VISUAL STUDY OF ARCHITECTURAL STRUCTURES WITH COMPUTERS

SURFACE STRUCTURES

MEMBRANES

BEAMS BEARING WALLS and SHEAR WALLS

FOLDED SURFACES

RIBBED VAULTING LINEAR and RADIAL ADDITIONS

parallel, triangular, and tapered folds

CURVILINEAR FOLDS

SHELLS: solid shells, grid shells

CYLINDRICAL SHELLS THIN SHELL DOMES

HYPERBOLIC PARABOLOIDS

TENSILE MEMBRANE STUCTURES

Pneumatic structures Air-supported structures

Air-inflated structures (i.e. air members)

Hybrid air structures

Anticlastic prestressed membrane structures Edge-supported saddle roofs

Mast-supported conical saddle roofs

Arch-supported saddle roofs

Hybrid tensile surface structures (possibly including tensegrity)

Everson Museum, Syracuse, NY, 1968, I. M. Pei

National Gallery of Art, East Wing, Washington, 1978, I.M. Pei

Boston Convention Center, Boston, 2005, Vinoly and LeMessurier

Incheon International Airport, Seoul.

2001, Fentress Bradburn Arch.

Delft University of Technology Aula Congress Centre, 1966, Bakema

Schlumberger Research Center, Cambridge, UK, 1985, Hopkins/ Hunt

MUDAM, Museum of

Modern Art,

Luxembourg, 2007,

I.M. Pei

Armchair 41 Paimio by Alvar Aalto, 1929-

33, laminated birchwood

MODELING OF SURFACE STRUCTURES Introduction to Finite Element Analysis

The continuum of surface structures must be divided into a temporary mesh

or gridwork of finite pieces of polygonal elements which can have various

shapes. If possible select a uniform mesh pattern (i.e. equal node spacing)

and only at critical locations make a transition from coarse to fine mesh. In the

automatic mesh generation, elements and their definitions together with

nodal numbers and their coordinates, are automatically prepared by the

computer.

Shell elements are used to model thin-walled surface structures. The shell

element is a three-node (triangular) or four- to nine-node formulation that

combines separate membrane and plate bending behavior; the element does

not have to be planar. Structures that can be modeled with shell elements

include thin planar structures such as pure membranes and pure plates, as

well as three-dimensional surface structures. In general, the full shell behavior

is used unless the structure is planar and adequately restrained.

Membrane and plate elements are planar elements. Keep in mind that

three-dimensional shells can also be modeled with plane elements if the

mesh is fine enough and the elements are not warped!

Planar elements: MEMBRANE: pure membrane behavior, only

the in-plane direct and

shear forces can be supported

(e.g. wall beams, beams, shear walls,

and diaphragms can be modeled

with membrane elements, i.e. the

element can be loaded only in its plane.

Planar elements: PLATE: pure plate behavior, for out-of plane

force action; only the bending moments

and the transverse force can be

can be supported (e.g. floor slabs,

retaining walls), i.e. the element can

only be loaded perpendicular to its

plane.

Bent planar elements: SHELL: for three-dimensional surface

structures, i.e. full shell behavior,

consisting of a combination of

membrane and plate behavior; all

forces and moments can be

supported (e.g. three- dimensional

surface structures, such as rigid shells,

vaults).

Solid elements

In general, the plane element is a three- to nine-node element for modeling

two-dimensional solids of uniform thickness. The plane element activates three

translational degrees of freedom at each of its connected joints. Keep in mind

that special elements are required when the Poissons ratio approaches 0.5!

An element performs best when its shape is regular. The maximum permissible

aspect ratio (i.e. ratio of the longer distance between the midpoints of opposite

sides to the shorter such distance, and longest side to shortest side for

triangular elements) of quadrilateral elements should not be less than 5; the

best accuracy is achieved with a near to 1:1 ratio. Usually the best shape is

rectangular. The inside angle at each corner should not vary greatly from 900

angles. Best results are obtained when the angles are near 900 or at least in the

range of 450 to 1350. Equilateral triangles will produce the most accurate results.

The accuracy of the results is directly related to the number and type of elements

used to represent the structure although complex geometrical conditions may

require a special mesh configuration. As mentioned above, the accuracy will

improve with refinement of the mesh, but when has the mesh reached its

optimum layout? Here a mesh-convergence study has to be done, where a

number of successfully refined meshes are analyzed until the results

converge.

Computers have the capacity to allow a rapid convergence from the initial

solution as based, for instance, on a regular course grid, to a final solution by

feeding each successive solution back into the displacement equations that is a

successive refinement of a mesh particularly as effected by singularities. Keep in

mind, however, that there must be a compromise between the required accuracy

obtained by mesh density and the reduction file size or solution time!

Finite element computer programs report the results of nodal displacements,

support reactions and member forces or stresses in graphical and numerical

form. It is apparent that during the preliminary design stage the graphical results

are more revealing. A check of the deformed shape superimposed upon the

undeflected shape gives an immediate indication whether there are any errors.

Stress (or forces) are reported as stress components of principal stresses in

contour maps, where the various colors clearly reflect the behavior of the

structure as indicated by the intensity of stress flow and the distribution of

stresses.

The shell element stresses are graphically shown as S11 and S22 in plane normal

stresses and S12 in-plane shear stresses as well as S13 and S23 transverse

shear stresses; the transverse normal stress S33 is assumed zero. The shell

element internal forces (i.e. stress resultants per unit of in-plane length) are the

membrane direct forces F11 and F22, the membrane shear force F12, the plate

bending moments M11 and M22, the plate torsional moment M12, and the plate

transverse shear forces V13 and V23. The principal values (i.e. combination of

stresses where only normal stresses exist and no shearing stresses) FMAX,

FMIN, MMAX, MMIN, and the corresponding stresses SMAX and SMIN are also

graphically shown. As an example are the membrane forces shown in Fig. 10.3.

The Von Mises Stress SVM (FVM) is identified in terms of the principal stress and

provides a measure of the shear, or distortional, stress in the material. This type of

stress tends to cause yielding in metals.

FMIN

FMAX

F11

F22

F12

F12

Axis 2

Axis 1

J4

J1

J3

J2

MEMBRANE FORCES

COMPUTER MODELING Define geometry of structure shape in SAP- draw surface structure contour using only plane

elements for planar structures.

click on Quick Draw Shell Element button in the grid space bounded by four grid lines

or click the Draw Rectangular Shell Element button, and draw the rectangular element by clicking

on two diagonally opposite nodes

or click the Quadrilateral Shell Element button for four-sided or three-sided shells by clicking on all

corner nodes

If just the outline of the shell is shown, it may be more convenient to view the shell as filled in

click in the area selected, then click Set Elements button, then check the Fill Elements box under

shells

click Escape to get out of drawing mode, click on the beam on screen go to Edit, then Mesh Shells

choose Mesh into, then type the number of elements into the X- direction on top, and then Z-direction

on bottom for beams or Y-direction on bottom for slabs; use an aspect ratio close to the proportions

of the surface element but less than the maximum aspect ratio of about 1/4 to 1/5, click OK, click

Save Model button

or for the situation where a grid is given and reflects the meshing, choose Mesh at intersection of

grids

to mesh the elements later into finer elements, just click on the Shell element and proceed as above.

adding new Shell elements: (1) click at their corner locations, or (2) click on a grid space as

discussed before

Define MEMBER TYPES and SECTIONS :

click Define, then click Shell Sections

click Add New Section button, then type in new name

go to Shell Sections, then define Material, then type thickness in Membrane and Bending box (normally the two

thicknesses are the same) in kip-ft if dimensions are in kip-ft

select Membrane option for beam action or Plate option for slab action or Shell option for bent surface structures,

then click OK, then click Save Model button

Define STATIC LOAD CASE

Click Static Load Cases, then assign zero to Self Weight Multiplier, then click Change Load, OK , or type DL in the

Load edit box (or leave LOAD1 then click the Change Load button, in other words self-weight is not set to zero

Type LL in the Load edit box then type 0 in the Self Weight Multiplier edit box, then click the Add New Load button

Assign LOADS

Single loads are applied at nodes.

Uniform loads act along mid-surface of the shell elements for membrane elements, in other words are applied as

uniformly distributed forces to the mid-surfaces of the plane elements that is load intensities are given as forces per

unit area (i.e. psi).

Assign joint loads

click on joint, then click on Assign

click at Joint Static Loads, then click on Forces, then enter Force Global Z (P for downward in global z-box), then

click Add to existing loads, then click OK

Assign uniform loads

select All, then click Assign, then click Shell Static Loads, then click Uniform

choose w (psf), Global Z direction ( i.e. Direction: Gravity), for spatial membranes project the loads on the horizontal

projection, then click OK

Assign loads to the pattern

click Assign, then select Shell Static Loads, and Select Pressure

from the Shell Pressure Loads dialog box select the By Joint Pattern option, then select e.g. HYDRO fro the drop-

down box, then type 0.0624 in the Multiplier edit box, then click OK.

MEMBRANES

BEAMS

BEARING WALLS and SHEAR WALLS

Atrium, Germanisches Museum, Nuremberg, Germany

1 K/ft

4'

40'

10 k

8'

2'

The maximum bending moment is,

Mmax = wL2/8 = 1(40)2/8 = 200 ft-k

The section modulus is,

S = bh2/6 = 6(48)2/6 = 2304 in3

The maximum shear stress (S12) occurs at the neutral axis at the supports,

fv max = 1.5(V/A) = 1.5(20000)/(6)48 =104 psi (0.72 MPa or N/mm2) 165 psi OK

The SAP shear stresses (c) are, S12 = 101 psi.

The maximum longitudinal bending stresses (S11) occur at top and bottom

fibers at midspan and are equal to,

fb max = M/S = 200(12)/2304 = 1.04 ksi (7.17 MPa or N/mm2) 1.80 ksi OK

The SAP longitudinal stresses (c) are, S11 = 1.046 ksi. Or, the maximum

stress resultant force F11 = 6.28 k, which is equal to stress x beam width =

1.046(6) = 6.28 k/inch of height.

1 K/ft

4'

40' a.

b.

c.

1.01 ksi

92 psi

10 k

8'2

'

30'

12'

10' 10' 10'

Pu= 500 k

R = 500 k R = 500 k

= 47.20 z =

0.9

h =

10.8

'

Hcu

Htu

Pu= 500 k

D u Du

strut: Hcu

tie: Htu

wd

wh

Mu

a. b.

BEARING WALLS and SHEAR WALLS

National Assembly, Dacca, Bangladesh, 1974, Louis Kahn

World War II bunker transformed into housing, Aachen, Germany

Documentation Center Nazi Party Rally Grounds, Nuremberg, 2001,

Guenther Domenig Architect

Dormitory of Nanjing University, Zhang Lei Arch., Nanjing

University, Research Center o0f Architecture

Shear-wall or Cantilever-column

LATERAL DEFLECTION OF SHEAR WALLS

LONG WALL CANTILEVER WALL

INTERMEDIATE WALL

10.5 k9 k/ft

10ft

10ft

25 k 25 k

a.L = 32'

h = 16' h

b.L = 8'

shallow beam

deep beam

Deep concrete beams

Shear Wall or Frame

Shear Wall Frame Shear Wall or Frame ?

Openings in Shear Walls

Very Large

Openings may

convert the Wall to

Frame

Very Small

Openings may not

alter wall behavior

Medium Openings

may convert shear

wall to Pier and

Spandrel System

Pier Pier

Spandrel

Column

Beam

Wall

Openings in Shear Walls - Planer

Shear Wall and Frame Behavior

Shear Wall and Truss Behavior

Shear Wall and Frame

Shear Wall Behavior Frame Behavior

Shear Wall Behavior Pier and Spandrel System Frame Behavior

D L

ww

= 0

.4 k

/ft

4 f

t4

ft

4 f

t4

ft

4 f

t

3ft

4 f

t

27

ft7 SP@ 3 ft = 21 ft

w = 1k/ft, w = 0.6 k/ft at roof and floor levels

LATERAL DEFLECTION OF WALLS WITH OPENINGS

PIER-SPANDEL SYSTEMS

Modeling Walls with Opening

Plate-Shell Model Rigid Frame Model Truss Model

In ETABS single walls are modeled as cantilevers and walls with openings as

pier/spandrel systems. Use the following steps to model a shear wall in ETABS:

Files > New Model > model outline of wall Edit grid system by right-clicking the model and use: Edit Reference Planes (or go to Edit >), Edit Reference Lines (or go to Edit >), and possibly Plan Fine Grid

Spacing (or go to Options > References > Dimensions/Tolerances Preferences)

Define as in SAP: Material Properties, Wall/Slab/Deck Sections, Static Load Cases, and Load Combinations

Draw the entire wall, then select the wall > Edit > Mesh Areas > Intersection with Visible Grids, then create window openings by deleting the respective panels.

Assign pier and spandrel labels to the wall: Assign > Shell Areas > Pier Label command and then the same process for Spandrel Label.

Assign the loads to the wall. Run the Analysis. View force output: go to Display > Show Member Forces/Stress diagram > Frame/Pier/Spandrel Forces > check Piers and Spandrels > e.g. M33

Design: Options > Preferences > Shear Wall Design > check Design Code, Start: Design > Shear Wall Design > Select Design Combo, then click Start

Design/Check of Structure.

Once design is completed, design results are displayed on the model. A right-click on one of the members will bring up the Interactive Design Mode form, then click

Overwrites, if changes have to be made.

THE STRUCTURE OF THE SKIN:

GLASS SKINS

Cologne/Bonn Airport, Germany,

2000, Helmut Jahn Arch., Ove

Arup USA Struct. Eng.

Cottbus University Library, Cottbus, Germany, 2005, Herzog & De Meuron

Max Planck Institute of

Molekular Cell Biology,

Dresden, 2002, Heikkinen-

Komonen Arch

Xinghai Square shopping mall, Dalian

Shopping Mall, Dalian

Sony Center, Potzdamer

Platz, Berlin, 2000, Helmut

Jahn Arch., Ove Arup USA

Struct. Eng

Shopping Center,

Chongqing

PLATES

SLABS

RETAINING WALLS

NIT, Ningbo

New National Gallery, Berlin, 1968, Mies van der Rohe

12' 12'

1 2 '

1 2 '

1 2

'

a. b.

c. d.

e. f.

2' 2' 8'

2'

2'

8'

6'

6'

6' 6'

a b c

d e f

12' 12'8'2' 2'

6'

6'

Investigate a square 6-in. (15 cm) concrete slab, 12 x 12 ft (3.66 x 3.66 m) in

size that carries a uniform load of 120 psf (5.75 kPa or kN/m2, COMB1),

that is a dead load of 75 psf (3.59 kPa) for its own weight (SLABDL taken

care by self weight) and an additional dead load 5 psf (0.24 kPa, TOPDL),

and a live load of 40 psf.(1.92 kPa, LIVE).

The concrete strength is 4000 psi (28 MPa) and the yield strength of the

reinforcing bars is 60 ksi (414 MPa). Solve the problem by using 2 x 2 ft

(0.61 x 0.61 m) plate elements.

Check the answers manually using approximations. Compare the various

slab systems that is study the effect of support location on force flow.

a. Assume one-way, simply supported slab action.

b. Assume a two-way slab, simply supported along the perimeter.

c. Assume the slab is clamped along the edges to approximate a continuous

interior two-way slab.

d. Assume flat plate action where the slab is simply supported by small

columns

at the four corners.

e. Assume cantilever plate action with four corner supports for a center bay

of 8x 8 ft (2.44 x 2.44 m).

Assume one-way, simply supported slab action.

Checking the SAP results according to the conventional beam theory:

The total slab load is: W = 0.120(12)12 = 17.28 k

The reactions are: R = W/2 = 17.28/2 = 8.64 k = wL/2 = 0.120(12/2) = 0.72 k/ft

or, at the interior nodes Rn= 2(0.72) = 1.44 k

The maximum moment is: Mmax = wL2/8 = 120(12)2/8 = 2160 lb-ft/ft

Checking the stresses, which are averaged at the nodes,

S = tb2/6 = 6(12)2/6 = 144 in.3

fb = M/S = 2(2160(12)/144) = 360 psi

According to SAP, the critical bending values of the center slab strip at mid-span

are:

M11 = 2129 lb-ft/ft, S11 = 354 psi

Assume a two-way slab, simply supported along the perimeter.

Checking the results approximately at the critical location at center of

plate according to tables (see ref. Timoshenko), is

Ms wL2/22.6= 120(12)2/22.6 = 764 lb-ft/ft

The critical moment values according to SAP are:

M11 = M22 = MMAX = 778 lb-ft/ft

Notice the uplift reaction forces in the corners causing negative

diagonal moments at the corner supports, M12 = -589 lb-ft/ft

Assume the slab is clamped along the edges to approximate a continuous

interior two-way slab. The critical moment values are located at middle

of fixed edge according to tables (ref. Timoshenko), are

Ms - wL2/20 = -120(12)2/20 = -864 lb-ft/ft

The critical moment values according to SAP are:

M11 = M22 = MMIN = -866 lb-ft/ft

a

d

b

e

c

f

b. DEEP BEAMS c. SHALLOW BEAMS a. WALL SUPPORT d. NO BEAMS

SLAB SUPPORT ALONG EDGES

#4 @ 12"

#3 @ 9"

15 ft12 in 12 in

#13 @ 305 mm

#10 @ 229 mm

4.57 m

305 mm

ETABS template SAFE template

Gatti Wool Factory, Rome, Italy, 1953, Pier Luigi Nervi

Schlumberger Research

Center, Cambridge, 1985,

Michael Hopkins

GI

GI

BM

BM

BM

16/2

4

16/2

4

12/24

12/24

12/24

34"

15"

15"

18"x18"

FOLDED SURFACES

RIBBED VAULTING

LINEAR and RADIAL ADDITIONS parallel, triangular, and tapered folds

CURVILINEAR FOLDS

Wallfahrtskirche "Mariendom" , Neviges, Germany, 1968, Gottfried Boehm

Neue Kurhaus, Aachen, Germany

Turin Exhibition Hall, Salone

Agnelli, 1949, Pier Luigi Nervi

St. Mary, Pirna, Germany, start of 16th century

St. Foillan, Aachen, Germany,

1958, Leo Hugot Arch.

SHELLS: solid shells, grid shells

CYLINDRICAL SHELLS

THIN SHELL DOMES

HYPERBOLIC PARABOLOIDS

Radiolaria, Buckminster Fuller dome,

skinned dome

Pantheon, Rome, Italy, c. 123 A.D.

Hagia Sofia, Constantinople (Istanbul), 535 A.D., Anthemius of Tralles and Isodore of Miletus

St. Peters (1590 by Michelangelo), Rome; US Capitol (1865 by Thomas U. Walther), Washington; Epcot

Center, Orlando, (1982by Ray Bradbury ) geodesic dome; Georgia Astrodome, Atlanta (1980);

Versuchsbau einer doppelt gekruemmtan Zeiss-Dywidag Schale (1.5 cm thick):

Franz Dischinger & Ulrich Finsterwalder, Dyckerhoff & Widmann AG, Jena, 1931

Hipodromo La Zarzuela, 1935,

Eduardo Torroja

Kresge Auditorium, MIT, 1955, Saarinen

Autobahnraststtte, Arch. & Ing.: Heinz Isler, Deitingen 1968

Earth sheltered building technology using concrete shells

Bubble Castle, Theoule, France, 2009, Designer Antti Lovag

Sydney Opera House, 1973, Jrn Utzon, Arup - Peter Rice

Eden Project, Cornwall, UK, 2001, Sir Nicholas Grimshaw , Anthony Hunt

Luce Memorial Chapel, Taichung, Taiwan, 1963, I. M. Pei

R2 = z2 + x2

Kimball Museum, Fort Worth, TX, 1972, Louis Kahn, August E. Komendant

Stadelhofen, Zurich, Switzerland, 1983, Santiago Calatrava

Shanghai Grand Theater, Shanghai, 1998, Jean-Marie Charpentier

College for Basic Studies, Sichuan University, Chengdu, 2002

CNIT Exhibition Hall, Paris, 1958, Bernard Zehrfuss Arch, Nicolas Esquillon Eng

P&C Luebeck, Luebeck, 2005, Ingenhoven und Partner, Werner Sobek

Iglesia Atlantida, Uruguy, 1960, Eladio Dieste

Calatrava

World Trade Centre Dresden,

1996, Dresden, nps + Partner

Railway Station

"Spandauer

Bahnhof, Berlin-Spandau, 1997,

Architect von

Gerkan Marg und

Partner, Scdhlaich

Bergermann

Railway Station "Lehrter

Bahnhof, Berlin, 2003, Architect von Gerkan Marg

und Partner, Schlaich

Bergermann

MUDAM: Futuro House (or UFO), 1968, Finland, Matti Suuronen

Garden Exhibition Shell Roof, Stuttgart, 1977, Hans Luz und

Partner, Schlaich Bergermann

St. Louis Airport, 1956,

Minoru Yamasaki, Anton

Tedesko, a cylindrical

groin vault

Dalian

Social Center of the Federal Mail, Stuttgart, 1989, Roland Ostertag

The Tunnel, Buenos Aires, Argentine, Estudio Becker-Ferrari

a. b.

a. b.

c. d.

x2 +y2 + z2 = R2

x2 +y2 + z2 = R2

Little Sports Palace, 1960 Olympic Games,

Rome, Italy, Pier Luigi Nervi,

Palazzetto dello Sport, Arch.: P. Luigi

Nervi & A.Vitellozz, Ing.: P. Luigi Nervi,

Rom, 1957

Biosphere, Toronto, Expo 67, Buckminster Fuller, 76 m, double-layer space frame

Reichstag, Berlin, Germany, 1999, Norman Foster Arch. Leonhardt & Andrae Struct. Eng

Museum of Hamburg History, Hamburg, 1989,

von Gerkan Marg, Partner,Sclaich Bergermann

Schlterhof Roof, German Historical Museum, Berlin, glazed grid shell, 2002,

Architect I.M. Pei, Schlaich Bergermann

Green house Dalian

National Grand Theater, Beijing, 2007, Paul Andreu

Allianz Arena, Munich, 2006, Herzog & Meuron, Arup

MUDAM, Museum of Modern Art, Luxembourg, 2006, I.M. Pei

Suspended models by Isler

Kresge Auditorium, MIT, Eero

Saarinen/Amman Whitney, 1955, on three

supports

a. b.

a. b.

Biosphere, Toronto, Expo 67, Buckminster Fuller, 76 m, double-layer space frame

Cylindrical grid with domical ends

Pennsylvania Station Redevelopment / James A. Farley Post Office, New York,

2003, SOM

Sydney Opera House, Australia, 1972, Joern Utzon/ Ove Arup

Project Cologne Mosque, 2008, Cologne, Germany, Paul und Gottfried Boehm

Georgia Dome, Atlanta, 1995,

Weidlinger, Structures such as the

Hypar-Tensegrity Dome, 234 m x 186 m

HYPAR TENSEGRITY DOME

Hyperbolic parabolid with curved

edges

Hyperbolic parabolid with straight

edges.

Flix Candela The Hyperbolic Paraboloid

The hyperbolic-paraboloid shell is doubly

curved which means that, with proper support,

the stresses in the concrete will be low and only

a mesh of small reinforcing steel is necessary.

This reinforcement is strong in tension and can

carry any tensile forces and protect against

cracks caused by creep, shrinkage, and

temperature effects in the concrete.

Candela posited that of all the shapes we can give to the shell,

the easiest and most practical to

build is the hyperbolic paraboloid. This shape is best understood as

a saddle in which there are a set

of arches in one direction and a

set of cables, or inverted arches,

in the other. The arches lead to an

efficient structure, but that is not

what Candela meant by stating

that the hyperbolic paraboloid is

practical to build. The shape also

has the property of being defined

by straight lines. The boundaries,

or edges, of the hypar can be

straight or curved. The edges in

the second case are defined by

planes cutting through the hypar surface.

z = (f/ab)xy = kxy

5/8 in. concrete shell, Cosmic Rays

Laboratory, U. of Mexico, 1951, Felix

Candela

Hypar umbrella structures, Mexico,

1950s, Felix Candela

Hypar roof for a

warehouse, Mexico,

1955, Felix Candela

More umbrella hypar by Felix

Candela

Iglesia de

la Medalla

Milagrosa,

Mexico City,

1955, Felix

Candela

Chapel Lomas de Cuernavaca,

Cuernavaca, Mexico, 1958, Felix Candela

Bacard Rum Factory, Cuautitln,

Mexico, 1960, Felix Candela

Los Manantiales, Xochimilco ,

Mexico, 1958, Felix Candela

Swimming Center, Hamburg-Sechslingspforte, 1967,

Niessen und Strmer, Jrg Schlaich

St. Marys Cathedral, Tokyo, Japan, 1963, Kenzo Tange, Yoshikatsu Tsuboi

Shanghai Urban Planning Center

EXPO-Dach Hannover, Arch.: Herzog und Partner, Ing.: Julius Natterer, 2000

Law Courts, Antwerp, Belgium,

2005, Richard Rogers, Arup

Schweinfurt bus shelter, Germany

a. b.

c.. d.

Heidi Weber Pavilion, Zurich (CH) - Le Corbusier Heidi Weber Pavilion, Zurich (CH) - Le Corbusier Heidi Weber Pavilion, Zurich (CH) - Le Corbusier

Heidi Weber Pavilion, Zurich (CH), 1963, Le Corbusier

Pompidou Museum II, Metz,

France, 2010, Shigeru Ban

Beijing National

Stadium, 2008, Herzog

and De Meuron Arch,

Arup Eng

BMW Welt, Munich, 2007, Coop Himmelblau

DG Bank, Berlin, Germany

2001, Frank Gehry, Schlaich

Glass Roof for DZ-Bank, Berlin, 1998, Schlaich Bergermann Struct. Engineers

MUDAM, Museum of Modern Art, Luxembourg, 2007, I.M. Pei

Tensile Membrane Structures

In contrast to traditional surface structures, tensile cablenet and textile

structures lack stiffness and weight. Whereas conventional hard and stiff

structures can form linear surfaces, soft and flexible structures must

form double-curvature anticlastic surfaces that must be prestressed (i.e.

with built-in tension) unless they are pneumatic structures. In other words,

the typical prestressed membrane will have two principal directions of

curvature, one convex and one concave, where the cables and/or yarn

fibers of the fabric are generally oriented parallel to these principal

directions. The fabric resists the applied loads biaxially; the stress in one

principal direction will resist the load (i.e. load carrying action), whereas the

stress in the perpendicular direction will provide stability to the surface

structure (i.e. prestress action). Anticlastic surfaces are directly

prestressed, while synclastic pneumatic structures are tensioned by air

pressure. The basic prestressed tensile membranes and cable net surface

structures are

Millenium Dome (365 m), London, 1999, Rogers + Happold

German Pavilion, Expo 67, Montreal, Canada, Frei Paul Otto and Rolf Gutbrod, Leonhardt + Andr

Olympic Stadium, Munich, Germany, 1972, Frei Otto, Leonhardt-Andrae

Soap models by Frei Otto

Sony Center, Potzdamer Platz, Berlin, 2000, Helmut Jahn Arch., Ove Arup

Traveling exhibition

TENSILE MEMBRANE STUCTURES

Pneumatic structures Air-supported structures

Air-inflated structures (i.e. air members)

Hybrid air structures

Anticlastic prestressed membrane structures Edge-supported saddle roofs

Mast-supported conical saddle roofs

Arch-supported saddle roofs

Hybrid tensile surface structures (possibly including tensegrity)

MATERIALS

The various materials of tensile surface structures are:

films (foils)

meshes (porous fabrics)

fabrics

cable nets

Fabric membranes include acrylic, cotton, fiberglass, nylon, and polyester. Most permanent large-scale tensile structures use fabrics, that is,

laminated fabrics, and coated fabrics for more permanent structures. In

other words, the fabrics typically are coated and laminated with synthetic

materials for greater strength and/or environmental resistance. Among the

most widely used materials are polyester laminated or coated with polyvinyl

chloride (PVC), woven fiberglass coated with polytetrafluoroethylene (PTFE,

better known by its commercial name, Teflon) or coated with silicone.

There are several types of weaving methods. The common place plain-

weave fabrics consists of sets of twisted yarns interlaced at right angles.

The yarns running longitudinally down the loom are called warp yarns,

and the ones running the crosswise direction of the woven fabric are

called filling yarns, weft yarns, or woof yarns. The tensile strength of the

fabric is a function of the material, the number of filaments in the twisted

yarn, the number of yarns per inch of fabric, and the type of weaving

pattern. The typical woven fabric consists of the straight warp yarn and

the undulating filling yarn. It is apparent that the warp direction is

generally the stronger one and that the spring-like filler yarn elongates

more than the straight lengthwise yarn. From a structural point of view,

the weave pattern may be visualized as a very fine meshed cable network

of a rectangular grid, where the openings clearly indicate the lack of shear

stiffness. The fact of the different behavioral characteristics along the

warp and filling makes the membrane anisotropic. However, when the

woven fabric is laminated or coated, the rectangular meshes are filled,

thus effectively reducing the difference in behavior along the orthogonal

yarns so that the fabric may be considered isotropic for preliminary

design purposes, similar to cable network with triangular meshes, plastic

skins and metal skins.

The scale of the structure, from a structural point of view,

determines the selection of the tensile membrane type. The

approximate design tensile strengths in the warp and fill

directions, of the most common coated fabrics may be taken as

follows for preliminary design purposes:

PVC-coated nylon fabric (nylon coated with vinyl):

200 400 lb/in (350 700 N/cm)

PVC-coated polyester fabric: 300 700 lb/in.(525 1226 N/cm)

PVC-coated fiberglass fabric: 300 800 lb/in.(525 1401 N/cm)

PTFE-coated fiberglass fabric: (e.g. Teflon-coated fiberglass)

300 1000 lb/in.(525 1751 N/cm)

Strength Properties

Samples taken from any roll will possess the following minimum ultimate

strength values.

Warp5700 N/50mmWeft (fill)5000 N/50mm

The 50mm width shall be a nominal width which contains the theoretical

number of yarns for 50mm calculated from the overall fabric properties.

(f) Design Life of Membrane

Membrane Properties

Poissons Ratio: ratio of

strain in x and y directions

Modulus of Elasticity (E)

E=stress/strain

(stress=force/area,strain=dL/L)

Bi-axial testing of every roll of raw goods.

Tensile only: no shear or compression

Strength (38.5 ounce per square yard PTFE coated Fibreglass Fabric)

Warp: 785 lb/in.

Fill: 560 lb/in.

Creep

Which Fabric do I Use? Easy!

There are five types of fabrics being used today for tensile fabric structures and they all have

special qualities. Below are descriptions of these fabrics, but there may be other fabrics that

are not listed here. These fabrics are (1) PVC coated polyester fabric, (2) PTFE coated glass

fabric, (3) expanded PTFE fabric, (4) Polyethylene coated polyethylene fabric, and (5) ETFE

foils.

PVC polyester fabric is a cost effective fabric having a 10 to 20 year lifespan. It has been

used in numerous applications worldwide for over 40 years and it is easy to move for

temporary building applications. Top films or coatings can be applied to keep the fabric clean

over time. It meets building codes as a fire resistive product and light translucencies range

between zero and 25%. PVC meets B.S 7837 for Fire Code. Typical woven roll width is 2.5

meters.

PTFE glass fabrics have a 30 year lifespan and are completely inert. They do not degrade

under ultra violet rays and are considered non combustible by most building codes. PTFE

meets B.S 476 Class 0 for fire code. They are used for permanent structures only and can

not be moved once installed. The PTFE coating keeps the fabric clean and translucencies

range from 8 to 40%. They are woven in approximately 2.35m or 3.0 meter widths.

ETFE foils are used in inflated pillow structures where thermal properties are important. The

foil can be transparent or fritted much like laminated glass products to allow any level of

translucency. Its fire properties lie somewhere between that of PTFE glass and PVC

polyester fabrics and it is used in permanent applications.

PVC glass fabrics are used for internal tensile sails, such as features in atriums, glare

control systems. Their maintenance is minimal and meet B.S 476 Class 0 for Fire Code.

LOADS

Tensile structures are generally of light weight. The magnitude of the roof

weight is a function of the roof skin and the type of stabilization used.

The typical weights of common coated polyester fabrics are in the range

of approximately 24 to 32 oz/yd2 (0.17 to 0.22 psf, 8 to 11 Pa). The roof

weight of a fabric membrane on a cable net may be up to approximately

1.5 psf (72 Pa). The lightweight nature of membrane roofs is clearly

expressed by the air-supported dome of the 722-ft-span Pontiac Stadium

in Michigan, weighing only 1 psf (48 Pa = 4.88 kg/m2).

Since the weight of typical pretensioned roofs is relatively insignificant,

the stresses due to the superimposed primary loads of wind (laterally

across the top and from below for open-sided structures), snow, and

temperature change tend to control the design. These loads may be

treated as uniform loads for preliminary design purposes and the

structure weight can be ignored. The typical loads to be considered are

snow loads, wind uplift, dynamic load action (wind, earthquake),

prestress loads, erection loads, creep and shrinkage loads, movement of

supports, temperature loads (uniform temperature changes and

temperature differential between faces), and possible concentrated loads.

The prestress required to maintain stability of the fabric membrane,

depending on the material and loading, is usually in the range of 25 to 50

lb/in (88 N/cm).

STRUCTURAL BEHAVIOR

Soft membranes must adjust their shape (because they are flexible) to the

loading so that they can respond in tension. The membrane surface must

have double curvature of anticlastic geometry to be stable. The basic

shape is defined mathematically as a hyperbolic paraboloid. In cable-nets

under gravity loads, the main (convex, suspended, lower load bearing)

cable is prevented from moving by the secondary (concave, arched, upper,

bracing, etc.) cable, which is prestressed and pulls the suspended layer

down, thus stabilizing it. Visualize the initial surface tension analogous to

the one caused by internal air pressure in pneumatic structures.

Suspended, load-carrying

membrane force

Arched, prestress

membrane force

f

f

wp

T2T2

T1 T1

w

Design Process

The design process for soft membranes is quite different from that for hard

membranes or conventional structures. Here, the structural design must be

integrated into architectural design.

Geometrical shape: hand sketches are used to first pre-define a geometry of the

surface as based on geometrical shapes(e.g. conoid, hyperbolic paraboloid)

including boundary polygon shape as based on functional and aesthetical

conditions.

Equilibrium shape: form is achieved possibly first by using physical modeling and

applying stress to the membrane (e.g. through edge-tensioning, cable-

tensioning, mast-jacking), where the geometry is in balance with its own

internal prestress forces, and then by computer modeling.

Computational shape: structural analysis is performed to find the resulting

surface shape due to the various load cases causing large deformations of

the flexible structure. The resulting geometry is significantly different from the

initially generated form; the biaxial properties of the fabric (elastic moduli and

Poissons ratios) are critical to the analysis. Not only the radius of curvature changes, but also the actual forces will be different.

Modification of surface shape

Cutting pattern generation of fabric membrane (e.g. linear patterning for saddle

roofs, radial patterning for umbrellas)

General purpose finite element programs such as SAP can only be used for the

preliminary design of cablenet and textile structures however the material

properties of the fabric membrane in the warp- and weft directions must be defined.

Special purpose programs are required for the final design such as Easy, a

complete engineering design program for lightweight structures by technet GmbH,

Berlin, Germany (www.technet-gmbh.com). The company also has second

software, Cadisi, for architects and fabricators for the quick preparation of initial

design proposals for the conceptual design of surface stressed textile structures

especially of saddle roofs and radial high-point roofs.

The spherical membrane represents a minimal surface under radial pressure,

since not only stresses and mean curvature are constant at any point on the

surface, but also because the sphere by definition represents the smallest

surface for the given volume. Some examples in nature are the sea foam, soap

bubbles floating on a surface forming hemispherical shapes, and flying soap

bubbles. The effect of the soap film weight on the spherical form may be

neglected.

Double Curvature

Large radius

of curvature

results in

large forces.

PNEUMATIC STUCTURES

Air-supported structures

Air inflated structures: air members

Hybrid air structures

Air-supported structures

high-profile ground-mounted air structures

berm- or wall-mounted air domes

low-profile roof membranes

Air-supported structures form synclastic, single-membrane structures, such

as the typical basic domical and cylindrical forms, where the interior is

pressurized; they are often called low-pressure systems because only a

small pressure is needed to hold the skin up and the occupants dont notice it. Pressure causes a convex response of the tensile membrane and suction

results in a concave shape.

The basic shapes can be combined in infinitely many ways and can be

partitioned by interior tensile columns or membranes to form chambered

pneus. Air-supported structures may be organized as high-profile ground-

mounted air structures, and berm- or wall-mounted, low-profile roof

membranes.

In air-supported structures the tensile membrane floats like a curtain on top of

the enclosed air, whose pressure exceeds that of the atmosphere; only a small

pressure differential is needed. The typical normal operating pressure for air-

supported membranes is in the range of 4.5 to 10 psf (0.2 kN/m2 to 0.5 kN/m2 =

0.5 kPa) or 2 mbar to 5 mbar, or roughly 1.0 to 2.0 inches of water as read from

a water-pressure gage.

pT = pR T = pR

US Pavilion, EXPO

70, Osaka, Davis-

Brody

Metrodome, Minneapolis, 1981, SOM

Examples of pneumatic structures

'Sleep and Dreams' Pavilion, 2006, Le Bioscope, France

'Spirit of Dubai' Building in front of Al Fattan Marine

Towers, Dubai, 2007

To house a touring exhibition

Using inflatable moulds and spray on polyurethane foam

Kiss the Frog: the Art of Transformation, inflatable pavilion for Norways National Galery, Oslo, 2001, Magne Magler Wiggen Architect,

Air inflated structures:

air members

Air inflated structures or simply air members, are

typically,

lower-pressure cellular mats: air cushions

high-pressure tubes

Air members may act as columns, arches, beams, frames, mats, and

so on; they need a much higher internal pressure than air-supported

membranes

Expo02 Neuchatel, air cussion, ca 100 m dia.

Roman Arena Inflated Roof, Nimes, France, removable

membrane pneu with outer steel, 1988, Architect Finn

Geipel, Nicolas Michelin, Paris; Schlaich Bergermann und

Partne.internal pressure 0.40.55 kN/m2

Roof for Bullfight Arena - Vista Alegre, 2000, Ayuntamiento de Madrid

Allianz Arena, Munich, 2005, Herzog and Pierre de Meuron, Arup

inflatable Ethylene Tetrafluoro Ethylene (ETFE) clad facade cushions

200'

15

'15

'

Hybrid air structures

Hybrid air structures are formed by a combination of the preceeding

two systems or when one or both of the pneumatic systems are

combined with any kind of rigid support (e.g. arch supported).

In double-walled air structures, the internal pressure of the main

space supports the skin and must be larger than the pressure

between the skins, which in turn, must be large enough to withstand

the wind loads. This type of construction allows better insulation,

does not show the deformed state of the outer membrane, and has a

higher safety factor against deflation. It provides rigidity to the

structure and eliminates the need for an increase of pressure inside

the building.

Fuji Pavilion, Expo 1970, Osaka,

air pressure 500..1000 mbar = 501000 kN/m2

Surface structures tensioned by cables and masts

are of permanent nature with at least 15 to 20 years of life expectancy (and

tents or other clear-span canvas structures which are often mass-

produced) have an anticlastic surface geometry, where the two opposing

curvatures balance each other. In other words, the prestress in the

membrane along one curvature stabilizes the primary load-bearing action

of the membrane along the opposite curvature. The induced tension

provides stability to form, while space geometry, together with prestress,

provides strength and stiffness.

The membrane supports may be rigid or flexible; they may be point or line supports

located either in the interior or along the exterior edges. The following organization

is often used based on support conditions:

Edge-supported saddle surface structures Arch-supported saddle surface structures Mast-supported conical (including point-hung) membrane structures (tents) Hybrid structures, including tensegrity nets

The lay out of the support types, in turn, results in a limitless number of new forms,

such as,

Ring-supported saddle roofs Parallel and crossed arches as support systems Parallel and radial folded plate point-supported surfaces Multiple tents on rectangular grids

The pre-tensioning mechanisms range from edge-tensioning systems (e.g.

clamped fabric edges) to cable-tensioning and mast-jacking systems. Since

flexible structures can resist loads only in pure tension, their geometry must reflect

and mirror the force flow; surface geometry is identical with force flow. Membranes

must have sufficient curvature and tension throughout the surface to achieve the

desired stiffness and strength under any loading condition. In contrast to traditional

structures, where stresses result from loading, in anticlastic tensile structures

prestress must be specified initially so that the resulting membrane shape can be

determined.

Tensile membranes can be classified either according to their surface form or to

their support condition.. Basic anticlastic tensile surface forms are derived from the

mathematical geometrical shapes of the paraboloid of revolution (conoid), the

hyperbolic paraboloid or the torus of revolution. In more general terms, textile

surface structures can be organized as,

Saddle-shaped and stretched between their boundaries representing orthogonal anticlastic surfaces with parallel fabric patterns

Conical-shaped and center supported at high or low points representing radial anticlastic surfaces with radial fabric patterns

The combination of these basic surface forms yields an infinite number of new forms

Anticlastic Shapes

Hyperbolic Paraboloid Double Ring Cone

Valley and Ridge Arch Support

Dorton (Raleigh) Arena, 1952,

North Carolina, Matthew Nowicki,

with Frederick Severud

Schwarzwaldhalle, Karlsruhe, Germany, 1954, Ulrich Finsterwalder + Franz Dischinger

Olympic Stadium, 1964, Tokyo, Kenzo Tange/ Y. Tsuboi

Ice Hokey Rink, Yale University, 1959, Eero Saarinen, Fred N. Severud

Dance Pavilion, Federal Garden Pavilion,

1957

Frei Otto

German Pavilion, Montreal EXPO 1967

Frei Otto, Rolf Gutbrod

Young Land, Japan 1968

K. Mori

Student Center, La Verne (CA) 1973

One of the first architectural applications of PTFE coated Fibreglass fabrics developed in 1972.

Fabric was tensile tested after 20 years at 70% fill/80% warp of original strength.

Ice Rink Roof, Munich,

1984, Architect

Ackermann und Partner,

Schlaich Bergermann

Schlumberger Research Center, Cambridge, UK, 1985, Hopkins/ Hunt

Haj Terminal, Jeddah, Saudi Arabia, 1982, SOM/ Horst Berger

Denver International Airport Terminal, 1994, Denver, Horst Berger/ Severud

Denver Airport

Analysis Program developed by William Spillers, NJIT

Stadium Roof, Riyadh, Saudi Arabia, 1984, Architect Fraser Robert, Geiger & Berger,

Nelson-Mandela-Bay-

Stadion , Port Elizabeth ,

South Africa, 2010,

Gerkan, Marg und

Partner

Moses Mabhida Stadion , Durban, South

Africa, 2009, Gerkan, Marg und Partner

Inchon Munhak Stadium, Inchon, South Korea, 2002, Adome, Sclaich Bergermann

Canada Place, Vancouver, 1986, Eberhard Zeidler/ Horst Berger

San Diego Convention Center, 1989, Arthur Erickson/ Horst Berger

IAA 95 motor show,

Frankfurt, BMW

Stellingen Ice Skating Rink Roof, Hamburg-Stellingen, 1994, Schlaich Bergermann

Ningbo

Max Planck Institute of Molekular Cell Biology, Dresden, 2002, Heikkinen-Komonen

Subway Station Froettmanning, Munich, 2005, Bohn Architect, PTFE-Glass roof

Cirque de Soleil, Disney World, Orlando, FL, 2000, FTL (Nicholas

Goldsmith)/Happold + Birdair

The prestress force must be large enough to keep the surface in

tension under any type of loading, preventing any portion of the

skin or any other member to slack because the compression

being larger than the stored tension. In addition, the magnitude of

the initial tension should be high enough to provide the necessary

stiffness, so that the membrane deflection is kept to a minimum.

However, the amount of pretensioning not only is a function of the

superimposed loading but also is directly related to the roof shape

and the boundary support conditions. The prestress required to

maintain stability of the fabric membrane, depending on the

material and loading, is usually in the range of

25 to 50 lb/in (44 to 88 N/cm).

Flexible structures do not behave in a linear manner, but resist

loads by going through large deformations and causing the

magnitude of the membrane forces to depend on the final position

in space.

For preliminary design of shallow membranes, all external loads (snow,

wind) can be treated as normal loads, are assumed to be carried by the

suspended portion of the surface, when the arched portion has lost its

prestress and goes slack. Also notice that at least one-half of the permitted

tension in the membrane is consumed by the initial stored tension.

T2 = Tmax = wR = wL2/8f

The design of the arched cable system or yarn fibers is derived, in general, from

the loading condition where maximum wind suction, ww, causes uplift and

increases the stored prestress tension, which is considered equal to one-half of

the full gravity loading, minus the relatively small effect of membrane weight. In

other words, under upward loading, the maximum forces occur in the arched

portion of the membrane

T1 = Tmax = (wp + ww)R =(wp + ww)L2/8f

COMB1

COMB2

COMB3

a. b.

COMB3

COMB2

COMB1

Form Finding Methodologies

There are three main methods used to find the equilibrium shape.

All lead to the same result, which is an minimum surface for a given

pre-stress, membrane characteristics, and edge and support

conditions. Modern programs can take into account structural

characteristics of supports, uneven loading, and non-linear

membrane characteristics.

For a constant membrane thickness taking into account the weight

of the membrane, no curved surface exists whereby all points on the

surface have equal tension. It is possible, however, to obtain a

curved surface where the shearing force at every point is zero.

An important component of design is the analysis of the equilibrium

surface, based on varying load scenarios. The final form the

designer chooses may vary from the equilibrium surface so as to be

optimized for estimated load extremes and considerations of on-site

construction and pre-stressing methods.

Pioneers of Computerized Form Finding

1965: Alistair Day introduces Dynamic Relaxation method for analysis, later refined by Micheal Barnes, J

Bunce, John Argyris, and David Wakefield (AS Day, An Introductions to Dynamic Relaxation The

Engineer, V219 1965.)

1969: Early work by Ove Arup on the analysis of hanging roofs. (AS Day, and J Bunce).

1970: David Geiger associates and M. McCormick: first computer analysis of a fabric membrane of the air

supported roof at US Pavilion at Expo in Osaka.

1969-71: Development of computer for form-finding for structures by Klaus Linkwitz (from work begun

in 1966) calling it The Stuttgart Direct Approach. Programmed on a CDC 6600 to design and analyze

the Olympic Roofs in Munich.(Linkwitz and HJ Schek, A New Method of Analysis of Prestressed Cable

Networks, IABSE, Amsterdam, 1972.

1971: First published form-finding method of membranes with specified prestress: Micheal Barnes

Pretensioned Cable Networks, Construction Research and Development Journal, Vol. 3, No. 1, 1971

1973: Interactive form finding program on an IBM Mainframe by Massimo Majowiecki at STM from

work deriving from 1970 thesis. Used to design the Coverture of the Rome stadium, Torin Stadium, and

Athen Stadium.

1974: HJ Schek introduces Force Density Method, now used by Geiger,and in commercialy available

programs Forten and Cadisi (Schek, Force Density Methods for Form Finding and Computation of

General Networks, Computer Methods in Applied Mechanics and Engineering, 1974)

1975: Ross Dalland: Cornell thesis on form finding and patterning.

1980: Robert Haber: Cornell thesis, Stiffness method kernel for Birdair Images program.

1980: Buro Happold Tensyl Program

1981:First Computer Patterning (?) by Birdair (Minneapolis Metrodome), 1981.(Source: Geiger).

Early 1980s: William Spillers pioneers advanced stiffness method with material with non-linear

properties used for the larger Berger/Geiger structures.

Expo at Osaka, 1970

Frei Ottos

MIT Thesis,

1962

Left: Minneapolis

Metrodome, 1981

Right: Robert Habers Cornell

thesis: Form Finding with

graphical results.

Modern Computer Programs Form

Finding

Patterning

Stress Analysis

Equilibrium Conditions

Soap bubbles are minimum surfaces

with uniform surface tension. Early

form finding work used 3D stereo-

photography of soap bubbles and

moiree methods.

Membrane Structures are optimized

for structural live loads and are

designed with a specified prestress,

which affects the equilibrium shape.

Support conditions, membrane

stiffness, and biaxial properties are

also factors in the final form.

Horst Bergers Grid Method

1. Start with a plan view with

a grid of nodes.

2. Enter starting elevation of

center node.

3. Compute for equilibium of

forces in sucessive

surrounding nodes.

4. Reiterate new z coordinates

into equilibrium equations.

5. Convert the isometric

shape to a geodesic shape by

rotating the coordinate system

to be orthagonal at each node.

5. Reiterate new x, y, z

coordinates into the

equilibrium equations.

Used to create initial geometric form based on a equilibrium of forces by

calculating values of the z coordinate using force balancing equilibrium equations.

Final geodesic form

(with added ridge cables)

Simple Preliminary Analysis

Radius

Chord Sag

Geometrically: R=(C2+4S)/8S and C=2Rsin(L/2R)

1. Calculate membrane tension for given pressure (T=P*R)

2.As membrane tension increases, membrane will stretch.

dL=T*L/wE (w=strip width). New length=L+dL

3. Iterate to find new radius based on new arc length.

4. Calculate new tension based on new new values.

Arc Length (L)

2-Dimensional Example

Moving to three dimensions requires solving

equations simultaneously.

Stress/strain is assumed to be linear.

Equations are geometrically non-linear.

Form finding is the process of determining

the equilibrium shape with a given pre-stress,

applied loads, and membrane properties.

Most form finding programs are based on a 3-node triangular

grid which is advantageous for subsequent patterning (though

John Hollyday and David Salmon researched multi-noded

elements at Cornell from 1983-1987).

Birdair Images Program

Basic

Parameters

Form Finding

Load Analysis

Patterning

Pattern 2000 (Birdair)

calculates lengths of

geodesic curves and

flattens triangles.

Tensys (David Wakefield)

Finding Form, Frei Otto and Bado Rasch, Edition Axel Menges, 1995

Tensile Structures, Volume 1 and 2. Edited by Frei Otto MIT Press, 1967, 1969

Calculation of Membranes, Frei Otto and R. Trostel, MIT Press, 1967

Engineering a New Architecture, Tony Robbin, Yale University Press, 1996

Soft Canopies-Details in Building, Martin Vandenberg, Academy Editions, 1996

FTL-Softness in Movement and Light, Academy Editions, 1997

Peter Rice-An Engineer Imagines, Peter Rice, Artemis, 1993

Soft Shells, Hans Joachim Schock, Birkhauser, 1997

Tensile Structures, Architectural Design Profile No 117, 1995

Ephermeral/Portable Architecture, Architectural Design, Vol 68 No.9/10, 1998

The Art of Structural Engineering, Alan Holgate, Edition Axel Meges, 1997.

Happold, The Confidence to Build, Derek Walker and Bill Addis, Happold Trust, 1997

Spatial Lattice and Tension Structures, John Abel et al, American Society of Civil Engineers, 1994

Membrane Designs and Structures in the World, Kazou Ishii, Shinkenchiku-sha Co, Ltd, 1999

Light Structures, Structures of Light, Horst Berger, Birkhauser, 1996

Structures, Dan Schodek, Prentice Hall, 2001

Digital Design and Production, Dan Schodek, Kenneth Kao, Draft Manuscript, 2000

The Structural Basis of Architecture, Bjorn Sandaker and Arne Eggen, Whitney Library, 1992

The Science of Soap Films and Soap Bubbles, Cyril Isenberg, Dover 1992

The Unique Role of Computing in the Design and Construction of Tensile Membrane Structures, American Society of Civil Engineers, New York, 1991