US3835599A

US3835599A - Roof construction

Info

Publication number: US3835599A
Application number: US00322531A
Authority: US
Inventors: D Geiger
Original assignee: Individual
Current assignee: Individual
Priority date: 1970-05-03
Filing date: 1973-01-10
Publication date: 1974-09-17
Anticipated expiration: 1991-09-17

Abstract

A roof comprises a ring conforming in plan to a closed curve, a plurality of sets of cables with the cables of each set extending in plan parallel to an axis of skewed symmetry of the closed curve, and a membrane attached to the cables which by means of inflation lifts the cables so as to pre-tension the roof and form a domed surface.

Description

United States Patent Geiger Sept. 17, 1974 ROOF CONSTRUCTION 3.233.376 2/1966 Naillon 52/80 [76] Inventor: Dav! Gelge" 788 Rverslde 3.744.191 7/1973 Bi r ci 52/2 New York, NY. 10032 FOREIGN TEN S O C O PA T R APPL] ATI NS [22] 1973 225.411 12/1968 U.S.S.R 52/2 [21] Appl. No.: 322,531 197.921 1/1966 ussR 1.037.628 7/1966 Great Britain .1 52/2 Related US. Application Data [63] Continuation of Scr. Nov 80.048. Oct. 12. 1970.

abandoned [30] Foreign Application Priority Data May 3, 1970 Japan 4537445 [52] US. Cl 52/2, 52/80, 52/63 [51] Int. Cl E04b U345 [58] Field of Search 52/2, 80, 63, 83, 167

[56] References Cited UNITED STATES PATENTS 1,302,182 4/1919 Lanchcster 52/2 2,355,248 8/1944 Stevens 52/2 Primary Examiner-Henry C. Sutherland Assistant E.\'ami11erHenry Raduazo Attorney, Agent. or FirmCurtis, Morris & Safford [57] ABSTRACT A roof comprises a ring conforming in plan to a closed curve, a plurality of sets of cables with the cables of each set extending in plan parallel to an axis of skewed symmetry of the closed curve, and a membrane attached to the cables which by means of inflation lifts the cables so as to pre-tension the roof and form a domed surface,

10 Claims, 12 Drawing Figures PAIENTEBSEH 1 1914 SHEET 1 OF 5 FIG. 1

6 X A R 0 v A M INVENTOR David H.Gei BYG ger A 6C1, WW TTORNEYS PAKNTED 7 I974 SHEET 3 BF 5 m GE INVENTOR David H.Geiger BY $17 5: l

944 @464, ORNEYS PAIENIED 3.835.599

saw u or 5 I I..." I ll,-

FIG. 6

' INVENTOR David H. Geiger ZT T SQQS Pmmznsm 1

m4 sum

5 or 5 FIG. 9

44 n uh, m =4;

FIG. 10

FIG. 12

FIG. H

-INVENTOR David H. Geiger BY QM M @GMA/ ATTORNEYS ROOF CONSTRUCTION This is a continuation, of application Ser. No. 80,048, filed Oct. 12, 1970 and now abandoned.

The invention relates to a roof construction and, more particularly, to a roof construction including a ring which restrains the remaining elements of the roof structure and in turn is subjected to internal stresses that are substantially compressive. The roof membrane is supported by means of a plurality of sets of cables, with the cables of each extending in plan parallel to an axis of skewed symmetry of the ring. The cables of the two sets are clamped together at their intersections and serve, by inflation of the space beneath them to sustain a membrane fastened to the cables, for example by means of webs under tension.

The invention will now be further described in terms of a presently preferred embodiment thereof and with reference to the accompanying drawings, in which:

FIG. 1 is a diagram of a closed curve to which the ring of the roof structure of the invention may conform, and illustrating certain properties of that curve useful in explaining the invention;

FIG. 2 is a diagram showing the family of superellipses (x/a) (y/b)" l, with a b but of the same values for all ellipses shown, and with m passing through values (including non-integral values) from less than unity up to infinity, for which latter value the ellipse takes the form of a rectangle circumscribing all ellipses of the family;

FIG. 3 is a further diagram useful in explaining the invention;

FIG. 4 is a plan diagrammatic view of the roof construction according to the invention shown in vertical section in FIG. 5;

FIG. 5 is a sectional view taken on the line 55 in FIG. 4 and showing in addition one form of supporting structure for the roof of FIGS. 4 and 5;

FIG. 6 is a sectional view at an enlarged scale through the ring of the roof of FIGS. 4 and 5 showing in addition the anchorage of a cable thereto and the fastening of the roof membrane thereto;

FIGS. 7 and 8 are views of clamping means to fasten together at their crossing points cables of the two sets of parallel cables in the roof of FIGS. 4 and 5; and

FIGS. 9 through 12 are diagrammatic representations of means for fastening the roof membrane to the ring and to the cables.

FIG. 1 shows two lines M and N that are not perpendicular which are axes of skewed symmetry for the closed curve 2 since any line L intersecting one of these axes (for example, line L intersecting axis N at B) is parallel to the other axis M and intersects the curve 2 at points C and A such that AB is equal to BC. In this manner closed curves having lines of skewed symmetry may be constructed. The curve 2 of FIG. 1 is shown by way of example, as an ellipse conforming to the usual equation (x/a) (y/b) l.

An example of a family of closed curves having this property is the family of superellipses (x/a)'" (y/b)" 1 shown in FIG. 2. Whatever the value of m, the ellipse can be circumscribed with a rectangle whose sides are perpendicular to the major and minor axes of the ellipse, this rectangle being tangent to the ellipse at the intersections of the major and minor axes with the ellipse. It is a property of the superellipse that the diagonals 8 and of the circumscribing rectangle are its axes of skewed symmetry, whatever the value of m.

This may be demonstrated by a coordinate transformation to the Q-vy axis where: x (m-Q) cos a. y (1 Q) cos a and a=tan' b/a.

In these expressions, illustrated with reference to FIG. 2, a is the angle between the major axis of the ellipses of that figure and the adjacent diagonal of the circumscribing rectangle. n is the length measured along the direction of one of those diagonals as a slant coordinate axis, and g is the length measured along the other diagonal of the circumscribing rectangle as another slant cord axis.

With cables placed so as to project in plan as straight lines parallel to the axes of skewed symmetry of the closed curve, such as the axes M and N of the ellipses of FIG. 2, the invention provides a roof structure in which the stresses are borne by a ring conforming, or conforming substantially, to the closed curve, which ring is substantially free of bending moments in the horizontal plane and which is to that extent funicular.

One roof construction according to the invention is shown in FIGS. 4 and 5. A ring 20 of elliptical shape rests upon a suitable foundation shown illustratively in FIG. 5 as a berm or earthwork ledge 22, of the same outline. Two sets of intersecting cables 24 and 26 (FIG. 4) each parallel in projection onto the horizontal plane to one of the diagonals 8 and 10 of the circumscribing rectangle 12, support a membrane 28 (FIG. 5) which may be, for example, of fabric or plastic. The mem brane covers the space enclosed within the ring 20 and is fastened thereto and to the

cables

24 and 26 by suitable means illustrated in FIGS. 6 and 9 to 12, to be described presently. In the embodiment of FIGS. 4 and 5 the roof is sustained by a supra-atmospheric pressure in the space 30 beneath it. A very small excess of pressure above atmospheric suffices for this purpose and can be maintained with pumping means, ingress to and egress from the space 30 being had through a lock not shown in FIG. 5.

In accordance with the invention, the

cables

24 and 26 are disposed to impose upon the ring 20 substantially zero bending moments in the horizontal plane. This is achieved by giving to the cables such lengths that with the roof load which they must support (whether an internal air pressure as in FIGS. 4 and 5, or a weight due to gravity), the cables will assume a configuration in vertical planes such that the horizontal component of the tensions therein at their anchor points with the ring will be those required to impose on the ring a substantially funicular loading.

If as in FIG. 4 the cables are in two

sets

24 and 26 of equally spaced cables parallel to the diagonal axes 8 and 10, the horizontal components of force to be exerted by each cable at its anchor points with the ring can be evaluated in terms of the stress along, i.e., tangent to the axis of the ring (FIG. 3), at the intersections of the diagonals 8 and 10 with the ring. Let this stress be denoted P an aggregate compressive stress in the ring on the assumption that the

elements

24 and 26 are cables under tension.

With the cables parallel to the axes 8 and 10 of skewed symmetry, let the horizontal component of the cable reactions H,- acting on the ring segment EF of FIG. 2 extending from the intersection with the ring at E of one diagonal 8 past the intersection therewith of the major axes to the intersection therewith at F of the other diagonal 10 be as shown in FIG. 3. The forces P acting at E and F are the internal forces. Structural symmetry is assumed about the major and minor axes 4 and 6 of the ellipse to which the ring 20 conforms. By considering the equilibrium of segments of this free body from E to H E to H etc, the horizontal reactions H H etc, can be determined in terms of P. Similarly, if there is considered the ring segment DE between the intersection with the ring at D of the opposite end of diagonal 10 and the point E already defined, and if there is determined the relationship between the horizontal component H, and the force P, one finds that the opposite ends of each cable in the set 26 exert the same horizontal reaction on the ring. This, however, is also the requirement for horizontal equilibrium of the cable net when forces acting on the cables are either vertical or symmetric with respect to a vertical axis if symmetry and the end points lie in the same horizontal plane.

With the horizontal components of tensions in the cables at their anchor points thus determined as a function of the axial force P in the ring at its intersections with the diagonals 8 and 10, a suitable shape for the roof in terms of the rise of the membrane above the plane of the ring and the consequent conformation of the cables themselves can be arrived at by considering vertical equilibrium at each of the cable intersection points. The material of the roof membrane and acces sory elements such as the cables themselves being known, the weight per unit projected area can be estimated. This gives the vertical force of gravity to be supported by air pressure within the enclosed space 30, re sulting in a net upward vertical force on the roof membrane. With this upward force and with an assumed value of P, one can compute the rise in the cables nec essary for the stress therein to have this vertical component and to have simultaneously the desired horizontal component at the anchor points of the cables. If the resultant three-dimensional shape for the roof is not the one desired, another value of the axial ring stress P can be assumed and another shape can be computed for the roof by the same process. In this way there can be found two roof shapes straddling the one desired, e.g. as to the degree of convexity thereof, and extrapolation between the two will yield the shape desired.

That is, one obtains a complete pattern or three dimensional specification of the geometry of the cable net the two sets of cables plus the tension at both ends of each cable and hence moreover the tensions in the successive segments of each cable. From this information it is possible for example to cut the cables and to mark them at intervals which will be spaced in service by the spacings which are to exist between adjacent intersections along each cable, due account being had in the process of marking for the differences between the uniform tension to which the cable is subjected in the shop and the non-uniform tensions along each cable in service, and due account being additionally had for other differences between shop and service conditions such as temperature.

The cables are thus available of suitable length and with markings at their points of intended crossover one with another and with end markings (46 in FIG. 6) to be made to coincide with the working points marked on the ring (45 in FIG. 6). The'cables can then be attached to the ring and clamped together. The roof membrane is then applied and the design load is imposed by inflation. with the consequence that within the margin of error accepted, the final shape for the roof and the loading of the ring will be that intended in the design.

The membrane 28 is secured to each of the cables by means of a web 42 and by means of lacing as shown in FIG. 6. The membrane is also secured to the ring by means ofa hold-down strip 44, suitably bolted or otherwise fastened through the membrane to the ring. This is also shown in FIG. 10, and FIGS. 11 and 12 illustrate a preferred method for attaching the webs 42 to the membrane.

The cables of the two sets are fastened to each other at their intersections by means of clamps which may take the form shown in FIGS. 7 and 8.

The clamp of FlGS. 7 and 8 comprises a pair of crossed split

steel pipe sections

50 and 51, the adjacent halves of the two sections being optionally welded to gether. Cables 24' and 26', one from each of the two sets of cables, lie in the split pipe sections which are held together with adequate available force of friction by a pair of U-bolts 52 and a plate 53. No interface material is provided between the cables and the pipe sections for improved holding power.

Details of one form of ring 20 are shown in the crosssectional view of FIG. 6. The entire roof, lightened by the upward force of inflation, rests on the ring 20 and thereby on the berm 22. The coefficient of friction between the ring and the berm must be of intermediate value, large enough to prevent the roof from sliding off but low enough to permit localized slippage of small ex tent rather than to cause failure to the berm. A single galvanized sheet as indicated at 33 in FlG. 6 on a smooth troweled concrete surface 32 of a ring-shaped slab resting on the berm is satisfactory.

The cables, of which one is shown at 40, pass through bores in the ring for anchorage to the ring. Provision is made for adjustment in length of the cable so that an index point 46 on the cable, determined in the preloading and marking process hereinabove described, will coincide with an index point 45 along the length of the bore. The index points 45 on the ring lie, with respect to plan, on the closed curve to which the ring basically conforms. Thus preferably this closed curve lies along the projection onto the horizontal of the center of gravity of the cross-sectional area of the ring.

I claim.

1. A non-circular and non-elliptical roof structure defining an enclosed occupiable building space comprising a continuous ring projecting in plan to a closed curve having a plurality of skewed axes of symmetry, a plurality of sets of structural members operatively connected to said ring with the structural members of each set extending in plan substantially parallel to a separate one of said skewed axes of symmetry, said structural members in one of said sets crossing the structural members of another of said sets at predetermined locations throughout said structure and being operatively interconnected to each other at substantially all of said crossing locations, and a membrane secured to said structural members whereby the stresses in said ring are substantially funicular for every selected closed curve in said family of curves and the stresses in said structural members are non-uniform.

2. A roof structure as defined in claim 1 wherein said closed curve is developed by use of the equation (x/a)" (y/bY" l where the value of m is selected as any number greater than 2.

3. A roof structure as defined in claim 1 wherein said skewed axes of symmetry coincide with the diagonals of the rectangle circumscribing said selected closed curve.

4. A roof structure as defined in claim 3 wherein said structural members are tension cables.

5. A roof structure as defined in claim 4 including a foundation conforming in plan substantially to the closed curve defined by said ring and a support plate between said ring and foundation providing a bearing surface frictionally engaged therebetween for supporting said ring on said foundation.

6. The roof structure as defined in claim 5 wherein said foundation includes an earthern berm extending around the outside of said ring and bieng inclined outwardly and downwardly therefrom along an incline conforming generally to the adjacent slope of the membrane to absorb positive wind pressures.

7. An air-supported cable roof structure defining an enclosed occupiable building space comprising, a peripheral support ring projecting in plan to a closed curve having a plurality of skewed axes of symmetry and developed by use of the equation (x/a)'" (y/b)"' 1, wherein m equals any number greater than 2 and the values of a and b are different; a plurality of sets of flexible tension cables operatively connected to said ring with the cables of each set extending in plan substantially parallel to a separate one of said skewed axes of symmetry, said cables in one of said sets crossing the cables of another of said sets at predetermined locations throughout said structure and being operatively interconnected to each other at substantially all of said crossing locations, and a flexible membrane secured to said cables on the side thereof opposite said occupiable building space thereby to enclose said cables within said occupiable building space, said cables and membrane being supported by pressurized air supplied to said structure, said cables having non-uniform stresses therein under uniform load conditions and said ring remaining in its closed curved configuration without deformation while the stress in said ring are substantially funicular.

8. A roof structure as defined in claim 7 wherein said skewed axes of symmetry coincide with the diagonals of the rectangle circumscribing said selected closed curve.

9. A roof structure as defined in claim 8 including a foundation conforming in plan substantially to the closed curve defined by said ring, said ring including a support plate secured thereto and having a low friction surface engaged with said foundation for supporting said ring thereon and to permit controlled movement therebetween.

10. The roof structure as defined in claim 9 wherein said foundation includes an earthen berm extending around the outside of said ring and being inclined outwardly and downwardly therefrom along an incline conforming generally to the adjacent slope of the membrane to absorb positive wind pressures

Claims

2. A roof structure as defined in claim 1 wherein said closed curve is developed by use of the equation (x/a)m + (y/b)m 1 where the value of m is selected as any number greater than 2.

7. An air-supported cable roof structure defining an enclosed occupiable building space comprising, a peripheral support ring projecting in plan to a closed curve having a plurality of skewed axes of symmetry and developed by use of the equation (x/a)m + (y/b)m 1, wherein m equals any number greater than 2 and the values of a and b are different; a plurality of sets of flexible tension cables operatively connected to said ring with the cables of each set extending in plan substantially parallel to a separate one of said skewed axes of symmetry, said cables in one of said sets crossing the cables of another of said sets at predetermined locations throughout said structure and being operatively interconnected to each other at substantially all of said crossing locations, and a flexible membrane secured to said cables on the side thereof opposite said occupiable building space thereby to enclose said cables within said occupiable building space, said cables and membrane being supported by pressurized air supplied to said structure, said cables having non-uniform stresses therein under uniform load conditions and said ring remaining in its closed curved configuration without deformation while the stress in said ring are substantially funicular.

10. The roof structure as defined in claim 9 wherein said foundation includes an earthen berm extending around the outside of said ring and being inclined outwardly and downwardly therefrom along an incline conforming generally to the adjacent slope of the membrane to absorb positive wind pressures.