surface structures, including sap2000

SURFACE STRUCTURES including SAP2000

Prof. Wolfgang Schueller

For SAP2000 problem solutions refer to “Wolfgang Schueller: Building

Support Structures – examples model files”:

https://wiki.csiamerica.com/display/sap2000/Wolfgang+Schueller%3A+Building+Su

pport+Structures+-

If you do not have the SAP2000 program get it from CSI. Students should

request technical support from their professors, who can contact CSI if necessary,

to obtain the latest limited capacity (100 nodes) student version demo for

SAP2000; CSI does not provide technical support directly to students. The reader

may also be interested in the Eval uation version of SAP2000; there is no capacity

limitation, but one cannot print or export/import from it and it cannot be read in the

commercial version. (http://www.csiamerica.com/support/downloads)

See also,

Building Support Structures, Analysis and Design with SAP2000 Software, 2nd ed.,

eBook by Wolfgang Schueller, 2015.

The SAP2000V15 Examples and Problems SDB files are available on the

Computers & Structures, Inc. (CSI) website:

http://www.csiamerica.com/go/schueller

https://wiki.csiamerica.com/display/sap2000/Wolfgang+Schueller:+Building+Support+Structures+-














http://www.csiamerica.com/support/downloads







http://www.csiamerica.com/go/schueller

Surfaces in nature

SURFACE STRUCTURES

- MEMBRANES

BEAMS

BEARING WALLS and SHEAR WALLS

- PLATES

slabs, retaining walls

- FOLDED SURFACES

RIBBED VAULTING

LINEAR and RADIAL ADDITIONS

parallel, triangular, and tapered folds

CURVILINEAR FOLDS

- 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 (including tensegrity)

- SHELLS: solid shells, grid shells

CYLINDRICAL SHELLS

THIN SHELL DOMES

HYPERBOLIC PARABOLOIDS

Slabs resisting gavity loads

Flat plate building

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

http://architectuul.com/architecture/new-national-gallery

Art Museum of Sao Paulo,

Sao Paulo, Brazil, 1968,

Lina Bo Bardi Arch

(prestressed concrete

beams)

Shear walls resisting wind

Cite Picasso, Nantere, Paris, 1977, Emile Aillaud Arch

Whitney Museum of American

Art, New York, 1966, Marcel

Breuer Arch

Everson Museum, Syracuse, NY,

1968, I. M. Pei Arch

Delft University of Technology Aula

Congress Centre, 1966, Jaap Bakema Arch

St. Engelbert, Cologne-Riehl, Germany, 1932,

Dominikus Böhm Arch

Design Museum, Nuremberg,

Germany, 1999, Volker Staab Arch

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

Stress contour of structural piping

Boston Convention Center, Boston, 2005, Vinoly and LeMessurier

Incheon International Airport, Seoul.

2001, Fentress Bradburn Arch.

MUDAM, Museum of

Modern Art,

Luxembourg, 2007,

I.M. Pei

Armchair 41 Paimio by Alvar Aalto, 1929-

33, laminated birchwood

Eames Plywood Chair, 1946,

Charles and Ray Eames

Designers

Panton Molded

Plastic Chair,

Denmark, 1960,

Verner Panton

Designer

Ribbon Chair, Model CL9,

Bernini, 1961, Cesare

Leonardi & Franca Stagi

designers

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!

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 Poisson’s 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.

LINE COMPONENT PLANAR COMPONENT SOLID COMPONENT

DISCRETE MODELCONTINUOUS MODELS

LINE ELEMENT TYPICAL PLANAR ELEMENTS TYPICAL SOLID ELEMENTSa. b.

c. d.

e.

Basics of Modeling

Possibilities for Modeling a Simple

Structure

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

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

National Gallery of Art, East Wing,

Washington, 1978, I.M. Pei Arch

ey

Fy F

yF

y

e.

ey F

yF

yb

d/2

1 2

Fy

Mp

d/2

Cp

Tp

3

d/2Bending Stresses

Glulam beams

Build-up wood beams

Equivalent stress distribution for typical singly reinforced concrete floor beams

at ultimate loads

Shear force resistance of vertical stirrups

Design of concrete floor structure (see Examples 3.17 and 3.18)

1 K/ft

4'

40'

10 k

8'

2'

(2) EXAMPLES: 12.1, 12.2

1 K/ft

4'

40' a.

b.

c.

EXAMPLE: 12.1: Beam membrane

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.

±1.01 ksi

92 psi

EXAMPLE: 12.1: Beam membrane

10 k

8'

2'

EXAMPLE 12.2: Cantilever beam

membrane

30'

12

'

10' 10' 10'

Pu= 500 k

R = 500 k R = 500 k

θ = 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.

EXAMPLE 12.3 Deep Beam; Flexural

Stress S11

Arbitrary membrane structure – S11 stresses –

displacements contour lines – displacements contour fill

BEARING WALLS and SHEAR WALLS

National Assembly, Dacca, Bangladesh, 1974, Louis Kahn

Wall behavior

World War II bunker transformed into housing, Aachen, Germany

Dormitory of

Nanjing

University,

Zhang Lei Arch.,

Nanjing

University,

Research Center

of Architecture

Seismic action

Shear-wall or Cantilever-column

LATERAL DEFLECTION OF SHEAR WALLS

Shear Wall and Frame

Shear Wall Behavior Frame Behavior

Shear Wall and Frame Behavior

Shear Wall and Truss Behavior

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'

Example 12.4: Effect of shear wall proportion

Long wall: axial stresses, shear stresses, bending stresses

From shallow to deep beam

shallow beam

deep beam

Deep concrete beams

Effect of shear wall proportion, S22 axial stresses, S12 shear stresses

S22 axial gravity stress – S12 wind shear stress – S22 flexural wind stress

EXAMPLE: 12.4: Bearing wall

Typical Long-wall structure

Typical shear wall structure

The behavior of ordinary shear

walls

Fig. 12.8, Problem 12.2: Stresses S22 (COMB1), S12 (COMB2), S22 (COMB3)

The response of exterior brick walls to lateral and gravity loading

The effect of lateral load action upon walls with openings

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 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

Problem: 12.3: Bearing wall with openings

LATERAL DEFLECTION OF WALLS WITH OPENINGS

PIER-SPANDEL SYSTEMS

Multiple Shear Panels

Shear Wall-Frame Interaction: Lateral Deflection (top), Wind Moments (bottom)

Modeling Walls with Opening

Plate-Shell Model Rigid Frame Model Truss Model

Truss model for shear walls

Rigid frame model

for shear walls

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 Arch

Max Planck Institute of Molekular Cell Biology, Dresden, 2002,

Heikkinen-Komonen Arch

Xinghai Square shopping mall, Dalian, China

Sony Center, Potzdamer

Platz, Berlin, 2000, Helmut

Jahn Arch., Ove Arup USA

Struct. Eng

Shopping Center,

Jiefangbei

business district,

Chongqing, China

PLATES

• SLABS

• RETAINING WALLS

A visual investigation of floor structures

Slab structures: the effect of

support and boundaries

Joist floor

Introduction to two-way slabs on rigid supports

Design of two-way slabs

on stiff beams

Flat slab building structures

Design of flat plates and post-tensioned slabs

Square and Round Concrete Slabs

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

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

SLAB SUPPORT ALONG EDGES

a

d

b

e

c

f

EXAMPLE: 12.5: Square concrete slabs

Punching shear

#4 @ 12"

#3 @ 9"

15 ft12 in 12 in

#13 @ 305 mm

#10 @ 229 mm

4.57 m

305 mm

Example 4.10 one-way slab cross section

ETABS template SAFE template

There are no slab templates in SAP2000 – planar objects must be modeled

Gatti Wool Factory, Rome,

Italy, 1953, Pier Luigi Nervi

Floor systems of Palace of Labor, Large

Sports Palace, Gatti Wool Factory,

Pier Luigi Nervi

Schlumberger Research

Center, Cambridge, 1985,

Michael Hopkins,

Anthony Hunt, Ove Arup

Dead + PT LC: vertical deflection plot of slab

GI

GI

BM

BM

BM

16

/24

16

/24

12/24

12/24

12/24

34"

15

"1

5"

18"x18"

EXAMPLE 4.10: Design of one-way slab

Retaining wall

Example of slab steel reinforcement layout

Example of steel

reinforcement layout

Ramp (STRAP)

FOLDED SURFACES

The folded surfaces of the following building cases many the early modern

period are constructed of reinforced concrete while most of the later periods are

of framed steel or wood construction (e.g. trusses)!

• RIBBED VAULTING

• LINEAR and RADIAL ADDITIONS parallel, triangular, and tapered folds

• CURVILINEAR FOLDS

Folded plate structure systems

Examples 7.1 and

7.2: slab action

Examples 7.1 and 7.2:

beam action

Triangular folded

plates

(1) Figs 7.6, 7.7, 7.8

Folded plate architecture

Saint John's Abbey,

Collegeville,

Minnesota, 1961,

Marcel Breuer Arch

American Concrete Institute Building (ACI), Detroit. Michigan, 1959, Minoru Yamasaki Arch

NIT, Ningbo

Neue Kurhaus, Aachen, Germany

Unesco Auditorium,

Paris, 1958, Marcel

Breuer, Pier Luigi

Nervi

Turin Exhibition Hall, Salone

Agnelli, 1949, Pier Luigi Nervi

St. Loup Chapel,

Rompaples VD,

Switzerland, 2008,

Danilo Mondada Arch

St. Foillan, Aachen, Germany,

1958, Leo Hugot Arch.

Wallfahrtskirche "Mariendom" , Neviges, Germany, 1972, Gottfried Boehm Arch

St. Gertrud, Cologne, Germany, 1965,

Gottfried Boehm Arch

St. Hubertus, Aachen, Germany, 1964,

Gottfried Böhm Arch

Riverside Museum,

Glasgow, Scotland, 2011,

Zaha Hadid Arch, Buro

Happold Struct. Eng

SHELLS: solid shells, grid shells

• CYLINDRICAL SHELLS

• THIN SHELL DOMES

• HYPERBOLIC PARABOLOIDS

Curvilinear Patterns

Surface classification 1

Surface classification 2

Arches as enclosures

Development of long-span roof structures

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);

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

Hagia Sofia, Constantinople (Istanbul), 537 A.D., Anthemius of Tralles and Isodore of Miletus

St. Mary, Pirna, Germany, 1616

Casa Mila, Barcelona, Spain, 1912,

Antoni Gaudi Arch (catalan vaulting)

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

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

Bent surface structures

UNESCO Concrete

Portico (conoid), Paris,

France, 1958, Marcel

Breuer, Bernard Zehrfuss,

Pier Luigi Nervi

Hipodromo La Zarzuela, 1935,

Eduardo Torroja

Kresge Auditorium, MIT, 1955, Eero Saarinen Arch,

Amman & Whitney Struct. Eng

Kresge Auditorium, MIT, Eero Saarinen/Amman

Whitney, 1955, on three supports

deflected structure under its own weight

Suspended models by

Heinz Isler

Autobahnraststätte, Deitingen,

Switzerland, 1968, Heinz Isler

http://www.baunetz.de/meldungen/Meldungen-Zum_Tod_von_Heinz_Isler_791805.html?backurl=http%3A%2F%2Fwww.baunetz.de%2Fmeldungen%2Findex.html&bild=3

Gartenhaus Center,

Zuchuil, Switzerland,

1962, Heinz Isler

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

Earth House Estate Lättenstrasse, Dietikon, Switzerland, 2012, VETSCH ARCH

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

Jubilee Church, Rom, Italy, 2000, Richard

Meier Arch, Ove Arup Struct. Eng.

Eden Project, Cornwall, UK, 2001, Sir

Nicholas Grimshaw Arch, Anthony Hunt

Struct. Eng

Shell surfaces in plastics

Basic concepts related to barrel shells

Barrels

Cylindrical shell beam structures

Vaults and short cylindrical shells

R2 = z2 + x2

Circular cylindrical surface

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

Komendant Struct. Eng

Shonan Christ Church,

Fujisawa, Kanagawa, Japan,

2014, Takeshi Hosaka Arch,

HITOSHI YONAMINE / OVE

ARUP Struct Eng

Stadelhofen, Zurich,

Switzerland, 1983,

Santiago Calatrava Arch

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 Struct. Eng

Cristo Obrero

Church,

Atlantida,

Uruguay, 1960,

Eladio Dieste

Arch+Struct Eng

World Trade Centre Dresden,

1996, Dresden, nps + Partner

Glass Roof for DZ-Bank, Berlin, 1998,

Schlaich Bergermann Struct. Eng

Railway Station

"Spandauer

Bahnhof“, Berlin-

Spandau, 1997,

Architect von

Gerkan Marg und

Partner, Scdhlaich

Bergermann

Greenhouse Dalian

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

Schlaich Bergermann

St. Louis Abbey Priory

Chapel, Missouri, 1962, Gyo

Obata of (HOK) and Pier

Luigi Nervi

St. Louis Airport, 1956, Minoru Yamasaki,

Anton Tedesko, a cylindrical groin vault

Ecole Nationale de Ski et d'Alpinisme

(ENSA), Chamonix-Mont Blanc, France, 1974,

Roger Taillibert Arch, Heinz Isler Struct. Eng.

Dalian

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

Schlaich Bergermann Struct. Eng

The Tunnel, Buenos Aires,

Argentine, Estudio Becker-Ferrari

Arch

Slab action vs beam action

From the joist slab to shell beam

Behavior of short

barrel shells

Long vs short

barrel shell

Behavior of long barrel shell

Rectangular beam vs shell beam

a. b.

c. d.

Transverse S22 stresses and longitudinal S11 stresses in short barrel shells

Pipe connected to plate - stress

contour of structural piping

Barrel shells with or without edge beams Various cylindrical shell types

Museum of Hamburg History Glass Roof, Hamburg, 1989,

von Gerkan Marg, Partner,Sclaich Bergermann

x2 +y2 + z2 = R2

surface geometry of spherical surface

x2 +y2 + z2 = R2

Don Bosco Church, Augsburg,

Germany, 1962, Thomas

Wechs Arch

MUDAM: Futuro House

(or UFO), 1968, Finland,

Matti Suuronen

Little Sports Palace, 1960 Olympic

Games, Rome, Italy, Pier Luigi Nervi

St. Rochus Kirche,

Düsseldorf,

Germany, 1954,

Paul Schneider-

Esleben Arch

National Grand Theater, Beijing, 2007, Paul Andreu Arch

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

Architect I.M. Pei, Schlaich Bergermann

Keramion, Frechen, Germany, 1971, Peter

Neufert Arch, Stefan Polónyi Struct. Eng.

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

Schlüterhof Roof, German Historical

Museum, Berlin, Germany, 2002, I.M.

Pei Arch, Schlaich Bergermann Struct.

Eng

Braced dome types

Dome structure cases

Major dome systems Membrane forces in a spherical dome shell due to

live load q

Membrane forces in a dome shell

due to self-weight w Dome shells on polygonal base

Schwedler dome (Example 8.6) Elliptic paraboloid

Junction of dome shell and

support structure

a. b.

a. b.

shallow and hemispherical shells

Cylindrical grid with domical ends

Allianz Arena, Munich, 2006,

Herzog & Meuron Arch,

Arup Struct Eng

Mineirão Stadium Roof, Belo

Horizonte, Brazil, 2012, Gerkan,

Marg + Gustavo Penna Arch,

Schlaich Bergermann Struct. Eng.

Climatron Greenhouse, St. Louis, 1960,

Murphy and Mackey Arch, Synergetics

Designers

Biosphere, Toronto, Expo 67,

Buckminster Fuller, 76 m,

double-layer space frame

Geodesic dome

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

Burnham Plan Centennial Eco-Pavilion, Chicago, 2009, Zaha Hadid Arch

Pennsylvania Station Redevelopment / James A. Farley Post Office, New York, 2003, SOM

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

Cologne Mosque, Cologne,

Germany, 2014, Paul und

Gottfried Boehm Arch

Case study of hypar roofs

Hyperbolic paraboloid

Hyperbolic parabolid with curved

edges

Hyperbolic parabolid with straight

edges.

Félix 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.

Hypar units on square grids

Membrane forces in basic hypar unit

Some hypar characteristics

Examples 8.9 and 8.10

z = (f/ab)xy = kxy The equation defining the surface of a

regular hypar

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

Zarzuela Racecourse Grandstand,

Madrid, 1935, Eduardo Torroja,

Carlos Arniches Moltó, Martín

Domínguez Esteban Arch, Eduardo

Torroja Struct Eng: overhanging

hyperboloidal sectors

More umbrella hypars

by Felix Candela

Iglesia de

la Medalla

Milagrosa,

Mexico City,

1955, Felix

Candela

Iglesia de la Virgen Milagrosa, Mexico City, 1955, Felix Candela

Chapel Lomas de Cuernavaca,

Cuernavaca, Mexico, 1958, Felix Candela

Bacardí Rum Factory, Cuautitlán, Mexico,

1960, Felix Candela

Los Manantiales, Xochimilco ,

Mexico, 1958, Felix Candela

Alster-Schwimmhalle, Hamburg-

Sechslingspforte, 1967, Niessen und Störmer

Arch, Jörg Schlaich Struct. Eng

The Cathedral of St. Mary of the Assumption, San Francisco, California, USA, 1971, Pietro

Belluschi + Pier-Luigi Nervi Design

St. Mary’s Cathedral, Tokyo, Japan, 1963, Kenzo Tange, Yoshikatsu Tsuboi

Shanghai Urban Planning Center,

Shanghai, China, 2000, Ling

Benli Arch

Law Courts, Antwerp, Belgium, 2005,

Richard Rogers, Arup Struct. Eng

Bus shelter, Schweinfurt,

Germany

a. b.

c.. d.

Intersecting shells

Other surface structures

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

Teepott Seebad,

Warnemünde,

Rostock, Germany,

1968, Erich Kaufmann

Arch, Ulrich Müther

Struct. Eng

Lehman College Art Gallery,

Bronx, New York, 1960,

Marcel Breuer Arch

Philips Pavilion, World's Fair, Brussels (1958), Le Corbusier Arch

Membrane forces - elliptic paraboloid

http://www.columbia.edu/cu/gsapp/BT/BSI/SHELL_MS/img0018.jpg

Multihalle Mannheim, Mannheim, Germany,

1975, Frei Otto Arch

TWA Terminal, JFK Airport, New York, NY,

1962, Eero Saarinen Arch, Amman and

Whitney Struct. Eng

EXPO-Roof, Hannover, Germany, 2000,

Thomas Herzog Arch, Julius Natterer Struct. Eng,

Japan Pavilion, Hannover Expo 2000,

2000, Shigeru Ban Arch

Centre Pompidou-Metz, 2010, France,

Shigeru Ban Arch

Pompidou Museum II, Metz,

France, 2010, Shigeru Ban

Sydney Opera House,

Australia, 1972, Joern

Utzon/ Ove Arup

Museum of

Contemporary

Art (Kunsthaus),

Graz, Austria,

2003, Peter

Cook - Colin

Fournier Arch

Wünsdorf Church, Wünsdorf, Germany, 2014,

GRAFT Arch, Happold Struct. Eng

Beijing National

Stadium, 2008,

Herzog and De

Meuron Arch, Arup

Eng

BMW Welt Munich, 2007, Coop

Himmelblau Arch, Bollinger und

Grohmann Struct. Eng

Heydar Aliyev Centre, Bakı, Azerbaijan, 2012,

Zaha Hadid Architects, Tuncel Engineering,

AKT (Structure), Werner Sobek (Facade)

Busan Cinema Center, Busan,

South Korea, 2012, CenterCoop

Himmelblau Arch, Bollinger und

Grohmann Struct Eng

DZ Bank auditorium, Berlin, Germany ,2001,

Frank Gehry Arch, Schlaich Bergemann

Struct. Eng

Museo Soumaya, Mexico City, 2011, Fernando Romero Arch,

Ove Arup and Frank Gehry engineering

Railway station

Spandau, Berlin,

Germany, 1998,

Gerkan, Marg Arch,

Schlaich, Bergemann

Alvin and Marilyn

Lubetkin House, Mo-Jo

Lake, Texas, 1972, Ant

Farm (Richard Jost, Chip

Lord, Doug Michels)

Endless House, 1958, Frederick Kiesler Arch

MUDAM, Museum of Modern Art, Luxembourg, 2007

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

Tensile membrane roof structures

Georgia Dome, Atlanta, 1995, Weidlinger,

Structures such as the Hypar-Tensegrity

Dome, 234 m x 186 m

http://www.columbia.edu/cu/gsapp/BT/DOMES/GEORGIA/geo-24.jpg

Millenium Dome (365 m),

London, 1999, Rogers +

Happold

Tent architecture

Hybrid tensile surface structures

Point-supported tents Edge supports for cable nets

Examples 9.9 and 9.10

German Pavilion, Expo ’67, Montreal, Canada, Frei Paul Otto and Rolf

Gutbrod, Leonhardt + Andrä Struct. Eng.

http://faculty.ksu.edu.sa/71200/pic15/German Pavilion, Expo %2767.jpg

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

Structural study model for the Munich Olympic Stadium (1972),

Behnisch Architekten, with Frei Otto

Soap models by

Frei Otto

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

2010 London Festival of ArchitecturePrice &

Meyers Arch

Rosa Parks Transit Center, Detroid, 2009,

Parson Brinkerhoff Arch

TENSILE MEMBRANE STUCTURES

Pneumatic structures


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


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

•Poisson’s 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

Poisson’s 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.

http://www.technet-gmbh.com/







Double Curvature

Large radius

of curvature

results in

large forces.

PNEUMATIC STUCTURES


Air – inflated structures: air members


Classification

of pneumatic

structures

Pnematic structures

Low-profile, long-span pneumatic structures

Effect of internal pressure on

geometry

Soap bubbles

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.

Traveling exhibition

Example 9.12

Effect of wind loading on

spherical membrane shapes

Air-inflated

members and

Example 9.14


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 don’t 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.

p

T = pR T = pR

EXAMPLE: 12.10 Air-supported cylindrical membrane

US Pavilion, EXPO

70, Osaka, Davis-

Brody

US Pavilion, EXPO 70, Osaka, Davis-

Brody Arch, Geiger – Berger Struct.

Eng.

Pontiac Metropolitan

Stadium , Detroit, 1975,

O'Dell/Hewlett & Luckenbach

Arch, Geiger Berger Struct.

Eng.

Metrodome, Minneapolis, 1982, SOM Arch, Geiger-Berger Struct. Eng