the ' angle between the normal to said surface and 5 . axis of revolution of said surface, r is the radius of said surface of revolution. 8 ' . The structure of claim 1 wherein said first arra and said second array each are c " at>les, each are bands, ar -10 bands and cables respectively, or are cables and bands respectively. 9. The structure of claim 1 . wlxerein one of said arrays is bands overlapped to form a membrane. 15 . 10. The structure of claim.1 wherein the members of said first array have a different modulus of elasticity t the members of said second array. 20 11- The structure of claim 1 wherein the members of said first array have a greater cross-sectional area than the members of said second array. 12. The structure of claim 10 wherein said members 25 said first array have a greater cross-sectional area than said members of said second array. 13*- The structure of claim 12 wherein said members said first array are bands and said members of said secon 30 array are cables. 14. The structure of claim 13 wherein said bands ar overlapped to form a membrane. 35 15. The structure of claim 1 wherein the members of each array are composed of the same material and said mat for each array independently i ' s selected from the group co sisting of metal, fabric, and reinforced cured resin. 16. In combination a tower and a compression column, said tower comprised of an ostensibly rigid, axi¬ symmetric geodesic network of two ' arrays of prestressed tensile members, the members of each array having opposed curvatures, said arrays attached to a lower and an upper contour element and geometrically arranged so that said network is torque-balanced within said contour elements, said contour elements being attached t ' o said compression column. 17. In combination a tower and a compression column, said tower comprised of an ostensibly rigid, . axisymmetric geodesic network of two arrays of prestressed tensile members, the members of each array having opposed curvatures, said arrays attached to a lower and an upper contour element and geometrically " arranged so that said network is torque-balanced within said contour elements, said compression column being fixed to the ground at its lower end, said upper contour element being attached to ' said column about its upper end and said lower contour element being attached to t.he ground. 18. Process for erecting a structure of tensile members wherein only two arrays of said members are required for obtaining ostensible rigidity of said structure, which comprises : a) attaching ends of said members -of a first array to an upper ring member ; b) positioning said array-attached upper ring member about the upper end of a central support column; c) attaching a first portion of the ends of said . members of said first array to a lower ring member having points of attachment for said first array and for said second array, and temporarily attaching said other portion of the ends of said members of said first array to the points of attachment of said second array on said lower ring member, said lower ring member being fixed at its location at the - said column; d) prestressing said first portion of said memb to induce a slightly prestressed network of said members; e) replacing and tensioning said other portion of said members of said first array with said second array and attaching said second array to said upper ring, and attaching and tensioning said other portion of said member of said first array to their point of attachment on said lower ring member, the members of each array having opposite curvat for forming an axisymmetric geodesic network, said first array forming an angle with respect the meridian of said surface, where α is greater than -90° but less than 0° from said meridian of said surface, said second array forming an angle β with respec to the meridian of said surface, where β is greater than but less than 0° rom said meridian, where | βj> |α| , said network being torque-balanced. 19- The process of claim 18 wherein said members of said first array are cables or bands, and said members of said second array are cables or bands. 20. The process of claim 19 wherein the members of one of said arrays are bands which are overlapped to form a membrane. 21. The process of claim 18 wherein for prestressing said arrays and for torque-balancing said network, the following equations are satisfied: σα = C c = s i nα = 3*1 1 sinβ c , , where σ α an< ^ σ β are the normal curvatures of said first an said second array, respectively, b and c define the sets o OMPI geodesic lines for said first and second array, respectively, on said surface of r where T and Tβ are the forces per unit increments du = 1 and dv = 1 of the coordinates of the members of said first array and said second array, respectively, where u and v are vectors of the members of said first array and said second array, respectively; and the first principal radius of curvature (R-, ) of said surface of revolution is: 2bcC [x i, -p 2 q 2 ] R, = C(p 2 - χ2) ( 2 -q 2 )l 3/2 where, θ is the angle between the normal to said surface and the axis of revolution of said surface, r is the radius of said surface of revolution, P 2 *= (c-b) 2 q 2 = (c+b) 2 x = C p sinθ = r sin (α+β), z is a distance measured along said axis, b = r sinα, and a = r sinβ. 22. The process of claim 21 wherein is between about -60° and -5° and 3 is between about 7° and 80°.

Inventors: GROOM J, KUZNETSOV E
Applicants: BATTELLE DEVELOPMENT CORP, BATTELLE DEV CORP
Application year: 1980
Main classification: E04H5
IPC codes: E04H5