FLEXURAL STRIUCTURE SYSTEMS BEAMS 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, (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 StructureSystems 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
FLEXURAL STRUCTURE SYSTEMS BEAMS There are infinitely many types of beams. They may be hidden or exposed; they may form rigid solid members, truss beams, or flexible cable beams. They may be part of a repetitive framing grid (e.g., parallel or two-way joist systems) or represent individual members. They may support ordinary floor and roof structures or span a stadium; they may form a stair, a bridge, or bridge-type buildings that span space; they distinguish themselves in material, construction, and shape. Beams may be not only common beams, but may be spatial members, such as folded plate and shell beams (e.g., corrugated sections), or space trusses. The longitudinal profile of beams may be shaped in funicular form in response to a particular force action, which is usually gravity loading; that is, the beam shape matches the shape of the moment diagram to achieve constant maximum stresses.
BEAMS may not only be the common, • planar beams
• spatial beams (e.g. folded plate, shell beams , corrugated sections • space trusses. They may be not only the typical rigid beams but may be flexible beams such as • cable beams.
The longitudinal profile of beams may be shaped as a funicular form in response to a particular force action, which is usually gravity loading; that is, the beam shape matches the shape of the moment diagram to achieve constant maximum stresses.
Beams may be part of a repetitive grid (e.g. parallel or two-way joist system) or may represent individual members; they may support ordinary floor and roof structures or span a stadium; they may form a stair, a bridge, or an entire building. In other words, there is no limit to the application of the beam principle.
The following slides represent:
1. Case studies as described above
presented in a
casual fashion
2. Basic beam mechanics including SAP2000 examples
The Parthenon, Acropolis, Athens, 448 B.C., Ictinus and Callicrates
Shanghai-Pudong International Airport, Paul Andreu principal architect, Coyne et Bellier structural engineers
Berlin
Breuer chair, 1928
Wassily chair, 1925, Marcel Breuer
Barcelona chair, 1929, Mies van der Rohe
Calder mobile, Hirschorn Museum, Washington, 1935
tizio table lamp, Richard Sapper, 1972
stationary tower cranes vs. mobile cranes
SIMPLE and CONTINUOUS FLOOR BEAMS
Atrium, Germanisches Museum, Nuremberg, Germany, 1993, me di um Architects
Incheon International Airport, Seoul, S. Korea, 2001, Fentress Bradburn Arch.
Renzo Piano Building Workshop, Genoa, Italy, 1991, Renzo Piano Arch
HDI-Gerling HQ, Hanover, Germany, 2010, Ingenhoven Arch, Werner Sobek Struct Eng
Petersbogen shopping center, Leipzig, 2001, HPP Hentrich-Petschnigg
Petersbogen shopping center, Leipzig, 2001, HPP Hentrich-Petschnigg
Petersbogen shopping center, Leipzig, 2001, HPP HentrichPetschnigg
TU Munich, Germany
Auditorium Maximum, TU Munich, 1994, Rudolf Wienands
CUMT, Xuzhou, China 2005
Chongqing Airport Terminal, 2005, Llewelyn Davies Yeang and Arup
Guangzhou Baiyun International Airport - 2, 2004, Parsons Brinckerhoff + URS Corporation (preliminary design) Arch + Struct. Eng
Potsdammer Platz, Berlin, 1998, Richard Rogers
Ningbo downtown, 2002, Qingyun Ma
Wanli University, Ningbo
Atrium, Germanisches Museum, Nuremberg, Germany, 1993, me di um Arch.
Pedestrian bridge over the Pegnitz Nuremberg
Cologne/Bonn Airport, Germany, 2000, Helmut Jahn Arch., Ove Arup USA Struct. Eng.
Marie-Elisabeth-Lüders-Steg, Berlin, 2003, Axel Schultes Arch
Ski Jump Berg Isel, Innsbruck, 2002, Zaha Hadid
Library University of Halle, Germany
Sobek House, Stuttgart, 2000, Werner Sobek
The New Renzo Piano Pavilion at the Kimbell Art Museum, Fort Worth, TX, 2013, Renzo Piano Arch
FM Constructive system, Elmag plant, Lissone, Milano, 1964, Angelo Mangiarotti Arch
Cable Works (Siemens AG), Mudanya, Turkey, 1965, Hans Maurer Arch
Moscone South (upper lobby), San Francisco, 1981, Hellmuth, Obata & Kassabaum
Philharmonie Berlin, 1963, Hans Scharoun Arch, Werner Koepcke Struct. Eng.
British Pavillion Sevilla Expo 92, Nicholas Grimshaw Arch
Museum of Anthropology, Vancouver, Canada, 1976, Arthur Erickson Arch
Modern Art Museum, Fort Worth, TX, 2002, Tadao Ando Arch, Thornton Tomasetti Struct. Eng
project by Eric Owen Moss Architects (EOMA)
Center for rhythmic gymnastics, Alicante, Spain, 1991, Enric Miralles Arch
Auditorium Parco della Musica, Rom, Italy, 2002, Renzo Piano Arch
Lufthansa Reception Building, Hamburg, 2000, Renner Hainke Wirth Architects
Oslo Opera House, Norway, 2007, Craig Dykers and Kjetil Trædal Thorsen Arch of Snohetta, Reinertsen Engineering ANS
National Museum of the American Indian, Washington DC, 2004, Douglas Cardinal, Johnpaul Jones Architects
Boston City Hall, Boston, Massachusetts, USA, 1968, Kallmann, McKinnell, & Knowles Arch, William LeMessurier Struct. Eng
Focus Media Center, Rostock, 2004, Helmut Jahn Arch, Werner Sobek Struct. Eng
Nelson Mandela Bay Stadium, Port Elizabeth, South Africa, 2009, GMP Architect (Berlin), Schlaich Bergermann and Partner
Shanghai Stadium, 1997, Weidlinger Assoc.
London Aquatic Center, 2012, Zaha Hadid Arch, Arup Struct. Eng.
Residence, Aspen, Colorado, 2004, Voorsanger & Assoc., Weidlinger Struct. Eng.
Barajas Airport, 0Rogers, Anthony Hunt Associates (main structure), Arup (main façade)
Dresdner Bank, Verwaltungszentrum, Leipzig, 1997, Engel und Zimmermann Arch.
National Gallery of Art, Washington, DC, 1978, I.M. Pei Arch
National Gallery of Art, East Wing, Washington, 1978, I.M. Pei
TGV Station, Paris-Roissy, 1994, Paul Andreu/, Peter Rice
Steel Tree House, Tahoe Donner, 2008, Joel Sherman
Arch
Fallingwater, Pittsburgh, 1937, Frank Llyod Wrigh Arch, Mendel Glickman and William Wesley Peters staff engineers
Everson Museum, Syracuse, NY, 1968, I. M. Pei Arch
Herbert F. Johnson Museum of Art, Cornell University, 1973, I. M. Pei
Super C, RWTH Aachen, Germany, 2008, Fritzer + Pape, Schlaich, Bergermann & Partner
Celtic Museum, Glauburg, Germany, 2011 designed by kadawittfeldarchitektur, Bollinger Grohmann Struct Eng
Centra at Metropark, Iselin, New Jersey, USA, 2011, Kohn Pedersen Fox Arch, DeSimone Struct. Eng
Rutgers Business School, Piscataway Township, New Jersey, USA, 2013, TEN Arch, WSP Cantor Seinuk Struct. Eng
Asma Bahçeleri Houses Office, Narlıdere, Izmir, Turkey, 2012, Metin Kılıç & Dürrin Süer Arch
VitraHaus, Vitra Campus, Weil am Rhein, Germany, 2009, Herzog & De Meuron Arch, ZPF Struct Eng
ING House , Amsterdam, The Netherlands, 2002, MVSA Arch, Aronsohn Struct. Eng
Orion Wageningen University , Bronland, Wageningen UR, 2013, Ector Hoogstad Arch, Aronsohn Struct. Eng.
Euram Building, Washington, 1971, Hartman-Cox Arch
Hyatt Regency, San Francisco, 1973, John Calvin Portman Arch
Tempe Municipal Building, Tempe, Arizona, 1970, Michael Goodwin Arch
DFDS Ferry and Cruiser Terminal, Hamburg, 1993, Alsop – Lyall with me di um Arch,
Documentation Center Nazi Party Rally Grounds, Nuremberg, 2001, Guenther Domenig Architect
German Museum of Technology, Berlin, 2001, Helge Pitz and Ulrich Wolff Architects
College for Basic Studies , Sichuan University, Chengdu, 2002
Chandigarh, India, 1952, Le Corbusier Arch
Looped Hybrid Housing, Beijing, 2008, Steven Holl Arch, Guy Nordenson Struct. Eng
Veteran's Memorial Coliseum, New Haven Connecticut, 1972, Kevin Roche Arch
Tokyo International Forum, 1997, Rafael Vinoly Arch, Kunio Watanabe Struct. Eng
INSTITUTE OF CONTEMPORARY ART, Boston Harbor, 2006, Diller Scofidio & Renfro of New York, 2006
The Tampa Museum of Art. Tampa, 2010, Stanley Saitowitz Office / Natoma Architects Inc., San , Walter P Moore,
MAXXI Art Museum, Rome, Italy, Zaha Hadid, 2010
Maxxi, the new museum of contemporary art, Rome, Italy, Zaha Hadid, 2009
MAXXI National Museum of XXI Century Arts, Rom, Italy, 2009, Zaha Hadid Arch, Anthony Hunts Struct. Eng.
School of Art and Art History, Universi Guy Nordenson Struct Eng ty of Iowa, Ames, 2006, Steven Holl Arch,
William J. Clinton Presidential Center, Little Rock, AR, 2004, Polshek Partnership
IVG Media Bridge, Munich, 2012, Steidle Arch, Burggraf + Reininger Struct Eng
Guthrie Theatre, Minneapolis, 2006, Jean Nouvel Arch, Ericksen & Roed Struct. Eng.
Phaeno Science Center, 2005, Wolfsburg, Zaha Hadid Arch, Adams Kara Taylor Struct. Eng
Porsche Museum, Stuttgart-Zuffenhausen , 2009, Delugan Meissl Arch, LAP Leonhardt, Andrä und Partner
Hirshorn Museum, Washington, 1974, Gordon Bunshaft/ SOM
Clam Shell House, Denver, Colorado, 1963, Charles Deaton Arch
Hotel Panorama, Oberhof, Thueringen, Germany
Uris Hall, Cornell University, Ithaca, 1973, Gordon Bunschaft (Skidmore, Owings & Merrill)
Beinecke Rare Book & Manuscript Library, Yale University, 1963, Gordon Bunshaft/ SOM
Beinecke Rare Book & Manuscript Library, Yale University, 1963, Gordon Bunshaft/ SOM
KAGAWA PREFECTURE GYMNASIUM, Takamatsu, Kagawa, 1964, Kenzo Tange
The building as a vertical cantilever beam
Eiffel Tower, Paris, 1889, Gustave Eiffel
Jin Mao Tower, Shanghai, 1999, SOM
Zhongguancun Financia Center, Beijing, 2006, Kohn Pederson Fox Arch
Shenzhen Stock Exchange HQ, Shenzhen, China, 2013, Rem Koolhaas of OMA, Ove Arup Struct. Eng.
World Trade Center proposal, New York, 2002, Rafael Vinoly
Hotel Tower, Macau, 2017, Zaha Hadid Arch, Buro Happold Struct. Eng
La Grande Arche, Paris, 1989, Johan Otto von Sprechelsen/ Peter Rice for the canopy
Basic beam mechanics including SAP2000 examples
Beams constitute
FLEXURAL SYSTEMS.
The frame element in SAP2000 is used to model axial truss members as well as beam-column behavior in planar and threedimensional skeletal structures. In contrast to truss structures, the joints along solid members may not be hinged but rigid. The loads may not be applied at the truss nodes but along the members causing a member behavior much more complicated than for trusses.
Beams cannot transfer loads directly to the boundaries as axial members do, they must bend in order to transmit external forces to the supports. The deflected member shape is usually caused by the bending moments. Beams are distinguished in shape (e.g. straight, tapered, curved), crosssection (e.g. rectangular, round, T-, or I-sections, solid or open), material (e.g. homogeneous, mixed, composite), and support conditions (simple, continuous, fixed). Depending on their span-to-depth ratio (L/t) beams are organized as shallow beams with L/t > 5 (e.g. rectangular solid, box, or flanged sections), deep beams (e.g. girder, trusses), and wall beams (e.g. walls, trusses, frames).
It is apparent that loads cause a beam to deflect. External loads initiate the internal forces: shear and moment (disregarding axial forces and torsion), deflection must be directly dependent on shear and moment. Typical beams are of the shallow type where deflection is generally controlled by moments. In contrast, the deflection of deep beams is governed by shear.
In the following discussion it is helpful to treat moment and beam deflection as directly related. Since the design of beams is primarily controlled by bending, emphasis is on the discussion of moments rather than shear.
Bending member types
examples of member cross-sections
Built-up wood beams
Composite wood-steel beams
REBARS TABLE B.3
ASTM standard reinforcing bars
Nominal Dimensions
Diameter in mm Bar Sizea (SI)b
Cross-Sect. Area in2 mm2
Weight Mass lbs/ft kg/m
#3
#10
0.375
9.5
0.11
71
0.376
0.560
#4
#13
0.500 12.7
0.20
129
0.668
0.944
#5
#16
0.625 15.9
0.31
199
1.043
1.552
#6
#19
0.750 19.1
0.44
284
1.502
2.235
#7
#22
0.875 22.2
0.60
387
2.044
3.042
#8
#25
1.000 25.4
0.79
510
2.670
3.973
#9
#29
1.128 28.7
1.00
645
3.400
5.060
#10
#32
1.270 32.3
1.27
819
4.303
6.404
#11
#36
1.410 35.8
1.56
1006
5.313
7.907
#14
#43
1.693 43.0
2.25
1452
7.650 11.380
#18
#57
2.257 57.3
4.00
2581
13.600 20.240
TABLE B.2 Typical allowable stresses of common materials for preliminary design purposes Approximate Allowable Stresses
Material
STEEL (carbon), A36 ksi (MPa) ALUMINUM ALLOY 6061-T6 ksi (MPa) CONCRETE 4000 psi (28 MPa) WOOD (small sections) psi (MPa) CLAY MASONRY f 'm = 2000 psi psi (MPa) SOIL
Compres s. Stresses, Fc
Tension Stresses, Ft
Bending Stresses, Fb
Shear Stresses, Fv
Bearing Stresses, Fcp
0.6Fy 22 (150)
0.6Fy 22 (150)
0.66Fy 24 (165)
0.4Fy 14.5 (100)
0.66Fy 24 (165)
0.6Fy 21 (150)
0.6Fy 21 (150)
0.6Fy 21 (150) circ. tubes: 24 (165)
12 (83)
21 (150)
0.25 f 'c 1000 (7.0)
1.6(f 'c)0.5 101 (0.7)
compress. 0.45 f 'c 1800 (12.0)
1.1(f 'c)0.5 70 (0.5)
0.3 f 'c 1200 (8.0)
1400 (10.0)
600 (4.1)
1200 (8.3)
160 (1.1)
500 (3.4)
0.2f 'm 400 (2.8)
28 (0.2)
compress. 0.33 f 'm 660 (4.5)
23 (0.16)
0.25f 'm 500 (3.4)
bearing pressure: Sand – gravel: 5200 psf = 36 psi (250 kPa) soft clay: 3000 psf = 21 psi (145 kPa)
Steel, A36 (≈Q235)
Compress. St. N/mm2 (MPa)
Tensile Stress N/mm2 (MPa)
Flexural Str. N/mm2 (MPa)
Shear Stress N/mm2 (MPa)
150
150
150
100
Rebars, A615Gr60 (≈HRB400)
Fy = 360
Concrete, 4000 psi (≈C30 )
7
0.7
12
0.5
Masonry
3
0.2
5
0.2
Wood
10
4
8
1
Approximate allowable stresses: the allowable stress design is used as a first simplified structural design approach
Dead loads
Live loads
Snow loads
Wind loads
kN/m2
kN/m2
kN/m2
kN/m2
Floors
4.00
3.00
#
#
Roofs
2.00
1.00
1.00
#
Walls
#
#
#
1.00
Typical preliminary vertical and horizontal design loads
The
FRAME ELEMENT for Flexural Systems
FLEXURAL SYSTEMS: BEAMS
BEHAVIOR of BEAMS FLEXURAL SYSTEMS: shallow beams, deep beams
BEAM TYPES LIVE LOAD ARRANGEMENT EFFECT of SPAN
LOAD TYPES and LOAD ARRANGEMENTS MOMENT SHAPE
DESIGN of BEAMS •
steel
•
concrete
FLOOR and ROOF FRAMING STRUCTURES
BEHAVIOR of BEAMS Beams, generally, must be checked for the primary structural determinants of bending, shear, deflection, possibly load effects of bearing, and lateral stability.
Usually
short beams
are governed by shear,
medium-span
beams
by flexure, and long-span beams by deflection. The moment increases rapidly with the square of the span (L2), thus the required member depth must also correspondingly increase so that the stresses remain within the allowable range.. The deflection, however, increases with the span to the fourth power (L4), clearly indicating that with increase of span deflection becomes critical.
On the other hand, with decrease of span or increase of beam depth (i.e. increase of depth-to-span ratio), the effect of shear must be taken into account, which is a function of the span (L) and primarily dependent on the cross-sectional area of the beam (A). Deflections in the elastic range are independent of material strength and are only a function of the stiffness EI, while shear and bending are dependent on the material strength.
The direction, location, and nature of the loads as well as the member shape and curvature determine how the beam will respond to force action. In this context it is assumed that the beam material obeys Hooke’s law and that for shallow beams a linear distribution of stresses across the member depth holds true.
For deep beams other design criteria must be developed. Only curved beams of shallow cross-section that makes them only slightly curved (e.g. arches) can be treated as straight beams using linear bending stress distribution.
Furthermore it is assumed that the beam will act only in simple bending and not in torsion; hence, there will be no unsymmetrical flexure. The condition of symmetrical bending occurs for doubly symmetrical shapes (e.g. rectangular and W shapes), when the static loads are applied through the centroid of their cross-section, which is typical for most cases in building construction.
Shallow beams vs deep beams
Wall beams
MOMENT SHAPE For general loading conditions, it is extremely helpful to derive the shape of the moment diagram by using the funicular cable analogy.
The single cable must adjust its suspended form to the respective transverse loads so that it can respond in tension. Under single loads, for example, it takes the shape of a string or funicular polygon, whereas under distributed loading, the polygon changes to a curve and, depending on the type of loading, takes familiar geometrical forms, such as a second- or third-degree parabola. For a simple cable, the cable sag at any point is directly proportional to the moment diagram or an equivalent beam on the horizontal projection carrying the same load. In a rigid beam, the moments are resisted by bending stiffness, while a flexible cable uses its geometry to resist rotation in pure tension. The various cases in the figure demonstrate how helpful it is to visualize the deflected shape of the cable (i.e. cable profile) as the shape of the moment diagram. The effect of overhang, fixity, or continuity can easily be taken into account by lifting up the respective end of the moment diagram.
FUNICULAR CABLE ANALOGY
Funicular cable analogy
SHALLOW BEAMS The general form of the flexure formula: fb = Mc / I = M/S Where I is defined as Moment of Inertia, a section that measures the size and "spread-outness" of a section with respect to an axis. Tables for standard steel and timber sections list two values for moment of inertia A strong axis value called Ixx, for the section bending in its strongest orientation. A weak axis value called Iyy, for the section bending in its weakest orientation. The general definition of section modulus: S = I/c Where c is the distance from the neutral axis to the extreme fiber of the section. Section modulus is also defined in terms of strong axis and weak axis properties: Sxx = Ixx / cxx , Syy = Iyy / cyy
CONTOURS of BENDING STRESS
CONTOURS of SHEAR STRESS
shallow beam
General Form of the Flexure Formula
For non-rectangular sections, there is a more general derivation of the flexure formula.
Internal forces at failure in reinforced concrete beam
SHEAR IN BEAMS
Shear causes a racking deformation, inducing diagonal tension and compression on mutually perpendicular axes. Shear failure in beams may manifest itself in several forms • Diagonal cracking (concrete). • Diagonal buckling (thin plates in steel beams). • Horizontal cracking (timber). In beams, the shearing stresses are maximum at the neutral axis because this is where the tension and compression resultants of the unbalanced moment create the greatest horizontal sliding action. Since maximum bending stresses occur at the extreme edge of a beam section while maximum shear stresses occur at the neutral axis, shear and bending stresses can be considered separately in design. They are uncoupled.
Deep concrete beams
Gravity force flow
BEAM TYPES: the Effect of Support Conditions Beams can be supported at one point requiring a fixed support joint (e.g. cantilever beams), at two points (e.g. simple beams, overhanging beams), and at several points (e.g. continuous beams). Beams may be organized according to their support types as follows: •
• • • • • •
simple beams cantilever beams overhanging beams hinge-connected cantilever beams fixed-end beams continuous beams simple folded and curved beams
A.
SIMPLE BEAMS
B.
OVERHANGING BEAMS: SINGLE-CANTILEVER BEAMS
C.
OVERHANGING BEAMS: DOUBLE-CANTILEVER BEAMS
2-SPAN CONTINUOUS BEAMS D.
3-SPAN CONTINUOUS BEAMS E.
F.
HINGE-CONNECTED BEAMS
G FIXED BEAMS
BEAM TYPES
The effect of different boundary types (pin, hinge, overhang, fixity, continuity, and free end) on the behavior of beams is investigated using the typical uniform loading conditions. It is known that a uniform load generates a parabolic moment diagram with a maximum moment of Mmax = wL2/8 at midspan. It is shown in the subsequent discussion how the moment diagram is affected by the various boundary conditions. In the following drawing the movement of the moment diagram is demonstrated in relation to various beam types.
moving the supports
Effect of boundary conditions on beam behavior
MEMBER ORIENTATION Is defined by local coordinate system
Typical: Moment 3-3, Shear 2-2 Each part of the structure (e.g. joint, element) has its own LOCAL coordinate system 1-2-3. The joint local coordinate system is normally the same as the global X-Y-Z coordinate system. For the elements, one of the element local axes is determined by the geometry of the individual element; the orientation of the remaining two axes is defined by specifying a single angle of rotation. For frame elements, for example, the local axis 1 is always the longitudinal axis of the element with the positive direction from I to J. The default orientation of the local 1-2 plane in SAP is taken to be vertical (i.e. parallel to the Z-axis). The local 3-axis is always horizontal (i.e. lies in the X-Y plane).
STEEL MEMBER PROPERTIES
CONCRETE MEMBER PROPERTIES
DESIGN
Modeling Steel Members using SAP2000 (see also Appendix A) SAP2000 assumes by default that frame elements (i.e., beams and columns) are laterally unsupported for their full length. But beams are generally laterally supported by the floor structure (Fig. 4.1). Therefore, assume an unsupported length of say Lb = 2 ft for preliminary design purposes, or when in doubt, take the spacing between the filler beams. For example, for a beam span of, L = 24 ft, assume an unbraced length ratio about the minor axis of Lb /L = 2 ft/24 ft = 0.083, or say 0.1; that is, take the minor direction unbraced length as 10% of the actual span length.
Lateral torsional buckling of steel beams
OVERHANGING BEAMS
Usually, cantilever beams are natural extensions of beams; in other words, they are formed by adding to the simple beam a cantilever at one end or both ends, which has a beneficial effect since the cantilever deflection counteracts the field deflection, or the cantilever loads tend to lift up the beam loads. The beam is said to be of double curvature, hence it has positive and negative moments. It is obvious that at the point of contraflexure or the inflection point (where the moment changes signs) the moment must be zero. For demonstration purposes, a symmetrical overhanging beam with double cantilevers of 0.35L span has been chosen. The negative cantilever moments at each support are equal to -Ms = w(.35L)0.35L/2 ≈ wL2/16 = M/2 The cantilever moments must decrease in a parabolic shape, in response to the uniform load, to a maximum value at midspan because of symmetry of beam geometry and load arrangement. We can visualize the moment diagram for the simple beam to be lifted up to the top of the support moments that are caused by the loads on the cantilever portion (i.e. moment diagrams by parts in contrast to composite M-diagrams). Therefore, the maximum field moment, Mf , must be equal to the simple beam moment, M, reduced by the support moment Ms. +Mf = M – Ms = wL2/8 – wL2/16 = wL2/16 = M/2
In general, with increase of span, the simply supported beam concept becomes less efficient because of the rapid increase in moment and deflection that is increase in dead weight. The magnitude of the bending stresses is very much reduced by the cantilever type of construction as the graphical analysis demonstrates. The maximum moment in the symmetrical double cantilever beam is only 17% of that for the simple beam case for the given arrangement of supports and loading! Often this arrangement is used to achieve a minimal beam depth for conditions where the live load, in comparison to the dead load, is small so that the effect of live load arrangement becomes less critical. As the cantilever spans increase, the cantilever moments increase, and the field moment between the supports decreases. When the beam is cantilevered by one-half of the span, the field moment at midspan is zero because of symmetry and the beam can be visualized as consisting of two double-cantilever beams. For this condition the maximum moment is equal to that of a simple span beam. A powerful design concept is demonstrated by the two balanced, double-cantilever structures carrying a simply supported beam; this balanced cantilever beam concept is often used in bridge construction. It was applied for the first time on large scale to the 1708-ft span Firth of Forth Rail Bridge in Scotland, 1890. The form of the balancing double-cantilever support structures is in direct response to the force flow intensity, in other words, the shape of the trusses conforms to that of the moment diagram.
DOUBLE CANTILEVER STRUCTURES
Firth of Forth Bridge (1708 ft), Scotland, 1890, John Fowler and Benjamin Baker
Gerber beam: hinge-connected cantilever beams
Nelson Mandela Bay Stadium, Port Elizabeth, South Africa, 2009, GMP Architect (Berlin), Schlaich Bergermann Struct. Eng.
International Terminal, San Francisco International Airport, 2001, SOM
International Terminal, San Francisco International Airport, 2001, SOM
International Terminal, San Francisco International Airport, 2001, SOM
LOAD TYPES and LOAD ARRANGEMENTS Beam loads can be arranged symmetrically and asymmetrically. Remember, for symmetrical beams with symmetrical loading, the reactions can be determined directly – each reaction carries one-half of the total beam load. Notice, the asymmetrical single load on a simple beam in Table A14 top, can be treated as a symmetrical load case plus a rotational load case. In other words, asymmetry of loading clearly introduces the effect of rotation upon the supports. Beam loads can consist of concentrated loads, line loads, and any combination of the two. Line loads usually are uniformly or triangularly distributed; occasionally they are of curvilinear shape. The various types of loads acting on a simple beam for symmetrical conditions by keeping the total beam load W constant are shown in the following drawing. We may conclude the following from the figure with respect to the shapes of the shear force and bending moment diagrams: • • • • • •
The shear is constant between single loads and translates vertically at the loads. The shear due to a uniform load varies linearly (i.e. first-degree curve). The shear due to a triangular load varies parabolically (i.e. second-degree curve). The moment varies linearly between the single loads (i.e. first-degree curve). The moment due to a uniform load varies parabolically (i.e. second-degree curve). The moment due to a triangular load represents a cubic parabola (i.e. third degree curve).
LOAD TYPES
LOAD ARRANGEMENT
LIVE LOAD ARRANGEMENT
DEAD LOAD (D)
LIVE LOAD 1 (L1)
LIVE LOAD 2 (L2)
LIVE LOAD 3 (L3)
LIVE LOAD 4 (L4)
PATTERN LOADING
In contrast to simply supported beams, for continuous beams and overhanging beams the arrangement of the live loads must be considered in order to determine the maximum beam stresses. Typical live load layouts are shown in the following figure. For example with respect to the critical bending moments of a 3-span continuous beam: • to determine the maximum field moment at mid-span of the center beam, the dead load case together with live load case L2 should be considered • to determine the maximum field moments of the exterior beams, the dead load case together with L3 should be taken, • to determine the maximum interior support moment, the dead load case with L4 should be used. For the preliminary design of a continuous roof beam, the uniform gravity loading may be assumed to control the design. It would be questionable to consider a critical live load arrangement for flat roofs where the snow does not follow such patterns, assuming constant building height and no effect of parapets, that is , assuming areas do not collect snow. Furthermore, the roof live loads are often relatively small in comparison to the dead load, as is the case in concrete construction, so the effect of load placement becomes less pronounced. Therefore, the beam moment usually used for the design is based on the first interior support and is equal to, M = wL2/10 This moment should also cover the effect of possible live load arrangement during construction at the interior column supports.
D, L1
L2
L3
L4
COMB1 (D + L1)
COMB2 (D + L2)
COMB3 (D + L3)
COMB4 (D + L4)
EFFECT OF SPAN
A.
SIMPLE BEAMS
B.
OVERHANGING BEAMS: SINGLE-CANTILEVER BEAMS
C.
OVERHANGING BEAMS: DOUBLE-CANTILEVER BEAMS
2-SPAN CONTINUOUS BEAMS D.
3-SPAN CONTINUOUS BEAMS E.
F.
HINGE-CONNECTED BEAMS
G FIXED BEAMS
1 k/ft
A. 1 k/ft
1 k/ft
1 k/ft
1 k/ft
B. I. C. 12 kft D.
12 kft
1 k/ft
J.
18 kft
1 k/ft K.
12 k
1 kft/ft E. F.
6k
2 k/ft
4k G.
H.
6k
4k
4k
L.
M.
2 k/ft
2 k/ ft
0.5 k/ft
Load Types and Boundary Conditions
1.5k/ft
N. O.
P = 97.87 k
M = 20.84 ft-k
Thermal beam loading
Mt = 10 ft-k
R
∆
- -2.5 2.5ftft-k
Mt
2.5 2.5ft-k ft-k
Torsional beam loading
M
Tapered beams
W8 x 10
W14 x 30
8'
a.
b.
c.
8'
W8 x 10
8'
W16 x 31
12'
W8 x 10
8'
12'
W14 x 30
16'
Tapered beam analysis
15.88" 7.89" 7.99" 7.99"/2 4" 13.84" 7.89" 5.95" 5.95"/2 2.98"
DESIGN of BEAMS In steel design, for the condition where a given member stress is checked that is the member input is known just assign the section to the member. But, when the member has to be designed the Automatic Steel Selection Feature in SAP will pick up the most economical member available from a list that has been pre-selected, i.e. for the conditions where the members are not known and an efficient solution must be found, more sections for the selection process have to be stored. The design results are based on default SAP2000 assuming, that the frame elements (i.e. beams and columns) are laterally unsupported for their full length. But beams are generally laterally supported by the floor structure. Therefore assume an unsupported length of say Lb = 2 ft for preliminary design purposes, or when in doubt, take the spacing between the filler beams (e.g. as 33% of the actual beam span). For example, for a beam span of L = 24 ft assume an unbraced length ratio about the minor axis of Lb/L = 2 ft/ 24 ft = 0.083 or say of 0.1, that is taking the minor direction unbraced length as 10% of the actual span length. The stress ratios in SAP represent the DEMAND/CAPACITY ratios as reflected by the various colors ranging from gray to red.
Concrete frame elements
can have the area of longitudinal and shear reinforcing steel automatically chosen for a selected section according to the selected design code. For normal loading conditions the program has built-in default loading combinations for each design code. For other special loading conditions the user must define design loading combinations. K-factors are calculated for concrete frame members, which are defined as type column under the frame section definition, reinforcement.
In concrete design you must define the frame section as a beam or column! Beams are not designed for axial forces. Treat one-way slabs as shallow, one-foot wide beam strips. In contrast to steel design, where SAP selects the least weight section from a list that has been pre-selected, in concrete design the area of the bars depends on the concrete section that is the STEEL RATIO (As/bd) or in SAP on the REBAR PERCENTAGE, As/bh.
Examples of rebar layout in concrete
18"
4"
be = 63“
a.
bw = 10“
bw = 10“
Concrete beam cross section
b.
be
d = 14.5"
h = 18"
d = 15.5"
4"
bw = 10“
a.
b.
NEGATIVE MOMENT @ SUPPORT
POSITIVE MOMENT @ MID-SPAN
Location of longitudinal reinforcement be
d = 15.5"
h = 18"
d = 14.5"
4"
bw = 10“
a. NEGATIVE MOMENT @ SUPPORT
b. POSITIVE MOMENT @ MID-SPAN
BM18x30
Low REBAR PERCENTAGE
BM16x28
Typical REBAR PERCENTAGE
BM14x24
High REBAR PERCENTAGE
Critical stirrup spacing: s = (1/0.061)0.22 = 3.61 in > ≈ 3 in
BM 14 x 24 in
BM 14 x 20 in
TBM 24 in deep
wD = 2 k/ft wL= 1.0 k/ft
Ps
wp 30"
Ps cosθ
e = 12" Ps L = 32'
PRESTRESSED CONCRETE BEAM Load balancing
18"
FLOOR and ROOF FRAMING STRUCTURES Whereas typical wood beams are rectangular solid sections, steel beams for floor or roof framing in building construction are the common rolled sections, cover-plated W-sections, open web steel joists, trusses, castellated beams, stub girders, plate girders, and tapered and haunched-taper beams. In cast-in-place concrete construction the beams form an integral part of the floor framing system. With respect to gravity loading they constitute T-sections (or L-sections for spandrel beams) with respect to positive bending along the midspan region, but only rectangular sections for negative bending close to the supports. Simple rectangular sections or inverted T-sections are also typical for precast concrete construction, where the slab may rest on the beams without any continuous interaction. There are numerous framing arrangements and layouts possible depending on the bay proportions, column layout, span direction, beam arrangement, framing floor openings, etc. A typical floor framing bay is shown to demonstrate the nature of load flow (i.e. hierarchy of members), and beam loading arrangements. It is shown how the load flows (and the type of loads it generates) from the floor deck (i.e. 1-ft slab strips) to the beams (or joists), to the girders, columns, and finally to the foundations.
Horizontal gravity force flow
FLOOR-ROOF FRAMING SYSTEMS
Floor framing systems
45 deg.
wS/2 M
M
wS/3
wS/2
S
L
M
M
(wS/3)(3 - (s/L)2)/2
Two-way slab action
Kaifeng, Xiangguo Si temple complex, Kaifeng
Beilin Bowuguan (Forest of Stelae Museum), Kaifeng
Force flow from typical floor framing bay
GI
3 Sp @ 8' = 24'
25'
BM
BM
BM
BM
GI
Beam design: The beam carries the following uniform load assuming the beam weight included in the floor dead load. w = wD + wL = 8(0.080) + 8(0.080)0.96 = 0.64 + 0.61 = 1.25 k/ft The maximum moment is, Mmax = wL2/8 = 1.25(25)2/8 = 97.66 ft-k The required section modulus is, Sx = Mx/Fb = Mx/0.66Fy = 97.66(12)/0.66(36) = 49.32 in.3 Try W18x35, Sx = 57.6 in.3, Ix = 510 in.4 (W460 x 52) The maximum live load deflection is within the allowable limits as shown, ΔL = 5wL4/(384EI) = 5(0.61/12)(25 x 12)4/ (384(29000)510) = 0.36 in. ≤ L/360 = 25(12)/360 = 0.83 in. Girder design: The girder weight is for this preliminary design approach ignored, it will have almost no effect upon the design of the beam. The girder must support the following reaction forces of the beams, P = [0.080 + 0.080(0.8)](25 x 8) = 28.80 k The maximum moment is, Mmax = PL/3 = 28.80(24)/3 = 230.40 ft-k The required section modulus is, Sx = Mx/0.66Fy = 230.40(12)/0.66(36) = 116.36 in.3 Try W18 x 71, Sx = 131 in.3 Notice, SAP uses a reduction factor of 0.96 therefore yielding a W18 x 76.
In ETABS, when floor elements are modeled with plate bending capacity (e.g. DECK section for steel framing), vertical uniform floor loads are automatically converted to line loads on adjoining beams or point loads on adjacent columns thereby evading the tedious task of determining the tributary loads on the floor beams as in SAP.
LOAD MODELING
A typical floor structure layout with a stair opening is investigated in the following figure in order to study asymmetrical loading conditions in addition to setting up beam loading. The floor deck spans in the short direction perpendicular to the parallel beams that are 8 ft (2.44 m) apart, as indicated by the arrows. Visualize the deck to act between the beams as parallel, 1-ft (0.31 m) wide, simply supported beam panels or as joists spaced 1 ft apart that transfer one-half of the deck loads to the respective supporting beams. The contributing floor area each beam must support is shaded and identified in the figure; it is subdivided into parallel load strips that cause a uniform line load on the parallel beams. However, beam B7 is positioned on an angle and hence will have to carry a triangular tributary area. The loading diagrams with numerical values are given for the various beams as based on a hypothetical load of 100 psf (4.79 kPa) including the beam weight; this load is also used for the stair area, but is assumed on the horizontal projection of the opening. Beam B1 is supported by beam B2 framing the opening; its reaction causes single loads on B2 and G2. Beam B2, in turn, rests on beams B3 and B4; its reactions are equal to the single loads acting on these two beams. Since most of the beams are supported by the interior girders, their reactions cause single load action on the girders, as indicated for G1, where the beam reactions from the other side are assumed to be equal to the ones for B5; the girder weight is ignored.
G1
G1
G4
BM5
BM5
BM5
BM5
21' 3 Sp @ 7' = 21'
BM2 BM4
3 Sp @ 7' = 21'
BM1 BM3
21'
BM5
BM5
G3
BM1 BM3
BM2
BM1
BM1
BM2
BM2 G2
P
8'
8' 24'
8'
R
20'
R
P
In concrete design you must define the frame section as a beam or column! Beams are not designed for axial forces. Treat one-way slabs as shallow, one-foot wide beam strips. • Define material and the concrete section (e.g. rectangular, T-section). • For the design of beams enter the top and bottom concrete cover in the text edit boxes. If you want to specify top and bottom longitudinal steel, enter reinforcement area for the section in the appropriate text edit box, otherwise leave values of zero for SAP2000 to calculate automatically the amount of reinforcing required. • First the load cases must be defined such as, D (dead load), L1 (live load 1), L2 (live load 2), L3 (live load 3), etc. according to the number of live load arrangements. • Click Define, then Analysis Cases and the load cases occur, highlight load case and click Modify/Show Case to check whether the load case is OK. In case of load factor design change the scale factor with the load factor (e.g. 1.2 D, 1.6 L). • Then define load combinations such as for a continuous beam for D + L: go to combinations and click Add New Combo button and define such as COMB1 (D + L1), COMB2 (D + L2), COMB3 (D + L3), etc. Change Scale Factor for combined load action such as 0.75(D + L + W or E) Check the strength reduction factors in Options/Preferences. • Assign C L E A R M E M B E R LE N G T H S, select member (click on member) then click on Assign, then Frame, then End Offsets: (total beam length -clear span, or support width of girder, for example)/2, then Offset Lengths • Click Analysis and check results • Click Design, then Concrete Frame Design then Select Design Combos and select combos, then click Start Design/Check of Structure. Start Design/ Check of Structure button, then select member, then right click, then choose ReDesign button, then check under Element Type: NonSway (for beams and laterally braced columns), or Sway Ordinary (for ordinary frames, laterally non braced columns). Click on member, then click right button of the mouse to obtain the Concrete Design Information, then highlight the critical location (e.g. support and center-span for longitudinal reinforcing, or support for shear reinforcing), then click Details to obtain the maximum moment and shear reinforcement areas which are displayed for the governing design combination by default
EXAMPLE 6.4:
Design of concrete floor framing
A 6-story concrete frame office building consists of 30 x 34-ft (9.14 x 13.11-m) bays with the floor framing shown in Fig. 6.13 The 6.5-in (165-mm) concrete slab supports 5 psf (0.24 kPa) for ceiling and floor finish, a partition of 20 psf (0.96 kPa), as well as a live load of 80 psf (3.83 kPa). The girders are 24 in (610 mm) high and 16 in wide (406 mm), whereas the beams have the same depth but are 12 in (305 mm) wide. Investigate a typical interior beam. The beam dead load is 1.81 klf (26.41 kN/m) and the reduced live load is 0.85 klf (12.40 kN/m). Use a concrete strength of fc' = 4000 psi (28 MPa ), fy = fys = 60 ksi (414 MPa ) and a concrete cover of 2.5 in.(63.5 mm).
1. Treat the typical interior span of the continuous beam as a fixed beam using the net span. 2. Model the intermediate floor beam (i.e. beam between column lines) as a continuous three-span beam fixed at the exterior supports. Consider live load arrangement. 3. Use the equivalent rigid-frame method by modeling the beam along the column lines as a continuous three-span beam to be framed into 18 x 18-in. columns and to form a continuous frame, where the ends of the 12-ft columns are assumed fixed. Consider live load arrangement. 4. Model six structural bays to design the beams using ETABS and then export the floor framing to SAFE to design the floor slab. . For this preliminary investigation, establish live load patterns for the design of the intermediate beams only, that is not for the beams along the column lines.
18"x18" GI 16/24
15'
15'
34'
16/24
BM
12/24
BM
12/24
BM
12/24
GI
24"
6.5" 16"
34'
lnet = 34 - 16/12 = 32.67'.
FIXED BEAM 1
1
1
A1
A1
A1
EQUIVALENT RIGID FRAME METHOD
12' 12'
lnet = 34 - 18/12 = 32.50'
h net = 12 24/12 = 10'
THREE-SPAN CONTINUOUS BEAM
h net = 12 24/12 = 10'
1
COMB1 = D + DS + L1
COMB2 = D + DS + L2 COMB3 = D + DS + L3
Floor Beam Grids The floor framing systems discussed till now consisted of one-dimensional resisting beams, in other words, the loads were carried by single beams in onedirectional fashion. However, when beams intersect loads may be transferred in two or more directions as is the case for beam grids. First let us investigate various cross beam layouts for floor framing shown in Fig. 7.22. The two left cases identify on directional beams, where either the short beams are supported by the long beam (left case) or the long beams are supported by the short beam, hence the structures are statically determinate. However, in the two other cases the beams are continuous and support each other; together they share the load and disperse the load in two-directional fashion, which makes the analysis statically indeterminate.
Ls
LL = L S = L
0.5P
P
P
P
0.06P
LL = 2LS
P
0.25P
0.5P
0.5P
0.44P
0.44P
0.25P
0.25P
0.25P
a.
0.5P
b.
0.06P
c.
The effect of beam continuity
d.
b.
a.
c.
rectangular and skew beam grids
d.
