Abstract:
A new adaptable grid or lattice structure identified by the coined term “polargrid” is disclosed. A polargrid may be designed or engineered to fill any space between two generally circular structures and where the ribs will intersect with both circular structures at a node. The intersection of a polargrid with a circular structure will not occur mid-rib. Rather, a polargrid can be designed to always have ribs terminate at an inner or outer circular structure at a node. Polargrids have particular applicability in cylindrical structures or when interconnecting concentric circular or cylindrical structures that may or may not be co-planar.

Description:
FIELD 
     Embodiments of the present invention are in the field of isogrids and, more particularly, a new form of such structures named polargrids. Polargrids have particular applicability in cylindrical structures or when interconnecting concentric circular or cylindrical structures. 
     BACKGROUND 
     In many structural contexts, isogrids are used to add strength to a structure while minimizing the addition of weight. An isogrid is a type of partially hollowed-out structure formed usually from a single metal plate or face sheet with triangular integral stiffening ribs. It is extremely light and stiff. The triangular pattern in an isogrid is very efficient because it retains rigidity while saving material and therefore weight. The term isogrid is used because the structure acts like an isotropic material, with equal properties measured in any direction, and grid, referring to the sheet and stiffeners structure. The location where the ribs intersect is referred to as a node. An image of an isogrid is shown in  FIG. 1 . 
     A deficiency with isogrids arises when interconnecting two generally concentric circular structures. As used herein, the term circular structure will include structures that are generally circular in structure, such as a multi-sided polygon that approximates a circle, as well as columnar structures that are circular or approximately circular in cross-structure. The problem is that the nodes do not align uniformly with the circumference of the circular structures. Rather, the intersection of the circular structure and the isogrid is often somewhere along the length of a rib but not at the end of a rib and, therefore, not at a node. Accordingly, individual ribs must be faceted to the circular structure or false nodes must be added where the partial rib intersects with the circular structure. In either case, more structure is added, more weight is added and the performance of the isogrid is less than optimal. In addition, more design and manufacturing work is needed and labor costs increase due to additional faceting required for adding false nodes.  FIG. 1  also illustrates how an isogrid intersects with a circular structure. As is readily apparent, very few nodes are located where the isogrid intersects with the outer circular structure. Although not shown in  FIG. 1 , isogrids similarly do not uniformly intersect with an inner circular structure at a node. At each location where a partial rib intersects the outer or inner circular structure, false nodes will need to be added in order to meet and maintain a structural design criteria. 
     One solution to the problem of interconnecting concentrically oriented circular structures is to utilize an orthogrid, which is a variant of the isogrid. An orthogrid uses rectangular rather than triangular openings. An orthogrid for circular structure is typically constructed in a radial pattern with a series of rectangles formed by radial and intersecting ribs increasing in area moving from the inner to the outer circular structure along an outwardly extending radial line. An orthogrid can be easier to manufacture and design than an isogrid, especially with circular structure, but an orthogrid is not isotropic in that it has different properties from different directions. An image of an orthogrid interconnecting two circular structures is shown in  FIG. 2 . As can be seen, the rectangles increase in size from the inner to the outer diameter. As a result, the ribs may need to be increased in size or shape at an outer radial position depending upon the intended loading. 
     SUMMARY 
     The foregoing problems are overcome by use of a new adaptable grid or lattice structure identified by the coined term “polargrid.” A polargrid may be designed or engineered to fill any space between two generally circular structures and where the ribs will intersect with both circular structures at a node. The intersection of a polargrid with a circular structure will not occur mid-rib. Rather, a polargrid can be designed to always have ribs terminate at an inner or outer circular structure at a node. One embodiment of a polargrid is shown in  FIGS. 3 and 4 .  FIG. 4  illustrates the polargrid of  FIG. 3  interconnecting two concentric circular structures. The polargrid was conceived for use with spacecraft. However, it may be used in other situations to interconnect two circular structures. For example, between the hulls of a submarine. In some embodiments, the inner circular structure may be a relatively small area, comprising very few or even a single node. The interconnected circular structures may or may not be co-planar. For example, the polargrid may be a dome or conical shape as illustrated in  FIGS. 43-48 . 
     In one embodiment, the polargrid structure comprises an inner and outer circular structure that are generally co-planar, a like number of nodes located at each inner and outer circular structure, a pair of curved spiral stringers extending from each node and interconnecting with separate nodes on the opposite circular structure, a node formed at each location where spiral stringers intersect, and one or more generally circular hoop stringers positioned between the inner and outer circular structures interconnecting nodes equally radially positioned from the inner or outer circular structures. This structure forms a plurality of circular rows of generally equilateral shaped triangles formed between the inner and outer circular structures, with the individual triangles generally decreasing in area from the outer circular structure to the inner circular structure. Each individual triangle is formed by three ribs. One rib is formed by a length of a hoop stringer and is known as a hoop rib. The other two ribs are formed by a length of a spiral stringer and are known as spiral ribs. The inner and outermost circular rows of triangles also have triangles formed by lengths of the inner and outer circular structures. 
     In a second embodiment, spiral ribs are formed between radially aligned inner and outer nodes. Triangles are formed between adjacent spiral ribs, starting at either the inner circle or outer circle. Triangles are formed keeping the interior angles approximately 60 degrees. The angles for one of the triangles will likely need to vary in order to maintain coincidence between the nodes and the spiral ribs. The process is repeated for adjacent spiral ribs until a completed polargrid is formed. 
     In an alternative embodiment, in situations where the ribs may be positioned too closely, such as when the inner circular structure has a relatively small diameter, one or more of the innermost hoops may be omitted due to space constraints. In such a situation, for example as illustrated in  FIG. 23 , the ribs in the innermost circular rows may need to be modified to accommodate additional load. Additional ribs, for example, radially extending ribs, may also be added to accommodate loading. 
     In a further alternative embodiment, for example as shown in  FIG. 49 , larger areas of triangles may be omitted to accommodate structure passing through the polargrid. Ribs may be added or remaining ribs may be modified to accommodate particularized loading. In a further embodiment of the present invention, for example as shown in FIGS.  41  and  42 , a skin may be added to either or both the upper and lower edges of the ribs for additional strength as may be required by the end use application of the polargrid. 
     A polargrid may be designed and constructed by a number of different methods. In a first method, an inner circular structure and outer circular structure are defined with an equal number of nodes on each. Each node on the inner circular structure is radially aligned with a node on the outer circular structure. First and second spiral stringers are interconnected to a single node on the inner circular structure. Each spiral stringer has generally the same curved shape. One example is seen in  FIG. 6 . The angle between the two spiral stringers is between 50 and 70 degrees and most preferably approximates 60 degrees. One example is seen in  FIG. 7 . The opposite ends of the two spiral stringers are connected to two separate nodes on the outer circular structure. Additional pairs of spiral stringers are connected to each node on the inner circular structure and separate nodes on the outer circular structure until all of the nodes have two spiral stringers connected thereto. The spacing on the outer circular structure between each pair of spiral stringers connected to a single node on the inner circular structure remains constant. The resulting structure forms a plurality of diamond shaped structures, each generally aligned such that a line extending between the two nodes that are furthest apart is a radial line, to both the inner and outer circular structures. The intersection of the spiral stringers defines a plurality of nodes. Hoop ribs are interconnected between nodes that are the same radial position from the inner and outer circular structures to thereby form generally concentric hoop ribs. The resulting structure comprises one or more circular rows of generally equilateral shaped triangles. One example is shown in  FIGS. 12-15 . If there are a plurality of circular rows, the area of each triangle in a circular row decreases as the circular rows move closer toward the inner circular structure. 
     According to a second method, an inner circular structure and outer circular structure are defined with an equal number of radially aligned nodes positioned on each circular structure. Radial lines are drawn between two adjacent nodes on the inner and outer circular structures. One example is shown in  FIG. 16 . The angle formed by those two radial lines is then bisected and a third radial line is positioned equidistant between the two previously located radial lines. The distance between the two adjacent nodes on the outer circular structure is then determined and that linear length is the length of a hoop rib for the outermost circular row of triangles. For example, see  FIG. 17 . Moving inwardly toward the inner circular structure, two spiral ribs are then constructed extending from the two nodes on the outer circular structure and terminating at the bisecting radial line. The length of the spiral ribs will be slightly less than the length of hoop rib such that the angle formed between the hoop rib and the two spiral ribs approximates 60 degrees. The same technique is then used to create triangles between each adjacent pair of nodes on the outer diameter circular structure. One example is shown in  FIG. 18 . The point at which the spiral ribs for each triangle meet creates a plurality of nodes equally radially spaced from the inner and outer circular structures. These nodes are then interconnected by hoop ribs to complete an outermost circular row of triangles. One example is shown in  FIG. 19 . Additional circular rows of triangles are constructed moving inwardly using the same protocol until the polargrid structure is completed. Examples are shown in  FIGS. 19-21 . 
     According to a third method for constructing a polar grid, radial construction lines are formed between radially aligned nodes on an inner and outer circle. Bisecting construction lines are added between the previously formed construction lines. One example is shown in  FIG. 25 . Starting at a node at the end of a first construction line on either the inner or outer circle, a line is drawn from the node to the next adjacent construction line where the angle between the first construction line and the drawn line determines the length of the drawn line. One example is shown in  FIG. 25 . The angle should approximate thirty degrees and is adjusted to meet design requirements. A second line is then drawn from the end of the first drawn line to the next adjacent radial construction line at approximately the same angle. One example is shown in  FIG. 26 . Further lines are drawn in a similar manner until the drawn lines interconnect a node on the inner and outer circles to form a spiral stringer. One example is shown in  FIG. 28 . The process is repeated from each inner and outer node moving circumferentially around the structure maintaining the overall shape of each spiral rib, recognizing that the angle between each leg may vary between 50 to 70 degrees until all of the spiral stringers are formed. One example is shown in  FIG. 30 . With the spiral stringers formed, circular hoop ribs are added interconnecting the nodes where the spiral stringers intersect. One example is shown in  FIG. 31 . The last line segment drawn for each spiral stringer may be tailored such that all of the end points of the final line segments fall coincident with the nodes on the inner or outer circle depending upon the direction of construction. 
     According to a fourth method for constructing a polargrid, radial ribs are formed between radially aligned nodes on an inner and outer circle. First and second adjacent radial ribs are selected. Starting at an outer or inner node on the first radial rib, a line is drawn from the node to the second radial rib such that the angle between the second radial rib and the drawn line is approximate to sixty degrees. A second line is then drawn from the end of the first line back to the first radial rib at an angle that also approximates sixty degrees. This forms a first substantially isosceles triangle between the adjacent radial ribs. Additional lines are drawn between the adjacent radial ribs to form additional adjacent isosceles triangles moving inwardly or outwardly as the case may be until a radial row of isosceles triangles is formed. One of the triangles may need to vary from pattern to maintain coincidence of the nodes with the radial ribs. The process is repeated for each adjacent pair of radial ribs. In this method, circular hoops are not required. However, as an alternative, an additional radial rib may be added between each existing pair of radial ribs, with the added rib interconnecting radial aligned nodes between the originally formed radial ribs to further strengthen the structure. These supplemental or added radial ribs do not interconnect the inner and outer circles. 
     The Summary is neither intended nor should it be construed as being representative of the full extent and scope of the present disclosure. The present disclosure is set forth in various levels of detail in the Summary as well as in the attached drawings and the Detailed Description and no limitation as to the scope of the claimed subject matter is intended by either the inclusion or non-inclusion of elements, components, etc. in this Summary. Moreover, reference made herein to “the present invention” or aspects thereof should be understood to mean certain embodiments of the present disclosure and should not necessarily be construed as limiting all embodiments to a particular description. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings, which are incorporated in and constitute a part of the specification, illustrate embodiments of the disclosure and together with the general description given above and the detailed description of the drawings given below, serve to explain the principles of these embodiments. 
         FIG. 1  is a partial view of an isogrid interconnected to an outer circular structure. 
         FIG. 2  is a perspective view of an orthogrid interconnecting two circular structures. 
         FIG. 3  is a perspective view of one embodiment of a polargrid according to the present invention. 
         FIG. 4  is a perspective view of the polargrid of  FIG. 3  interconnecting two concentric circular structures in a spacecraft. 
         FIG. 5  is a partial top plan view of one embodiment of a polar grid, with accompanying labels. 
         FIG. 6  is a top plan view of two spiral ribs interconnecting separate nodes of an outer circular structure to a common inner node of an inner circular structure. 
         FIG. 7  is an enlarged view of the inner diameter node and spiral ribs shown in  FIG. 6 . 
         FIG. 8  is a top plan view of the embodiment of  FIG. 6  with four additional spiral ribs added to the structure. 
         FIG. 9  is an enlarged view of the inner circular structure, inner nodes and spiral stringers of  FIG. 8 . 
         FIG. 10  is a top plan view of the embodiment of  FIG. 6 , further illustrating a full complement of spiral stringers extending between the inner circular structure and the outer circular structure for a polargrid having 40 inner and outer nodes. 
         FIG. 11  is an enlarged view of a portion of the embodiment of  FIG. 10 . 
         FIG. 12  is a top perspective view of the embodiment of  FIG. 10 , further showing circular hoops connecting nodes formed at the intersection of spiral stringers of the embodiment shown in  FIG. 10 . 
         FIG. 13  is an enlargement of a portion of the structure shown in  FIG. 12 . 
         FIG. 14  is a top perspective view of a polargrid made according to the present invention and interconnecting two circular structures. 
         FIG. 15  is a top plan view of the embodiment of  FIG. 14 . 
         FIG. 16  is a partial top plan view of an inner and outer circular structure with two radial lines drawn between radially corresponding inner nodes and outer nodes and a third radial line bisecting the angle between the two radial lines. 
         FIG. 17  is an enlarged view of a single triangle formed according to one method of the present invention. 
         FIG. 18  is a top plan view of a fully completed outer circular row according to a first method of the present invention. 
         FIG. 19  is an enlarged view of a triangle formed in a second circular row of the embodiment shown in  FIG. 18 . 
         FIG. 20  is a top plan view of the embodiment of  FIG. 18 , further showing a second circular row adjacent and inward of the first circular row depicted in  FIG. 18 . 
         FIG. 21  is a top plan view of a completed polargrid interconnecting an inner and outer circular structure according to a first method of the present invention. 
         FIG. 22  is an alternative embodiment of that shown in  FIG. 21 , wherein the two innermost circular rows are removed and replaced with an alternative design. 
         FIG. 23  is a partial enlarged view of the innermost circular rows of the embodiment depicted in  FIG. 22 . 
         FIG. 24  is a top plan view of an inner and outer circular structure with radial lines interconnecting radially aligned nodes. 
         FIG. 25  is an enlarged view of a portion of the inner circular structure of  FIG. 24  with the innermost line segment of a spiral rib illustrated interconnecting adjacent radial lines. 
         FIG. 26  is an enlarged view of a portion of the inner circular structure of  FIG. 24  with a second line segment illustrated interconnecting the first line segment of  FIG. 25 . 
         FIG. 27  is an enlarged view of a portion of the inner circular structure of  FIG. 24  with two spiral ribs illustrated. 
         FIG. 28  is an enlarged portion of the inner and outer circular structures of  FIG. 24 , showing a single spiral rib constructed from interconnected line segments. 
         FIG. 29  is an enlarged portion of the inner and outer circular structures of  FIG. 24  showing symmetrically opposed spiral ribs emanating from a common inner node. 
         FIG. 30  is a top plan view of the inner and outer circular structures of  FIG. 24 , populated with a complete set of spiral ribs. 
         FIG. 31  is a top plan view of the inner and outer circular structure of  FIG. 30 , with circular hoops added to complete a polargrid. 
         FIG. 32  is a top plan view of an inner and outer circular structure having 40 nodes each, showing a single spiral stringer extending to the origin. 
         FIG. 33  is a top plan view of an inner and outer circular structure. 
         FIG. 34  is a top plan view of the embodiment of  FIG. 33  showing radially extending lines between radially aligned nodes. 
         FIG. 35  illustrates the creation of a first triangle at the outermost location between two adjacent radial lines according to a third method of the present invention. 
         FIG. 36  shows a complete set of triangles formed between adjacent radial lines according to a third embodiment of the invention. 
         FIG. 37  shows two radial rows of triangles created according to the third method of the present invention. 
         FIG. 38  is a top plan view of a completed polargrid made according to the third method of the present invention. 
         FIG. 39  shows an alternative embodiment of the polargrid shown in  FIG. 38 . 
         FIG. 40  is a partial enlarged view of the polargrid illustrated in  FIG. 39 . 
         FIG. 41  is a top perspective view of a polargrid formed with a surface skin. 
         FIG. 42  is a partial enlarged perspective section view of the embodiment of  FIG. 41 . 
         FIG. 43  is a top plan view of a conical shaped polargrid. 
         FIG. 44  is a front elevation view of the embodiment of  FIG. 43 . 
         FIG. 45  is a perspective view of the embodiment of  FIG. 43 . 
         FIG. 46  is a top plan view of a dome shaped polargrid. 
         FIG. 47  is a top perspective view of the embodiment of  FIG. 46 . 
         FIG. 48  is a front elevation view of the embodiment of  FIG. 46 . 
         FIG. 49  is an alternative embodiment of a polargrid made according to the present invention. 
     
    
    
     It should be understood that the drawings are not necessarily to scale. In certain instances, details that are not necessary for an understanding of the disclosure or that render other details difficult to perceive may have been omitted. It should also be understood that in some instances, details have been added, such as details relating to methods of construction, including for example construction lines and dimensions, to assist in explaining the methods and structures of the preferred embodiments described herein. It should be understood, of course, that the claimed invention is not necessarily limited to the particular embodiments illustrated herein. 
     DETAILED DESCRIPTION 
     With reference to  FIGS. 3-5 , a polargrid  10  comprises a structure that interconnects to two generally circular concentric structures. The inner circular structure  12  may or may not be co-planar with the outer circular structure  14 . A plurality of outer diameter nodes  20  are located on or proximate the outer circular structure  14  and a plurality of inner diameter nodes  22  are formed on or proximate the inner circular structure  12 . A plurality of spiral stringers  24  extend between the inner diameter nodes  22  and the outer diameter nodes  20 . For clarity, examples of spiral stringers  24  are more easily seen in  FIGS. 6-10 . Nodes  26  are formed at the locations where the spiral stringers  24  intersect. A plurality of hoop stingers  28 , generally circular in shape, interconnect nodes  26  of equal radial position relative to the inner and outer diameter nodes. For clarity, the hoop stringers  28  may be more easily seen in  FIG. 12 . As is illustrated, a number of triangles  30  are formed as a result of this pattern. The individual leg of triangle  30  formed by a portion of a hoop stringer  28  is called a hoop rib  32  and the other two legs of each triangle are formed by a portion of a spiral stringer  24  is called a spiral rib  34 . As further shown, a generally diamond shaped polygon  36  is formed by adjacent triangles sharing a common hoop rib  32 . In this embodiment, the diamond pattern is oriented such that the greater node to node length is oriented radially. The radial node to node length is denoted L 1  and the transverse node to node length is denoted L 2 , which is the length of a hoop rib  32 . 
     There are numerous ways to design and construct a particular polargrid structure. A number of factors are taken into account. These include the load that will be applied to the structure, both overall and localized due to the placement of particular equipment, the cross-sectional shape of the spiral ribs  34  and hoop ribs  32 , whether any open areas will need to be created within the lattice structure for equipment pass-throughs such as conduits and/or cabling, and critical rib buckling margins. Other factors will be known to and appreciated by those of skill in the art upon reading the present disclosure and understanding embodiments of the inventions. More open space or fewer ribs allow better radiative cooling of and reduced vibro-acoustic environments for equipment packages connected to or located adjacent to the polargrid structure. The area of the polargrid structure and the load capacity will vary by application, and the designs of the embodiments described herein may vary to meet the requirements of each application. Increased node spacing generally means less load may be applied to the polargrid, assuming rib size and structure remains constant, but creates a polargrid with a lower triangle density to allow the polargrid to attach to an inner circular structure having a reduced diameter. In constructing a polargrid structure, it is most preferable to maintain the individual triangles  30  in the general form of an equilateral triangle, although the inner angles may vary as required by the application. Generally speaking, the inner angles in the triangles may vary between approximately 50 and 70 degrees. 
     One method of constructing a polargrid lattice structure will now be described in connection with  FIGS. 6-15 . As depicted, an inner circle  40  and outer circle  42  represent concentric inner and outer circular structures  12  and  14 , structures between which a polargrid structure will be constructed. For purposes of this example, there are forty outer diameter nodes  20  and forty inner diameter nodes  22 . Line  44  in  FIGS. 6-8  illustrates that an inner and outer node are radially aligned. In  FIG. 6 , two spiral stringers  24  are shown connected to a single inner node  22 . The angle alpha (α) separating the spiral stringers  24  at the inner node  22  is approximately 67 degrees. Any angle near 60 degrees may be selected to allow a change in the size or shape of the grid lattice. Each spiral stringer  24  has the same variable radius spiral curvature that originates from the coaxial center of the inner and outer circles  40  and  42 . Recognizing that an inverse relationship exists between angle α and the ratio of L1/L2, angle α may be varied from 60 degrees to tailor the L1/L2 ratio, and thereby the resultant aspect ratio of the diamond patterns  36 , as appropriate to satisfy design requirements. Preferably angle α will be between approximately 50 degrees and 70 degrees. Alternatively, although not shown, the angle α may approach 180 degrees. In  FIGS. 8 and 9 , additional spiral stringers  24  have been added to adjacent inner nodes  22 . Because there are 40 inner and outer nodes  20  and  22  in this example, each node is separated from adjacent nodes by a 9 degree angle. As more easily seen in  FIG. 9 , the diamond pattern  36  is positioned with the larger node to node distance L 1  oriented radially. As shown in  FIG. 10 , the remaining spiral stringers  24  are added to each of the inner nodes resulting in a spiral rib pattern formed by the spiral stringers  24 . The location at which two spiral stringers  24  intersect forms a node  26 . As illustrated in  FIGS. 12 and 13 , individual hoop ribs  32  are connected between the nodes  26  that are at the same radial position from either the inner circle  40  or outer circle  42 . Adjacent hoop ribs  32  interconnect to define hoop stringers  28 . Adjacent hoop stringers  28  define circular rows  48 . As shown, there are 8 circular rows  48  formed by adjacent hoop stringers  28  and interconnecting the spiral stringers  24 . As best seen in  FIG. 13 , each triangle  30  is formed by two spiral ribs  34  and a hoop rib  32 .  FIGS. 14 and 15  show a completed polargrid interconnecting two circular structures. 
     A second method of constructing a polargrid structure will now be described with reference to  FIGS. 16-21 . As illustrated in  FIG. 16 , an inner circle  40  and outer circle  42  define generally concentric structures between which the polargrid structure will be connected. A polargrid will also work with slightly non-concentric structures, but will require additional design effort for implementation. A first radial line  44 A and a second radial line  44 B interconnect aligned adjacent outer nodes  20  and inner nodes  22 . A further radial line  50  bisects the two radial lines  44 A and  44 B. Starting at the outer circumference or outer circle  42 , circular rows  48  are formed moving radially inwardly until the polargrid is completed. One could also start at the inner circumference and work outwardly. The anticipated load to be applied to the polargrid will determine a minimum number of outer nodes  20 . Turning to  FIG. 17 , the number of outer nodes  20  will determine the length of the individual hoop ribs  32  at the outer diameter. The line  32 A represents a hoop rib interconnecting outer nodes  20 A and  20 B. Because the triangle  30  is intended to approximate an equilateral triangle, the length of the hoop rib  32 A then controls the length of the spiral ribs  34 A and  34 B. 
     As shown in  FIG. 17 , two outer nodes  20 A and  20 B are illustrated. As a first step, the distance between nodes  20 A and  20 B is measured. The linear length X between nodes  20 A and  20 B is the length of a hoop rib  32 A. The length of each spiral rib  34  will be the same as or a fraction of the length of the hoop rib. When constructing circular rows  48 , the distance between each pair of adjacent nodes will be X. The length of a spiral rib  34  can be less than, equal to, or greater than X. For load critical areas and a more isotropic structure, the length of the spiral rib  34  is set equal to or close to X. If the length of a spiral rib  34  is set greater than X a larger L1/L2 ratio will elongate the diamond pattern  36  in a more radial direction, thereby creating a structure stiffer in the radial direction and softer in torsion/shear. If the length of the spiral ribs  34  is less than X, the L1/L2 ratio will decrease to make the structure stiffer in torsion/shear but softer in the radial direction. In this example, a 0.95 factor is used to determine Y, such that Y is 95% the length of X. This factor should preferably approach 1 to minimize deviation for forming equilateral triangles  30  where the angles are approximately 60 degrees between ribs. As shown in  FIG. 17 , spiral ribs  34 A and  34 B are approximately 95% of the length between nodes  20 A and  20 B (0.95X). When the triangle  30  is constructed, the angle alpha (α) between the two spiral ribs  34 A and  34 B approaches 60 degrees. In this particular example, a is approximately 63.5 degrees. The remainder of the triangles  30  in the outer circular row are constructed in the same manner. The result is shown in  FIG. 18 . As a result, a first hoop stringer  28 A may be formed by interconnecting nodes  26 . The outer circular structure may also be considered a hoop stringer, in which case it would be the first hoop stringer  28 A and the reference in  FIG. 18  to first hoop stringer  28 A would then be changed to second hoop stringer  28 B. 
       FIG. 19  illustrates the construction of the next innermost circular row  48 B. Nodes  26 A and  26 B are formed by adjacent triangles  30 A. These nodes  26 A and  26 B define a distance X 1  which is the length of a hoop rib  32 C in the new circular row  48 B. In this example, a factor of 0.97 is utilized for calculating the length Y of a spiral rib  34 . Accordingly, the length of spiral ribs  34 C and  34 D is 0.97X. The angle beta β between hoop rib  32 C and spiral rib  34 C approaches 60 degrees, and in this example is approximately 59 degrees. The angle alpha (α) between spiral ribs  34 C and  34 D approaches 60 degrees and in this example is approximately 62 degrees. The second circular row  48 B is completed by constructing adjacent triangles in the same manner as constructing triangles to form the first circular row  48 A. As a result, nodes  26 C and  26 D are created which, in turn, define the formation of a third hoop stringer  28 B. 
       FIG. 20  discloses a polargrid having a first circular row  48 A and a second circular row  48 B. As should be appreciated, additional circular rows are constructed until the polargrid is completed as is shown in  FIG. 21 . The innermost circular row  48 H is completed by constructing the inner circular structure  12  at the innermost diameter. To ensure all spiral stringers and hoop nodes are coincident, the next closest circular rows are adjusted by tailoring the multiplication factor for X until the innermost circular row  48  defines triangles  30  that adequately approximate equilateral triangles. The coincidence achieved in this example relates to having the inner and outer nodes in perfect radial alignment. If such radial alignment is not desired or required by design, then tailoring the angle can be omitted since the spiral stringers will intersect any radius. As should be appreciated to those of skill in the art, this construction process may be programmed using known equations. 
     In some circumstances the nodes at the innermost diameter may become too densely packed, particularly if the diameter of the inner circular structure  12  is relatively small. As shown in  FIG. 22 , embodiments of the present design are flexible. For example, every other inner diameter node  22  may be skipped in the formation of the innermost circular row  48 I. In this manner, generally equilateral triangle construction is still maintained. As illustrated best in  FIG. 23 , triangles  30 E approximate equilateral triangles. As a further option, depending upon the load to be received by the polargrid structure, radial ribs  62  may be added to distribute loads from spiral stringers in circular row  48 J and to further strengthen the innermost circular row  48  or, for that matter, at any location in the polargrid where additional support is needed, such as proximate cut-out portions associated with pass-throughs. 
     A third method of constructing a polargrid according to embodiments of the present invention is illustrated in  FIGS. 24-31 .  FIG. 24  shows an inner circle  12  and outer circle  14  with forty inner nodes  22  and forty outer nodes  20 . Radial construction lines  50 A-I interconnect radially aligned nodes and bisect these interconnecting lines. Turning to  FIG. 25 , as a first step, and starting at either an inner node or an outer node, a single line  52  is drawn from the selected node  22  to the adjacent radial line  50 B. The angle α determines the length of the line  52 . In this example, a is 33.5 degrees as this is one-half of 67 degrees. The angles are selected, and may be adjusted, for design requirements. The preferred method results in triangles approaching equilateral shape. As illustrated in  FIG. 26 , a second line  54  is drawn from the end of the first line  52  to the next adjacent radial line  50 C at the same angle α (33.5° as the first line  52 . As shown in  FIG. 27 , the angle β between the two segments  52  and  54  is 177.5 degrees. The remaining line segments that complete the spiral stringer between the inner and outer circular structures are added in the same manner. This methodology forms a spiral stringer  60  by these and the following spiral ribs that are formed in the same manner as ribs  52  and  54 . The last line segment may be tailored such that all of the end points of the final line segments fall coincident with the nodes on the inner or outer circle depending upon the direction of construction. A completed spiral stringer  60 A is shown in  FIG. 28 . As illustrated in  FIG. 29 , a mirrored spiral stringer  60 B is formed opposite the radial line  50 A. The process is repeated until all of the spiral ribs are formed interconnecting the inner and outer circles  12  and  14  as shown in  FIG. 30 . Here, because there are forty nodes, adjacent nodes are 9 degrees apart.  FIG. 31  illustrates a completed polargrid  10  where hoop ribs  34  interconnect nodes created by intersecting spiral stringers  60 . As shown in  FIG. 32 , the spiral stringer  60  diverges away from a constant arc radius. The radius increases as one moves further from center. Stated differently, this causes the spiral stringer  60  to converge toward the coaxial center point—essentially resulting in a logarithmic spiral not an arc. This technique may be used to create polar grids of varying shapes using other logarithmic spirals as design circumstances dictate. 
     A fourth method of constructing a polargrid according to embodiments of the present invention will now be described. This fourth method is illustrated in  FIGS. 33-38 .  FIG. 33  shows an inner circular structure  40  and an outer circular structure  42  with nodes  22  and  20 , respectively, formed on each circle. In this example forty nodes are located on the inner and outer circles  22  and  20 .  FIG. 34  illustrates radial ribs  76  extending between radially aligned outer nodes  20  and inner diameter nodes  22 . As a first step, and with reference to  FIG. 35 , a pair of adjacent radial ribs  76 A and  76 B are selected and define a radial row  68 . With nodes  20 A and  20 B also selected, angle α and angle β are, respectively, less than and greater than 60 degrees by 4.5 degrees. 4.5 degrees is chosen in this example to create generally equilateral triangles, but may be varied as appropriate to satisfy design requirements. Angle gamma γ is approximately 60 degrees. The legs  78  of the triangle  30 A are defined by the selection of the angles α and β. Angle delta δ is equivalent to angle α. The next inner triangle  30 B is formed in the same way with the same angles α, β, γ and δ. The process is repeated for the remaining triangles in the radial row  68  of triangles  30 C- 30 K. Relative angles for  FIG. 36  are shown in the below 
     
       
         
               
               
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Triangle 
                 Angle α 
                 Angle β 
                 Angle γ 
                 Angle δ 
               
               
                   
               
             
             
               
                 30A 
                 — 
                 — 
                 60° 
                 64.5° 
               
               
                 30B 
                 64.5° 
                 55.5° 
                 — 
                 — 
               
               
                 30C 
                 — 
                 — 
                 60° 
                 64.5° 
               
               
                 30D 
                 64.5° 
                 55.5° 
                 — 
                 — 
               
               
                 30E 
                 — 
                 — 
                 60° 
                 64.5° 
               
               
                 30F 
                  62.54°  
                 55.5° 
                 — 
                 — 
               
               
                 30G 
                 — 
                 — 
                   61.96° 
                 64.5° 
               
               
                 30H 
                 64.5° 
                 55.5° 
                 — 
                 — 
               
               
                 30I 
                 — 
                 — 
                 60° 
                 64.5° 
               
               
                 30J 
                 64.5° 
                 55.5° 
                 — 
                 — 
               
               
                 30K 
                 — 
                 — 
                 60° 
                 64.5  
               
               
                   
               
             
          
         
       
     
     In this example, one triangle  30 F is allowed to vary from pattern to maintain coincidence of the nodes with the radial ribs. The location within the radial row  68  of triangles  30  for the varied triangle may vary as required by application. In this example triangle  30 F is altered. Triangle  30 F is altered to create coincident intersections of the radial stringers  24  and hoop nodes  26 . Any of the triangles may be altered to achieve this objective. Slight non-coincidence could also result in an acceptable structure, but the resulting structure may be less efficient as more material may need to be added to handle slight load path offset.  FIG. 37  shows an adjacent second row of triangles  30  formed by the same method defining radial rows  68 A and  68 B.  FIG. 38  depicts a completed polargrid using the fourth methodology. As illustrated, spiral stringers  24  connect with one-half of the inner and outer nodes  20  and  22 . Every other inner and outer node  20  and  22  are interconnected only by a radial stringer  76 .  FIGS. 39 and 40  show an alternative embodiment comprising two radial rows  80 A and  80 B formed between an adjacent pair of radial ribs  76 . The process is the same as described above in connection with  FIGS. 35 and 36 , but every inner and outer node  20  and  22  is used. Because the process is repeated 40 times (because all 40 nodes in the example are used), the density of triangles  30  is doubled compared to that of  FIG. 38 .  FIG. 40  shows an enlarged portion of the polargrid of  FIG. 39  further illustrating that additional radial ribs  82  may be added to nodes  26  to further strengthen the structure. Here, the additional radial ribs  82  do not interconnect the inner and outer circular structures  12  and  14 , but may do so to suit design requirements. This creates substantially equilateral triangles from the diamond shaped patterns that exist and are illustrated in  FIG. 39 . 
     The polargrid lattice structure of the present invention may be created with skins  90  to provide additional strength as shown in  FIGS. 41 and 42 . In addition, the polargrid structure is useful in conic and domed shapes as shown in  FIGS. 43-45  and  46 - 48 , respectively. 
       FIG. 49  shows a modified polargrid structure with open areas  92  accommodating pass through of equipment and other structure, and modifications made to the two outermost circular rows  48 A and  48 B to accommodate loading. Radial ribs  66  may also be added at an outer diameter similar to those shown in  FIG. 23  with respect to inner circular rows. 
     The various embodiments, methods and resulting polargrid structures have been described herein in detail and create substantially equivalent polargrid structures. Such polargrid structures are capable of being designed and constructed using other methods and of being practiced or of being carried out in various ways as will be readily understood by those of skill in the art upon review of the present disclosure. Such modifications and alterations of those embodiments as will occur to those skilled in the art upon review of the present disclosure are within the scope and spirit of the claimed invention, as set forth in the following claims. In addition, it is to be understood that the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of “including,” “comprising,” or “having” and variations thereof herein is meant to encompass the items listed thereafter and equivalents thereof as well as additional items.