# Patent application title: Frameless space structure

##
Inventors:
Michael M. Davarpanah (Lancaster, CA, US)

IPC8 Class: AE04B132FI

USPC Class:
52 801

Class name: Static structures (e.g., buildings) compound curve structure

Publication date: 2009-12-03

Patent application number: 20090293374

## Abstract:

A construction module comprises a plurality of shaped plates connected to
each other at selected angles and orientations to form the constructions
module. The plates have two or more different shapes, and may be arranged
with respect to each other so as form a three dimensional construction
module whereby external forces acting on the module are balanced by
internal tensile and compressive stresses in symmetrical plates acting in
opposite directions. Also provided is a frameless structural system
comprising a plurality of such construction modules, the plurality of
construction modules being fastened to each other to form the structural
system.## Claims:

**1.**A construction module comprising a plurality of shaped plates connected to each other at selected angles and orientations to form the constructions module.

**2.**A construction module as claimed in claim 1 wherein the plates have two or more different shapes.

**3.**A construction module as claimed in claim 1 wherein the plates are arranged with respect to each other so as form a three dimensional construction module whereby external forces acting on the module are balanced by internal tensile and compressive stresses in symmetrical plates acting in opposite directions.

**4.**A construction module as claimed in claim 1 comprising a plurality of elongate plates having long edges and end edges, the plates being joined to each other along their common long edges.

**5.**A construction module as claimed in claim 4 further comprising end plates joined to the elongate plates along the end edges.

**6.**A construction module as claimed in claim 1 wherein the plate shape and angle connection thereof is based on one or more of the following parameters: module rotation angle, module end connection angles, module rotation arm, module width, load bearing plates projected end widths dimensions, main load bearing plates projected mid width dimension, and main loading bearing plates intersection angle.

**7.**A construction module as claimed in claim 1 further comprising at least one connection means for connecting a construction module to an adjacent construction module.

**8.**A construction module as claimed in claim 1 wherein the module is a three dimensional structural unit which carry and transfer loads in three dimensions using the plates as load bearing structures.

**9.**A frameless structural system comprising a plurality of construction modules, each construction module comprising a plurality of shaped plates connected to each other at selected angles and orientations to form the constructions module, the plurality of construction modules being fastened to each other to form the structural system.

**10.**A frameless structural system as claimed in claim 9 wherein the construction modules are bolted to each other, lower nodules being bolted to a foundation.

**11.**A frameless structural system as claimed in claim 9 comprising a chain of construction modules bolted to each other.

**12.**A frameless structural system as claimed in claim 9 wherein construction modules are bolted to each other to form a structure comprising an arch; a shell; and a sectionalized system.

**13.**A frameless structural system as claimed in claim 9 wherein lateral support is provided by adjacent construction modules.

**14.**A frameless structural system as claimed in claim 9 wherein lateral support is provided using braces for bracing outer plates of a module so as to act in opposite directions.

**15.**A frameless structural system as claimed in claim 9 constructed based in calculations in accordance with one or more of the following variables: Modular width; Main Load Bearing Plates Intersection Angle; Modular Thickness; Maximum Allowable Compressive Stress; Maximum allowable Tensile Stress; and Maximum Allowable Shear Stress.

**16.**A frameless structural system as claimed in claim 9 wherein the system is one of the following: a completed structure; roof structure; dome; high rise; bridge; dam system; and tunnel system.

## Description:

**FIELD AND BACKGROUND OF THE INVENTION**

**[0001]**This invention relates to a frameless, preferably folded plate, space structure. More particularly, the invention relates to frameless folded plate space structural systems which use folded plate space modular(s) as the building block(s) of a structural system and may be used to create structures without the need for load bearing columns, frames or other structural support members, in order to maintain the integrity of the structure.

**SUMMARY OF THE INVENTION**

**[0002]**This invention has several main components including the folded plate space modular, and frameless folded plate space structural systems (FFPSSS). The system applications may include:

**TABLE**-US-00001 Frameless Folded Plate Space Structures FFPS-S Frameless Folded Plate Space Roof Systems FFPS-RS Frameless Folded Plate Space Domes FFPS-D Frameless Folded Plate Space High Rises FFPS-HR Frameless Folded Plate Space Bridge Systems FFPS-BS Frameless Folded Plate Space Dam Systems FFPS-DS Frameless Folded Plate Space Tunneling Systems FFPS-TS

**[0003]**The space modular is designed based on the folded plate structural concept, and can be made from any material, including plastics. Space modules are, in accordance with one aspect of the invention, made from thin plates of structural plastic, and these can be as thin as 1/8 inch or less, specifically configured and shaped so that a plurality of such plates, whether of the same shape or different shapes, can be located adjacent one another in a selected manner to create a folded plate space modular.

**[0004]**A single folded plate space modular is used with other identical or different shaped modules to form a frameless folded plate space structural system. Frameless folded plate space structural system(s) can be assembled to form a complete structure or a component of a structure such as frameless plate space roof structures.

**[0005]**Frameless folded plate space structures offer advantages of conventional structures and a lot more. Unlike conventional structures in which the load on the structure is transferred onto a framing component of the structure, the load on frameless space structures is carried by the entire shell of the structure. This particular design may eliminate the need for framing. The framing is indeed the largest and heaviest component of any structure.

**[0006]**Folded plate spaced structures have numerous usages in private, commercial and public sectors. They may be ideal for school cafeterias and lunch shelters, gathering rooms, swimming pool enclosures, gyms, shade structures, green houses, warehousing, light industries, hangars, to name just a few examples.

**[0007]**Lightweight, easy and fast construction makes the space structures of the invention additionally advantageous for fast track projects. A 2000 square foot structure can be assembled by a small crew in a couple of days. The modular space structures of the invention can evenly distribute the load to the foundation thereby eliminating the need for large foundation due to concentrated framing loads.

**[0008]**Roof modules are typically installed on the exterior walls of the structure. They are only supported by the exterior walls and there is no need for any roof framing or interior supports. Furthermore, they are water proof and insulated.

**[0009]**Flexible metal molds (or jigs) can be designed and fabricated which with some modifications can take the angles and sizes of several size modules. All modules (including wall, ceiling, roof) may be typically the same and follow the same formulas. The same mold can be used to fabricate different modules.

**[0010]**Folded plate solar modules can be fabricated by incorporating solar cells into the structural plastic plates. Solar modules are like the other modules in that they have all architectural and structural characteristics of the modules. The number of solar modules used in any structure will typically depend upon the electrical and other needs and requirements of the structure.

**[0011]**Preferably, each module is monolithically fabricated to act as one structural unit. Forces on the module typically result from dead load, live load, wind and seismic movements and these are balanced by the tension and compression stresses in the plates. Stress in a structure is an internal resistant to an external force. It is the sum of these stresses in the plates which holds the modules in equilibrium.

**[0012]**The modules of the invention (including wall, ceiling and roof) are typically, but not necessarily, the same. Flexible molds can be adjusted to form the angles and measurements of different modules. This design flexibility offers structures which can practically satisfy all architectural needs of any projects.

**Fabrication**

**[0013]**The modules preferably need to be monolithical. The plates preferably need to have fixed connections to transfer the stresses in the plates. There are two primary fabrication methods.

**[0014]**(1) Molding: Selected material is molded into the shape of the modular. Structural Plastic can be molded by injection, press or vacuum methods. Flexible wood and metal Molds are being fabricated which with some modifications can take the angles and sizes of several modular.

**[0015]**(2) Continuous Connection: Pre-cut and angled structural plates are assembled in the above molds and connected to each other with stainless steel or aluminum, or other material, profiles in the shape and angles of the module. The connections are combinations of mechanical and chemical attachments and these may include rivets, bolts, glue and epoxy, as examples only.

**Connections**

**[0016]**The modules may be connected to each other and the foundation by flexible bolted connections. The connections allow free movement of the individual modules as well as the entire structure.

**[0017]**In a foundation by a flexible bolted connection, the module can freely move in all directions. The flexible connections absorb a big percentage of pressure on the module which may result from seismic movements, wind and other loads on the structure. In connecting of the modules to each other, the entire structure can freely move in the direction of the load. The pressure on the structure is preferably absorbed by the connections reducing the load on individual modules.

**BRIEF DESCRIPTION OF THE DRAWINGS**

**[0018]**In the drawings:

**[0019]**FIG. 1 shows various views of a folded plate space structure or modular in accordance with one aspect of the invention (SM-1);

**[0020]**FIG. 2 shows various views of pieces of a folded plate space structure or module in accordance with one aspect of the invention, including individual plate configurations and dimensions (SM-1A);

**[0021]**FIG. 3 shows various views of types of folded plate space modulars and information relating thereto (SM-2);

**[0022]**FIGS. 4, 5 and 6 show various views of a folded plate space modular including structural calculations and formulas, maximum moment calculations, and outer plates lateral forces and movements (SM-3B, SM-3 and SM-3A);

**[0023]**FIGS. 7, 8, 9 and 10 show various views of and calculations for a Frameless Folded Plate Structural System (FFPSS), including Loads/reactions Structural Formulas and Sample calculations (SYS-1), Wind/Reactions Structural Formulas and Sample Calculations (SYS-2), Shear/Moment Diagrams and Formulas (SYS-3), and Actual Moment Calculations (SYS-4);

**[0024]**FIGS. 11, 12 and 13 show various views and calculations of a frameless folded plate space structure (FFPSS), including: Type-3 Space Structure, Plans, Elevations (SS-1A-1); Front Elevation, Space Wall modules (SS-1A-2); and Structure and Modules Perspective (SS-1A-3);

**[0025]**FIGS. 14 and 15 show details relating to Type-2, frameless folded plate space structures in Standard Sizes (SS-2A) and System Flexibility (SS-3A);

**[0026]**FIGS. 16, 17, 18, 19 and 20 show a details relating to a frameless folded plate space structure, including: Structural Calculation Formulas (SS-1S-1); Sample Structural Calculations (SS-1S-2); Front Elevation Structural Formulas (SS-1S-3); Wall Modules Geometric Design & Sections (SS-4A); and Wall Modules Structural Formulas (SS-4B);

**[0027]**FIGS. 21, 22 and 23 show views and details of a Frameless Folded Plate Space Roof System (FFPSRS), including: Type-2 Roof Structure, Plans and Elevations (RS-1A-1); Roof Plans and Details (RS-1A-2); and Roof Structural Calculation Formulas (RS-1S-1);

**[0028]**FIGS. 24 and 25 show views and details of a Frameless Folded Plate Space Super Dome (FFPSSD), including Plans, Sections. Seat and Floor Deck Details (SD-1A), and Plans. Structural Modules and Sections (SD-1S);

**[0029]**FIG. 26 shows views and details of a Frameless Folded Plate Space Mid and High Rise (FFPS-HR), including

**Typical High Rise**, Plan, Section, Detail (HR-1A);

**[0030]**FIGS. 27, 28 and 29 show views and details of a Frameless Folded Plate Space Bridge (FFPSB), including: Bridge, Sections and Details (BS-1); Ramp Modular, Plan, Sections and Details (BS-2); and Elevations and Sections (BS-3);

**[0031]**FIG. 30 shows views and details of a Frameless Folded Plate Space Dams (FFPSD), including plans, elevations and Anchoring Details (D-1A); and

**[0032]**FIG. 31 shows views and details of a Frameless Folded Plate Space Tunneling Methods (FFPSTM), including Rock Tunneling Method, RTM, Sections and Details (TM-1).

**DETAILED DESCRIPTION OF THE INVENTION**

**[0033]**The invention comprises a specially designed and configured space module. The invention is also for a structure which is comprised of such a space module which may be used with other similar and/or differently configured space modules, the various space modules being designed and attached to each other in a manner which results in a structure that is strong, easy to construct and has various other advantages and benefits as will be described herein.

**[0034]**Reference is made to FIG. 1 of the drawings which show a top view and side view respectively of a typical space module 10. Although typical, it is just one of many a great range of shapes, dimensions and configurations of space module 10 which can be made in accordance with the invention. Indeed, the benefits and advantages of the invention lie in the fact that a space module 10 of selected, customized and desired configuration can be created to suit a specific application. The space module 10 now briefly described with reference to FIG. 1 should therefore be seen as representative only.

**[0035]**The space module 10 is generally of overall rectangular shape and is made up of longitudinal panels 12, 14, 16 and 18. The panels are angled with respect to each other, and their relationship changes over the length of the panel. This can best be seen by reference to section A-A, section B-B and section C-C in this figure. End panels 20 and 22 and provided. In FIG. 1(b) four sets of such panels are shown, two joined end to end and two joined along common side edges. The panels are arranged at selected angles and shapes calculated to have optimal strength and ability to be joined with other similar of differently configured panels so as to form a structure.

**[0036]**Other figures in this application show different panels configured for different purposes, and further details relating thereto are discussed below.

**Examples**

**[0037]**In the description and disclosure below, a range of examples of applications of the frameless space structure of the invention are set forth, together with reference to drawings and formulae which may be used in calculating and determining the shape, size and other characteristics of space structures and their assembly in order to achieve suitably designed structures.

**[0038]**Brick was used by early Romans as the Building Block for Arch Shape Structural Systems. The Arch Shape Structural System was used to develop several applications including Arch Shape Structures and Roofs, Domes, Arch Shape Bridges, Arch Shape Tunnels, and Arch Shape Dams.

**[0039]**In one embodiment, the Frameless Folded Plate Space Structural system of the invention uses the folded Plate Space Module, as the building block of the new system. The system has numerous applications including Frameless Folded Plate Space Structures, Roof Systems, Domes, Future Pressurized Colonies, High Rises, Bridge Systems. Dam Systems, and Tunneling Systems.

(1) Space Module

**[0040]**The Space Modular is designed based on the Folded Plate Structural Concept. Space Modules are monolithic Structural Units made of thin structural plates, placed in various planes and angles to form the shape of the space modules. Forces on the modular resulting from Dead Loads, Live Loads, Wind and Seismic Movements are balanced by internal Tensile and Compressive Stresses in symmetrical plates acting in opposite directions. Stresses in a structure are internal resistance to external forces. It is the sum of these stresses which holds the Space Modules in equilibrium.

**[0041]**Space Modules in accordance with one aspect of the invention are three dimensional structural units, carrying and transferring loads in three dimensions. It is the three dimensional structural characteristic of the modules and using the plates as structural members and stresses in the plates to counterbalance the external forces which provide the space module with its unique features. Plates are placed, supported and involved in a manlier to carry their maximum loading capacities and reach their maximum allowable stresses.

**[0042]**The Space Module of the invention is designed to be supported on all four sides. Bottom and Top modules in the chain provide the longitudinal support and adjacent modules or symmetrical outer plates of the same modular acting in opposite directions provide the lateral Support.

1-A: Geometric Design

**[0043]**There are a number of Architectural and Structural factors and considerations to consider when designing and selecting a Space Module of the invention. FIG. 1, or SM-1, illustrates a type of Space Modular of the invention which combines two similar sections each made of four pairs of Symmetrical Plates acting in opposite directions. Each Modular has two End Connection Plates and two side connection plates.

**[0044]**FIG. 1(a) shows the side view of an atypical modular. Rotation Angle A, End Connection Angles A1L, A1H and Rotation Arm a are variables and are selected based on the Longitudinal Cross section of the system and positioning of each modular in the system. The Desired Span, Middle and End Height values of each modular among other structural and architectural requirements of the system dictates the selection of these values.

**[0045]**FIG. 1(b) illustrates the top view of a Space Modular. Detail shows Total width of the Modular W, width of each section W

_{5}, Projected end widths of Load Bearing Plates W

_{2}, W

_{4}and projected middle width value of W

_{3}. In Standard Shape Space Modules, the value of W

_{3}is half of the section width W

_{5}.

**[0046]**1-A-a: Design Variables

**[0047]**The first step in design of the space modular is the selection of following variables:

**TABLE**-US-00002 Modular Rotation Angle, Angle A Modular End Connection Angles Angles, A

_{1}l A

_{1}h Modular Rotation Arm, Dimension a Modular Width, Dimension W Load Bearing Plates Projected End Widths Dimensions, W

_{2}, W

_{4}Main Load Bearing Plates Projected Mid Width Dimension, W

_{3}Main Loading Bearing Plates Intersection Angle, Angle γ

**[0048]**1-A-b: Variable Selection and Considerations

**[0049]**1-A-b-1: Modular Rotation Angle A, Modular Connection Angles A

_{1}l and A

_{1}h

**[0050]**Rotation and connection angles are calculated based on the architectural characteristics of the system and positioning of each modular. Designers should select these angles, among other considerations, with emphasis on limiting the number of typical modules needed to design the system. Designers should choose angles which would best form the required longitudinal cross section while minimizing the number of typical modules.

**[0051]**1-A-b-2: Modular Rotation Arm a

**[0052]**Among architectural factors which designers need to consider when selecting the Length of Rotation Arm a as outlined above, Designers should consider and take into consideration the structural impact of increasing the length of Rotation Arm a. An increase in length of rotation Arm a reduces the number of modules required to form the required cross section but proportionally decreases the Maximum allowable Moment and Load Bearing capacity of the Modular.

**[0053]**1-A-b-3: Modular Width W, Section Width W

_{5}, Load Bearing Plate Projected End Widths W

_{2}, W

_{4}Load Bearing Plate Projected Mid Width W

_{3}.

**[0054]**Modular Width W is independent but should be chosen proportional to Modular Rotation Arm a. A ratio close to 1/2 is preferred for W/a is a reasonable starting point. Increase of the total width of the modular increases the maximum Allowable Moment a Modular can carry.

**[0055]**Section Width W

_{5}is chosen based on the structural requirement of the modular. An increase in the Section Width W

_{5}, proportionally increases the Load Bearing Plate Widths and effective structural depths. Designers should use a reasonable number of sections in each modular to achieve the required load requirements while maintaining reasonable Plate Widths.

**[0056]**Plate Projected End width W

_{2}is proportional to the end width of Load Bearing Plates (Dimensions f

_{1}and f

_{2}). An increase in Dimension W

_{2}increases the above f

_{1}and f

_{2}dimensions, End Plate Connection Plate Sizes and Shear Force capacity of the modular.

**[0057]**1-A-b-4: Plate Projected Mid Width W

_{3}

**[0058]**In standard Shape Modules, Dimension W

_{3}is half the dimension W

_{5}. In tapered modules, W

_{3}is half the dimension of section at the middle.

**[0059]**1-A-b-5: Main Load Bearing Intersection Angle γ

**[0060]**FIG. 2 shows individual plates in each section, their dimensions and Plate intersection Angles. Main Load Bearing Plates Intersection angle γ is the main structural variable of the modular. This is the angle of intersection of the Main Load Bearing Plates and an imaginary Plane passing through the rotation Arm a and the base of the Modular (see section X-X, FIG. 2). An increase in Angle of γ increases all structural capacities of the modular including Maximum allowable Moment and Shear Force. As structural formulas will show, increase of the Angle γ is generally the most economical way to increase the maximum allowable moment of the Modular.

**[0061]**1-A-c: Geometric Formulas and Calculations

**[0062]**FIG. 2 includes geometric formulas. Formulas calculate Space Modular Dimensions, Plates Dimensions and Plate Intersection Angles.

**[0063]**Variables:

**TABLE**-US-00003 A Modular Rotation Angle A

_{1}Modular End Connection Angle a Modular Rotation Arm W

_{2}, W

_{4}Load Bearing Plates Projected End Widths W

_{3}Main Load Bearing Plates Projected Mid Width γ Main Loading Bearing Plates Intersection Angle

**[0064]**Equations:

**α 1 = α 2 + W 4 2 ##EQU00001## B = tan - 1 [ ( W 4 α ] ) d = W 2 cos B p = W ? cos B d 1 = d cos γ p ? = p cos γ d 2 = d × tan γ p 2 = p × tan γ α = tan - 1 ( α 1 × W 4 × tan γ W 4 2 + α 2 ) k = W 2 × tan B k 1 = W 2 × tan B c 1 = W 2 × ( tan γ cos B - tan B × tan α ) ##EQU00001.2## c = W 2 W 2 × c 1 ##EQU00001.3## C ? = A ? + α ##EQU00001.4## C = A - 2 × α ##EQU00001.5## l ? = c ? × ( sin ( 90 ° - α ) sin ( A ? + α ) ) ##EQU00001.6## l = c × ( sin ( 90 ° - α ) sin ( A 2 - α ) ) ##EQU00001.7## f = l 1 2 + W 2 2 ##EQU00001.8## r = l 2 + W 3 2 ##EQU00001.9## e = c × sin ( 90 ° - α ) ##EQU00001.10## g = l × tan ( C 2 ) ##EQU00001.11## n 1 = c 1 × ( sin ( 90 ° - A 1 ) sin ( A 2 + α ) ) ##EQU00001.12## n = c × ( sin ( 90 ° - A / 2 ) sin ( C / 2 ) ) ##EQU00001.13## l = a cos α + n + n 1 ##EQU00001.14## γ 1 = tan - 1 ( W ? e ) ##EQU00001.15## γ 2 = 2 × tan - 1 ( g W ? ) ##EQU00001.16## γ ? = 180 ° - tan - 1 ( g 1 W 2 ) ##EQU00001.17## ? indicates text missing or illegible when filed ##EQU00001.18##**

1-B: Space Modular Shapes, Types and Identifications

**[0065]**1-B-a: FC Brick Shapes

**[0066]**FIG. 3 illustrates the two major shape FC Bricks, Standard and Tapered. Standard Shape FC Bricks have a uniform width. End and Mid Widths of Standard Shape FC Bricks are the same. There are several Standard Shape FC Bricks including: Folded Plate Space Modular; Folded Plate Wall Modular; and Folded Plate Deck and Ceiling Modular.

**[0067]**Tapered FC Bricks have tapered shapes with variable End and Mid Widths. Tapered FC Bricks are used for round sections including Dome and Super Dome Structures. End and mid Widths are calculated based on the radius each one turns. FIGS. 16 to 18 show a Dome structures. As the drawing shows, the end and mid Widths are calculated based on the exterior circle which each one forms.

**[0068]**1-B-b: FC Brick Types

**[0069]**FC Brick Type illustrates the Shape and the number of sections each FC Brick is made of. FIG. 3 shows Standard and Tapered Shape, also identified as Type 3 FC Bricks, Types S3 and T3.

**[0070]**I-B-c, Identifications, Catalogue and Part Numbers

**[0071]**Catalogue and Part numbers represent Type and Major FC Brick dimensions including: Rotation Angle, A; Rotation Arm, a; End Connection Angles, A

_{1}h and A

_{il}, Brick Width, W; and Main Load Bearing Plates Intersection Angle, γ.

1-C: Space Modular Structural Calculations

**[0072]**Space modules are three dimensional structural units. They carry and transfer loads in three dimensions, using the plates as load bearing structural members. FIG. 3 illustrates different Loading Conditions to which a Space Modular can be subjected.

**[0073]**1-C-a: Loading Conditions

**[0074]**FIG. 6 illustrates different Loading Conditions which a Space modular can be subject to: Reactions, Moment and Shear.

**[0075]**1-C-a-1: Loading Causing Positive Longitudinal Moment

**[0076]**FIG. 6(a) shows a Space modular subject to Downward Loading condition with upward reactions at end connection points (End Connection Plates). This Loading Condition results in positive moment on the space Modular and inward forces on the outer side Plates due to internal stresses in the outer plates. Positive moment causes Compressive Stresses above the Neutral Axis X-X and Tensile Stresses below the Neutral Axis. It is the sum of Internal Compressive and Tensile Torques which balances the External Moment. Maximum Shear happens at the connection points and is equal to the Resultant of the Reactions at the connection. Inward Forces on the outer symmetrical Plates are either balanced by the adjacent modules' outer plates acting in opposite directions or by a compression member (Braces) tying the outer symmetrical plates of the same Modular acting in opposite directions.

**[0077]**1-C-a-2: Loading Causing Negative Longitudinal Moment

**[0078]**FIG. 6(e) shows the Modular subjected to upward loading condition with downward Reactions. This Loading Condition results in negative moment on the Space modular and outward Forces on the outer Plates due to internal stresses. Negative Moment on the Modular causes Tensile Stresses above Neutral Axis X-X and Compressive Stresses below Neutral Axis X-X. Maximum Shear happens at the connection points and is equal to the Resultant of Reactions at the connection. Outward forces on the outer symmetrical Plates are either balanced by the adjacent module's outer Plates acting in opposite directions or by a tension member (braces) tying the outer Plates of same modular acting in opposite directions.

**[0079]**1-C-a-3: Lateral Loading

**[0080]**This Loading Condition results in positive lateral moment on the space Modular, inward Forces on the plates subject to Lateral Loading and outward forces on the opposite outer plates. Positive lateral moment on the modular causes Compressive stresses in the plates to the loading side of Y-Y axis and tensile stresses in the plates on the other side of Y-Y axis. Maximum shear happens at the connections equal to the resultant of the reactions at the connections.

**[0081]**1-C-b: Structural Formulas

**[0082]**1-C-b-1: Centroid and Neutral Axis

**[0083]**Centroid of a plane surface is a point that corresponds to the center of gravity of a very thin homogeneous plate of the same area and shape. Neutral Axis of a section is a line through the Centroid of the section. The equation of moments is used to locate the Neutral Axis. If an area is divided into a number of parts, Statical Moment of the area with respect to an axis is equal to the sum of Statical Moments of the parts with respect to the same axis.

**[0084]**Statical Moment of a plane area with respect to an axis is the product of the area times the distance of the Centroid of the area to the axis. Section A-A, in FIG. 5A, illustrates a section through the center of the space modular. If the distance between the Neutral Axis of the entire cross section and the uppermost surface is Lm1, the equation of moments about the uppermost surface is:

**L m**1 = n . o . p × r × l 2 + r 1 2 n . o . p × r + 2 r 1 ##EQU00002## l m 2 = l - l m 1 ##EQU00002.2##

**[0085]**1-C-b-2: Moment of Inertia

**[0086]**Moment of inertia can be defined as the sum of products obtained by multiplying all the infinity small areas a by the square of their distances z to the Neutral Axis or: I=a×z

^{2}. It can be shown that the moment of inertia of a rectangular cross section of width b and depth d through its base is I=b×d

^{3}. It can also be shown that if I be the moment of inertia of a section, having an area A about the Neutral Axis, I

_{x}, the moment of inertia of the same section with respect to an axis X-X which is parallel to the Neutral Axis at h distance is I

_{x}=I+A×h

^{2}.

**[0087]**Referring to Section A-A, in FIG. 5, the Moment of Inertia of the Space Modular I

_{x}with thickness t about Neutral Axis X-X is the sum of Moment of Inertia of parts of the same modular about the same axis, is therefore

**r**2 = r × l m 1 l ##EQU00003## r 3 = r - r 2 ##EQU00003.2## I x = n . o . p × t 3 × ( r 2 3 + r 3 3 ) + 2 × t × [ r 1 3 12 + r 1 × ( l m 1 - r 1 2 2 ) 2 ] ##EQU00003.3##

**[0088]**1-C-b-3: Maximum Allowable Longitudinal Moment

**[0089]**Moment Formula

**M**= f m × l c ##EQU00004##

**is used in the design of Space Modules in which**: M is Maximum allowable Bending Moment: f

_{m}is Maximum Allowable Unit Stress in the fiber farthest from the Neutral Axis; c is Distance of the fiber farthest from the Neutral Axis; and I is Moment of Inertia about the Neutral Axis.

**[0090]**Section A-A, in FIG. 5, shows the modular cross section at the middle of the Modular. As previously explained, it is the sum of internal torques of the stresses which resist the external longitudinal moment. Under positive external moment, the area above the Neutral Axis is in compression and the area below the Neutral Axis is in tension. Or if

**M**= ( + ) ##EQU00005## M xc = f c × I x Lm 1 ##EQU00005.2## M xt = f t × I x Lm 2 ##EQU00005.3## and ##EQU00005.4## M ( max ) ≦ M xc ##EQU00005.5## M ( max ) ≦ M xt ##EQU00005.6## M = ( - ) ##EQU00005.7## M xc = f c × I x Lm 2 ##EQU00005.8## M xt = f t × I x Lm 1 ##EQU00005.9## and ##EQU00005.10## M ( max ) ≦ M xc ##EQU00005.11## M ( max ) ≦ M xt . ##EQU00005.12##

**[0091]**1-C-b-4: Maximum Allowable Lateral Moment

**[0092]**Referring to section A-A, in FIG. 5, Lateral Moment is balanced by the moments of the stresses about Neutral Axis Y-Y, and since the Space Modular is symmetrical about Neutral Axis Y-Y, then

**M yc**( max ) = f c × I y × 2 W ##EQU00006## M yt ( max ) = f t × I y × 2 W ##EQU00006.2##

**and M**

_{y}(max) is the lesser of the two values or

**M**

_{y}(max)≦M

_{yc}(max)

**M**

_{y}(max)≦M

_{yt}(max)

**[0093]**1-C-b-5: Shear Calculations

**[0094]**After the Space Modular is designed for flexure, it should be investigated for shear. Space Modular has a tendency to fail by shear by the fibers that slide past each other both vertically and horizontally. Shearing stresses are not equally distributed over the cross section but are greatest at the Neutral Axis and are zero at the extreme fibers. Maximum allowable vertical shear V is the product of the cross sectional area A and maximum allowable shearing stress ν or V=ν×A. Maximum shear force happens at the connections and is equal to the sum of reactions.

**[0095]**Section B-B, FIG. 5, shows the Modular cross section at the connections points. Maximum shear force is the product of total cross sectional area and maximum allowable shear stress or

**V**=ν×t×(N.O.P×f

_{1}+N.O.P×f

_{2}+2r

_{1})

**[0096]**1-C-b-6: Buckling

**[0097]**Buckling is the failure of Space Modular at a concentrated load or at reactions due to compression stresses or

**R**

_{max}=t×(N.O.P×f

_{1}+N.O.P×f

_{2}+2r

_{1})f

_{c}

**[0098]**1-C-b-7: Outer Plates Lateral Forces Calculations

**[0099]**Sheet SM-3B illustrates a modular subject to negative and positive moments. FIG. 5(a) shows the modular subject to positive moment, Inward lateral forces and outer plates inward movement. As the detail shows, the lateral force itself is not uniform. It is zero at the ends and maximum at the middle of the modular. The sum of lateral forces at any point along the outer plates is the projection of the Internal Longitudinal Forces in the plates due to internal stresses. If F

_{il}represents the Internal Longitudinal Force in the plates, it can be shown that the sum of lateral forces F

_{l}at any point along the outer plate is

**F l**= F il × Cos ( Tan α W 3 - 1 ) . ##EQU00007##

**[0100]**Internal longitudinal forces F

_{il}is the product of the stresses in the plates, effective depth, 1 ml and the thickness of the plates. The maximum force happens under maximum allowable positive and negative moments causing maximum allowable compressive and tensile stresses. The following formulas should be used to find the maximum internal longitudinal forces.

**[0101]**If M=(+)

**[0102]**The upper portion of the plates are under compressive stress which causes an Internal Longitudinal Compressive Force, F

_{ilc}

_{-}- of

**F ilc**= F c × L m 1 × t 2 . ##EQU00008##

**[0103]**And Maximum Lateral Inward Force, F

_{li}of

**F li**= F ilc × Cos ( Tan α W 3 - 1 ) = F c × L m 1 × t × Cos ( Tan α W 3 - 1 ) 2 ##EQU00009##

**[0104]**If M=(-)

**[0105]**The upper portion of the plates are under tensile stress which cause an Internal Longitudinal Tensile Force, F

_{ilt}of

**F ilt**= F t × L m 1 × t 2 . ##EQU00010##

**[0106]**And Maximum Lateral Inward Force, F

_{li}of

**F li**= F ilt × Cos ( Tan α W 3 - 1 ) = F t × L m 1 × t × Cos ( Tan α W 3 - 1 ) 2 ##EQU00011##

**[0107]**1-C-b-8: Outer Plates Lateral Support (Braces, Connections), Selection

**[0108]**The lateral force is either provided by adjacent sections or in case of single section systems, it is provided by braces.

**[0109]**Multiple Sections: In case of multiple sections, the total allowable tensile forces of connections, bolts should be more than the total Lateral Inward Force described in previous chapter.

**[0110]**Single Sections: In case of single section systems, the total allowable tensile and compressive forces of the braces should be more than the total Lateral Inward or outward forces described in previous chapter.

**[0111]**1-C-c: Sample Calculations

**[0112]**FIG. 6 illustrates structural calculations for Space Modular FCB-2S-16×16×8'-0''×60°. As calculations show, the Space Modular can support a maximum positive moment of 618,130 ft.lb and a maximum negative moment of 1,229,502 ft.lb and a maximum shear force of 478,000 lbs. These are the principle values which designers will need to use when designing the system.

(2) Frameless Folded Plate Space Structural System

**[0113]**Frameless Folded Plate Space Structural System uses the Space Modular as the core of the system. Typical Modules, FC Bricks are bolted together by the end plates to form a longitudinal chain. Each FC Brick is supported longitudinally by bottom and top bricks, eliminating the need for any intermediate supports. The system is only supported by the two end FC bricks bolted to the foundation (Exterior Supports). Lateral support is provided by adjacent Bricks. or in case of single section structures, lateral support is provided by outer plates of the same modular acting in opposite Directions. The number and kinds of bricks to form the chain depends on the architectural, civil and structural characteristic of the system.

2-A: System, Structural Description

**[0114]**Frameless Folded Plate Space Structural System combines and expands four main structural systems: Arch System; Shell System; Sectionalized System; and Folded Plate. The end system is an Arch, Shell, Folded Plate, and Sectionalized System. Each system has numerous advantages which will be briefly explained but it is the folded Plate structure of the FC Bricks and new extended sectionalized concept which practically provides unlimited structural capabilities for the system.

**[0115]**Monolithic Dome and Shell Structures, with all their advantages, have their limitations. As the span of the structure grows, Moment on the structure proportionally increases. A structure can fail due to several factors including Moment, Shear and buckling. However, Moment is often the main reason a structure collapses. The system in accordance with one aspect of the invention prevents accumulation of the moment by dividing the cross section into several sections. Each section is a Folded Plate Unit.

**[0116]**FIG. 7 illustrates a "Type 10" System longitudinal cross section, made of 10 typical modules bolted together by end plates. Cross Sections show the behavior of the system and each individual modular subject to three moving loads on the system. Moments at the bolted connections are zero and each individual modular is subject to negative and positive moment as the loads move along the cross section. Moving Loads create positive moments on the modules directly carrying the loads causing downward curvature of the Modules and negative moments on the rest of the modules in the system causing upward curvature of the modules.

**[0117]**Among other structural characteristics and capabilities of the FC Brick is the structural capability of the FC Brick to resist negative and positive moments which make the system possible and unique. Under normal loading conditions, a modular is held in place by top and bottom modules acting as main supports, just like a structural unit in conventional structures is supported by columns or walls. The moment which each FC Brick is subject to is the moment caused by the loads directly on the brick and the moment caused by the resultant of the reactions at each end on the connections. However, unlike conventional structures in which reactions are positive causing positive moments on the modular, reactions on a connection in this system can be negative causing negative moment in the brick.

**[0118]**Maximum Shear Force on the FC Brick is equal to the upward or downward resultant reaction Forces.

**[0119]**Most Space Structures are made of several sections. Each section is bolted to the adjacent sections providing the necessary lateral support. The resultant inward or outward forces on the outer plates are the sum of dead and live load on the outer plates, and the longitudinal forces in the plates result from stresses in the plates. Maximum longitudinal forces in the plates happen at the point of maximum moment on the modular.

**[0120]**In the case of single section structures, lateral support is provided by symmetrical outer plates acting in opposite directions. This is done by compression and tension members (braces) tying the two outer plates together.

2-B: Architectural, Civil Design

**[0121]**Preferably, the optimum numbers and kinds of Typical Space Modules are selected to form the required cross section.

**[0122]**2-B-a: Geometric Design Considerations

**[0123]**FIG. 8 illustrates a "Type 10", Space System. The type 10 Space system includes ten FC Bricks. The section shows the height and clear span of the entire Space System, and ends and mid span projected angles of each individual FC Bricks in the system, Angles α1 to α10

**[0124]**These angles should be selected to minimize the use of different kinds of units while maintaining the cross sectional requirement. Selection of rotation Arm a, should also be based on cross sectional requirement and optimum number of typical Space Modules to form the cross section.

**[0125]**2-B-b: Geometric Calculations and Modular Selection

**[0126]**2-B-b-1: Variable Selections

**[0127]**The first step of selection is to select the number of modules, Modular Rotation Arms and Modules Projected angles α

_{1}to α

_{n}.

**TABLE**-US-00004 Number of Modules nom Modules' Projected Angles α

_{1}to α

_{nom}Rotation Ann a

**[0128]**Then the following variables are selected

**TABLE**-US-00005 Rotation Angle <A Connection Angles <A

_{1}l and <A

_{1}h Modular Width W Widths W5, W3, W2

**[0129]**2-B-b-2: Modular Rotation & Connection Angle Formulas

**A**1 l ( mod 1 ) = ∝ 1 A mod 1 = 180 ° - ∝ 2 - ∝ 1 A 1 h ( mod 1 ) = 90 ° + ∝ 3 - ∝ 2 2 A 1 l ( mod 2 ) = A 1 h ( mod 1 ) A mod 2 = 180 ° - ∝ 3 - ∝ 2 A 1 h ( mod 2 ) = 90 ° + ∝ 5 - ∝ 4 2 A 1 l ( mod 5 ) = 90 ° + ∝ 9 - ∝ 8 2 A ( mod 5 ) = 180 ° + ∝ 10 - ∝ 9 A 1 h = 90 ° - ∝ 10 ##EQU00012##

2-C: Structural Design

**[0130]**The aim of structural calculation is to select the optimum space modular for the system.

**[0131]**2-C-a: Variable Selections

**[0132]**Variables can be grouped in two categories.

**[0133]**2-C-a-1: Variables Governed by Structural Codes and Project Specifications:

**[0134]**The following Variables are chosen as per structural codes and Project Specifications:

**TABLE**-US-00006 Design live Load DLL Design Wind Pressure DWP

**[0135]**2-C-a-2: Space Modular Structural Variables

**TABLE**-US-00007 Main Load Bearing Plates Intersection Angle Angle gamma Modular Thickness t Maximum Allowable Compressive Stress fc Maximum allowable Tensile Stress ft Maximum Allowable Shear Stress ν

**[0136]**2-C-b: Dead Load, Live Load

**[0137]**2-C-b-1: Loads, Reactions, Calculation

**[0138]**The first step to analyze the Space Structures is to calculate the Reactions due to Deal Load, Live Load and Wind Pressure. FIG. 9, titled Load/Reaction Formulas shows a two dimensional longitudinal cross section of the system. The drawing shows loads on each modular, distribution of loads and reactions. Generally, for each section to remain in equilibrium, the section needs to be supported by the remaining section in the system. Each modular is supported and supports the rest of modules in a chain form. Each connection point has an uplift reaction force equal to half of the load on that section of the chain and a downward load equal to half of the load of the rest of the chain. The sections are assembled by simple bolt connections, dividing the span of the structure into several sub-spans preventing accumulation of moments.

**[0139]**FIG. 7 illustrates a "Type 10" Space System. The system includes 10 typical FC Bricks, supported by end Modules. The drawing shows a longitudinal cross section of the Space System. The System has eleven connections, A, B, C, . . . K. Each connection provides an upward reaction force equal to half of the total load of the section to the left for the modular to the left and applies a force, equal to half of the total load of the section to the right on the modular. The same connection provides an upward Reaction Force equal to half of the total load of the section to the right for the Modular to the right, and applies a force equal to half of the section to the left on the modular.

**[0140]**For instance. Connection C provides an upward Reaction Force Of r

_{3}y for modular 2, equal to half of the total load of the section to the left and applies a downward force of W

_{3}, on the modular, equal to half of the total load of the section to the right.

**r**? y = - ( LL 1 + DL 1 ) + ( LL 2 + DL 2 ) 2 ##EQU00013## and ##EQU00013.2## w 4 = - r ? = ( LL 1 + DL 1 ) + ( LL 2 + DL 2 ) 2 ##EQU00013.3## w ? = TDL + TLL - [ ( LL 1 + DL 1 ) + ( LL 2 + DL 2 ) ] 2 ##EQU00013.4## and ##EQU00013.5## r 4 y = - w ? = - TDL + TLL - [ ( LL 1 + DL 1 ) + ( LL 2 + DL 2 ) ] 2 ##EQU00013.6## ? indicates text missing or illegible when filed ##EQU00013.7##

**[0141]**Similarly, at connection point D:

**w**5 = TDL + TLL - [ ( LL 1 + DL 1 ) + ( LL 2 + L 2 D ) + ( LL ? + DL ? ) ] 2 ##EQU00014## r ? = - w 5 = - TDL + TLL - [ ( LL 1 + DL 1 ) + ( LL 2 + DL 2 ) + ( LL ? + DL ? ) ] 2 ? indicates text missing or illegible when filed ##EQU00014.2##

**[0142]**FIG. 9 illustrates the reactions formulas for all connections in the system. The sum of Upward Reaction and Downward Forces (r

_{3}y+W

_{3}), (r

_{4}y+W

_{4}), . . . at the connection points for each modular is the resultant force on the modular.

**[0143]**2-C-b-2: Maximum Moment Calculations

**[0144]**FIG. 7 illustrates three longitudinal cross section of the system, Moments and the modular's curvature tendency due to three moving loads on the system. As the sections show, the modular subject to live load, where the resultant of the forces (r

_{ny}+W

_{n}) and (r

_{n}1y+W

_{n}1) on the end of modular, at the connections, are positive, is subject to positive moment. Modules which are subject to negative resultant forces at either end, at connection points, are under negative moment.

**[0145]**FIG. 9 illustrates the cross section of a "Type 10" space system including ten FC Bricks with rotation arms of 20 feet and Dead Loads of 5 tons, subject to three loads of 80 tons on Modules 1, 2 and 3. FIG. 9 also shows each individual modular in the system, including Live Load, Dead Load and end forces at the connections. The equation of moment about the middle of the modular can be as follows:

**M a**( middle ) = ( r 1 y + W 1 ) × a + ( LL 1 2 + DL 1 2 ) × a 2 M a ( middle ) = ( - 102.5 + 42.5 ) × 20 + ( - 80 / 2 - 5 / 2 ) × 20 / 2 = - 1 , 625 ft - tons = - 3 , 250 , 000 ft - lb s Modular 1 M a ( middle ) = ( r 2 y + W 2 ) × a + ( LL 2 2 + DL 2 2 ) × a 2 M a ( middle ) = ( + 102.5 - 42.5 ) × 20 + ( - 80 / 2 - 5 / 2 ) × 20 / 2 = 775 ft - tons = + 1 , 550 , 000 ft - lb s Modular 2 M a ( middle ) = ( r 4 y + W 4 ) × a + ( LL 3 2 + DL 3 2 ) × a 2 M a ( middle ) = ( + 60 - 85 ) × 20 + ( - 80 / 2 + - 5 / 2 ) × 20 / 2 = - 925 ft - tons = - 1 , 850 , 000 f t - lbs Modular 3 M a ( middle ) = ( r 6 y + W 6 ) × a + ( LL 4 2 + DL 4 2 ) × a 2 M a ( middle ) = ( + 17.5 - 127.5 ) × 20 + ( - 5 / 2 ) × 20 / 2 = 2 , 225 f t - tons = - 4 , 450 , 000 f t - lbs . Modular 4 ##EQU00015##

**[0146]**The same formulas can be written for each modular under different loading conditions and the maximum negative and positive moments applied to each modular can be calculated. Depending on the loading conditions, maximum moments and signs change. Modules are selected based on the worst loading conditions.

**[0147]**2-C-b-3: Actual Shear and Buckling Force Calculations

**[0148]**Actual shear Forces on each unit occur at the ends of the Modular, equal to the sum of reactions and loads at the connections. Maximum Shear force is the bigger of the two resultant forces at each end or V

_{a}(max.sub.)=(r

_{ny}+W

_{n}), or V

_{a}(max.sub.)=(r

_{n}1y+W

_{n}1). Maximum Buckling occurs at the supports and the maximum actual buckling force is equal to the total reactions at the supports, equal to R

_{1}y and R

_{3}y.

**[0149]**2-C-c, Wind Calculations

**[0150]**2-C-c-1: Wind Pressure and Reaction Calculations

**[0151]**FIG. 8 shows a two dimensional longitudinal cross section of the system. The drawing shows wind loads on each modular, distributions of loads and reactions. The drawing also shows the formulas for calculation of reactions at each connection. Reaction calculations are similar to Live Load/Dead Load reaction calculations described in previous chapter 2-C-b-1.

**[0152]**2-C-c-2: Maximum Moment Calculations

**[0153]**Moment calculations for each modular is similar to calculations previously described in chapter 2-C-b-2.

**[0154]**2-C-c-3: Shear Calculations

**[0155]**Shear calculations are similar to calculations previously described in chapter 2-C-b-3.

**[0156]**2-C-c-4: Modular Selection

**[0157]**Modular selection is based on the maximum moment for each modular. Maximum moment is the maximum moment on each modular based on Live/Dead Load, wind Pressure or a combination thereof.

**[0158]**2-C-d: Modular Verification

**[0159]**2-C-d-1: Actual Moment Calculations

**[0160]**Previous chapters calculated the maximum moment at the center of the modular. The maximum moment does not necessarily happen at the center. Maximum moment happens at a point where the resultant moments on each side of the point are equal.

**[0161]**2-C-d-2

**[0162]**FIG. 10 shows a "type 2", Space modular. Tile drawing illustrates formulas for calculating the maximum allowable moment for different sections of the modular. The drawing also shows the process of verification of the modular.

(3) Applications

**[0163]**Frameless Folded Plate Space Structural Systems have numerous applications and usages. Applications have been grouped into eight separate main categories, which are intended to be representative and not definitive.

3-A: Frameless Folded Plate Space Structures--FFPSS

**[0164]**Simple Frameless Folded Plate Space Structures are made of assembling several similar sections of a Frameless Folded Plate Space Structural System with required cross section in a row to achieve the required depth (Length). Numerous advantages of FFPSS to conventional structures make them ideal for construction of all kinds and types of structures including Residential, commercial, Agricultural and industrial buildings. Their light weight, easy delivery, fast construction, and structural capabilities make them unique for fast track or Emergency Shelter construction.

**[0165]**3-A-a: Types

**[0166]**FIG. 11 illustrates a simple "Type 3-5" Space Structure. Type 3-5, FFPSS is made of five sections of "Type-3", Frameless Folded Plate Space Structural Systems in a row with a total of fifteen FC Bricks. The types of the systems and number of sections used depend on a number of variables including the required building cross sections and building size. Each section is bolted to the adjacent sections and the foundation forming a folded plate arch shell structure. More complex structures are constructed by using several different systems with different cross sections.

**[0167]**3-A-b: Front and Back Elevations, Space Wall Modules

**[0168]**FIG. 2 illustrates the front elevation and a partial longitudinal cross section of a Type 3 FFPS Structure. Front and Back Elevations are enclosed with Space wall modules bolted to the side connection plates of the outer modules and the foundation. Wall modules are manufactured in several standard and custom made sizes. FIG. 13 illustrates the perspective of a Type 3 FFPS-Structure, typical Space modules and the End sections. End Sections (End modules) are specially designed with side connection plates which extend to form the openings for typical wall modules. Wall Modules can be removed to form the rough openings to receive gates and/or windows.

**[0169]**3-A-c: Standard Sizes

**[0170]**FIG. 14 shows a simple Type 2 Space Structure, cross section, and front elevation. FIG. 14 also illustrates several standard size Type 2 Space structures.

**[0171]**3-A-d: System Flexibility

**[0172]**FIG. 15 illustrates some of the system flexibilities. The drawing shows the different structures which can be made from FC-Bricks used in a Type, 3-15, Space Structure. Type 3-15, FFPS Structures is made of 45 FC-Bricks which can be used to construct three units of Type, 3-5 FFPS Structures, or five units of Type 2-3 FFPS Structure and five units of Type, 1-3 FFPS Roof Structures, or can be used to construct a Type, 5-15 FFPS Structures by adding an additional two FC-Bricks to each section.

**[0173]**3-A-e: Structural Design

**[0174]**3-A-e-1: Shell Calculations

**[0175]**Structural calculations for Space structures follow the basic calculations and formulas described above. FIG. 16 illustrates a Type 3 Space Structure, Loads, and Reactions formulas. FIG. 17 shows the moment formulas and sample calculations for a Type 3 Space Structure made with three FC-Bricks (W=8', a=16'). FIG. 17 illustrates the maximum actual moment for both unloaded and loaded (Live & wind) conditions and maximum allowable negative and positive moments for a FCB-S2-16-16-8-60°, Part No. 138°-81°-81°. The resultant of lateral forces due to internal stresses, Dead Load and Live Load, Wind Pressure are balanced by adjacent sections. Side connection calculations and design should follow the formulas previously described in section 1-C-b-8.

**[0176]**3-A-e-2: End Sections and Wall Modules Calculations

**[0177]**End sections: End sections should be treated as laterally unsupported. End sections should be braced like a single section system, previously described in section 1-C-b-8. In most structures, a total of three braces is adequate to provide the required lateral support.

**[0178]**Wall Modules: FIG. 18 shows the cross section of a Space Structure, Wall Modules and the Reactions. FIG. 18 also illustrates the formulas for calculating the maximum moments on wall modules. FIGS. 19 and 20 illustrate Space wall modules' architectural and structural calculations. Reaction WL5 is one of the lateral forces on the outer plates of the end sections which need to be considered when designing the End Section Braces.

3-B: Frameless Folded Plate Space Roof Structures, FFPS-RS

**[0179]**Frameless Folded Plate Space Roof Structures are made of sections of Frameless Folded Plate Space Structural Systems with the desired cross section to achieve the required size. They may be installed on exterior walls or on main structural framing eliminating the need for any additional framing, sheathing, roofing and insulation.

**[0180]**3-B-a: Types

**[0181]**FIG. 21 shows a Type 2 Space Roof System and a Type 3 Space Structure. FIG. 22 illustrates a Type 3-5 Space Roof System. The type 3-5 Roof system is made of five sections of the Type 3 FFPS-Structural System. FIG. 12 also shows the connection details for a typical Roof System.

**[0182]**3-B-b: Structural Design

**[0183]**Structural calculations for Space Roof Systems follow the basic calculations and formulas described above. FIG. 22 illustrates a Type 2 Space Roof System, Loads, and Reaction Formulas. Moments can be calculated as herein. Side connections calculations and design should follow the formulas previously described in section 1-C-b-8. End sections should be treated as laterally unsupported. End sections should be braced, as needed for a single section system, described in section 1-C-b-8. In most Roof Systems, a total of three braces is adequate to provide the required lateral support.

3-C: Frameless Folded Plate Space Domes, FFPS-D

**[0184]**3-C-a, Geometric Design

**[0185]**Frameless folded plate Space Domes are Dome, Shell, Sectionalized structures. They look like an upside down bowl, cut horizontally and vertically into several layers and Grids.

**[0186]**3-C-a-1: Shell

**[0187]**FIG. 24 shows a typical Space Dome Structure with exterior outside diameter of OD-1. Section A-A in this figure shows a section through the Dome. As the section shows, the Dome is horizontally cut into several layers: SM-A, SM-B, SM-C, etc. Each layer represents a circular row of typical Tapered Space Modules bolted to each other to form a certain circular foot print with outside Diameters of OD1, OD2, etc. Outside diameters, ODs, can easily be calculated by deducting each layer's horizontal projection from OD-1.

**OD**3=OD1-First Layer Total Horizontal Projection

**OD**-5=OD3-2

^{nd}Layer Total Horizontal Projection, etc.

**[0188]**FIG. 22(b) shows the top view of the same Dome. As the top view shows, the Dome has been divided into Vertical Grids. Vertical Grid Lines divide each layer into typical, similar size sections: SM-1A, SM-2A, SM-3A, . . . ; SM-1B, SM-2B, SM-3B, . . . . Each section is a tapered space modular with bottom, mid and top total widths, W

_{l}, W

_{M}and W

_{H}equal to:

**W**

_{AL}=OD1/Number of Grids

**W**

_{AM}=OD2/Number of Grids

**W**

_{AH}=W

_{BL}=OD3/Number of grids

**[0189]**3-C-a-2: Tapered Modular

**[0190]**Tapered modules should be designed using the following sets of variables:

**TABLE**-US-00008 Modular Exterior Connection Angles Angles C

_{1}L C

_{1}H Modular Exterior Rotation Angle Angle C Modular Connection Arm J

**[0191]**3-C-b: Architectural Design

**[0192]**FIG. 25 illustrates a Super Dome. The drawing shows the design possibilities and capabilities. All seating Platforms, offices, walkways can be constructed using FC-Bricks, Space Ceiling/Floor modules hung from the shell itself, eliminating any need for interior supports or walls.

**[0193]**3-C-c: Frameless Folded Plate Space Future Pressurized Colonies, FFPS-FPC

**[0194]**Unique Structural Capabilities of space domes make them ideal for future pressurized structures. Space Domes can carry the same loads regardless of direction of the loads. All formulas developed treat the loads as vectors. Direction of the loads basically changes the direction of the reactions and moments.

3-D: Frameless Folded Plate Space High Rises, FFPS-HR

**[0195]**Frameless Folded Plate Space High Rises are constructed by using Space wall Modules and Space Ceiling Modules as Load Bearing Walls and Ceilings. Structural capabilities of the Space System reduces the number of interior load bearing walls, providing architectural capabilities which conventional structures can not economically provide.

**[0196]**3-D-a: Architectural Design

**[0197]**FIG. 26 illustrates a typical FFPS High Rise and connection details. FFPS-High Rises have two main components, Space Wall Modules and Space Ceiling Modules. Space wall modules act as load Bearing walls, supporting the Space Ceiling Modules.

**[0198]**3-D-b: Structural Calculations

**[0199]**Structural calculations follow the structural calculations described in System Structural Design and Space Modular Structural calculations. Lateral support for each section (walls and ceilings) is provided by adjacent sections.

3-E: Frameless Folded Plate Space Bridge Systems, FFPS-Bs

**[0200]**FFPS-Bridge Systems with their numerous advantages including Light Weight, Easy Delivery and Erection can be used as temporary or permanent bridges. System flexibility offers any required cross section and span.

**[0201]**3-E-a: Type

**[0202]**FIG. 27 illustrates a Type 10. Bridge System. The type 10 Bridge System is made of total ten Space Modules, including two Space Ramp Modules. FIG. 28 shows the top, longitudinal section and cross section of a typical Space Ramp Modular. Space Ramp Modules are similar to the rest of the modules with plates extending down to the required grades. FIG. 29 shows a Type 10-2, Space Bridge System. The number 2 represents the number of sections.

**[0203]**3-E-b: Civil & Geometric Design

**[0204]**Geometric and Civil design of Space Bridges are similar to the System design described herein.

**[0205]**3-E-c: Structural Design

**[0206]**Structural Design of FFPS-Bridge Systems is similar to System Structural Design described in previous chapters. Lateral Support for the Bridges is provided by a Bridge Deck Framing System. FIG. 29 shows a cross section of a bridge and the Deck Framing. Deck Framing is made of fiber glass plates in two directions, providing lateral support for the Bridge System and support for Bridge Decking.

3-F: Frameless Folded Plate Space Dam Systems, FFPS-DS

**[0207]**FFPS-Dam System are made of sections of FFPS-Structural systems with desired cross section to achieve the desired height. FIG. 30 illustrates the plan view and elevation of a Space Dam. The drawing shows the support locations and anchoring systems.

**[0208]**3-F-a: Structural Design

**[0209]**FFPS-Dam is calculated as per structural calculations described herein. Live Load on each section varies and is proportional to the height of the Dam. Each section needs to be designed to independently carry the live load. Since the live load on each section varies, the lateral force resulting from internal stresses also varies. The entire structure needs to be laterally supported by a system of continuous bracings. Section A-A in FIG. 30 shows the lateral continuous bracing and the attachment detail.

3-G: Frameless Folded Plate Space Tunneling Systems, FFPS-TS

**[0210]**FFPS-SS can be used for tunneling purposes for both Rock Tunneling and Shield Tunneling Methods. Tunnel sections are made of sections of FFPS-Structural System with the desired cross section installed after the drilling is done. In the case of Rock tunneling, each section can be assembled in whole or parts and erected after the drilling is done. FIG. 31 shows a section of a tunnel. Sections can be erected and temporarily anchored before the slab and foundation is poured. The space between the sections and rock can be pump filled with pea gravel. In the case of Shield Tunneling, Sections can installed as the shield system moves.

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