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

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

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

Art Museum of Sao Paulo,
Sao Paulo, Brazil, 1968,
Lina Bo Bardi Arch
(prestressed concrete
beams)

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 MODEL
CONTINUOUS MODELS
LINE ELEMENT TYPICAL PLANAR ELEMENTS TYPICAL SOLID ELEMENTS
a. 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 Fy Fy
e.
ey Fy Fyb
d/2
1 2
Fy
Mp
d/2
Cp
Tp
3
d/2
Bending Stresses

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=0.9h=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

National Assembly, Dacca, Bangladesh, 1974, Louis Kahn

World War II bunker transformed into housing, Aachen, Germany

Dormitory of
Nanjing
University,
Zhang Lei Arch.,
Nanjing
University,
Research Center
of Architecture

Shear-wall or Cantilever-column

LATERAL DEFLECTION OF SHEAR WALLS

Shear Wall and Frame
Shear Wall Behavior Frame Behavior

LONG WALL CANTILEVER WALL
INTERMEDIATE WALL
10.5 k 9 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

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

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

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 FrameShear 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=0.4k/ft
4ft4ft4ft4ft4ft3ft4ft
27ft
7 SP@ 3 ft = 21 ft
w = 1k/ft, w = 0.6 k/ft
at roof and floorlevels
Problem: 12.3: Bearing wall with openings

LATERAL DEFLECTION OF WALLS WITH OPENINGS
PIER-SPANDEL SYSTEMS

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

Introduction to two-way slabs on rigid supports

Design of two-way slabs
on stiff beams

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 BEAMSa. WALL SUPPORT d. NO BEAMS
SLAB SUPPORT ALONG EDGES

a
d
b
e
c
f
EXAMPLE: 12.5: Square concrete slabs

#4 @ 12"
#3 @ 9"
15 ft
12 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"15"
18"x18"
EXAMPLE 4.10: Design of one-way slab

Example of slab steel reinforcement layout

Example of steel
reinforcement layout

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

Saint John's Abbey,
Collegeville,
Minnesota, 1961,
Marcel Breuer Arch

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

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

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

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

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

Basic concepts related to barrel shells

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

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.

Social Center of the Federal Mail, Stuttgart, 1989, Roland Ostertag Arch,
Schlaich Bergermann Struct. Eng

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

From the joist slab to shell beam

Behavior of short
barrel shells
Long vs short
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

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

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

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

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.

Membrane forces in basic hypar unit

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

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

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 (Façade)

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

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

Hybrid tensile surface structures

Point-supported tents Edge supports for cable nets

German Pavilion, Expo ’67, Montreal, Canada, Frei Paul Otto and Rolf
Gutbrod, Leonhardt + Andrä Struct. Eng.

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
T2
T2
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.

Double Curvature
Large radius
of curvature
results in
large forces.

PNEUMATIC STUCTURES
 Air-supported structures
 Air – inflated structures: air members
 Hybrid air structures

Classification
of pneumatic
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.

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

Surface Structures, including SAP2000

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