AXIAL STRUCTURE SYSTEMS including TRUSSES using 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, (1) The Design of Building Structures (Vol.1, Vol. 2), rev. ed., PDF eBook by Wolfgang Schueller, 2016, published originally by Prentice Hall, 1996, (2) 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
Structure Systems & Structure Behavior INTRODUCTION TO STRUCTURAL CONCEPTS SKELETON STRUCTURES • • • • •
Axial Systems Beams Frames Arches Cable-supported Structures
SURFACE STRUCTURES • • • • •
Membranes: beams, walls Plates: slabs Hard shells Soft shells: tensile membranes Hybrid tensile surface systems: tensegrity
SPACE FRAMES LATERAL STABILITY OF STRUCTURES
L I NE E L E M E NT S
AXIAL STRUCTURE SYSTEMS
TENSILE MEMBERS
COMPRESSIVE MEMBERS
BEAMS
FLEXURAL STRUCTURE SYSTEMS
FLEXURAL-AXIAL STRUCTURE SYSTEMS
BEAM-COLUMN MEMBERS FRAMES
S UR F A CE E L E M E NT S
TENSILE MEMBRANES SOFT SHELLS MEMBRANE FORCES
PLATES SHELLS
SLABS, MEMBRANE BENDING and TWISTING
RIGID SHELLS
SKELETON STRUCTURES PLANAR STRUCTURES Axial force systems
• TRUSSES • STAYED STRUCTURES
Flexural force systems • BEAMS
Flexural-axial force systems • FRAMES
• ARCHES
SPATIAL STRUCTURES • SPACE FRAMES
• CABLE STRUCTURES • COMPOSITE STRUCTURES
Examples of AXIAL
STRUCTURE SYSTEMS include, for instance,
• trusses
• compression-tension roof enclosure systems
• lateral bracing of frames • suspended glass walls • battered piles
• polyhedral domes • space frame structures • cable-supported structures: e.g. beams, roofs
• air-supported structures, air members • etc.
AXIAL STRUCTURE SYSTEMS primarily in compression and tension The following structure types are investigated: •
Columns
• Beam Columns • Cables
• Trusses • etc.
Simple compression-tension enclosure systems
The building response to load intensity
Some roof support structures
Examples of horizontalspan roof structure systems
Balance of forces
COLUMNS COMPRESSION/ TENSION MEMBERS
BEAM – COLUMN Intersections
Paper Mill, Mantua, Italy, 1979, Pier Luigi Nervi Struct. Eng
UNESCO Headquarters, Paris, France, 1958, Marcel Breuer, Bernard Zehrfuss, Pier Luigi Nervi
FM Constructive system, Elmag plant, Lissone, Milano, 1964, Angelo Mangiarotti Arch
Theatre Erfurt, Erfurt, Germany, 2003, Joerg Friedrich Arch
Theater Erfurt (2003), Joerg Friedrich Arch, foyer
Bundeskanzleramt, Berlin, 2001, Axel Schultes Arch
Marie-Elisabeth-Lüders-Haus, Berlin, 2003, Stephan Braunfels Arch
Paul-Löbe-Haus, Berlin, 2002, Stephan Braunfels Arch
New Beijing Planetarium, 2001, AmphibianArc – Nanchi Wang
School of Architecture, Bond University, Brisbane, Australia, 2014, Peter Cook (CRAB) Arch
Paper Concert Hall, L’ Aquila, Italy, 2011, Shigeru Ban Arch
TU Munich, Munich, Germany
Museum of Art, Bonn, Germany,1993, Axel Schultes Arch
Crematorium Baumschulenweg, Berlin, Germany, 1998, Axel Schultes Arch, GSE Saar Enseleit Struct Eng
Pinakotek der Moderne, Munich, Germany, 2002, Stephan Braunfels Arch
Canadian Embassy, Washington, 1989, Arthur Erickson Arch
Xinghai Square shopping mall, Dalian, China
Guangzhou Baiyun International Airport, 2004, Parsons
Atrium, Germanisches Museum, Nuremberg, Germany, 1993, me di um Arch
Interchange Terminal Hoenheim-Nord, Strassbourg, 2002, Zaha Hadid
Undulating viewing platform, Ludian, Yunnan, China, 2014, John Lin +Olivier Ottevaere Architects
Wanli University, Ningbo, China
Centre Céramique, Maastricht, The Netherland, 1999, Jo Coenen Arch
Haans Office, Tilburg, The Netherlands, 1993, Jo Coenen Arch
Modern Wing at the Art Institute of Chicago, Chicago, Illinois, 2009, Renzo Piano Arch
The Luxembourg Philharmonie, Luxemboug, 2007, Portzamparc Arch
The Aluminum Forest ,Utrecht, Netherlands, 2001, M.de Haas Arch
Kanagawa Institute of Technology Workshop , Kanagawa, Japan, 2007, Junya Ishigami + Associate
Leonardo Glass Cube, Bad Driburg 2007, 3deluxe Arch
The Netherlands Architectural Institute, Rotterdam, 1993, Jo Coenen Arch
Kunstmuseum Wolfsburg, Wolfsburg, Germany, 1994, Peter P.Schweger Arch
Indianapolis International Airport, Aerodesign Group Arch, 2008
Sendai Mediatheque, 2-1, Kasugamachi, Aobaku, Sendai-shi 980-0821, Japan, 2001, Toyo Ito Arch, Mutsuro Sasaki Struct Eng
Park Güell, Santa Coloma de Cervello, Spain, 1908, Antoni Gaudi Arch
column shapes
Members in compression have the potential to buckle: to suddenly lose the ability to carry load my moving laterally with respect to the load. In some cases, a member may buckle about the strong axis direction (based on a strong axis bending shape), or a weak axis direction, as shown below. The allowable stress for each is calculated using appropriate k, L, and r properties for the direction.
Column formula: material failure (short column), inelastic buckling (intermediate column), elastic buckling (long column)
The load which causes a member to buckle elastically depends on the following member properties: The unbraced length: L The cross section size and "spread-outness": I The material stiffness: E The end conditions are also important, since they change the effective length of the member. This is accounted for by an "effective length factor", denoted by k.
The following table shows effective length factors:
Effective Length Factor, K • • • •
0.5
To account for “Axial-Flexural Buckling” Indicates the “total bent” length of column between inflection points Can vary from 0.5 to Infinity Most common range 0.75 to 2.0
1.0
0.5 - 1.0
2.0
1.0 -
K = 0.5
K = 0.7
K=1
K=1
K=2
K=2
(a)
(b)
(c)
(d)
(e)
(f)
SAP2000 examples of basic elastic buckling modes
(1) Fig. 3.21 & (2) Fig. 4.5, Effective Length Factors
Sagrada Familia, Barcelona, 1882 -2028(?), Antonio Gaudi
Elephant Temple, Bombay
Expo Dach, Hannover EXPO 2000, Thomas Herzog, Julius Natterer
Turmrestaurant Bierpinsel, BerlinSteglitz, 1976, Ralf Schüler + Ursulina Schüler-Witte Arch
Parkland Mall, Dalian, China, 1999
San Francisco Federal Building, 2007, Thom Mayne of Morphosis
Tokyo International Forum,,1997, Rafael Vinoly Arch, Kunio Watanabe Struct. Eng
Altmarkt Galery, Dresden, Germany, 2002, Manfred Schomers + Rainer Schürmann Arch
Science and Technology Museum Shanghai, 2002, RTKL/Arup
Guangzhou Baiyun Internationl Airport, 2004, Parsons (Asia Pacific) + URS Greiner Arch, T.Y. Lin Struct. Eng.
Ningbo Air Terminal
Civic Center, Shenzhen , 2004, Li Mingyi Arch
Science Museum Principe Felipe, Valencia, Spain, 2001, Santiago CalatravaArch
BEAM COLUMNS INLINED COLUMNS, FRAMES
Typical beamcolumn loading
Beam-column interaction
What is Slenderness Effect Moment Amplification
e
P
P
Capacity Reduction
I e
II C
P
D = f(Mc)
C
M
I. Mc = P.e Short Column
II : Mc = P(e + D) Long Column
Column Capacity (P-M)
e = Mu/Pu Pu
Pu
Mu
Region 1: max axial compression
Po
e Pu фPn e = Mu/Pu Mn/Pn a.
Pu Pu
min
Region 2: compression controls e
Pn, Axial compression
Pn max = 0.80Po (tied)
Balanced strain condition
Pb eb
Region 3: tension controls
Mo Mb Mn, bending moment
b.
Fig. 4.11 Concrete Column Behavior Under Combined Bending and Axial Loads
Looped Hybrid Housing, Beijing, 2008, Steven Holl Arch
Green roofs, Vancouver Civic Centre West, LMN + DAQ/MCM, 2009
Vancouver Civic Centre West, LMN + DAQ/MCM, 2009
Beijing
Downtown Ningbo, China, 2002, Qingyun Ma Arch
Dresdner Bank, Verwaltungszentrum, Leipzig, 1997, Engel und Zimmermann Arch
CABLES TENSION MEMBERS
Golden Gate Bridge, San Francisco, 1937, Joseph Strauss and Irwing Morrow
Bollman Truss Railroad Bridge, Savage, Maryland, 1869, Wendel Bollman Eng.
Sunniberg Bridge, between Klosters and Serneus, Switzerland, 2005, Christian Menn Designer
Paper Mill, Mantua, Italy, 1979, Nervi Struct..Eng.
Sundial Bridge, Redding, CA, 2006, Santiago Calatrava
Experiments with structure, Iakov Chernikhov Arch, Russian Constructivism, 1925-1932
NY Times, New York, 2004, Renzo Piano Arch; X-bracing of high-rise building
Building stability and slope stability anchors Hopewell Center, Wanchai, Hong Kong, Gordon Wu & Assoc., Ove Arup Eng.
Berlin Stock Exchange, Berlin, Germany, 1999, Nick Grimshaw
House (World War 2 bunker), Aachen, Germany
Godzilla House, Seoul, South Korea, Chae-Pereiera Architect
Cable-supported structures
Yountville Town Center, Yountville, CA, 2011, Siegel+Strain-Arch, Coastland Engineering
Structures primarily in tension
Funicular tension lines, or pressure lines in the inverse
Single-strut and multi-strut cablesupported beams
Kempinski Hotel, Munich, Germany, 1997, H. Jahn/Schlaich:
Auditorium Paganini, Parma, Italy, 2001, Renzo Piano Arch
Landeshauptstadt München, Baureferat, Georg-Brauchle-Ring, Munich, Germany, Christoph Ackerman
Saibu Gas Museum for natural Phenomen-art, Fukuoka, 1989, Shoei Yoh + Architects
Hongkong Bank, Honkong, 1985, Foster + Arup
Pompidou Center, Paris, 1977, Piano and Rogers
Centre George Pompidou, Paris, 1977, Piano & Rogers
Fondation Avicienne (Maison de l'Iran), Cité Internationale Universitaire, Paris, 1969, Claude Parent + Moshen Foroughi et Heydar Ghiai Arch
Fondation Avicienne (Maison de l'Iran), Cité Internationale Universitaire, Paris, 1969, Claude Parent + Moshen Foroughi et Heydar Ghiai Arch
Petersbogen shopping center, Leipzig, 2001, HPP HentrichPetschnigg Arch
Tower Bridge House, London, 2006, Richard Rogers Arch
Airport Munich Hangar 1 (153 m), Munich, 1992, Günter Büschl Arch, Fred Angerer Struct. Eng
Patcenter, Princeton, USA, 1984, Richard Rogers Arch, Ove Arup Struct Eng
Shanghai-Pudong International Airport, 2001, Paul Andreu principal architect, Coyne et Bellier structural engineers
Milleneum Bridge, London, 2000, Foster/Arup
The Munich Airport Business Center, Munich, Germany, 1997, Helmut Jahn Arch
Sony Center, Potzdamer Platz, Berlin, 2000, Helmut Jahn Arch., Ove Arup Struct. Eng
Munich Olympic Stadium, 1972, Günther Behnisch, Frei Otto
Cable-supported structures
Space needle, Hirshhorn, Washington, 1968, Kenneth Snelson
SPHRERICAL ASSEMBLY OF TENSEGRITY TRIPODS
TRUSSES COMPRESSION-TENSION MEMBERS
Palladio's Trusses, 1580
TRUSSES are typical examples of axial structure systems . Because of their simplicity of behavior they provide an ideal introduction to computer analysis. Trusses are composed of frame elements ,which are modeled as straight lines connecting two joints I and J which are also called nodes. It is assumed that the members in trusses are pinconnected and subject only to joint loads, hence only axial internal member forces are generated in the truss. For determinate structures the effect of material and member sizes is in the first introduction to structural software disregarded (i.e. use using either elements with zero moments of inertia or using default setting), since member stiffness has no effect on the magnitude of internal member forces, however deflection results can not be used.
Application of trusses ranges from the small scale of a joist to the large scale of a deep truss supporting a stadium roof. They are used as roof and bridge structures and as wind bents that is vertical cantilevers, which brace high-rise skeleton structures. Trusses may replace any solid element such as beams, columns, arches, or frames. From a structure system point of view, they may be classified as, Truss cantilevers, truss beams of various profiles (flat, tapered, pitched, curved, crescent, etc.) Truss arches Truss frames forming single or multi-bay structures The typical truss profiles for roofs are flat, pitched (e.g. triangular, trapezoidal, hip, gambrel) or curved. Trusses may be organized according to the arrangement of members and according to behavioral considerations as:
Trussed structures primarily in compression and tension
Examples of trussed structures
Fig. 7.24 Trussed Frames (The Vertical Building Structure by Wolfgang Schueller)
Warehouse Magazzini Generali, Chiasso, 1925, Robert Maillart Arch and Eng
Project, Nationaltheater, Mannheim, 1953, Ludwig Mies van der Rohe Arch
Abgeordneten Bridge, Berlin, Stephan Braunfels Arch, Setzpfandt Struct Eng; Marie-Elisabeth-Lüders-Haus
Red-Bull-Arena, Leipzig, Germany, 2006, Wirth+Wirth, Glöckner Architekten, Körber, Barton, Fahle
Allianz Arena, Munich, Germany, 2005, Herzog & de Meuron Arch, ARUP Struct Eng for roof
Roof structure of Olympia Stadium, Berlin, 2012, von Gerkan, Marg Arch, Schlaich Bergermann Struct Eng
Arena “Auf Schalke”, Gelsenkirchen, Germany, 2001, Hentrich, Petschnigg Arch, Klemens Pelle, Struct Eng
Quingpu Pedestrian Bridge, Shanghai, 2008, Pedro Pablo Arroyo Alba Arch, Bridge Structures Dept. of Tongji University
A380 Lufthansa Maintenance Hangar, Frankfurt/Main Airport , 2007, van Gerkan Marg Arch, Schlaich Bergermann Struct. Eng.
Internationales Congress Centrum Berlin,1979, Ralf Schüler + Ursulina Schüler-Witte Arch,
George Washington Bridge Bus Station , New York, 1963, Pier Luigi Nervi
new Trade Fair Center, parking garage across A8, Stuttgart, 2007, Wulf & Partners
Trade Fair Centre, parking garage, Stuttgart, 2007, Wulf & Partners
Gund Hall, Harvard U., Cambridge, 1972, John AndrewsArch, William LeMessurier Struct Eng
Picture Window House, Shizuoka, Japan, 2002, Shigeru Ban Arch
Daimler Chrysler, Potsdamer Platz, Berlin, 2002, Richard Rogers Arch
Potsdamer Platz Office and Commercial Building, Berlin, Germany , 2000, Helmut Jahn Arch
Schulhaus Leutschenbach , Zürich, 2009, Christian Kerez Arch
Medellin Sports Coliseum, Medellin, Colombia, 2010, Giancarlo Mazzanti + Felipe Mesa Arch, Nicolás Parra+Daniel Lozano Struct Eng
Hancock Tower, Chicago, 1970, Bruce Graham + Fazlur Khan / SOM
Burj-Al-Arab Hotel, Dubai, United Arab Emirates, 1998, Tom Wright Arch (W.S. Atkins), W.S.Atkins Struct Eng
Citic Bank HQ, Hangzou,China, 2015, by Norman Foster+Partners, East China Architectural Design Institute Struct. Eng
8 Chifley, Syney, Australia, 2013, Rogers Stirk Harbour + Partners, Arup Eng
San Francisco International Airport, International Terminal, 2001, SOM
Bush Lane House, London, 1976, Arup
The Leadenhall Building, London, 2010, Rogers Stirk Harbour + Partners, Arup
Bank of China, Hong Kong, 1990, I.M. Pei Arch, Leslie E. Robertson Struct. Eng.
Core bracing example
Highrise Beijing, 2006, bracing example
Fiduciary Trust Building, Boston, 1977, The Architect’s Collaborative
Sony Tower (AT&T Building), New York, 1984, Johnson/Burgee Arch, Leslie Robertson Struct Eng
Onterie Center, Chicago, Ill, 1986, SOM Arch+Struct Eng
Citycorp Center, New York, 1977, Stubbins Arch, William LeMessurier Struct Eng
Eccentrically braced frame
Hongkong Bank, Hong Kong, 1985, Foster/Arup
staggered truss system
Godfrey Hotel, Chicago, Ill, 2014, Valerio Dewalt Train Associates Arch, Structural Affiliates International
Centre Georges Pompidou, Paris, 1977, R. Piano + R. Rogers Arch, P. Rice (Arup) Struct. Eng
TKFC Yum! Center, Louisville Ky, 2010, Populous Arch, , Walter P Moore Struct. Eng.
Merzedes-Benz Zentrale, Berlin, 2000, Lamm, Weber, Donath und Partner
Holocaust Memorial Museum, Washington, 1993, James Ingo Freed
Säynätsalo Town Hall, Finland, 1993, Alvar Aalto Arch
NOVARTIS CAMPUS, FABRIKSTRASSE 14 , Basel, Switzerland, 2009, JOSE RAFAEL MONEO Arch, Werner Sobek Struct. Eng
Veteran's Memorial Coliseum, New Haven Connecticut, 1972, Kevin Roche Arch
De Brug Unilever, Rotterdam, 2007, JHK Arch
Centra at Metropark, Iselin, NJ, 2011, Kohn Pedersen Fox Arch, DeSimone Struct Eng
Pennsylvania State, Millennium Science Complex, University Park, PA, 2010, Rafael Viñoly Arch, Thornton Tomasetti Struct Eng
POPS, Arcadia, Oklahoma, 2008, Elliott + Associates Architects
Library Gainesville, FL
Bordercrossing Aachen-Lichtenbusch, Belgium, 2006
British Airways hangar, Munich
BOK Center, Tulsa, Oklahoma, 2008, Cesar Pelli Arch, Thornton Tomasetti Struct. Eng
TU Stuttgart, Germany
Valeo - Auto - Electric Motoren GmbH, Bietigheim – Bissingen, Germany, Germany, Christoph Ackermann Arch
Museum Marta, Herford, 2005, Frank Gehry
Petersbogen shopping center, Leipzig, 2001, HPP Hentrich-Petschnigg
Chongqing Jiangbei International Airport Terminal, 2005, Llewelyn Davies Yeang Arch, Arup Struct Eng
Serpentine Joists: Lindhout Architects Headquarters, Brighton, MI, 2008
Skijump tower, Oberhof, Thueringen, Germany
INTRODUCTION TO TRUSS ANALYSIS
Basic triangular and rectangular trusses
Simple trusses are formed by the addition of triangular member units and can be further subdivided into regular, irregular (e.g. fan trusses), and subdivided truss systems often used for bridge trusses to give adequate support to the deck (Fig. 6.2, 6.3). Compound trusses are formed by addition of simple trusses. These trusses are not necessarily composed of triangles, indicating that stable trusses can be generated by figures other than triangles (Fig. 6.6)
Complex trusses are neither simple nor compound. Special methods of analysis must be applied to these trusses even if they are statically determinate, since more than three members are attached to each joint, and sections cut through at least four members result in more unknowns than available equations at the location to be investigated (Fig. 6.7).
A.
B.
C.
D.
E.
F.
G.
G.
BASIC TRIANGULAR TRUSSES
Common simple truss types are identified in Fig. 6.3. It is demonstrated in the drawings that the arrangement of the members determines which of the members are in tension and in compression under uniform gravity loading along the top chord.. Pratt truss (a.) with diagonals in tension and the verticals in compression, Howe truss (b.) using compression diagonals and vertical tension members, Modified Warren truss (c., d.) in which the diagonals are alternately in tension and compression, Lattice or double Warren truss (e.) can be visualized as two superimposed single Warren trusses, K-truss (f.) in which the top half behaves similar to a Howe truss, while the bottom half behaves like a Pratt truss. X-truss (g.) often designed as hinged frame truss cross-braced with tensile rods, where the bars are not connected at their point of intersection. Since the diagonals cannot carry compression, the truss reduces to a Pratt truss under the given loading. Irregular truss (h.): for the shown truss the diagonal bars cross each other without being connected at their point of intersection. The truss reduces to a Warren truss under the given loading. Other examples of irregular simple trusses include fan trusses. Composite, truss-like structures where struts resist compression and ties tension (for discussion see Fig. 11.2)
a.
b.
c.
d.
e.
f.
g.
g.
SIMPLE RECTANGULAR TRUSSES
From a point of view of optimum weight, the truss configuration should reflect the funicular shape due to the critical external loading so that the chords carry all the loads and the web members are zero and are primarily used for the secondary asymmetrical loading case and for lateral bracing of the compression chord. One may also want to consider the constant-force design of trusses, where the force flow along the top and bottom chords is constant. Several examples of efficient truss forms as related to gravity loading, are shown in Fig. 6.4. A truss should be curvilinear in response to uniform load action (e.) with a funicular top chord arch and constant-stress bottom chord; for a fish belly truss the situation is opposite. However, in the ideal form, the vertical web members should be arranged in a radial fashion so that that their extensions intersect at a concurrent point, the center of the circle forming the top (or bottom) arch. For this situation, the forces are constant in the arched top chord and almost constant in the radial web members and the bottom chord (e.). A truss should be lens-shaped, if the loads are shared by the arched top and bottom chords in compression and tension respectively (f.). A truss should be pitched for point loading. In other words, the truss should be triangular with respect to a single load (a.), trapezoidal for two loads (d.), and of gambrel profile for three single loads (c.). A triangular truss should have a funicular bottom chord, if the web columns transfer loads from the top chord to the tensile bottom chord (b.).
a.
b.
c.
d.
e.
f.
FUNICULAR TRUSSES
a.
b.
c.
d.
d.
f. FAN TRUSSES
Compound Trusses When several simple trusses are connected to each other, they are called compound trusses (Fig. 5.8). Trusses may be connected by: by three nonparallel bars whose axes cannot cross a common point, by a single member and common joint, by replacing members of the main truss with secondary trusses.
Compound trusses can be analyzed by using a combination of the methods of sections and joints. For the third truss type, however, remove the secondary truss members and replace them with fictitious members to form the main truss. First figure the reactions and force flow of the secondary trusses, and then apply the reactions as external loads to the main truss.
a.
b.
d.
c.
g. e.
f.
COMPOUND TRUSSES
Complex Trusses Trusses, which cannot be classified as simple or compound trusses are called complex trusses. Complex trusses may have any member configuration and any number of support conditions as long as Eq. 5.1 is satisfied for statically determinate trusses and they are stable. To check the computer solution of complex, determinate trusses manually may not be simple, since joints generally have more than three unknowns. In other words, using the method of joints, the equilibrium equations for several joints must be set up, and then the equations must be solved simultaneously. Another method of analysis is to reduce the complex truss to a stable simple truss by removing a member and substituting it somewhere else to form a simple truss, called the method of substitute members. For fast approximation purposes of parallel chord trusses, however, use the beam analogy by assuming that the moments are carried by the flanges and the shear by the web members.
a.
b.
c.
d.
e.
f.
COMPLEX TRUSSES
The distribution of force flow depends not just on the arrangement of the web members but also on the truss shape. Comparing the basic truss profiles of flat, pitched, and curved trusses under uniform gravity loading, one may conclude: The curvilinear profile can be considered optimal since it is nearly funicular for the given loading. The chords of the flat truss are only used efficiently in the midspan range thereby suggesting the benefit of the shape for larger spans where bending controls.
For the triangular truss the chords are only efficiently used at the supports where the shear is maximum, indicating the advantage of the shape for shorter spans where shear beam action controls. The trapezoidal truss falls between the parallel and triangular trusses.
Response of ordinary trusses to uniform gravity loading
Concurrent force system using METHOD OF JOINTS for analysis
The cabin of Mac Dunstan and Linda Grob, in a hillside near Seattle, 2008
Investigate a basic 15-ft (4,57-m) high, 30-ft (9,14-m) span, triangular, hinged truss unit using SAP2000. Apply vertical, horizontal, or combined single loads of 1-k = 4,45kN at the joints as indicated on the drawing; assign zero to self-weight. Disregard the effect of material and member sizes (i.e. use default setting), since member stiffness in determinate structures has no effect on the magnitude of internal member forces, however do not use deflection results. Study the load flow effect due to change of: GEOMETRY: profile, crown location, roof slope, inclination of bottom chord, etc. LOAD ARRANGEMENT: load location, load direction SUPPORT LOCATION and ORIENTATION (i.e. rotate supports) Start with the basic symmetrical regular truss, and then reshape the unit and run the case analysis, and so on. Use a 5x5-ft (1,524x1,524-m) grid to construct the layout of the truss. Show the axial force flow with numerical values, and show the reaction forces. Study the relationship of member tension and compression so you can develop a feeling for the structure and predict the direction of the force flow. Check manually (graphically or analytically) the computer results of member forces and reactions for at least half the cases.
a. c.
b. d.
5' 5'
Structural software, West Point Bridge Designer, version 4.1.1, which was developed by Colonel Professor Steve Ressler at the U.S. Military Academy, West Point, NY. (bridgecontest.usma.edu/index.htm).
a. b. c. d.
4' 4'
a. b. c. d.
4' 4'
COMPUTATION OF AXIAL MEMBER FORCES Generally, two methods can be used: Method of Joints and Method of Sections The internal member forces at mid-span are only checked. Therefore, a segment of the truss is investigated by cutting an imaginary section through the truss, (i.e. method of section) and looking only to the left free-body. The unknown internal forces in the top, bottom and diagonal members are designated as Nt and Nb with the assumed direction as shown in the previous free-body that is acting in compression towards the members, and Nd acting in tension away from the members. Vertical equilibrium of forces gives the magnitude of the normal force Nd in the diagonal member, ΣV = 0 = 2.5 - 2(1) - (Nd /)1, or Nd = 0.707 k (C) Rotational equilibrium about the top joint D at mid-span yields the magnitude of the bottom chord force Nb. ΣMD = 0 = 2.5(16) -1(16) - 1(8) - Nb(8), or Nb = 2.0 k (T) Rotational equilibrium about the bottom joint J yields the magnitude of the top chord force Nt. ΣMJ = 0 = 2.5(8) -1(8) - Nt(8), or Nb = 1.5 k (C) Check: ΣH = 0 = 2.0 - 1.5 - (0.707/)1, OK The magnitude of the other members in the free-body can be obtained now by the method of joints.
1k
1k 2.0 k
c
70 0.
0.5 k
7 k 1.5 k i 2.5 k
METHOD OF SECTIONS for analysis
Dr.Frame program, the structural programs of Dr. Software, www.drsoftware-home.com
William J. Clinton Presidential Center , Little Rock, Ark, 2004, James Polshek
Seoul National University Museum, 2006, Rem Koolhaas
Cardinals Stadium, Glendale, Ariz, 2006, Peter Eisenman Arch, Walter P Moore+Buro Happold (roof) Struct. Eng
Rafael del Pino Auditorium, Madrid, Spain, 2008, Rafael de La-Hoz Arch
SPACE FRAME STRUCTURES
Three-dimensional structures may be organized as follows: • Spatial frameworks, such as derricks, building cores, towers, guyed structures • Linear space truss members, such as beams and arches • Single-layer, three-dimensional frameworks, which are folded or bent latticed surface structures, such as folded plate planar trusses, polyhedral dome-like structures and other synclastic and anticlastic surface structures. They obtain their strength through spatial geometry, that is, their profile.
• Multi-layer, three-dimensional, space frames, which are generated by adding polyhedral units to form three-dimensional building blocks. In contrast to single-layer systems, the multi-layer structure has bending stiffness and does not need to be curved; familiar examples are the flat, double-layer space frame roofs and the subtensioned floor/roof structure
trestle table by Carlo Mollino (1948)
Ningbo Airterminal
San Siro Stadium, Milan, Italy, (1925 originally) 1990 renovation, Giancarlo Ragazzi, Henry Hoffer, Leo Finzi E
National Motor Museum, Beaulieu, Hampshire, England, UK, 1972, Leonard Man asseh + Ian Baker Arch
Post Tower, Bonn, Germany, 2003, Murphy/Jahn Arch, Werner Sobek Struct Eng
Petersbogen shopping center, Leipzig, 2001, HPP HentrichPetschnigg
Beijing Capital International Airport Terminal 2, 1999
San Francisco International Airport, International Terminal, 2001, SOM
Leicester University, Engineering Building, 1959, Stirling and Gowan Arch
Polyhedral Roof Structures
Single-layer space frame roofs
Force equilibrium in space
Atrium, Germanisches Museum, Nuremberg, Germany,1993, me di um Arch
Common polyhedra derived from cube
Support conditions for three-dimensional structures
The basic three-dimensional structure
1k e 15' z y
a
c
15' 1k
y x
15' d
b
20'
15'
Pyramidal Roof Structure
Spatial Tree Unit: Axial Force Diagram
Problem 1, Braced building core
Flat space-frame roofs
Generation of space grids by overlapping planar networks
Platonic solids
Double-layer space frame types
Common space frame joints
Slab analogy and slab support
Structural behavior of double-layer space frames
Space Grid Truss – Axial Force Flow
Two-Way Space Truss Beam Grid
US Air Force Hangar Projekt, 1951, Konrad Wachsmann Arch
Stuttgart Airport, Terminal 1, Germany, 1991, von Gerkan Marg Arch
Trees
concept of tree geometry Tree geometry
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Example of Space Frame Structures: tree construction in SAP2000
tree construction
National Air and Space Museum, Washington, 1976, Gyo Obata of Hellmuth, Obata and Kassabaum
Us Airline Terminal, O’Hare International Airport, Chicago, 1987, Helmut Jahn
B&B Italia Offices, Novedrate, Italy, 1973, Studio Piano & Rogers
New Trade Fair Parc Luxembourg and Train Station, Luxembourg-Kirchberg, 2010, Pohl Arch, Knippers Helbig Struct Eng
Ice and Swim Stadium Lentpark, Cologne, Germany, 2012, Schulitz Arch, Arup Struct Eng
Eco House Prototype, Malibu , CA, 2009, Peter Jon Pearce
Eco House Prototype, Malibu , CA, 2009, Peter Jon Pearce Arch
Olympic Stadium, Mexico City, 1968, Felix Candela Arch
Geodesic dome
Eden Project, Cornwall, UK, 2001, Nicholas Grimshaw Architects, Anthony Hunt Struct Eng
Examples of terraced housing
Ramot Polin housing, Jerusalem, Israel, 1975, Zvi Hecker
Habitat '67, Montreal, 1967, Moshe Safdie Arch, August E. Komendant Struct Eng
Danciger Building, Mechanical Engineering Building, Technion, Haifa, Israel, 1966, Zvi Hecker + Alfred Neumann Arch
Beijing National Stadium roof (Bird’s Nest), 2008, Herzog and De Meuron Arch, Arup Struct. Eng.
The 313-m (1027-ft) span Beijing National Stadium roof (Herzog and De Meuron Arch, and Arup Eng,) is saddle-shaped with an elliptical building footprint. The concept of the dome form is a “bird’s nest”, where the interwoven mesh of tubular steel members appear to be arranged in a chaotic, random manner. This irregular, grid-like, threedimensional space frame structure, however, only looks like random, in reality it consists of a primary members based on 24 column points at ground level spaced at regular intervals around the elliptical footprint. Truss columns of roughly pyramidal shape supporting a regular series of interwoven trusses that span tangentially to the central roof opening across the stadium. Secondary, diagonal members are placed along the perimeter for the staircases and are arched across the roof down to the other side. Finally, a tertiary group of infill members is added as required by the aesthetics of the façade. The façade is in-filled with translucent ETFE panels or openings are left, to allow natural ventilation through the public concourse, into the stadium, and through the central opening in the roof structure.
Beijing National Swimming Center (Water Cube), 2008, Arup Arch
The 177x 177 x 31 m (581 x 581 x 102 ft) Beijing National Swimming Center (PTW Arch and Arup Eng., 2008) known as the “Water Cube” celebrates the transparency and dynamics of water bubbles. It has column-free spans up to 121 m (396 ft) in either direction. The development of the subdivision of space is derived from the arrangement of organic cells, mineral crystals and the natural formation of soap bubbles. The enclosing structure consists of a polyhedron space frame, but the arrangement of the cells is irregular and appears random and organic although it is mathematically rigorous and repetitious; because when an all space-filling regular polyhedral system is cut at an arbitrary angle it appears to be random. The space frame is made up of slender steel pipes and joints; it is clad with translucent plastic foil air pillows that look like bubbles. The skin’s material is ethylene tetrafluoroethylene (ETFE) a species of Teflon, which is designed to react to changing light conditions thereby creating amazing visual effects. The building skin has excellent insulation properties and creates a greenhouse effect capturing the energy from the sun for heating and lighting.
Inflatable Yorkshire Diamond Pavilion, Oslo Norway, 2009, Various Architekts AS, Tom Gam Struct Eng
Culver City, LA, 1990, Eric Owen Moss Arch
