Building Structures
Fundamentals of Crossover Design
Revised Edition
By Nawari O. Nawari and Michael Kuenstle
University of Florida
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Companion Material
xiii
Acknowledgmentsxv
Dedicationxvii
Chapter 1 Introduction
1
Natural Structures
2
Man-Made Structures
3
5
Building Structures
Structural Systems
10
Foundations12
Structural Walls
13
Slabs15
Framework15
Truss Systems
19
Arches and Vaults
19
21
Plates and Shells
Tensile Structures
23
24
Hybrid Systems
Study of Buildings and Structures
28
Building Structural Classifications
29
Superstructure30
Substructure32
Form, Space, and Function
33
Chapter 2 Forces on Buildings
41
Introduction41
Loads on Buildings
41
Forces47
Force Systems
49
Distributed Forces
49
External and Internal Forces
54
contents
Prefaceix
Resultant of Concurrent Forces
Resolution of Forces
Finding Resultant of Forces Using their Rectangular Components
Chapter 3 Equilibrium of Building Structures
55
65
68
71
Introduction71
Moment of a Force
73
Moment of a Couple
77
Equivalent Forces
78
Resultant of Parallel Forces
78
Equilibrium of a Structural Element
82
Free Body Diagram (FBD)
84
Reactions at Supports and Connections
86
Chapter 4 Load Path: Vertical Forces
97
Introduction97
Tributary Areas
105
One-Way and Two-Way Spanning Systems
107
Stacked vs. End Framing
113
Openings on Floors and Roofs
114
Framing Levels
115
Chapter 5 Load Path: Lateral Forces and Stability
127
Introduction127
Wind Loads
128
138
Calculations of Wind Forces
Approximate Method
142
Seismic Loads
143
Fundamental Period
145
Building Configuration Effects
147
Summary151
Calculation of Seismic forces
152
Lateral Resisting Systems
158
Lateral Resisting Frame Systems
158
Braced Frames
158
Moment Resisting Frames
160
Shear Walls
162
Floor and Roof Diaphragm Action
165
Arrangement of Lateral Resisting Systems
165
Chapter 6 Structural Elements: Cables
175
Introduction175
Fundamentals176
Anchorage and load path
176
Stability180
Analysis182
Design Notes
194
Chapter 7 Structural Elements: Arches
197
Introduction197
Fundamentals200
Modern Arches
202
Stability209
Analysis209
Design Notes
219
Chapter 8 Structural Elements: Trusses
221
Introduction221
Fundamentals223
Load Path
226
Lateral Stability
228
Analysis228
Joints under Special Loading Conditions
239
The Method of Sections
242
247
Design Notes
Chapter 9 Shape Factors: Properties of Sections
251
Introduction251
Centroids of Areas
251
255
Moment of Inertia
Radius of Gyration
257
Section Modulus
258
Chapter 10 Structural Materials: Strength and Behavior
259
Introduction259
Concept of Stress
259
Shear Stress
260
Bearing Stress
261
Bending Stress
262
Torsion Stress
263
Concept of Deformation
264
Strain265
Stress-Strain Relationship
265
Thermal Strain and Stresses
271
Design Notes
277
Chapter 11 Structural Elements: Beams
279
Introduction279
Analysis285
Relationships among Distributed Load (w), Shear Force (V), and Bending Moment (M)
298
Determinant and Indeterminate Beams
308
Conclusions310
Stresses in Beams
315
Section Modulus
318
Shear Stresses
320
Shear Stresses in Other Sections
322
Vertical and Horizontal Shear Stresses
324
Summary326
Beam deformation
327
Deflection Formula
329
Analysis Conclusions
334
Stability335
Beam Grid Systems
337
Beam Design
340
Design Notes
349
Chapter 12 Structural Elements: Columns
355
Introduction355
Analysis357
Slenderness Ratio
360
Stability362
372
Design Notes
Chapter 13 Vaults and Domes
370
Introduction370
Fundamentals381
Domes386
Stability388
References
391
Appendix A
393
Units of Measurement
Appendix B
395
Wood Section Properties
Appendix C
397
Wide-flange steel sections properties
Index
407
PReFAce
This book strives to elucidate the principles that bridge between building
structures and architectural design. This is what we call crossover design in
architectural structures. The book represents our view of how to critically engage the subject and how to understand building structures as an integral component of architectural space-making strategies and building design concepts.
This approach to building design is a merger of architecture and structures,
and represents a point of view that has been recognized by both academia and
practice, and in particular, with the work of Professor Wolfgang Schueller, as
displayed in his excellent books The Design of Building Structures and Building
Support Structures.
One of the primary focuses of this book is to introduce the fundamental
aspect of the structural behavior of the elements within various architectural
forms, using simple hand calculation techniques with a minimum of mathematics while determining the preliminary design of a structural member with
reasonable accuracy. The mathematics is deliberately kept to a basic level so
that the main emphasis on behavioral and conceptual concerns is not concealed behind complex analytical approaches. At the same time, it deals with
the subject in a qualitative and practical manner that introduces this matter by
means of many illustrations in the form of photographs of buildings and structural diagrams to reinforce and extend the understanding of the mathematical
equations and calculations.
We tried also to highlight the embedded concepts from different related
fields such as building construction and engineering to explore the relationships between structural behavior and architectural design ideas at different
scales. This engagement of crossover design in architectural structures depends
mainly on two principles:
∙ Deep understanding of the fundamental behavior and concepts of building structures as related to architectural design and other related fields.
∙ Use of simple mathematical formulas and equations to analyze and verify
structural design.
PReFACe
ix
The need for these fundamentals is even more crucial nowadays where digital
technology has had its biggest impact in the designs of buildings with seemingly infinite variations of structural materials, member shapes and spans, and connection
types that result in remarkably fluid, complex, and curvilinear forms that are more
easily accomplished because of rapid advances in integrated digital technologies.
Students can easily get lost in these technologies in terms of viability of their design
development, specifically on how structural behavior and systems influence their
design ideas.
Currently, architectural students in the design studios are concerned primarily
with artistic expressions and philosophical description, independent of the building as an organism and how it is constructed. Structure is minimally discussed and
presented in their work. They apparently are not motivated by the current way
of conveying structural concepts and design processes. The purely mathematical approach of the classical engineering schools is not effective in architectural
and building construction colleges. Thus, students of these schools are driven to
consider themselves as artists with less interest in scientific and engineering principles. However, all artists must acquire mastery of the technology of their chosen
medium, particularly those who choose buildings as their means of expression.
The structure of a building is the framework that preserves its integrity in response to external and internal excitations. It is a massive object that must somehow
be incorporated into the architectural program. It must therefore be given a form
that is compatible with other aspects of the building. Many fundamental issues associated with the function and appearance of a building, including its overall form,
the pattern of its fenestration, the general articulation of solid and void within it,
and even, possibly, the range and combination of the textures of its visible skins are
affected by the nature of its structure. The structure also influences programmatic
aspects of a building’s design because of the capability of the structure to organize
and determine the feasibility of pattern and shape of internal and external spaces.
On the other hand, engineering students are well trained in understanding
advanced calculus and numerical methods for analyzing and designing building
structures—they know how to set up the analytical model and solve equations to
get solutions. But they lack the understanding of the overall structural behavior
of the building and its connection to other architectural and construction details,
and thus may use an abstract mathematical and analytical model that imperfectly
simulates reality.
The relationship between structure and architecture is therefore a fundamental
aspect of the art of building. It sets up conflicts between the technical, scientific,
and artistic agendas that architects and engineers must resolve. The method in
which the resolution is carried out is one of the most tested criteria of the success of a building design. This book focuses on providing the fundamentals to help
architects and engineers achieve a successful resolution to such conflict.
Not only do we see that our approach provides the fundamentals of architectural
structures design and as a means to developing an intuitive understanding of how
building structures work and how their forms and arrangement make sense, but
our approach also enables more conceptual linking and integrating architectural
and structural engineering principles.
x Building Structures
In this edition of Building Structures, we strove to achieve that combination of
accuracy, clarity, and presentation that have always been our objectives. To this
end, we have relied on the support and advice of students and colleagues who are
sensitive to any shortcomings of the book. It is hoped that it will enable students
and members of the profession to gain a better understanding of the relationship
between structural design and architectural design. We conclude, therefore, with
an earnest invitation to our readers to join us in advancing this never-ending project. Please send us corrections and suggestions of any kind. Your contributions,
sincerely welcome, may be most conveniently addressed to Nawari O. Nawari at
[email protected] or Michael Kuenstle at [email protected]. We will receive your
responses to this book with respect and heartfelt gratitude.
N. Nawari
University of Florida, Gainesville
M. Kuenstle
University of Florida, Gainesville
Preface xi
coMPAnIon MAteRIAL
C
omplementary material to this book is provided in the Analyzing Building
Structures: An Exercise and Solutions Manual, which will be published
in the fall of 2012. The book contains exercises and solved problems that will
facilitate comprehension of essential information in this book. The book will
also help improve proficiency in many aspects of building design.
CoMPAnion MAteRiAL
xiii
AcKnoWLeDGMentS
W
e would like to express our gratitude to many students who have assisted in making this publication. They are too numerous for all to be
mentioned individually, but special thanks and appreciation are due to the
following:
Daun Jung for doing many 3D drawings and giving helpful suggestions and
feedback.
Audrey Gutierrez for the cover image, many 2D drawings’ details, and great
effort in reviewing illustrations.
Greatly appreciated is the help of Luis P. Delfin, who was a second-year civil
engineering student when the first editon of this book was published.
We would also like to thank the faculty, staff, and students of the School of
Architecture at the University of Florida for the frequent assistance and helpful
discussions, particularly the Dean of the College of Design, Construction and
Planning, Prof. Chris Silver; Prof. Martin Gold; Prof. Wolfgang Schueller; Prof.
Nancy Clark; and Prof. John Maze.
We also wish to thank those other students and colleagues whose names
we have not mentioned, but who have also supported us through critical and
constructive feedback.
We are, of course, indebted to the staff of University Readers, Inc. & Cognella
Academic Publishing for the editorial and production supervision of the book.
Finally, we would like to thank our families for their patience and support.
ACKnoWLeDGMents
xv
DeDIcAtIon
this work is dedicated to our parents.
DeDiCAtion
xvii
Learning objectives: to foster a deeper understanding of what
structures really are and how they can be assessed within the
context of buildings and architectural design strategies.
1
IntRoDUctIon
I
n general, building structures are deceptively complex. At their best, they
connect us with the past and represent the greatest legacy for the future.
They provide shelter, encourage productivity, embody our culture, and certainly play an important part in life on the planet. In fact, the role of buildings
is constantly changing. Buildings today are life support systems; communication and data terminals; centers of education, justice, and community; and so
much more. They are incredibly expensive to build and maintain and must
constantly be adjusted to function effectively over their life cycle. The economics of building depend largely on proper design and understanding of its
structural components.
A structure in general can be defined in many different manners, all of
which, however, center on the single theme “the arrangement of individual
components (or particles) in a definite (or random) pattern to form the whole.”
Structure is an integral part of nature; it establishes order. It relates various entities or all the elements of a whole, displaying some pattern of organization. It
occurs at any scale, ranging from the molecular structure of material to the laws
of the universe. Everything has structure, even if we have not yet recognized it.
Societies are structured to properly function (e.g., structure of the government:
legislative, executive, judicial branches); language has structure; and the interrelationship of plants and animals with their environment represents another
example of structure in the ecosystem.
According to their origination, there are two categories of structures: the
natural and the man-made objects. All material objects in nature and technique
manifest themselves through the form specific to them. Form in the realm of
the corporeal is the distinctive distribution of the structural substance in three
dimensions. It is geometric.
The material forms in nature and technique perform each in a distinct
manner; they fulfill functional requirements. Functions in this context are not
only the mechanical and instrumental, but also the biological, semantic, and
psychological, or simply the substance-preserving causes and effects.
The specific function is tied to the specific form. Thus, if form is encroached
upon or annihilated, the functions will likewise be afflicted. The preservation
intRoDUCtion
1
of form, therefore, is prerequisite for the perpetuation of functions in the building
environment.
Each structural form, i.e., the object as represented by that form, is inevitably
exposed to the action of environmental forces. Force actions originate first from the
function of the architectural object, second from the characteristics and articulation of the substance, and finally from the conditions of the surroundings. That is
to say, for the existence of an architecture and of its form, it is prerequisite that the
object can bear those forces. It rests upon its capability to cope with forces of various kinds, to resist them. The reliability that grants this capability is the structural
system.
In short, structural systems are the very preservers of the functions of the building environment in natural and man-made systems.
Natural Structures
Everything around us can be visualized as having a structure to it. In fact, this
process of arranging the constituents to form the entity can occur at the molecular
as well as the cosmic level. Man-made structures provide analogies, parallels, and
similarities with the structures in the realm of nature. This seems rational: In his
endeavor to mold the environment to suit his aims, man has forever used nature as
a model. Science and technology emerge from the exploration of nature.
Structural order in nature occurs in all natural forms, ranging from shells to
honeycomb cells, leaf structures to spider webs and soap bubbles. These natural
structures have evolved into their most efficient forms in response to various environmental forces.
The correlation of the natural and man-made structures, however, is based less
upon the proximity of man and nature, but rather upon two basic identities:
∙∙ Both structure categories serve the purpose of safeguarding material forms in
their continuance against acting forces.
∙∙ Both structure categories fulfill that purpose on the basis of the identical
physical laws of mechanics.
In terms of the mechanical process: Natural and man-made structures both affect a
redirection of oncoming forces in order to preserve a definite form that stands in a
definite relation to the functional requirements. Both execute this identically on the
basis of the two principles: flow of forces (load paths) and state of equilibrium. Due
to this underlying and instrumental identity, the structures of the natural objects
are legitimate model comparatives in the design of architectural structures. They
are foremost important sources for learning about the linkage of function, form,
and structure.
The essential cause for differences between the two structure families, both as
physical reality and as notion, is given by the disparity of their origination as shown
in Figure 1.1 below.
2 Building Structures
Natural Structures
Manmade Structures
growth, mutation,
evolution
planning, design,
construction
fusion, fissures,
and decay
cracks, distresses,
demolition
Figure 1.1: Comparison between Natural and Man-Made Structures.
The rudimentary divergences of the two structure categories are intensified
further by the heteronomy of the constituent fabric. The structural forms in nature,
although presenting infinite illustrative material for the multiple ways of structural
behavior and showing ways for the optimization, are not apt to be literally transferred to man-made structures.
However, as integrated forms for both the object’s function and the management
of forces, the structures of nature present classical directives and ideal examples for
efforts in the building design to resolve the existing separation of the technical systems: building structure, space enclosure, services, and communication. Foremost,
they show the great design potential contained in the development of synergetic
structural forms.
Our world and universe are filled with infinitely many examples of naturally occurring or non-man-made structures. The next figures illustrate the gradual degree
of complexity of naturally occurring structural forms and functions, from molecular to cosmic levels (Figure 1.2).
Animals have also been responsible for the creation of some of the most spectacular structural forms on Earth. No one has pinpointed exactly how they have
been able to muster such ingenuity in building their habitats. Images in Figure 1.3
below illustrate such remarkable construction feats.
Man-Made Structures
Ever since humans mixed straw with mud to create a shelter from environmental
elements found in their everyday lives, we have been on a quest to build more with
less (i.e., materials and cost). In the process, we have been able to achieve feats
Introduction 3
(a) Water
(b) Propane molecule
(d) iceberg
(c) Diamond
(e) naturally occurring arch
(f) solar system
Figure 1.2: examples of natural structures.
4
BUiLDinG stRUCtURes
(g) Free-standing trees
of unbelievable stature and construction that is nothing short of miraculous.
We invented an almost inexhaustible number of new building shapes through
transformation and arrangement of basic building elements, through analogies
with nature, the human body, animals, insects, crystallography, machines, flow
forms, and many others.
Figures below illustrate the gradual evolvement of man’s tireless efforts to
build.
(a) A bird's nest
Building Structures
Buildings in general consist of the structural systems, the skin (exterior envelope), the ceilings, the partition walls, mechanical systems to control interior
climate, and electrical and plumbing systems. Structures make spaces within
(b) A spider's web
Figure 1.3: AnimalMade Structures.
Figure 1.4: Examples of Man-Made Structures.
(a) Brick pyramid, Sakkara, Egypt (3900 B. C.)
(b) Sphinx and pyramid, Giza, Egypt (3700 B. C.)
(c) Karnak Temple, (1530 B. C.)
Introduction 5
(d) Parthenon, Athens, Greece (447 B. C.)
(e) Straw and mud huts, Africa
(f) Pure ice igloos
(g) Roman Aqueduct, Segovia, Spain (1st century A. D.)
(h) Pantheon Dome, Rome, Italy (longest dome span for 17 centuries) (126 A. D.)
6 Building Structures
(i) Hagia Sophia, Istanbul, Turkey (532–537)
(j) Eiffel Tower, Paris, France (1887–1889)
(k) Sydney Opera House, Sydney, Australia (1957–1973)
(l) The Bird's Nest: A monument of the
Olympics, a symbol of modern Beijing, China
(2008)
(m) Chrysler Building, New York (1930)
(n) The Burj Tower is the world's tallest
structure, Dubai, United Arab Emirates (2010)
Figure 1.4: Examples of Man-Made Structures (continued).
Introduction 7
a building possible; they give support to the material. Whereas the structures hold
the building up and provide integrity, the exterior skin provides a protective shield
against the outside environment, and the partitions form interior space organizers.
The support structures in buildings are normally integrated with other systems
and sometimes it is difficult to separate them. For example, nonstructural spaceenclosing elements and structure do not have to be separable; for instance, masonry
walls can be a bearing wall, a shear wall, and a partition wall, as in many apartment
and hotel buildings.
The design of building structures involves a number of complex interactive
processes. These may include a range of environmental factors, be they cultural
or physical, to the building organism itself, which must properly function. There
are distinct building characteristics referring to building form, function, material,
economy, and the processes of construction and fabrication. The aspects defining
architecture range from a purely subjective nature perceiving the building as art to
the rational considerations based on an organized body of scientific knowledge and
technology. The various factors influencing building design in general and building
structure in particular include the following criteria:
∙∙ Architectural program
∙∙ Building perspective (e.g., political, social, legal restrictions, topography, geology, orientation, sustainability, existing urban fabric)
∙∙ Aesthetic experience, and massing (architectural massing is the act of composing and manipulating 3D forms into a unified, coherent architectural
configuration)
∙∙ Organization and planning of spaces
∙∙ Enclosure and materials
∙∙ Activity and functioning system (e.g., circulation)
∙∙ Mechanical, Electrical, and Plumbing (MEP) systems
∙∙ Fire protection and security
∙∙ Construction techniques and fabrication.
Building and structure are inseparable and intimately related to each other. The structure of a building is the framework that preserves its integrity in response to external
and internal excitations. Structure makes the artistic expression of a building and
spaces within the building possible. Next, we will explore the different elements of
building structures that are fundamental to the crossover design in building.
A building structure is a collection of elements that work together as a framework
to satisfy the above-mentioned criteria, and safely transfers applied gravitational
and lateral loads to the supporting foundations (Figures 1.5 and 1.6).
Several kinds of basic structural elements are found in buildings. Each embodies
a different type of structural behavior. Buildings are usually assemblies of a number
of different kinds of structural elements. The six basic elements types are as follows:
∙∙ Cables;
∙∙ Arches;
∙∙ Trusses;
8 Building Structures
Slab: Concrete on
metal deck
Figure 1.5: Typical Building
Structure.
Foundation
Gusset Plate
Brace
Beam
Column
Figure 1.6: Components of a
Typical Building Structure.
∙∙ Beams;
∙∙ Columns;
∙∙ Shells and plates.
Structure is the primary and solitary instrument for generating form and space
in architecture. Due to this function, structure becomes the essential means
for shaping the material environment of architectural design. Architectural
structure, in its relationship to architectural form, nevertheless commands an
infinite scope for interpretation. Structure can completely be hidden by the
Introduction 9
building form; it can as well become the building form itself. Architectural structure
personifies the creative intent of the designer to unify form, function, material, and
forces. Structure thus presents an aesthetic, inventive medium for both shaping
and experiencing buildings. Thus, architectural structures determine buildings in
fundamental ways: their origination, their being, their consequence. Accordingly,
among the formative forces of architectural design, architectural structure ranks as
an absolute norm.
Consequently, developing structural models, i.e., basic architectural structure
design, is an integral component of any architectural design. Hence, the prevalent
differentiation of structural design from architectural design—as to their objectives,
their procedures, their ranking, and for that matter, their performers—is unfounded
and in contradiction to the cause and idea of architecture. The differentiation of
architectural design and structural design has dissolved into integrated building
design. Thus, a structure is an artifact expressing one of the many aspects of human
creativity, but it is an artifact that cannot be created without a deep respect for the
laws of nature.
Architectural structure relies on the discipline exerted by the laws of natural
sciences. Consequently, building structure must satisfy minimum requirements
that generally:
∙∙ Resist gravity, lateral, and other external and internal excitations;
∙∙ Conform to the architectural requirements and those of the user or owner, or
both;
∙∙ Facilitate, as appropriate, the service systems, such as heating, ventilation,
and air conditioning; horizontal and vertical cabling; and other electrical and
mechanical systems;
∙∙ Have adequate resistance to fire;
∙∙ Enable the building, foundation, and ground to interact properly;
∙∙ Are economical.
Structural Systems
The purpose of architecture, past and present, is to provide and interpret space for
satisfying a number of requirements and functions, which is achieved through the
shaping and organizing of material form.
The material form is subjected to forces that challenge the endurance of form,
and thus threaten its very purpose and meaning. The threat will be warded off in
that acting forces are redirected into courses that don’t encroach upon form and
space (see Figure 1.7). The mechanism effectuating this is called structural system.
Redirection of forces is the cause and essence of structural system. Therefore,
structural systems are inherently deeply rooted in architecture. The interface
between the two, which is referred to as the crossover design, is critical for any
building design. The specific relationship between architecture and structural
10 Building Structures
Figure 1.7: Crossover Design in Building Structures.
system, however, whereby the one incorporates the other, varies greatly throughout
architectural history. In contemporary architecture, we often encounter buildings
whose structures are of major interest for architectural expression, as well as others
that display a predominant connection between structural form and its impact on
architectural space and expression.
In a broad sense of defining structural systems, they are considered to be any
collection of structural elements arranged in specific configurations for the purpose of producing the most efficient medium that organizes space to fulfill program
requirements, and safely transfers applied gravity and lateral loads in any given
structure to the ground. Thus, there is no limit to structural systems or building
shapes and forms, ranging from boxy to compound hybrid, to organic and crystalline shapes. Most conventional buildings are derived from the rectangle, triangle,
circle, trapezoid, cruciform, pinwheel, letter shapes, and other composite figures
usually formed from rectangles.
Structural systems can be categorized into three main groups:
1.Solids;
2.Frameworks of simple elements;
3.Surfaces.
Typically, combinations of those groups, particularly the first three, which encompass a huge and highly competitive variety of systems available for the designer to
choose from, are selected for the construction of any specific building project. In
general, the bases of such selections are
∙∙ Economy;
∙∙ Special structural requirements;
Introduction 11
Figure 1.8: Main Categories of Structural Systems
∙∙ Problems of design and construction; and
∙∙ Materials and scale limitations.
Realistically, however, in the majority of construction projects, the overriding criterion for selecting
systems is that of economy.
The following are a brief listing and examples of the various traditional and innovative structural
systems encompassed under the three main groups mentioned above.
Foundations
Solid and monolithic elements of foundations include footings (single, strip, and mat), slab-ongrade, micro-piles, large piers, and abutments (Figure 1.10). They represent the last structural
components of the load path in a building. Their main purpose is to safely discharge all applied
loads to the supporting soil or rocks without distress or excessive movement.
Frames and Trusses
12 Building Structures
Load-bearing walls
One-way slab element
Cable suspension element
Beam element
Two-way slab element
Hyperbolic paraboloid
Arch and vault elements
Dome
Figure 1.9: Basic Structural Elements.
Structural Walls
Structural walls function as supports for either horizontal spanning systems (i.e., as
bearing walls) or as stabilizing elements for the lateral bracing of structures (i.e., as
Introduction 13
shear walls). Depending on the way they are constructed, walls could be classified
as either solid or framed elements (see Figure 1.11). Examples of such systems can
be found in any building project such as supports for floor and roof structural subsystems. Examples of structural walls acting as lateral bracing elements also abound
(Figure 1.12), but particularly notable are shear walls in skyscrapers. Another function of walls is to retain soils and waters (i.e., retaining walls, seawalls, and dams).
Figure 1.10: Types of Building
Foundations.
(a) Single footing
(b) Combined footing
(c) Pile foundation
14 Building Structures
Slabs
Examples of structural slab elements are shown in Figure 1.13 below.
Framework
Structural framework is simply a combination of some basic structural elements
such as beams, columns, and cables. The classical and most basic manner in which a
framework is constructed is based on the column-and-beam system. It is composed
Figure 1.11: Structural Walls.
Introduction 15
Figure 1.12: Shear Walls.
of two essential elements, the posts (i.e., typically vertical compression members,
columns) and the beams (i.e., typically horizontal elements transferring applied
loads through shearing forces and bending moments). As such, this system could
expand to form the three-dimensional skeletal frame of a building structure.
Figure 1.13: Structural Slab Systems.
16 Building Structures
Examples of such systems are numerous. In New York City, for example, the
American architectural firm of Shreve, Harmon and Lamb selected a structural
steel frame system for the construction of the Empire State Building. The skyscraper, completed in 1931, has 102 stories of office space and stands at a height of
1,250 ft (Figure 1.16).
Figure 1.14: Structural Framework.
(a) One-way frames
(b) Two-way systems
(c) Space frames
Introduction 17
(d) Diagrid framing systems
Figure 1.15: Types of Building Frames.
18 Building Structures
Arches and Vaults
Normally arches and vaults are composed of a curved rigid surface with a certain
thinness, being capable of taking compressive and bending forces. The main objective of the arch is to span horizontally using a structural system, which develops
Figure 1.16: Empire State Building, New York.
Truss Systems
Truss systems are perhaps the most efficient structural framework used for horizontal spanning in roofs and floors as well as for bridges. Trusses rely exclusively on
straight members framed together either through bolted, welded, or combinationtype joints.
The advantage of trusses lies in the lightness and openness of their structure
in allowing required electrical cables/conduits and mechanical pipes/ducts to go
through them without requiring additional dedicated space. Due to their relatively
lightweight and rigid triangularly based framework, trusses are capable of spanning
very large distances.
Introduction 19
Figure 1.17: Simple Trusses.
Figure 1.18: Wood Truss.
Figure 1.19: Typical Steel Truss System.
predominantly internal compression forces. A temporary structure is erected in
order to hold the arch solid elements while being placed, starting at the ends of the
arch.
Once both sides of the arch are completed, the last voussoir (i.e., keystone) is
inserted in place and the temporary shoring is removed. If expanded in one direction, arches form tunnel vaults. If crossed at 90 degrees, two vaults will produce
a groin vault. The Romans and the Greeks displayed remarkable mastery of the
engineering and building of such systems. The city of Rome, Italy, is still filled with
many salient structures containing some of the most beautiful arches, such as the
20 Building Structures
Figure 1.20: Surface Structures—Arches.
Figure 1.21: Surface Structures—Vaults.
Pantheon (124 AD) and the Colosseum’s amphitheater (80 AD).
Plates and Shells
Plates and shells are classified as surface structures
due to their very high ratio of surface area to thickness. They encompass slabs, panels, folded plates,
and shells with various simple, as well as complex,
Figure 1.22: The Pantheon (143 ft. wide).
Introduction 21
geometric surface configurations (spherical, hyperboloids, etc.). Architectural history is filled with cases of those forms. Figures 1.23a, b, and c show illustrations of
folded plates and shells structures.
(a) Surface structures—folded plates
(b) Surface structures—shells
(c) Surface structures—shells
22 Building Structures
(d) L'Oceanografic (Valencia, Spain)
Figure 1.23: Plates and Shells Structural Systems.
Tensile Structures
Membrane structure represents one of the oldest forms of structural systems.
Membrane structures use cable as their main support system. It is a simple architectural reality that one of the most economical ways to span a large distance is with
cable. This, in turn, derives from the unique physical fact that steel in tension is
several times stronger than steel in any other form of loading. Cables are relatively
light, and unlike beams, arches, or trusses, they have virtually no rigidity or stiffness.
Figure 1.24: Surface Structures—Membrane.
Architects and engineers have been successful in using such systems whenever
a large span is desired. Completed in 1999 and spanning 1,198 ft. (365 m), the
Millennium Dome in Greenwich, England, is a contemporary example of the
cable-suspended tent system.
Introduction 23
Figure 1.25: Millennium Dome (London, UK)
Figure 1.26: Golden Gate Bridge (San Francisco, CA)
Due to its elegant design and structural efficiency, cable-suspended construction has also been mandatory when it comes to large span bridges. Again, a very
visible case of such a system is the Golden Gate Bridge in San Francisco, California.
Hybrid Systems
Any combination of the systems above constitutes a hybrid system. The fact that
humans have had an eternal quest to “span more with less” has led to continual efforts by designers and builders to create systems whereby materials and geometrics
were used to optimize the performance of structures as well as the construction
process.
Examples on the use of such innovative systems in today’s construction are
abundant. They include, but are not limited to:
∙∙ Framed Domes
∙∙ Truss-Based Arches
∙∙ Framed Plates and Shells
Framed Domes
Frame-based arrangements (see Figure 1.27), otherwise known as geodesic domes,
were made famous by R. Buckminster Fuller back in the 1960s; they use multiple,
very stable triangularly shaped units, which, when framed side by side, form a shelllike structure. Architects Murphy and Mackey in St. Louis, Missouri, implemented
this framing concept in their design of the Climatron, a geodesic dome used in the
Missouri Botanical Gardens.
24 Building Structures
(a) Framed domes
(b) Geodesic domes
Figure 1.27: Examples of Framed Domes.
An interesting example in the recent contemporary structure is the United
Kingdom Biodome. This massive biodome will be located at the Chester Zoo, will
cover 172,000 square feet, and will be home to a number of animals, including a
band of gorillas, a troop of chimpanzees, several okapi (solitary giraffe-like creatures), flocks of birds, etc. The arched form of the biodome will be created via a
diagonal grid structure referred to as a gridshell—that is, a framework formed by
hollow steel members, each up to 400 mm in depth and 150 mm in width. The
members will be arranged in a triangular grid pattern and clad in a series of lightweight, largely transparent ethylene tetrafluoroethylene (ETFE) pillows that will be
stretched between the steelwork and held in tension.
Truss-Based Arches
Modern arches and vaults are being achieved using structural systems such as
framed truss members (Figures 1.28 and 1.29) forming the arch without resorting to
solid elements. In 1998, Tate and Snyder architects employed steel trusses, formed
into arches, to support the curved roofing system in the main lobby of Terminal D
of McCarran International Airport in Las Vegas, Nevada.
Framed Plates and Shells
There are many cases where plates and shells have been successfully incorporated
into the architectural design. Truss-supported floor and roof plates (i.e., slabs):
Abundant examples of the use of such a system are in existence due to the economy
Introduction 25
Figure 1.28: Arched Trusses.
Figure 1.29a: Arched Trusses.
Figure 1.29b: Arched Trusses.
Figure 1.29c: Arched Trusses.
of the configuration as well as the convenience of leaving utilities duct space available for other use.
Building structures can be made from different materials. The most commonly
used building materials for structural systems include:
∙∙ Wood Framing;
26 Building Structures
Figure 1.30: A Trussed Hyper-Plate.
∙∙
∙∙
∙∙
∙∙
∙∙
∙∙
∙∙
Figure 1.31: A Trussed Cylindrical Shell.
Heavy Timber Framing;
Hot-Rolled Steel Framing;
Cold-Formed Steel Framing;
Cast-in-Place Concrete Framing;
Precast Concrete;
Masonry Structures;
Membrane Structures.
The following figures show examples of these systems.
(a) Wood framing structure
(b) Heavy timber framing
(c) Cold-formed steel framing
Introduction 27
(d) Cast-in-place concrete framing
(e) Reinforced concrete building
(f) Membrane structures
Figure 1.32: Types of Structural Framing Systems.
Study of Buildings and Structures
Buildings and structures are directly related to one another. The external excitation
that acts on buildings will result in internal forces within buildings. The forces flow
through the structural members to the subsurface, requiring foundations to act
as discharge structures to the supporting soils. The members must be strong and
stiff enough to resist the internal forces. In other words, building structures must
provide the necessary strength and stiffness to resist the vertical forces of gravity
and the lateral forces from wind and earthquakes, and guide those forces safely
to the ground. In addition to strength and stiffness, stability is a requirement for
building structures to maintain their shapes and forms.
Architecture embodies ineffable yet sensible, aesthetic, and functional qualities
that merge from a number of domains such as space, form, and structure. The particular connections that exist between structures and architecture are referred to
as the crossover design in building structures or architectural structures. The field
investigates the following main issues:
∙∙ What purpose does the structure serve in architectural design?
∙∙ What requirements govern the conditions determining its overall shape, materials, detailed form, sustainability, economy, and safety?
∙∙ In what ways do these conditions relate to one another?
The field encompasses also a number of other distinct but related areas such as
28 Building Structures
∙∙
∙∙
∙∙
∙∙
Programmatic Aspects of Buildings;
Statics;
Strength of Materials;
Structural Planning and Design.
To understand the fundamentals of the crossover design in building structures
in a wider sense as being part of an architectural context also means seeing their
forms as space-organizing and defining elements, as devices that control the inflow
and quality of natural ventilation and light, enhance the soundscape and acoustic
quality, that reflect contemporary sustainability concerns, or any number of other
functional requirements. Hence, architectural structures can serve many purposes
concurrently to providing supports, which need to be kept in mind, not only to
enable a more profound understanding of the development of structural forms, but
also to undertake an appropriate and enlightening evaluation of structures within
architectural design context.
Building Structural Classifications
As described previously, crossover designs in building structures range from those
conceived of as pure support systems (i.e., structural efficiency), to those designed
to act as visual images (i.e., aesthetic experiences). Typically these multi-aspects of
the structural systems are not wholly separate from one another. Instead, they tend
to blend and their divisions to blur so that certain formal features of a structure may
both be explained by science and engineering, and also be understood in light of
their architectural spaces and the establishment of architectural expressions.
In studying building structures, structural systems can be further categorized
into the following:
a. Superstructure
b. Substructure
Introduction 29
Figure 1.33: Main Subsystems of Building Structures.
Superstructure
The vertical extension of the structure above the foundation consists of:
∙∙ Shell: Roof, exterior walls, doors, and windows.
∙∙ Structure: System required to support the shell of the structure, as well as
the interior floors, walls, and partitions, and to transfer the loads safely to the
substructure. It includes, for example, columns, beams, load-bearing walls,
and floor and roof structures.
The purpose of the superstructure in buildings may be summarized into:
Strength: It must be stable and strong enough (i.e., provide necessary
strength) to keep the building up under any type of load action,
so it does not collapse either on a local or global scale (e.g., due
to buckling, instability, yielding, fracture, etc.). The superstructure makes the building and spaces within the building possible;
it gives support to other building systems.
Serviceability: It must be durable and stiff enough to control the functional
performance, such as excessive deformation, vibrations, and
drift.
30 Building Structures
Order: It functions as a spatial and dimensional organizer besides identifying assembly or construction systems.
Form Presenter: It defines the spatial configuration, reflects other meanings,
and is part of aesthetics, i.e., aesthetics as a branch of artistic
philosophy.
The superstructure is, in turn, composed of two main supporting systems:
a. Vertical resisting system;
b. Lateral resisting system.
Figures 1.34a and 1.34b illustrate examples of vertical and lateral resisting structural
systems.
Figure 1.34a: Vertical and Lateral Structural Systems.
Introduction 31
Figure 1.34b: Vertical and Lateral Structural Systems.
Substructure
It is the lowest subdivision of the structures and is referred to as its foundation
structure. It must be designed to both accommodate the form and layout of the
superstructure above and respond to varying conditions of soil, rock, and water
below. Similar to the superstructure, substructure must also satisfy strength and
serviceability requirements. It is commonly divided into the following groups
∙∙ Shallow foundation (single footing, combined footing, trip footing, mat foundation)
∙∙ Deep foundation (driving piles, drilled piles)
∙∙ Retaining structures
32 Building Structures
These substructure elements are most commonly constructed of reinforced concrete (see Figure 1.10). As compared to the design of the superstructure, additional
consideration must be given to concrete substructure elements due to permanent
exposure to potentially harmful materials, less precise construction tolerance, and
even the possibility of unintentional mixing with soil and groundwater.
Form, Space, and Function
The structuring of building spaces is highly dependent upon the choice of a structural system and its particular articulation on the practical function that is associated with it. The general impact of structural systems on architectural design can
be summarized into the following areas:
∙∙
∙∙
∙∙
∙∙
Formal and spatial composition of the building;
Scale and proportions of forms and spaces;
Functional partitioning of spaces according to purpose and use;
Access and circulation to the vertical and horizontal paths of movement
through a building;
∙∙ Integration with the nature and built environment (ventilation, lighting,
acoustics, etc.);
∙∙ Sensory and cultural characteristics of the building site.
The composition and orchestration of the structural systems influence design in
various ways. In general, there are four fundamental ways in which building structures can correlate to the form of an architectural design:
∙∙
∙∙
∙∙
∙∙
Exposing the structural systems;
Partially exposing the structural systems;
Hiding the structural systems;
Celebrating the structural systems.
Structural Expressions: Exposing Structural
Systems
The structural systems in this case are the architecture. The history of contemporary
buildings is filled with examples of this category. A simple example is represented
by Rowell Brokaw Architects’ building in downtown Eugene, Oregon (Figure 1.35).
Introduction 33
Figure 1.35: Exposing Structural Steel System, Rowell Brokaw Architects Building in Downtown Eugene, Oregon.
34 Building Structures
Structural Expressions: Hiding Structural Systems
Designers may simply decide to have the structural form as subordinate to the
outward architectural form and want freedom of expression for the shell, without
considering how the structural system might aid or hinder formal decisions (see
Figure 1.36a). A good example is the house designed by Studio Daniel Libeskind
(Architectural Record, 2011). The floor plan and section are shown in Figures 1.36b
and 1.36c, respectively. The actual structural form is entirely hidden by the bronze
and stainless steel panels cladding the exterior, which change in the tones and hues
from dark copper to purple to dark brown, depending on the position of the observer
and the time of day, as do the panels’ alternation from a shimmering reflectiveness
to a matte opacity. The stainless steel elements are mounted on plywood structural
insulated panels (SIPs), with the entirety supported on a steel frame consisting of
four angular arches.
Figure 1.36a: House by Studio Daniel Libeskind, Connecticut, 2010.
Introduction 35
Figure 1.36b: Building Section Plan (House by Studio Daniel Libeskind, Connecticut, 2010).
Figure 1.36c: Floor Plan (House by Studio Daniel Libeskind, Connecticut, 2010).
36 Building Structures
Structural Expressions: Partially Exposing the
Structural Systems
Often, the structural systems are partially exposed. This represents an intermediate
manifestation between the two previously mentioned expressions.
Structural Expressions: Celebrating the Structural
Systems
Rather than being merely exposed, the structural systems can be exploited as a
design feature, celebrating the form and materiality of the structure. There are also
those structures that dominate by the sheer forcefulness with which they express
the way they resolve conflict between technical and artistic programs.
They often become iconic symbols due to their striking aesthetic experience. To
name some examples: the Eiffel Tower, the Sydney Opera House, L’Oceanogràfic
(Valencia, Spain), the Rosa Parks Transit Center (in Detroit, Michigan) (see Figures
1.37 and 1.38), and the French Art Gallery at the Centre Pompidou-Metz, France
(Figures 1.39 and 1.40). An interesting example also is the Spanish Pavilion for the
Expo 2010 in Shanghai, China (Figure 1.41). It has a complex basketlike structure
woven from lightweight steel and wicker. The Spanish Pavilion displays a good
example for the intense dialogue between architecture and the development of the
underlying structural system.
Figure 1.37: L'Oceanografic (Valencia, Spain)
Introduction 37
Figure 1.38: The Rosa Parks Transit Center, a new multimodal facility in downtown Detroit, Michigan, 2009.
Figure 1.39 (above) and 1.40 (right): The French Art Gallery at the Centre PompidouMetz, France.
38 Building Structures
Figure 1.41: Spanish Pavilion for the Expo 2010, in Shanghai, China.
Introduction 39
Learning objectives: to understand types and nature of
loading conditions a structure may be exposed to and their
effects and modeling as forces represented mathematically
as vectors.
2
T
he external loads that act on buildings will result in internal forces within
buildings. The forces flow through the structural members to the subsurface, requiring foundations to act as discharge media to the supporting
soils. The structural elements must be strong and stiff enough to resist the
internal forces. In other words, building structures must provide the necessary strength and stiffness to resist the vertical forces of gravity and the lateral
forces from wind and earthquakes and guide those forces safely to the ground.
In addition to strength and stiffness, stability is a necessary requirement for
building structures to maintain their shapes and forms.
Therefore, one of the first steps taken before one can properly design any
building structure is to understand the relationships between applied and
induced loads on buildings and the behaviors of such systems. In order to
achieve that objective, we need to understand the types of loading conditions a
structure may be exposed to during its life expectancy.
This chapter introduces common gravitational and lateral loads and their
modeling as forces represented mathematically (i.e., idealized) as vectors. It
also addresses the translational and rotational effects of forces on building
structures.
loadS oN BuildiNgS
FoRceS on BUILDInGS
iNtroductioN
Structures in nature and technique serve the purpose of not only controlling
their own object weight but also of receiving additional loads (forces). This
mechanical action is what is termed “Resistance and Stability.” The essence of
the bearing process, however, is not the rather overt action of receiving loads,
but the internal operating process of transmitting them. Without the capability
of transferring and discharging loads, a structure cannot bear its own (dead)
load, much less additional (live) external loads.
FoRCes on BUiLDinGs
41
The structure, thus, functions in three subsequent operations:
∙∙ Load reception;
∙∙ Load transfer;
∙∙ Load discharge.
Types of loads or excitations on building structures can be attributed to many
different sources. Figures 2.1a and 2.1b show causes and conditions of various
buildings’ excitations.
Static Loads
These are gravity-type forces/loads that are applied slowly to the building, which
result in gradual deformations in the structure. They include
∙∙ Dead loads: Static fixed loads that are relatively permanent in character, such
as the building structure and other permanently attached building elements.
They normally act vertically downward. Self-weight of building materials is a
good example. Table 2.1 gives the unit weight of different building materials.
∙∙ Live loads: Transient and moving loads in or on the building, such as occupants and furnishings. Table 2.3 shows typical values of live loads.
∙∙ Snow: Loads due to snow.
∙∙ Rain: Forces induced by rainfall.
∙∙ Settlement forces: Loads caused by the movement of the foundation of the
building.
∙∙ Soil and water pressure: Loads resulting from ground or water pressure.
∙∙ Thermal stresses: Forces induced by thermal expansion or contraction.
Figure 2.1a: Examples of Building Loads.
42 Building Structures
Figure 2.1b: Examples of Building Loads.
Dynamic Loads
These are forces/loads applied to a structure that are time dependent, i.e., change
their magnitude and direction with time. They normally introduce vibration and
other time-dependent deformations. For building structures, the main loads of that
nature are
∙∙ Wind Loads: They normally have resonant effects.
∙∙ Seismic Loads: These are earthquake-generated impacts on a building structure.
Forces on Buildings 43
Table 2.1: Average unit weight of building materials
Material
lb/ft3 (pcf )
Metals
Table 2.2 Average dead loads of structural components
Component
lb/ft2 (psf )
Roofs
Asphalt shingles
2.0
Steel (hot rolled)
490
Cement asbestos shingles
4.0
Aluminum (cast)
165
3-ply and gravel
5.5
5-ply and gravel
6.5
Concrete
Plain
145
Corrugated metal: 20 US std. gauge
1.7
Reinforced
150
Corrugated metal: 28 US std. gauge
0.8
Lightweight
100
Lumber sheathing (1 inch)
2.5
Brick
Clay and concrete
100–130
Plywood sheathing (1 inch)
3.0
Wood
Wood
35
Concrete slab (per inch of thickness)
12.5
Hollow-core concrete planks
45–50
Floors
Soil
Dry clayey soil
63
Plywood (1 inch)
3.0
Moist clayey soil
110
Steel decking
2-10
Sand and gravel
120
Brick (4 inch)
40.0
Brick (8 inch)
80.0
Concrete Masonry Units (CMU) (4 inch)
30.0
Concrete Masonry Units (CMU) (8 inch)
55.0
Wood Studs (2 x 4 @ 16 in o.c.)
8.0
44 Building Structures
Walls
Table 2.3: Typical values of live loads
Component
lb/ft2 (psf )
Libraries (stacks)
150
Office buildings (offices)
60
Apartment (corridors)
80
Office buildings (lobbies)
100
Apartment (public rooms)
100
Residential dwellings (floors)
40
Apartment (private rooms)
40
Dining rooms and restaurants
100
Stores, retails (first floor)
100
Schools (classrooms)
40
Stores, retails (upper floors)
75
Schools (corridors)
80
Theaters (aisles, corridors)
100
Stairs
100
Theaters (balconies)
60
Dance halls
100
Theaters (floors)
60
Garages (public)
100
Theaters (stage floors)
150
Gymnasiums
100
Hospitals (operating rooms, laboratories)
60
Hotels (corridors)
100
Hospitals (patient rooms)
40
Hotels (guest rooms)
40
Hospitals (corridors above first floor)
80
Hotels (public rooms)
100
Roofs (flat, pitched, curved)
20
Libraries (reading rooms)
60
Roofs (roof gardens or assembly purposes)
100
Atypical Loads
In addition to the above-mentioned loads, buildings can be subjected to abnormal
or accidental loads, which generally have a low probability of occurrence and need
special design approaches. Some examples of these loads include:
∙∙
∙∙
∙∙
∙∙
∙∙
Blast or explosion loads
Tornado
Tsunami
Terrorism and vandalism attacks
Collisions, impact loads (e.g., caused by vehicle, ship, aircraft).
Forces on Buildings 45
Example 2.1
Determine the average load on a typical floor for a 10-story office building having:
--------
Plan dimension of 100 ft x 200 ft; typical floor height
Live load per typical floor is Steel framing and walls load per floor Lightweight concrete on metal deck MEP (mechanical, electrical, and plumbing)
Ceiling, finish Partitions
Total dead load per floor
Total live load per floor
= 12 ft
= 60 psf
= 20 psf
= 45 psf
= 10 psf
= 5 psf
= 15 psf
= 95 psf
= 60 psf
The roof live load is 30 psf and the roof dead load can be assumed to be equal to 70
psf. Find also the total building weight and its density.
Solution
The average load per floor = floor (dead load + live load) x area
= (60 psf + 95 psf ) (100 ft x 200 ft) = 3,000,000 lb or 3,000 kips.
The total building load = loads of the roof + loads of the floors
= (30 psf + 70 psf ) (100 ft x 200 ft) + (3,000,000 lb) x 9
= 2,000,000 lb + 27,000,000 lb = 29,000,000 lb or 29,000 kips.
The density of the building can be estimated as = Total load/volume of the building
= 2,900,000 lb/(100 ft x 200 ft x 120 ft) = 12.1 pcf.
Flow of Forces
Aside from their space- and form-defining functions, structural systems, in essence, exist to resist forces that result from the two types of loads mentioned in the
previous section, namely static and dynamic loads.
Flow of forces in a building structure involves the systematic process of determining loads and support reactions of individual structural members as they, in
turn, affect the loading of other structural elements. This process is called LOAD
PATH or the FLOW OF FORCES (see Figures 2.2 and 2.3). It is the basic conceptual
46 Building Structures
Figure 2.2: Flow of Forces in a Simple Structure.
Figure 2.3: Flow of Forces in a Wood.
image for the design of an architectural structure; it is its
fundamental idea. As a path of forces, it is also a measure for the economy of the structure. In general, the
shorter the load path to its foundation and the fewer
elements involved in doing so, the greater the economy
and efficiency of the structure.
Forces
Figure 2.4: A Force of 10 lbs at Point A.
A force is defined as an action of one body on another,
characterized by its
-- point of application;
-- magnitude;
Forces on Buildings 47
Figure 2.5: Tension and Compression Forces.
-- line of action; and
-- sense.
For example, Figure 2.4 shows a pull force of 10 lb acting on Object A. In this
case, the point of application is A, the magnitude is 10 lb, the line of action is inclined 30 degrees from the horizontal line, and lastly the sense of it is tension.
Figure 2.5 illustrates more examples of tension and compression forces.
In studying forces in structure, a force must be represented graphically by the
components shown in Figure 2.6 as follows:
48 Building Structures
Figure 2.6: Representation of a Force Graphically.
Force Systems
Structures are generally subjected to various combinations of forces. Forces viewed
collectively are generally referred to as a force system. Force systems are often
identified by the types of system on which they act:
-------
Collinear;
Coplanar, parallel;
Coplanar, concurrent;
Noncoplanar, parallel.
Non-coplanar, concurrent
Non-coplanar, non-concurrent
The following diagrams illustrate these systems of forces for the forces F1, F2,
and F3.
Figure 2.8 below depicts practical examples of force systems in building
structures.
Distributed Forces
Distributed forces are loads that cover an area (2D, i.e., two-dimensional elements)
such as slabs and walls or a linear element (1D, i.e., one-dimensional elements) like
beams or columns. The following figures illustrate different types of distributed
loads in building elements.
(i) 2D-Uniformly distributed loads: These are generally distinct by a constant
load intensity (i.e., uniform) across the surface to which it is applied. They are
Forces on Buildings 49
Figure 2.7: Force Systems.
Figure 2.8 (right) depicts practical examples of force systems in building structures.
50 Building Structures
Figure 2.8b: Sign Support Structure (coplanar,
parallel force system).
Figure 2.8a: Forces on a Retaining Wall (nonconcurrent, coplanar).
Figure 2.9a: Uniformly Distributed Load on a Building Floor.
Forces on Buildings 51
Figure 2.9b: 2D-Linearly Distributed Load on Structural Wall.
Figure 2.9c: Nonlinearly Distributed Load on a Building Floor Slab.
52 Building Structures
Figure 2.9d: A Steel Beam Subjected to a Uniform Distributed Load, w.
Figure 2.9e: Triangular Distributed Load.
Figure 2.9f: Trapezoidal Distributed Load.
Forces on Buildings 53
also often called area loads. The commonly encountered type includes the
dead weight of a floor with a constant thickness and live Q on building floors
and roofs (see Figure 2.9a). Units used are for force per unit area such as lb/
ft2 (psf), kips/ft2 (ksf), N/m2 (Pa), and kN/ m2 (kPa).
(ii) 2D-Non-uniformly distributed loads: These types of loads can be linearly
distributed or nonlinearly distributed. Examples include wind pressure on
walls, water and soil pressure on retaining walls, and snow loads on roofs
(Figures 2.9b and 2.9c). In these types of distributions, force intensities could
vary in one direction or two directions.
(iii) 1D-Uniform loads: These are forces per unit length of the element (Figure
2.9d). They are also known as lineal or linear loads. Normally, they are defined for beams, columns, and walls. Units for such load distributions are
lb/ft, kip/ft, N/m, and kN/m.
(iv) 1D-linearly distributed loads: In contrast to 1D-uniformly distrusted loads,
in these types of distributions, force intensities vary linearly in one direction
only, as shown in Figures 2.9e and 2.9f.
External and Internal Forces
The forces we defined so far represent external actions on the buildings. They are
entirely responsible for the external behavior of the building, i.e., move or stay at
rest. On the other hand, internal forces are forces that hold together the particle
forming the structural element. For instance, consider the truss structure for the
bridge shown in the photos below. The external load on the truss is induced by the
traffic, and the internal forces within each member of the truss are the result of
these external forces as shown in Figure 2.10.
Figure 2.10: Bridge Truss.
54 Building Structures
Figure 2.11: Joint Details of the Truss Showing the Internal Forces in Each Member.
Figure 2.12a: External forces.
Figure 2.12b: Internal forces.
Internal forces can be exposed by passing a section through the structural
members. Figure 2.12a represents a truss with external forces, whereas Figure 2.12b
shows the internal forces in each member of the truss.
Resultant of Concurrent Forces
Experimental evidence shows that the combined effect of two forces may be represented by a single resultant force. For example, in Figure 2.6, two people are
pulling a boat in two different directions. The boat will move as a consequence of
the resultant of these two forces in a third direction. In other words, these two pull
Forces on Buildings 55
Figure 2.13: A Tent Pole is Fixed by Two Tension Ropes.
Figure 2.14a: Two Students Pulling a Toolbox.
Figure 2.14b: The Resultant Force R of the Two Students’ Pull.
56 Building Structures
(tension) forces can be replaced by a single force that has the same effect of these
two forces, i.e., the resultant force (see Figures 2.13 and 2.14).
In Figure 2.9, a tent pole is positioned by two tension ropes. The resultant of
these two tension forces is the compression force acting vertically on the pole.
In Figure 2.10, the resultant force R of two tension forces exerted by the students
is equivalent to the diagonal of a parallelogram that contains the two forces in adjacent legs. Therefore, forces do not obey the rules of addition defined in ordinary
arithmetic or algebra. For example, two forces acting at a right angle to each other,
one of 4 lb and the other of 3 lb, add up to a force of 5 lb, not to a force of 7 lb.
Thus, forces are vectors possessing magnitude and direction, which add according to the parallelogram law (Figure 2.15).
To generalize the parallelogram law to determine the resultant of more than two
forces, consider a point A of a structural element acted upon by several coplanar
forces, i.e., several forces contained in the same plane. Since the forces considered
here all pass through A, they are also said to be concurrent.
Figure 2.15: Parallelogram Law.
Figure 2.16b: Polygon Rule.
Figure 2.16a: Several Concurrent Coplanar Forces.
Forces on Buildings 57
The resultant of the forces acting on A may be obtained by the repeated application of the parallelogram law. The process is known as the polygon rule.
Since the use of the polygon rule is equivalent to the repeated application of the
parallelogram law, the vector R thus obtained represents the resultant of the given
concurrent forces, i.e., the single force that has the same effect on the particle A as
the given forces.
The polygon rule can be applied graphically by using the tip-to-tail method to
obtain resultant forces. Consider the column loaded by the forces A, B, and C, as
shown in Figure 2.17.
The resultant (R) of the forces A, B, and C action at the top of the column can
be obtained graphically using the Tip-to-Tail method as illustrated in Figures 2.18a,
2.18b, and 2.18c.
It is important to note that in the tip-to-tail graphical method, forces are drawn
to scale but not necessarily in any particular sequence.
Figure 2.17: Column Subjected to Three Forces: A, B, and C.
58 Building Structures
Figure 2.18a: Tip-to-tail Graphical Method for Determining the Resultant.
Figure 2.18b: Tip-to-tail Graphical Method.
Figure 2.18c: Tip-to-tail Graphical Method.
Forces on Buildings 59
Example 2.2
The top end of a column is subjected to three pulling forces, as shown in Figure
2.19. Using the graphical tip-to-tail method, determine the resultant of forces A, B,
and C (magnitude and direction).
Figure 2.19: Column Subjected to Three Forces.
Solution
To use the graphical method, you have basically two options:
i. Using traditional drafting tools, including an engineering scale.
ii. Digitally, using AUTOCAD or SKETCHUP or REVIT.
(i) Manual Drafting
Step 1:
For such problems, the use of grid paper is usually helpful. You begin by redrawing
the forces at an appropriate scale to fit on an 8.5” x 11” grid sheet, a scale of 1 in = 8
lb is suggested. Then set the origin at the intersection of a grid and draw the x- and
y-axes.
Step 2:
Draw a line along the x-axis in the same sense of the force C and then scale a distance of 2.75 inches (22/8 = 2.75).
60 Building Structures
Figure 2.20: Force Polygon.
Step 3:
Establish the slope for force B on the grid paper as shown, then using a pair of
drawing triangles, transfer the line of action of the force B to the head of force C.
Measure a distance of 4.25 inches along that line to form the head of force C. Now,
establish the slope of force A, then transfer it to the head of force B. Measure a
distance of 3.25 inches along the line of action of force A.
Step 4:
To determine the resultant, construct a line from the tail of force C (origin) to the
tip of force A. Then, scale the length of the line to obtain the magnitude of the
resultant R, and using the protractor, measure the slope to establish the direction
of the resultant.
R = 6.25 in x 8 = 50 lb.
Slope = 53°
Figure 2.21. Force Polygon Using AutoCAD.
Forces on Buildings 61
(ii) Digital Drafting—Using AutoCAD
Step 1:
Changing units to decimal makes things easier, because there is no reason to use a
scale (just draw at 1:1). Enter “units” at the command line and change from architectural to decimal if not already set to decimal.
Step 2:
Convert the slope of 15 rise and 8 run into degrees as follows:
Slope = tan-1(15/8) = α = 61.93°
Repeat this for 5 rise and 12 run: slope = tan-1(5/12) = 22.62°
3.25 in
22.62o
R = 6.25 in
3.25 in
4.25 in
53.1°
61.93°
2.75 in
Step 3:
Draw vectors tip-to-tail with separate lines, as you would on paper. To draw a vector at a specific angle and distance, click the first line point and then type @F<angle.
For example, enter @34<61.93 for 34 lb at 61.93° from the x-axis. Remember that
in AutoCAD, 0° angle is to the right (positive x-axis), 90° is up (positive y), 180° left,
and 270° is down, so angles can be calculated from those reference angles, such as
300° would be 30° from the negative y-axis.
Step 4:
When the resultant vector R is drawn, its angle and length (force) can be determined with the distance command, properties palette, or list command.
R = 50 lb and its slope = 53°
(iii) Digital Drafting—Using REVIT Structure
Step 1:
Open a new Revit Structure 2011 project. Make sure you are on the home tab. On
the model panel, click on the model line and start drawing a horizontal line of 22 ft
(using 1 ft = 1 lb) to represent the 22-lb force.
62 Building Structures
Step 2:
Convert the slope of 15 rise and 8 run into degrees as follows:
Slope = tan-1(15/8) α = 61.93°
From the end of the line drawn in step 1, draw a line at 61.93° from the horizontal
measuring a length (34/8 = 4.25 in). This line represents the force, 34 lb.
Step 3:
Repeat step 3 for 5 rise and 12 run: slope = αtan-1(5/12) = 22.62°
From the end of the line drawn in step 1, draw a line at 22.62° from the horizontal
measuring a length of (26/8 = 3.25 in). This line represents the force, 26 lb.
Step 4:
When the resultant vector R is drawn, its angle and length (force) can be determined with the measuring tools from the “Annotate” tab.
R = 50 lb and its slope = 53°
Example 2.3
A tent stake is subjected to two forces P and Q at A. Determine their resultant.
Figure 2.22: A Tent Stake Subjected to Two Forces.
Solution
(i) Manual Drafting
Step 1:
You begin by redrawing the forces at an appropriate scale to fit on an 8.5”x11”in
grid sheet, a scale of 1 in = 10 lb is suggested. Then set the origin at the intersection
of a grid and draw the x- and y-axes as shown below.
Forces on Buildings 63
Step 2:
Using a protractor, measure an angle of 20° and draw a line in the same sense of the
force P and then scale a distance of 4 inches (50/10 = 5 in).
Step 3:
At the tip of force P (point B), using a protractor, measure an angle of 45° and draw
a line in the same sense of the force Q and then scale a distance of 6 inches (100/10
= 10 in).
Step 4:
To determine the resultant, construct a line from the tail of force P (origin) to the
tip of force Q. Then, scale the length of the line to obtain the magnitude of R, and
using the protractor, measure its slope to establish the direction of the resultant: R
= 147 lb, and α= 35°.
Figure 2.23: Force Polygon for Example 2.2.
(ii) Digital Drafting
Step 1:
Changing units to decimal makes things easier, because there is no reason to use a
scale (just draw at 1:1). Enter “units” at command line and change from architectural to decimal if not already set to decimal.
64 Building Structures
Step 2:
Draw vectors tip-to-tail with separate lines, as you would on paper. To draw a
vector at a specific angle and distance, click the first line point and then type @
F<angle. For example, enter @50<20 for 50 lb at 20° from the x-axis. Remember that
in AutoCAD, 0° angle is to the right (positive x-axis), 90° is up (positive y), 180° left,
and 270° is down, so angles can be calculated from those reference angles, such as
300° would be 30° from the negative y-axis.
Step 3:
When the resultant vector is drawn, its angle and length (force) can be determined
with the distance command, properties palette, or list command.
R = 146.84 lb and its slope = 35.04°
Resolution of Forces
We have seen that two or more forces acting on a structural member may be
replaced by a single force that has the same effect on that member. Conversely,
a single force F acting on a member at a point may be replaced by two or more
forces that together have the same effect on the member. These forces are called the
components of the original force F, and the process of substituting them for F is
called resolving the force F into components.
Clearly, for each force F there exists an infinite number of possible sets of components. Sets of two components, Fx and Fy, are the most important as far as practical applications are concerned. These components are known as the Rectangular
Components, since the parallelogram drawn to obtain the two components is
a rectangle, and Fx and Fy are called rectangular components or the x- and ycomponents of the force.
The values of these rectangular components are given by:
Figure 2.24: Components of a Force.
Forces on Buildings 65
Fx = F cos θ
Fy = F sin θ (2.1)
Other important formulas relating the rectangular components to the resultant
are
F = F x2 + F y2 (2.2)
and the angle describing the direction of the resultant is given by:
i = tan- 1
Fy
Fx
(2.3)
Example 2.4
The top of a column is subjected to a force of 80 lb, as shown in Figure 2.25.
Determine the rectangular components of the force F.
Figure 2.25: Example 2.3.
Solution
Step 1:
The rectangular components are given by Equation (2.1):
Fx = F cos θ
66 Building Structures
Fy = F sin θ
Step 2:
Substituting 80 lb for F and 40° for θ, we have:
Fx = 80 lb cos40° = 63.1 lb
Fy = 80 lb sin40° = 51.4 lb
Example 2.5
The staircase shown in Figure 2.26 is carrying a load of 200 lb that is inclined at 45°
from the horizontal line. Determine the x- and y-components of the load and draw
their directions.
Figure 2.26: Staircase Supporting an Inclined Force.
Solution:
Step 1:
The rectangular components are given by:
Fx = F cos θ
Fy = F sin θ
Step 2:
Substituting 250 lb for F and 45° for θ, we have:
Forces on Buildings 67
Fx = 200 lb cos45° = 141.42 lb
Fy = 200 lb sin45° = 141.42 lb
Step 3:
To draw the components, notice the sense of the force and the angle between the
force and the x-axis (see Figure 2.27).
Figure 2.27: X- and y-Components of the Inclined Lload.
Finding Resultant of Forces Using their
Rectangular Components
The resultant of forces acting at a point on a structural member can be obtained by
first resolving all of these forces into their x- and y-components. Then, the resultant
can be obtained by simply adding the respective x- and y-components of each of
these forces to determine the resultant x- and y-components:
Rx = F1x + F2x + F3x and Ry = F1y + F2y + F3y
or, generally:
68 Rx = ∑Fx and Ry = ∑Fy(2.4)
Building Structures
And the resultant is determined from: R =
R 2x + R 2y (2.5)
Then, corresponding direction of the resultant is i = tan- 1 Ry (2.6)
Rx
Example 2.6
The roof truss of the building shown is subjected to a vertical dead load of 4 kips and
a wind load of 5 kips inclined at 30° at point A, as shown in Figure 2.28. Determine
the direction and magnitude of the resultant force acting at point A.
Figure 2.28: Forces Acting on a Roof Truss.
Solution
Step 1:
The rectangular components of any force F are given by:
Fx = F cos θ
Fy = F sin θ
Forces on Buildings 69
The resultant components are obtained from adding the respective rectangular
component of each force:
Rx = F1x + F2x and Ry = F1y + F2y(2.7)
Step 2:
Substituting 5 kips for F1 and 30° for θ, we have:
F1x = (5 kips) cos30° = 4.330 k
F1y = (5 kips) sin30° = 2.5 k
The second force (4 kips) F2 is parallel to the y-axis, therefore it does not have an
x-component:
F2x = 0 k
F2y = 4.0 k
Step 3:
Substituting in Equation 2.7 to obtain the resultant components:
Rx = F1x + F2x = 4.330 + 0 = 4.33 kips and
Ry = F1y + F2y = 2.5 + 4.0 = 6.5 kips
The resultant force R is given by:
R = R 2x + R 2y = (4.33) 2 + (6.5) 2 = 7.81kips
The direction of the resultant R is determined from:
70 i = tan- 1
Building Structures
Ry
= tan- 1 6.5 = 56.33°
Rx
4.33
Learning objectives: to understand the meaning of static
equilibrium of forces and its application in analyzing building
forces and stability.
3
E
quilibrium refers essentially to a state of rest or balance. The fundamental
requirement of equilibrium is concerned with the guarantee that a building or any of its parts, will not move. Any system that obeys Newton’s laws of
motion with the conditions that it does not accelerate and has zero velocity is
said to be in Static Equilibrium. Certain elementary conditions that will ensure the equilibrium of a simple system can be easily visualized in the popular
games displayed in Figures 3.1 and 3.2.
State of
Equilibrium
eQUILIBRIUM oF BUILDInG
StRUctUReS
iNtroductioN
Figure 3.1: state of equilibrium in a tug-of-War Game.
eqUiLiBRiUM oF BUiLDinG stRUCtURes
71
Figure 3.2: States of Equilibrium in a Seesaw Game.
72 Building Structures
On the other hand, Figures 3.3a and 3.3b show that the state of static equilibrium
of the building is violated by extreme wind pressure effect, while 3.3c shows the
effect of ground settlement upon the equilibrium of buildings.
Moment of a Force
Quantitatively, the moment of a force with respect to a reference point is equal to
the product of the force and the perpendicular distance of the force from the point.
The units of a moment are the units of a force and distance expressed as: kips-ft
or ft-kips; lb-ft or ft-lb. In SI units, moment units can be N.m or m.N; kN.m or
m.kN.
Figure 3.3a: Building Equilibrium is Desecrated by Wind Pressure.
Equilibrium of Building Structures 73
Figure 3.3b and c: Building Equilibrium is Desecrated by Wind Pressure (b) and Foundation Settlement (c).
Figure 3.4: Rotational Moment Applied to a Bolt.
74 Building Structures
Figure 3.5: Definition of a Moment of a Force.
Example 3.1
Consider the seesaw shown in Figure 3.6 below. When the seesaw is in static
equilibrium (i.e., balanced) the moment produced by the left-side force equals the
moment produced by the right-side force.
Solution:
The moment produced by the left-side force = F1 x d1 (counterclockwise rotation)
The moment produced by the right-side force = F2 x d2 (clockwise rotation)
At equilibrium, we have F1 x d1 = F2 x d2
Figure 3.6: Balanced Seesaw.
Equilibrium of Building Structures 75
Example 3.2
A cantilevered steel beam is subjected to a force of 2 kips applied at the end of the
beam as shown in Figure 3.7. Determine the moment of this force about the fixed
end of the cantilever.
Figure 3.7: Cantilevered Steel Beam.
Solution:
Moment of a force about a point = Force x Distance
(3.1)
F × d = (2k) x (5ft)= 10k–ft
Example 3.3
Two equal and opposite forces (F=100 lb) act on a beam as shown. Determine the
total bending moment M, due to the two forces, about A, B, and C.
76 Building Structures
Figure 3.8: Beam Loaded with a Couple of Forces.
Solution:
Moment of forces about point A = ∑ Force x Distance
= ∑F × d = (100k) x (1ft) – (100k) x (3ft) = –200 k–ft
Moment of forces about point B = ∑ Force x Distance
= ∑F × d = –(100k) x (1ft) – (100k) x (1ft) = –200 k–ft
Moment of forces about point C = ∑ Force x Distance
= ∑F × d = –(100k) x (3ft) + (100k) x (1ft) = –200 k–ft
Moment of a Couple
From the previous example, it can be concluded that two forces F and -F having
the same magnitude, parallel lines of action, and opposite sense are said to form a
couple. Couples have pure rotational effects on a body with no capacity to translate
the body in the vertical or horizontal direction.
The moment of a couple M, is computed as the product of the force F times the
perpendicular distance d between the two equal and opposite forces:
M=Fxd
The moment of a couple is a constant value and is independent of any specific reference point.
Equilibrium of Building Structures 77
Equivalent Forces
Two systems of forces are equivalent (i.e., have the same effect on a structural element) if we can transform one of them into the other by means of one or several of
the following:
∙∙
∙∙
∙∙
∙∙
∙∙
Replacing two forces acting on the same point by their resultant.
Resolving a force into two components.
Canceling two equal and opposite forces acting on the same point.
Attaching to the same point two equal and opposite forces.
Moving a force along its line of action.
Resultant of Parallel Forces
Parallel forces do occur frequently in building structures and finding their resultant
and its location will simplify the analysis. For example, in a typical steel framing
system, girders support beams or joists. Since all these beams or joists are parallel,
they produce parallel reaction forces acting on the girder as shown in Figure 3.9
below.
Figure 3.9: Parallel Beams’ Reactions Supported by a Girder.
78 Building Structures
The equivalent resultant R of the forces F1, F2, and F3 must produce the same
translational tendency as forces F1, F2, and F3, as well as the same rotational effect.
Since forces by definition have magnitude, direction, sense, and a point of application, it is necessary to establish the exact location of the resultant R from some
given reference point.
Location of the resultant R is obtained by employing the principle of moment.
Simplifying Figure 3.9 by drawing the free body diagram of the girder is shown in
Figure 3.10.
Figure 3.10: Girder supporting two Parallel Beam Reactions.
By summing moments of the forces F1, F2, and F3 about point C, we have:
∑MC: -F1 × a - F2 × (a + b) - F3 × (a + b + c) = -R × (x)
(3.2)
Solving for the distance x, we have:
x = F1 ×a + F2 × (a + b) + F3 (a + b + c)
R
(3.3)
The magnitude of the resultant of parallel forces is simply the algebraic sum of
their magnitudes:
R = F1+ F2 + F3
(3.4)
eqUiLiBRiUM oF BUiLDinG stRUCtURes
79
Figure 3.11: A Wood Frame Supporting a Metal Sign.
Example 3.3
In the wood frame shown below, determine the resultant force action on the beam
and its location.
Solution:
Step 1:
Draw the beam with the applied parallel loads as shown below:
Figure 3.12: Wood Beam Supporting Two Parallel Loads.
Step 2:
The resultant R of parallel forces is simply the algebraic sum of these forces,
R = 8 k + 8 k = 16 k
Step 3:
The location of the resultant force can be determined by summing moments of
the forces about either point A or B. However, in this case, since the loading is
80 Building Structures
symmetrical about the resultant, force will be effective at the center of the beam, as
shown in Figure 3.13.
Figure 3.13: Resultant of Two Parallel Loads.
Figure 3.14: Building Elevation Showing Wind Forces.
Example 3.4
A four-story building is subjected to wind forces concentrated at each of the floor
levels, as illustrated in Figure 3.14. Solve for the resultant wind force and its location above the ground.
Equilibrium of Building Structures 81
Solution:
Step 1:
The magnitude of the resultant force R of the parallel wind forces F1, F2, F3, and F4 is
R = F1 + F2 + F3 + F4 = 2.5 + 5 + 7 + 10 = 24.5 k
Step 2:
The location of the resultant R is obtained by taking the moment of forces about
point A:
R . y = F1 . 10 + F2 . 20 + F3 . 30 + F4 . 40
(24.5 k) y = (2.5 k) (10 ft) + (5 ft) (20 ft) + (7 k) (30 ft) + (10 k) (40 ft)
y = 30 ft
Equilibrium of a Structural Element
Referring to the tug-of war example, the state of static equilibrium can be defined at
the situation of a deadlock (i.e., resultant force, R = 0) as follows:
Figure 3.15: A Deadlock Condition in a Tug-of-War Game.
-F1 – F2 + F3 + F4 = 0(3.5)
And generally for a collinear force in the x-direction, the static equilibrium
condition can be expressed mathematically as:
∑Fx = 0(3.6)
Generally, for a two-dimensional system, the mathematical requirements to
establish equilibrium can be stated as:
82 Building Structures
Rx = ∑Fx = 0; Ry = ∑Fy = 0;
M = ∑Mi = 0
(3.7)
Where,
Rx = Resultant force in x-direction
Ry = Resultant force in y-direction
∑Mi = Summation of moments about any point i
Figure 3.16: Concrete Column.
Example 3.5
The concrete column shown is subjected to two horizontal forces, as shown in
Figure 3.16. Determine the force and moment resultant reactions at the base of the
column at equilibrium.
Solution:
Step 1:
Resultant of forces in the x-direction, Rx = 100 lb + 120 lb = 220 lb
Step 2:
Resultant moment MR at the base =
∑MB = (120 lb) (10 ft + 10 ft) + (100 lb) (10 ft) = 3,400 lb-ft
3400lb - ft
= 3.4k - ft
1000
Equilibrium of Building Structures 83
Free Body Diagram (FBD)
An essential step in solving equilibrium problems involves the drawing of free body
diagrams (FBD). This is the essential key to studying the mechanics of architectural
structures. Everything in structural mechanics is reduced to forces in FBD. This
method of simplification is very efficient in reducing an apparently complex building structure into a concise force system.
The following examples illustrate the establishment of FBDs for different building structures.
Figure 3.17: Steel Girder Supporting Beams.
Figure 3.18: Free Body Diagram (FBD) for the Structure Shown in Figure 3.17.
84 Building Structures
Example 3.6
A girder supporting two floor beams.
Example 3.7
A crane carrying 5.0 kips weight plus its own weight of 2 kips.
Figure 3.19: Crane Carrying Load.
Figure 3.20: Free Body Diagram (FBD) of the Crane Shown in
Figure 3.19.
Example 3.8
Two edge beams (B1 and B2) supporting a reinforced concrete slab.
Figure 3.22: Free Body Diagram (FBD) of Beams B1 and B2.
Figure 3.21: A Simple Reinforced Concrete Frame.
Equilibrium of Building Structures 85
Reactions at Supports and Connections
The following table illustrates reactions produced at different supports and connections along with their analytical model symbol for two-dimensional structural
members.
Table 3.1: Reaction and support types
In the table above, reactions in group I are equivalent to a force with a known
line of action.
Whereas in group II, reaction is of unknown magnitude and direction. Or for
simplicity, this force can be resolved into two rectangular components (x- and ycomponents), which reduces this case into two unknown forces (Rx and Ry).
86 Building Structures
In group III, reactions are equivalent to a force with unknown magnitude and
direction and a moment. By resolving the unknown force into its rectangular
components in a similar way to that of group III, the fixed support produces two
unknown translational reaction (Rx and Ry) and one rotational moment reaction
(M).
The next table shows figures of some examples of building support structures
and their idealization for analysis:
Table 3.2:
Equilibrium of Building Structures 87
88 Building Structures
Equilibrium of Building Structures 89
Example 3.9
For the beam shown below, loaded with an inclined force F, draw its Free Body
Diagram (FBD) and show all of the reactions at the supports.
Figure 3.23: A Simply Supported Beam Loaded with a Force F.
Solution:
Figure 3.24: Free Body Diagram (FBD) of the Beam AB.
Example 3.10
A roof wood beam is subjected to an inclined load of 500 lb, as illustrated in Figure
3.25.
Figure 3.25: Roof Beam.
90 Building Structures
Solution:
Step 1:
Draw the FBD of the roof wood beam:
Figure 3.26: FBD for a Roof Wood Beam Supporting 500 lb.
Step 2:
Resolve the force 500 lb into its rectangular components, i.e., the x- and
y-components:
The inclination of the force 500 lb is given by a rise of 4 and a run of 3. Using the
Pythagorean theorem for a right-angle triangle, we have
Fx = 3 (500lb) = 300lb
5
Fy = 4 (500lb) = 400lb
5
Now, draw these components on the FBD of the beam as shown below:
Figure 3.27: FBD for a Roof Wood Beam Showing the x- and y-Components.
Equilibrium of Building Structures 91
Step 3:
Solve static equilibrium equations.
Rx = ∑Fx = 0: therefore, Ax - 300 lb = 0, or Ax = 300 lb
Similarly for the y-components
Ry = ∑Fy = 0: → Ay+By – 400 lb = 0
Using symmetry, the reactions Ay = By = 400lb = 200 lb
2
Example 3.11
A cantilevered steel beam is supported at midpoint by a cable, as shown in Figure
3.28. Draw the FBD of the beam ABC and solve for the unknown reactions at A.
Figure 3.28: Cantilevered Steel Beam.
92 Building Structures
Solution:
Step 1:
Draw the FBD of the cantilevered steel beam:
Figure 3.29: Free Body Diagram of the Steel Beam ABC.
Step 2:
Resolve the tension force in cable BD T into its rectangular components, i.e., the
x- and y-components:
The slope of the cable can be obtained from the geometry given. Using the
Pythagorean theorem for right-angle triangles, we have
10 (T) = T
10 2
2
Ty = 10 (T) = T
10 2
2
Tx =
Step 3:
Solve static equilibrium equations.
First, consider summing the moments of all forces about A:
∑MA = 0:
500 lb (20 ft) – Ty (10 ft) = 0
Ty = 10,000/10 = 1,000 lb
Substituting for Ty from step 2, we have
T = 1, 000lb or
2
T = √2 (1000 lb)=1414.2 lb
Second, consider the equilibrium of forces in the x-direction:
Equilibrium of Building Structures 93
Rx = ∑Fx = 0: → Tx – Ax =0 or Ax = Tx
Substituting for Tx from step 2, we have
Ax = T =
2
2 (1000lb)
= 1000lb
2
The third equilibrium equation is the summation of forces in the y-direction:
Ry = ∑Fy = 0: → Ay – (500 lb) – Ty = 0
Substitute for Ty from step 2, we get
Ay – (500 lb) – T
2
Ay – (500 lb) –
= 0 or
2 (1000lb)
= 0 → Ay = 1,500 lb
2
Example 3.12
A structural steel framing elevation for a two-story building is subjected to lateral
and vertical loads as shown in Figure 3.30 below. Draw the FBD and determine the
horizontal and vertical reactions at support A.
Figure 3.30: Loads on a Two-Story Structural Steel Frame.
94 Building Structures
Solution:
Step 1:
Draw the FBD of the two-story framing elevation as illustrated below.
Figure 3.31: FBD of the Steel Frame.
Step 2:
Consider the equilibrium of forces in the x-direction:
Rx = ∑Fx = 0: → (5 k) + (2 k) + Ax = 0
Which yields, Ax = -7 kips
Second, consider summing the moments of all forces about B:
∑MB = 0:
-(Ay)(35 ft) – (2 k)(15 ft) – (5 k)(25 ft) + (10.5 k)(35 ft/2) + (5.25 k)(35 ft/2) = 0
Which gives, Ay = -3.45 kips
Equilibrium of Building Structures 95
Learning objectives: to understand vertical load paths, framing systems, and their relationships in building structures.
4
A
building is a cellular aggregate of spaces that must be dimensionally coordinated so that it can be constructed. This dimensional network forms
patterns of a certain order. The key to understanding that order lies in the
nature of the structural support systems.
Buildings basically consist of the support structure, the exterior envelope,
the ceilings, and the partitions. Structure makes spaces within a building possible—it gives support to the material. Whereas the structure holds the building up, the exterior envelope provides a protective shield against the outside
environment, and the partitions form interior space dividers. Most buildings
consist of horizontal planes (floor and roof structures), the supporting vertical planes (columns, walls, frames, etc.), and the foundations. The horizontal
planes tie the vertical planes together to achieve some type of a rigid 3D effect,
and the foundations make the transition from the building to the ground possible. Keep in mind, however, that structure not only occurs on the large scale
of the building, which tends to be more of an organizational nature, but also on
the small scale of the detail, which is more metaphorical and physical and on a
more human scale.
Although the structure’s primary responsibility is that of support to transfer
loads to the ground, it also functions as a spatial and dimensional organizer.
Should the designer decide to expose the structure rather than hide it behind
skin in order to articulate its purpose, then the structure may also enrich the
quality of space. The designer may treat the structure not just in the minimal sense as support, but superimpose other layers of meaning to enrich its
expression.
The structure resists the vertical action of the gravity loads; that is its own
weight, as well as the nonpermanent live or occupancy loads. It also resists
the horizontal force action of wind and earthquakes; in other words, it must
guarantee lateral stability of the building.
The horizontal and vertical structural building members must transmit the
external and internal loads to the ground. The load path is the course that loads
LoAD PAtH: VeRtIcAL FoRceS
iNtroductioN
LoAD PAtH: veRtiCAL FoRCes
97
travel from, where it acts to where it is resisted. The load path may be short and
direct, or long and indirect and suddenly interrupted, causing a detour. The paths
the loads may take along horizontal and vertical building planes depend on the
structure layout, which must respond to the functional organization of the building where the columns and walls may help to separate and reinforce the spaces to
allow for different activities. Loads (forces) travel along load paths, and the analysis
method is often referred to as load tracing.
Figure 4.1 illustrates the vertical load path of different building structures. In
Figure 4.1a, the weight of the slab is transmitted to the vertical walls and then to the
foundation. In 4.1b, the decking is supported by joists resting on walls. Figure 4.1c
shows the load path from the slab to the beams and then to the supporting walls.
The vertical of forces in 4.1d travel from the decking to joists and then transmit
their reactions to the supporting beams. The beams, in turn, transfer vertical loads
to the walls. The loads in Figure 4.1e flow from the slab to the floor joist and then
to the floor beams. The beams transmit their loads to girders, which are resting on
columns. The final destination of all vertical loads is the foundation structure and
the supporting soils.
98 (a)
(b)
(c)
(d)
Building Structures
(e)
(g) Forces flow in curved shell
(f) Forces flow in conical shell
Load Path: Vertical Forces 99
(h) Forces flow in cross vault
(k) Forces flow in ribbed dome
Figure 4.1: Flow of Vertical Forces.
100 Building Structures
Knowledge of the mechanisms for directing forces in other directions or to other
structural members is the basic requisite for analyzing and designing structures.
The theory underlying the possibilities of how to redirect forces is the core of the
knowledge on structures and is the basis for a systematic process in architectural
structures.
To develop such knowledge, consider the load path in a structural system given
in Figures 4.2a and 4.2b. The direct load path is represented by Figure 4.2a, where
forces are transported immediately to the ground. The load path system in Figure
4.2b is not direct with the least overall transport distance.
Figures 4.2c to 4.2f show various arrangements of structural members to improve load transmission and support effectiveness.
(a)
(b)
Load Path: Vertical Forces 101
(c)
(d)
(e)
(f)
Figure 4.2: Understanding Load Path.
102 Building Structures
Figure 4.3a shows the load path in a fan structure that is normally used in steel
and wood construction. The load-carrying capacity can be improved by shortening
the load path. as shown in Figure 4.3b. The treelike branched structure in Figures
4.3c and 4.3d are even more effective as the buckling lengths of the compression
members are reduced.
(a)
(b)
(c)
(d)
Figure 4.3: Examples for Variation of Load Paths.
Load Path: Vertical Forces 103
The examples shown in Figures 4.2 and 4.3 can also be seen in natural plant support structures. They do compare very well with engineered branched structures.
Both direct forces with minimum detours (e.g., in bushes and shrubs). The form
and structure of trees is predetermined by genetic coding, which ensures that, in
the case of the tree, the trunks and branches form closed mesh (bracing).
In a typical building structure, various arrangements of vertical support structures are possible. For instance, Figures 4.4 and 4.5 depict vertical forces path from
floor to beam to columns, and finally to foundation elements.
The flow of forces in a building structure does not pose problems as long as the
object form follows the direction of the acting forces. In the case of gravitational or
lateral loads, such a situation would exist if structure is connected in the shortest
and most direct route with the point-of-load discharge, i.e., the ground foundation
as shown in the previous examples in Figures 4.1, 4.3, and 4.4. A problem, however,
will arise, when the flow of forces does not take such a direct route but has to accept
detours.
Figure 4.4a: Vertical Flow of Forces in Wood Structures.
104 Building Structures
Figure 4.5: Vertical Flow of Forces in Steel Structure.
Tributary Areas
Tributary area is area supported by a single structural member. In a typical building
construction, the slab or decking is supported by a number of beams. Thus, each
beam is going to support a part of the decking or slab area. That part is called the
tributary area. Figures 4.6 and 4.7 below show the tributary area for each single
beam.
Fundamental to understanding vertical load path is the determination of the
tributary (contributing) area, i.e., the load area that each supporting beam will
carry. Notwithstanding the simplicity of this concept, its visualization is frequently
the first error made in designing a structural system.
Figure 4.8 shows steel open-web joists supporting the roof metal deck. The
tributary area for each joist is illustrated in the framing plan (Figure 4.8b).
The following examples show various tributary areas for beam and truss support
systems.
Load Path: Vertical Forces 105
Figure 4.6: Tributary Area and Load Path.
Figure 4.7: Tributary Area for Each Beam.
Figure 4.8b: Framing Plan.
Figure 4.8a: Steel Structural System.
106 Building Structures
Figure 4.9: Tributary Areas Change with the Spacing of Supporting Beams.
One-Way and Two-Way Spanning Systems
One-way span action may be assumed for a rectangular diaphragm having the
proportions of 1:2 or greater, even if it is also supported along the short sides
(Figure 4.10). On the other hand, square slabs with the proportions less than 1:2
and properly reinforced in both directions are considered to span in two ways and
carry load to each of the four supporting beams. The loads may be considered to be
distributed in the two perpendicular directions to the supporting beams, according
to the tributary areas formed by the intersection of 45° lines extending from the
columns, as shown in Figures 4.11 and 4.12.
Load Path: Vertical Forces 107
Figure 4.10: One-Way Spanning Slab.
Figure 4.11: Load Distribution in Two-Way Spanning Slab.
108 Building Structures
(b) Steel framing plan showing load distributions.
(c) Loading transferred to beams B7 and B11.
Figure 4.12: Load Distribution in Two-Way Spanning Slab.
Load Path: Vertical Forces 109
Example 4.1
In Figure 4.13, trace the load path from the upper support system to the foundation. Assume total area load of 55 psf.
Solution:
Beam B1:
Tributary area = 10 ft x 15 ft = 150 ft2
Total load = 55 psf x 150 ft2 = 8,250 lb
Load per linear foot, w = 8,250 lb/15 ft = 550 lb/ft
Figure 4.13b: Beam B1
Figure 4.13a: Loading on B1
Reaction (Resistance) at each end of the beam = uniform load x span of the beam/2
= 550 lb/ft x 15 ft/2 = 4,125 lb
Beam B2:
Tributary area = 5 ft x 15 ft = 75 ft2
Total load = 55 psf x 75 ft2 = 4,125 lb
Load per linear foot, w = 4,125 lb/15 ft = 275 lb/ft
Reaction (Resistance) at each end of the beam
110 Building Structures
Figure 4.13d: Beam B2.
Figure 4.13c: Loading on B2.
= uniform load x span of the beam/2
= 275 lb/ft x 15 ft/2 = 2,062.5 lb
Girder G1:
Loads are transferred from each end of
Beam B1 and Beam B2 to girder G1:
B1 = 4,125 lb
B2 = 2,062.5 lb
As shown in Figure 4.13f below:
Reaction (Resistance) at each end of girder G1 = (B2 + B1 + B2)/2
= (2,062.5 + 4,125 + 2,062.5)/2 = 4,125 lb.
Column C1:
Loads are transferred from the girder G1 to the end of the column C1, as illustrated in Figures 4.13h and 4.13g.
Load Path: Vertical Forces 111
Figure 4.13e: Girder G1.
Figure 4.13f: Loads on Girder G1.
112 Building Structures
Figure 4.13h: Columns.
Figure 4.13g: Load on Column
C1.
Stacked vs. End Framing
The following diagrams show the framing options as related to the way beams and
girders are connected. Beams can be stacked over girders, as illustrated in Figures
4.13a and 4.13b. The load tracing for such condition is given in Example 4.1.
Figures 4.14c and 4.14d depict the end framing condition where the beams are
framed into the girder or sometimes into the column.
Load Path: Vertical Forces 113
Figure 4.14: Stacked vs. End Framing Condition.
Openings on Floors and Roofs
It is important to recognize the openings in floors and roofs when tracing loads on
structural elements. The arrows shown in Figure 4.15a indicate the slab floor system
spans from beam B1 to B2 and B2 to B3. Therefore, part of beam B2 is loaded only
from one side, whereas the other part carries load from both sides of the beam, as
114 Building Structures
Figure 4.15a: 3D View of Structural Framing with Opening.
Figure 4.15b: Structural Framing Plan of Figure 4.15a.
illustrated in Figures 4.15a and 4.15b. Also note that since the slab spans parallel
and not perpendicular to the joist (J1), the joist carries negligible vertical load.
Framing Levels
Early in a project’s structural design phase, an initial assumption is made by the
designer about the path across which forces must travel as they move throughout
the structure to the foundation. The load path can be created from a single-level,
double-level, or triple-level framing, as shown in Figure 4.16 below.
One-Level Framing
Although it is not a common framing system, precast hollow-core concrete planks
or heavy-timber-plank decking can be used to span between closely spaced bearing
walls or beams. Spacing of the supports (the distance between bearing walls) is
based on the span capability of the concrete planks or timber decking.
Load Path: Vertical Forces 115
Two-Level Framing
This is a very common floor system that uses a regular, relatively closely spaced
series of secondary beams (called joists) to support a deck. The decking is laid perpendicular to the joist framing. Span distances between bearing walls and beams
affect the size and spacing of the joists’ relatively long spans between bearing walls.
Lighter deck materials such as plywood panels can be used to span between the
closely spaced joists.
(a) One-level framing.
(b) Two-level framing.
Three-Level Framing
When bearing walls are replaced by girders or trusses spanning between columns,
the framing involves three levels. Joist loads are supported by beams, which transmit their reactions to girders or trusses. Each level of framing is arranged perpendicular to the level directly above it. Buildings requiring large open floor areas, free
of bearing walls and with a minimum number of columns, typically rely on the span
capability of joists supported by beams, girders, and/or trusses. The spacing of the
columns and the layering of the beams and girders establish the regular bays that
subdivide the space.
116 Building Structures
The structural framing, if exposed, can contribute significantly to the architectural expression of buildings. Short joists loading relatively long beams yield shallow
joists and deep beams. The individual structural bays are more clearly expressed.
Considerations should include the materials selected for the structural system, the
span capability, and the availability of material and skilled labor. Standard sections
and repetitive spacing of uniform members are generally more economical.
(c) Three-level framing.
Figure 4.16: Relationship between Vertical Supports and Framing
Levels.
Example 4.2
A structural steel framing for a commercial building is given in Figure 4.17 with a
staircase opening. Figure 4.17b depicts the structural steel framing plan.
The floor deck and floor joists span the short direction normal to the parallel
beams that are spaced at 8 ft on center. Assume a total dead load of 120 psf, including the beams’ self-weights and a live load of 80 psf. This load is also assumed for
the staircase but it is assumed on the horizontal projection of the opening. Trace
the vertical loads from the deck to the columns.
Load Path: Vertical Forces 117
Figure 4.17a: Isometric View of the Structural Framing System.
Figure 4.17b: Framing Plan.
Figure 4.17c: 3D Loading Diagram.
118 Building Structures
Solution:
Typical tributary areas for beams are shown in Figure 4.17b. Please note that
beam B7 is not forming a right angle with the other beams; thus it carries a triangular tributary area. Beam B2 is supported by beam B3 framing the stair opening,
and therefore its reaction causes a single load on beam B3 and girder G2. Beam B3,
in turn, rests on beams B1 and B4.
Since most of the beams are supported by the interior girders, their reactions
cause a point load action on the girder. A 3D free body diagram of the structure is
shown in Figure 4.17c.
Figure 4.18: Beam B2.
Beam B2:
Total load = dead load + live load
= 120 psf + 80 psf = 200 psf
Tributary area = 8 ft x 14 ft=112 ft2
Total load = 200 psf x 112 ft2 = 22,400 lb
Load per linear foot, w =22,400 lb/14 ft= 1600 lb/ft
A shortcut for the above calculation is to use the tributary width of 8 ft to come
up with the load per liner foot as follows:
Tributary width = 8 ft
Load per linear foot = load per square foot x tributary width = 200 psf x 8ft =
1600 lb/ft
Convert the load per linear foot to kips = (1600 lb/ft)/1,000= 1.6 k/ft
Reaction (Resistance) at each end of the beam = uniform load x span of the
beam/2
= (1.6 k/ft x 14)/2 = 11.2 kips
Load Path: Vertical Forces 119
Figure 4.19c: FBD of Beam B1.
120 Building Structures
Beam B3:
Reaction at each end = 11.2 k 2 = 5.6 kips
Beam B1:
Loads per linear foot up to the opening = 200 psf x 8 ft = 1600 lb/ft = 1.6 k/ft
Loads per linear foot along the opening = 200 psf x 4 ft = 800 lb/ft = 0.8 k/ft
Plus loads from staircase:
200 psf x 8 ft = 1600 lb/ft = 1.6 k/ft
Total load per liner foot = 2.4 k/ft
The FBD of beam B1 with the resultant of the distributed loads is given in Figure
4.19c. To determine the reaction R1, sum the moments of forces about reaction R2:
(14.4 k)(17 ft) + (5.6 k)(14 ft) + (22.4 k)(7 ft) – R1(20 ft) = 0
R1= 24.0 kips
To determine the reaction R2, consider summing the forces in the vertical
direction:
R1 + R2 = 14.4 k + 5.6 k + 22.4 k = 42.4 k
→ R2 = 42.4 k – 24.0 k = 18.4 k
Beam B4:
Loads per linear foot up to the opening = 200 psf x 8 ft = 1600 lb/ft = 1.6 k/ft
Loads per linear foot along the opening = 200 psf x 4 ft = 800 lb/ft = 0.8 k/ft
The FBD of beam B4 with the resultant of the distributed loads is given in Figure
4.20b.
To determine the reaction R1, sum the moments of forces about reaction R2:
(4.8 k)(17 ft) + (5.6 k)(14 ft) + (22.4 k)(7 ft) – R1(20 ft) = 0
R1= 15.84 k
To determine the reaction R2, consider summing the forces in the vertical
direction:
R1 + R2 = 4.8 k + 5.6 k + 22.4 k = 32.8 k
→ R2 = 32.8 k – 15.84 k = 16.96 k
Load Path: Vertical Forces 121
Figure 4.20a: Beam B4.
Figure 4.20b: Beam B4.
Beam B5:
Load per linear foot = 200 psf x 8 ft = 1600 lb/ft
Reaction (Resistance) at each end of the beam
= uniform load x span of the beam/2
= (1.6 k/ft x 20)/2 = 16.0 kips
Figure 4.21: Beam B5.
122 Building Structures
Figure 4.22a: Beam B6.
Figure 4.22b: Beam B6—Uniform Load.
Beam B6:
The tributary area for this beam consists of two
areas, namely a rectangular and triangular on each side of the beam. This will result
in a trapezoidal
load distribution, as shown in Figure 4.22.
Load per linear foot due to the rectangular tributary
Area = 200 psf x 4 ft = 800 lb/ft = 0.8 k/ft (Figure 4.22a).
Load per linear foot due to the triangular tributary area varies from zero at one
end of the span to a maximum value at the other end = 200 psf x 4ft = 800 lb/ft =
0.8 k/ft (Figure 4.22c).
Beam reactions due to the uniform load are equal:
R1 = R2 = (0.8 k/ft)(20)/2 = 8 kips
To determine beam reactions due to the triangular load, first find the resultant
force of the loading as follows:
W = Area of the triangular loading = (1/2)(20 ft)(0.8 k/ft) = 8 kips
The location of the resultant is at (2/3) x (span) = (2/3) (20 ft) = 13.33 ft from the
left support.
Load Path: Vertical Forces 123
R1
R2
Figure 4.22c: Beam B6—triangular load.
Beam reactions due to the triangular load can be determined from considering
the equilibrium equations for the free body diagram in Figure 4.22c:
Sum moments of forces about the left support:
R2(20) – 8(13.33) = 0
R2 = 5.34 kips
The left reaction can be determined from summing forces in the vertical
direction:
R1 + R2 = 8; therefore, R1 = 8.0 k – 5.34 k = 2.66 kips
Total reaction R1 = 8 k + 2.66 k = 10.66 K and total reaction R2= 8 k + 5.34 k =
13.34 k.
Beam B7:
The tributary area for this beam consists of a triangular load, as given in Figure
4.22d below:
To find out the reactions at the supports, sum moments of forces about the left
support:
R2(20) – 8(13.33) = 0
R2 = 5.34 kips
The left reaction can be determined from summing forces in the vertical
direction:
124 Building Structures
Figure 4.22d: Beam B7—triangular load.
R1 + R2 = 8; therefore,
R1 = 8.0 k – 5.34 k = 2.66 kips
Girder G1:
The FBD of girder G1shows the loads from beams B4 and B5 (Figure 4.23).
To determine the reactions R2, consider the summation of moments about the
left reaction of the girder:
R2 (32 ft) – (16 k)(8 ft) – (15.84 k)(16 ft) = 0 → R2 = 11.92 k;
R1 = (16 k) + (15.84 k) – 11.92 k = 19.92 k
Figure 4.23: Loads on Girder G1.
Load Path: Vertical Forces 125
Figure 4.24: Loads on Girder G2.
Girder G2:
Sum the moments about the right reaction of G2:
-R1(40 ft) + (13.34 k)(32 ft) + (16 k)(24 ft) + (16.96 k)(16 ft) + (11.2 k)(8 ft) = 0
R1 = 29.3 k
R2 = (13.34 k) + (16.0 k) + (19.963 k) + (11.2 k) – 29.3 k = 28.2 k
Columns:
Figure 4.25 summarizes the forces’ flow in the supporting members.
Figure 4.25: Loads Transferred to Columns.
126 Building Structures