Beam Structures including sap2000

FLEXURAL
STRIUCTURE SYSTEMS
B E A M S
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

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

LINEELEMENTSSURFACEELEMENTS
FLEXURAL STRUCTURE
SYSTEMS
FLEXURAL-AXIAL STRUCTURE SYSTEMS
TENSILE MEMBERS
COMPRESSIVE
MEMBERS
BEAMS
BEAM-COLUMN
MEMBERS
FRAMES
TENSILE MEMBRANES
PLATES
MEMBRANE FORCES
SOFT SHELLS
SLABS, MEMBRANE BENDING and TWISTING
AXIAL STRUCTURE
SYSTEMS
SHELLS RIGID SHELLS

FLEXURAL STRUCTURE SYSTEMS
B E A M S
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

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

Petersbogen shopping center, Leipzig, 2001, HPP Hentrich-Petschnigg

Petersbogen shopping center, Leipzig,
2001, HPP Hentrich-Petschnigg

Petersbogen shopping center,
Leipzig, 2001, HPP Hentrich-
Petschnigg

Auditorium
Maximum, TU
Munich, 1994, Rudolf
Wienands

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

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.

Library University of Halle, Germany

Ski Jump Berg
Isel, Innsbruck,
2002, Zaha
Hadid

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

Fallingwater, Pittsburgh,
1937, Frank Llyod Wrigh Arch,
Mendel Glickman and William
Wesley Peters staff engineers
Arch

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

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.

William J. Clinton Presidential Center, Little Rock, AR, 2004, Polshek Partnership

Guthrie Theatre, Minneapolis, 2006, Jean Nouvel Arch, Ericksen & Roed Struct. Eng.

Phaeno Science Center, 2005, Wolfsburg, Zaha Hadid Arch,
Adams Kara Taylor Struct. Eng

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

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 three-
dimensional 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), cross-
section (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.

examples of member
cross-sections

TABLE B.3
ASTM standard reinforcing bars
Nominal Dimensions
Bar Sizea (SI)b
Diameter
in mm
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
REBARS

TABLE B.2
Typical allowable stresses of common materials for preliminary design
purposes
Approximate Allowable Stresses
Material Compres
s.
Stresses,
Fc
Tension
Stresses,
Ft
Bending
Stresses,
Fb
Shear
Stresses,
Fv
Bearing
Stresses,
Fcp
STEEL
(carbon),
A36
ksi (MPa)
0.6Fy 
22 (150)
0.6Fy 
22 (150)
0.66Fy 
24 (165)
0.4Fy 
14.5 (100)
0.66Fy 
24 (165)
ALUMINUM
ALLOY
6061-T6
ksi (MPa)
0.6Fy 
21 (150)
0.6Fy 
21 (150)
0.6Fy 
21 (150)
circ. tubes:
24 (165)
12 (83) 21 (150)
CONCRETE
4000 psi
(28 MPa)
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)
WOOD
(small
sections)
psi (MPa)
1400
(10.0)
600 (4.1) 1200 (8.3) 160 (1.1) 500 (3.4)
CLAY
MASONRY
f 'm = 2000
psi
psi (MPa)
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)
SOIL
bearing pressure: Sand – gravel: 5200 psf = 36 psi (250 kPa)
soft clay: 3000 psf = 21 psi (145 kPa)

Approximate allowable stresses: the allowable stress design is used as a first
simplified structural design approach
Compress.
St. N/mm2
(MPa)
Tensile Stress
N/mm2
(MPa)
Flexural Str.
N/mm2
(MPa)
Shear Stress
N/mm2
(MPa)
Steel, A36 (≈Q235) 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

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 ELEMENTfor 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.

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.

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

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 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.
SHEAR IN BEAMS

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

BEAM TYPES
A.
B.
C.
D.
E.
F.
SIMPLE BEAMS
OVERHANGING BEAMS: SINGLE-CANTILEVER BEAMS
OVERHANGING BEAMS: DOUBLE-CANTILEVER BEAMS
2-SPAN CONTINUOUS BEAMS
HINGE-CONNECTED BEAMS
FIXED BEAMS
G

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.

Effect of boundary conditions on beam behavior

MEMBER ORIENTATION
Is defined by local coordinate system
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).
Typical: Moment 3-3, Shear 2-2

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.

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

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).

LIVE LOAD ARRANGEMENT
DEAD LOAD (D)
LIVE LOAD 1 (L1)
LIVE LOAD 2 (L2)
LIVE LOAD 3 (L3)
LIVE LOAD 4 (L4)

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.

COMB1 (D + L1)
COMB2 (D + L2)
COMB3 (D + L3)
COMB4 (D + L4)

EFFECT OF SPAN
A.
B.
C.
D.
E.
F.
SIMPLE BEAMS
OVERHANGING BEAMS: SINGLE-CANTILEVER BEAMS
OVERHANGING BEAMS: DOUBLE-CANTILEVER BEAMS
HINGE-CONNECTED BEAMS
FIXED BEAMS
G

Load Types and Boundary Conditions
I.
J.
K.
L.
M.
N.
O.
A.
B.
C.
D
E.
F.
G.
H.
18 kft
12 k
6 k6 k
4 k 4 k 4 k
2 k/ft
2 k/ft
2 k/ ft 0.5 k/ft
12 kft
1.5k/ft
1 kft/ft
1 k/ft
.
12 kft
1 k/ft
1 k/ft1 k/ft
1 k/ft 1 k/ft
1 k/ft

M = 20.84 ft-k
P = 97.87 k
Thermal beam loading

2.5 ft-k
- 2.5 ft-k
Mt = 10 ft-k
Mt M
R∆
- 2.5 ft
2.5 ft-k
Torsional beam loading

W8 x 10 W14 x 30
W8 x 10
W8 x 10
W16 x 31
W14 x 30
15.88"
7.89"
7.99"
7.99"/2
4"
13.84"
7.89"
5.95"
5.95"/2
2.98"
a.
b.
c.
8'8' 8'
8' 16'
12'12'
Tapered beam analysis

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

4"
18"
be = 63“
bw = 10“bw = 10“a. b.
Concrete beam cross section

h=18"
be
b.
POSITIVE MOMENT @ MID-SPANNEGATIVE MOMENT @ SUPPORT
4"
bw = 10“
a.
d=15.5"
d=14.5"
h=18"
be
b.
POSITIVE MOMENT @ MID-SPANNEGATIVE MOMENT @ SUPPORT
4"
bw = 10“
a.
d=15.5"
d=14.5"
Location of longitudinal reinforcement

Typical REBAR PERCENTAGEBM16x28

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

PRESTRESSED CONCRETE
BEAM Load balancing
Ps cosθ
Ps
e = 12"
Ps
L = 32'
18"
30"
wD = 2 k/ft
wL= 1.0 k/ft
wp

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.

L
S
45 deg.
wS/2
M
M
M
M
wS/2
wS/3
(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

3 Sp @ 8' = 24'
25'
BM
BM
BM
BM
GI
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.

BM1
BM2
BM1
BM2
G 2
BM3
G1
BM2
BM1
BM1
BM2
G3
BM5
G4
3 Sp @ 7' = 21' 21'
3Sp@7'=21'21'
G 1
BM3
BM4
BM5
BM5
BM5
BM5
BM5

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.

GI
GI
BM
BM
BM
16/24
16/24
12/24
12/24
12/24
18"x18"
15' 15'
34'

34'
16"
24"
6.5"
1
1
A1
lnet = 34 - 16/12 = 32.67'.
lnet = 34 - 18/12 = 32.50'
12'12'
hnet
=12-
24/12=10'
hnet
=12-
24/12=10'
1
A1
1
A1
EQUIVALENT RIGID FRAME METHOD
THREE-SPAN CONTINUOUS BEAM
FIXED BEAM

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 one-
directional 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=2LS
a. b. c. d.
PP P P
0.25P0.5P
0.5P
0.5P
0.25P
0.25P
0.25P
0.06P
0.06P
0.44P0.44P
LL = LS = L
0.5P
The effect of beam continuity

a.
b.
c. d.
rectangular and skew beam grids

Beam Structures including sap2000

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