Note : Les descriptions sont présentées dans la langue officielle dans laquelle elles ont été soumises.
This invention relates to a roof construction formed from multiple
hyperbolic paraboloid units. More particularly, it relates to a roof con~
struction in which the multiple units are joined together at contiguous
surfaces and are supported by both vertical members and horizontally dis-
posed and tensioned or compressed members.
Roof constructions employing hyperbolic paraboloid units are
known in the prior art. In factr applicant is ~he inventor of a laminated
hyperbolic paraboloid unit that is described and claimed in U.S. Patent No.
3,653,166.
A hyperbolic paraboloid unit makes efficient use of materials by
relying on form or shape for strength, rather than on mass or depth of
bending members. Specifically, a hyperbolic paraboloid unit contains two
sets of parabolic curves, which, in plan view, extend in the diagonal
directions of the unit. One set of parabolas is concave downwardly and
the other set in concave upwardly, and a uniform load on the unit is
carried in the two diagonal directions by the series of parabolas. The
set of parabolas that is concave downwardly carries its load in axial
compression, as in an arch, while the set of parabolas that is concave
upwardly carries its load in tension, as in a cable.
A hyperbolic paraboloid unit, or shell, can be divided into four
quadrants; each quadrant having a shell field that includes the two sets
of parabolas in the diagonal directions. Most preferably, the edges of
each quadrant are provided with stiffening members, and the quadrants are
connected together through tha stiffening members to form the complete
hyperbolic paraboloid unit.
For many small building constructions it may be practical to em-
ploy a single hyperbolic paraboloid unit for the entire roof span. This
is the case when it is easy to ship the ~mit with at least the quadrants
fully assembled. If the quadrants are complete it is a fairly easy matter
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' to assemble the unit on site by merely connecting the quadrants together.
~owever, it i5 not desirable to employ a single unit to form an entire
xoof span when the span is so large that it is impractical to ship the
unit with at least the quadrants in a fully assembled state. On site
completion of the quadrants is a difficult and time consuming task. In
the latter discussed situation, it is preferred to form the span from
multiple units that are of a size permitting easy shipment of fully
assembled quadrants. It is also desirable to form a roof construction with-
out using interior vertical supports. Obviously, by omitting such supports
the area under the roof span will be unobstructed and can be most effec-
tively'utilized. Roof constructions that are free of interior vertical
columns are referred to as "free span" roof constructions.
A free span roof construction formed of multiple hyperbolic para-
boloid units is employed in the athletic facility at the Pratt Institute
in Brooklyn, New York. This roof construction is primarily a three-
hinged arch that depends upon long sloping compression struts to transmit
the load from the interior area of the roof to peripheral vertical but-
tresses. Although the sloping strut arrangement creates a high vaulted
' exterior, which may be desirable for some installations, it does so by
transmitting a large horizontal component of load to the vertical buttres-
ses. The heavy buttresses and strong foundation necessary to support
these large loads are expensive to construct. Furthermore, the sloping
struts are formed from heavy members because they are required to carry
large loads, and these heavy struts are also quite expensive to use in
roof constructions. The roof construction of aspects of the present in-
vention does not require the use of heavy sloping struts, and can employ
lighter and less expensive vertical supports than those employed in the
above described three-hinged arch arrangement.
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By one broad aspect of this invention a roof construction is
provided comprising at least a pair of juxtaposed hyperbolic paraboloid
units, each unit comprising four quadrants, each quadrant having a pair
of spaced, stiffened horizontal edges and a pair of spaced, stiffened
sloping edges, the sloping edges sloping downwardly from an upper level
to a lower level, two sloping edges of the first unit being contiguous
to two sloping edges of the second unit and the other sl~ping edges of
the units being non-contiguous, the stiffened horizontal edges all lying
. in a top, horizontal plane; the juxtaposed units being joined together
at respective contiguous sloping edges thereof to form a multiple unit
roof span that-includes both horizontally extending edges and sloping
edges, a plurality of vertical members supporting the roof span at the
lower ends of only the non-contiguous sloping edges~ and first horizon-
tally disposed and tensioned members connecting the lower ends of the
sloping edges that are unsupported by vertical members with the lower
ends of sloping edges that are supported by vertical members, the first
tensioned members all lying in a lower horizontal plane, whereby a free
span roof construction is provided.
By a variant thereof, the joined units are defined by an outer
~0 periphery, and wherein the vertical members are positioned only at the
periphery.
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if ~!~Y another variant, one of the pair of spaced h~rizontal
edges in a quadrant is disposed at right angles to the other of the pair
of spaced horizontal edges.
By yet another variant, the lower hoxizontal plane is spaced
below the top horizontal plane by a distance equal to the vertical dis-
tance between the upper level of a sloping edge and the lower level of
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a sloping edge.
sy another aspect of this invention the above aescribed roof
construction includes second horizontally disposed and tensioned members
connecting the lower edges of:respective sloping edges that are supporred
by vertical members.
. By a variant thereof, the first horizontally disposed and
tensioned members are positioned at right angles to the second horizontal~
ly disposed and tensioned members.
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The roof construction of aspects of this invention includes multi-
ple hyperbolic paraboloid units that are joined together at contiguous
surfaces thereof to provide the desired roof span. The multiple unit
roof span includes both horizontally extending edges and sloping edges,
and vertical members to support the roof at the ends of only some of the
sloping edges. Preferably the vertically supported ends of the sloping
members are all adjacent the periphery of the roof span to provide a free
span construction. Horizontally disposed and tensioned or compressed mem-
bers connect ends of sloping edges that are unsupported by vertical members
with ends of sloping edges that are supported by vertical members to
transmit horizontal forces between the connected ends. Most preferably,
an end of each sloping edge is in force transmitting communication with a
` hori~ontally disposed and tensioned or compressed memoer.
The hyperbolic paraboloid units cooperate with the horizontally
tensioned or compressed members to provide a statically determinate truss.
This permits large roof spans to be supported without the use of heavy
and extensive buttresses, foundations and sloping struts of the type
which must be employed in the high vaulted, three-hinged arch construction
discussed earlier.
A hyperbolic paraboloid unit, as defined in this application,
includes four quadrants and each quadrant has a shell field of hyperbolic
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paraboloid configuration. In other words, each quadrant has a double
curvature which permits loads to be transferred to supports entirely by
direct forces so that all of the material in the cross-section of each
quadrant of the unit is uniformly stressed. Most preferably, each of the
quadrants consists of the hyperbolic paraboloid shell field and stiffen-
ing members, such as edge beams, connected to all four edges. Each hyper-
bolic paraboloid unit (i.e., a set of four quadrants) is assembled by
connecting the quadrants together through the stiffened edges. The or-
ientation of the quadrants within the units can be varied to provide
different shapes and configurations as desired. Most preferably, the
hyperbolic paraboloid units employed in this invention are the laminated
wood constructions disclosed in my issued U.S. Patent No. 3,653,166.
Most preferably the horizontally extending edges in the multiple
unit roof span are formed by stiffening members, e.g. edge beams, and
these edges lie in a common horizontal plane. The horizontally disposed
and tensioned or compressed m~mbers are in a different plane than the
stiffened horizontal edges and these tensioned or compressed members
cooperate with the stiffened edges and the shell fields to`provide the
statically determined truss. In other words, the stiffened horizontal
ed~es of the roof span provide one chord of the truss, the horizontally
tensioned or compressed members connected with the ends of sloping edges
provide a second chord of the truss and the sloping edges and the shell
fields provide connecting webs between the two chords to complete the
truss arrangement.
In accordance with aspects of this invention, the hyperbolic para-
boloid units can be connected together in two horizontal directions to
form large roof spans that do not require internal vertical supports~
Although high vaulted, three-hinged arch constructions are not formed in
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aecordanee with this invention, different construetions can be formed, such
as cantilever and continuous truss eonstructions.
In the accompanying drawins,
Fig. 1 is an isometric view of a roof construetion in aecordance
with an embodiment of this invention;
- Fig. 2 is a side elevation view of the roof construction shown in
Fig. l;
Fig. 3 is a bottom view of the roof construetion shown in Fig. 1
with parts broken away to show details of eonstruetion;
Fig. 4 is an isometric view of a roof construction employing two
hyperbolie paraboloid units and showing the shell field foree distribution
when the roof is subject to vertieal loading;
Fig. 5 is an isometric view of the construction of Fig. 4 showing
the load distributlon that provides for a statically determinate truss
arrangement.
Although specifie terms are used in the following deseription for
the sake of elarity, these terms are intended to refer only to the par-
tieular strueture of one embodiment of this invention seleeted for illus-
tration in the drawings.
Referring to Fig. 1, the roof eonstruetion 10 is formed ofmultiple hyperbolie paraboloid units joined together to provide the de-
sired roof span. In the embodiment shown for illustration, six sueh
units 12, 14, 16, 1~, 20 and 22 are employed. Preferably these units are
identieal, and are of the type deseribed in V.S. Patent 3,653,166. For
purposes of eompleteness, I will briefly deseribe the construetion of the
hyperbolie paraboloid unit 12.
Referring to Figs. 1 and 3, the hyperbolie paraboloid unit 12 in-
eludes four quadrants 12a, 12b, 12e and 12d. Eaeh of these quadrants is
formed of two plywood layers that are laminated together, and the edges
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of each quadrant are stiffened, such as by edge beams formed from sized
lumber. The edqe beams are laminated to the surfaces of each quadrant, and
the quadrants are connected together through the edge beams by bolts,
screws or the like, as is schematically indicated at 24 in Fig. 3.
Each quadrant of the unit 12 has a pair of stiffened horizontal
edges and a pair of stiffened sloping edges. The horizontal edges of each
quadrant are joined to horizontal edges of adjacent quadrants so that
these joined edges lie in a common horizontal plane. The joined horizontal
edges of the unit 12 are shown at 26, 28, 30 and 32 (Figs. 1 and 3) and
they all lie in the top horizontal p]ane of the roof construction 10. Ad-
jacent hyperboIic paraboloid units are joined together through contiguous
sloping edges. For example, the adjacent hyperbolic paraboloid units 12
and 14 are joined together through their contiguous sloping edges as indi-
cated at 34 and 36 (Figs. 1 and 3). The adjacent hyperbolic paraboloid
units are joined together so that all of the stiffened horizontal edges
lie in the same horizontal plane. In the roof construction 10, the hori-
zontal stiffened edges all lie in the top horizontal plane that includes
the joined edges xepxesented at 26, 28, 30 and 32.
Referring to Figs. 1 - 3, vertical support members 38 are
positioned about the periphery of the roof construction for supporting the
horizontal span formed from the hyperbolic paraboloid units. These verti-
cal supports 38 are connected to the ends of sloping edges o~ the hyper-
bolic paraboloid units, but not in the interior of the roof construction
10 (Fig. 3). In other words, the roof constxuction 10 is a free span con-
struction in which vertical columns or supports are positioned only about
the periphery.
The free span of roof construction 10 is made possible by employing
horizontally positioned members 40, e.g. multi-strand cables, that are
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tensioned between and connected to adjacent free ends of the sloping edges
of the hyperbolic paraboloid units 12, 14, 16, 18, 20 and 22. These hori-
zontally tensioned members cooperate with the shell field and the stiffened
horizontal edges of the hyperbolic paraboloid units to establish a stati-
cally determinate truss. If desired, either some or all of the horizontally
tensioned members can be provided with adjustment means, such as turnbuckles
42, to permit adjustment of the tension.
The roof construction 10 of an aspect of this invention is a
statically determinate truss provided by top ana bottom chords connected
together through diagonal web members. The top chord is provided by the
stiffened horizontal edges that lie in the top horizon*al plane of the roof
construction 10, as described earlier. The bottom chord is provided by the
horizontally positioned tensioned members ~0, and the connecting webs are
provided by the sloping edges and the shell fields of the hyperbolic para-
boloid units. Since the stiffened edqes are actually part of the hyperbolic
paraboloid quadrants or shells, these shells actually are used to form one
of the chords as well as the connecting web members.
In accordance with an aspect of this invention, the roof span can
be varied, as desired, by adding hyperbolic paraboloid units in the two
hori~ontal directions indicated by arrows 44 and 45 (Fig. 1). In this man-
ner large areas can be spanned without the necessity of utilizing interior
columns, and without the necessity of employing excessively large hyperbolic
paraboloid ~mits that require complex field assembly operations.
In order structurally to design the truss arrangement, the shell
stresses for the individual hyperbolic paraboloid units are added to any
stresses that are developed by the truss action, and the structural elements
are proportioned to carry the maximum total stresses imposed upon the system.
In view of the fact that the roof construction is formed from hyperbolic
paraboloid units, the stresses carried by the members due to shell and
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truss action are axial tension or compression, without any bending. Thus,
all of the members employed in the roof construction can be designed pri-
marily as axially loaded elements, wh~ch provides for simplicity of design
Referring to Figs. 4 and 5, an explanation of the loading en-
countered in a roof construction lOA of an aspect of this invention that
employs two hyperbolic paraboloid units 50 and 50A will now be described.
It should be understood that a similar analysis can be employed in roof
constructions including more than two hyperbolic paraboloid units, for ex-
ample, the six unit roof construction 10 shown in Figs. 1 - 3. In fact,
the analysis of the construction lOA shown in Fig. 4 can actually be
viewed as an analysis of the loading encountered in the two hyperbolic
paraboloid units 12 and 14 of the roof construction lO. Note that the
hyperbolic paraboloid units 12 and 14 provide a two unit span that is
supported by four vertical members 38 at the periphery thereof ~Fig. 3).
m ese four members each provide a vertical reaction force of the type in-
dicated at Ll, L5, Ll5 and LlI of the roof construction lOA (Fig. 4).
Also, it should be noted that vertical supports are not provided at the
ends of the joined sloping edges 34 and 36 of the units 12 and 14 in the
roof construction 10 (Fig. 3). ~is corresponds to the omission of verti-
cal reaction forces at L3 and L13 in the roof construction lOA. According-
ly, the roof construction 10 shown in Figs. 1 - 3 can be considered as a
combination of three of the units lOA shown in Figs. 4 and 5. Therefore
the loading analysis of the unit lOA is, with only slight modification,
equally applicable to the analysis of the six unit roof construction 10.
Referring specifically to Fig. 4, the roof construction lOA in-
cludes the two hyperbolic paraboloid units 50 and 50A that are joined to-
gether through sloping edges indicated at 52, 54 respectively. The units
50 and 50A are identical and are the same as the hyperbolic paraboloid units
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employed in the roof construction 10. As can be seen in Fig. 4, the hori-
zontal stiffened edges that are joined togehter all lie in a common top
- horizontal plane. As explained earlier, external vertical reaction forces,
e.g. those provided by vertical support columns 38 are established at the
ends of the sloping edges, indicated at Ll, L5, Lll and L15. Also as ex-
plained earlier, there is no external vertical reaction forces provided at
the ends L3 and L13 of the joined sloping edges of the units 50 and 50A.
Horizontally tensioned members, e.g. cables, are in force transmitting re-
lationship between the adjacent ends of the sloping edges. Specifically,
tensioned cable members are connected between Ll-L3, L3-L5, L5-L15, L15-
L13, L13-L3, L13-Lll and Lll=Ll. These-tensioned cable members provide
the bottom chord in the statically determinate truss arrangement. As ex-
plained earlier, the top chord is provided by the stiffened horizontal
edges that are disposed in;the top horizontal plane, and the two chords
are connected together through the sloping edges and the shell fields of
the hyperbolic paraboloid units.
The shell action stresses are indicated by the various arrows
shown in Fig. 4. The arrows designated "T" indicate regions-in axial ten-
sion; the arrows designated "C" indicate regions in axial compression; and
the arrows of the type designated 56 indicate the shear force that is
transmitted from the shell field of each quadrant to the stiffened edges.
The conditions shown in Fig. 4 are encountered under uniform vertical load-
ing of the roof construction. Vnder this vertical loading, the various
shell fields are under axial compression "~" in the set of upwardly con-
cave parabolas (e.g., along diagonal Lll-U7) are are in axial tension "T"
in the set of downwardly concave parabolas (e.g., along diagonal U6-U12).
The load imposed upon each shell field is transnlitted to the sloping edges
of the quadrants, such as the sloping edges U8-L13 and U12-L13. When the
ends of the various sloping edges are provided with an exterior vertical
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reaction, such as that which can be provided by a vertical column, the
sloping edges are generally axially loaded in compression. For example,
the sloping edges U6-Lll and V12-Lll are loaded in compression since an
external vertical reaction is provided at their junction.
By eliminatlng the use of interior vertical columns in a roof
construction of the type shown in Figs. 1 - 3, there will be a number of
sloping edges that will not be provided with an external vertical reaction
force at their ends. In Fig. 4, this is represented by the omission of ex-
ternal vertical reactions at L3 and L13. Because there are no external
vertical reactions at L3 and L13, the sloping edges U2-L3, U4-L3, U12-
L13 and U14-L13 are all stressed in tension, rather than in compression.
; The sloping surfaces U8-L13 and U8-L3 (the joined edges oE the hyperbolic
paraboloid units 50 and 50A) are stressed in compression since the force in
these joined edges can be balanced by the other sloping edges that meet at
L3 and L13. In order to provide a statically determinate truss it is im-
portant to balance the tensilè stresses that are set up in the various
sloping edges~ For example, the tensile stress built up in the sloplng
edge U12-L13 must be balanced at U12.
Referring to Fig. 5, the manner in which the various loads and
forces are balanced by the truss arrangement will now be described in con-
nection with unit 50. The tensile stress in V12-L13 can be regarded as an
additional load at U12, and that load is designated by arrow 58. The
sloping edge V12-Lll must then be additionally loaded in compression, as
indicated by arrow 60, and the shell field from U6 to U12 must pick up an
additional tensile stress from the truss action, in addition to the shell
action tensile stress depicted in Fig. 4. The combined tensile stress
from the shell and truss action is schematically represented by the
curved arrow 62. In addition, the horizontal stiffened connection between
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U7-U12 takes added compression as indicated by arrow 64 to balance the load
at U12. At U6 the added tensile stress in the shell field indicated by
i arrow 62 is balanced by an added compression of the stiffened sloping edge
U6-Lll, as indicated by arrow 66, and by an added compression in the
stiffened horizontal edges U6-U7 and U7-U8, as indicated by the arrows 68,
70, respectively.
rom the above analysis it can be seen that the various stiffened
edges and the shell field take up the shell action axial stresses plus the
additional axial stresses induced by the truss action when some external
vertical reactions are eliminated from the roof construction. This permits
the roof constructions of aspects of this invention to be formed with large
spans without the use of internal vertical support members.
It is within the scope of aspects of this invention to vary the
configuration of the roof construction. For example, the roof construction
10 tFigs~ 1-3) can be inverted so that the horizontally stiffened edges of
the hyperbolic paraboloid units form the bottom chord of the truss. In
this variant the member 40 will constitute the top chord of the truss, and
the top and bottom chords will still be connected by the sheil field.
As used throughout this application, all references to hyperbolic
paraboloid units refer to units having four quadrants, each of a hyperbolic
paraboloid configuration. ~lowever, reference in the claims to the use of
a plurality of multiple hyperbolic paraboloid units does not preclude the
possibility ~that at least some of the units will contain less than four
quadrants, provided that at least two of the units contain four quadrants.
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