Building Structures Fundamentals of Crossover Design

Revised Edition

By Nawari O. Nawari and Michael Kuenstle

University of Florida Cover image digitally rendered by Audrey M. Gutierrez.

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ISBN: 978-1-62131-045-7 (pbk) / 978-1-60927-366-8 (br) Preface ix s

Companion Material xiii

Acknowledgments xv

Dedication xvii

Chapter 1 Introduction 1

Natural Structures 2 content Man-Made Structures 3 Building Structures 5 Structural Systems 10 Foundations 12 Structural Walls 13 Slabs 15 Framework 15 Truss Systems 19 Arches and Vaults 19 Plates and Shells 21 Tensile Structures 23 Hybrid Systems 24 Study of Buildings and Structures 28 Building Structural Classifications 29 Superstructure 30 Substructure 32 Form, Space, and Function 33

Chapter 2 Forces on Buildings 41

Introduction 41 Loads on Buildings 41 Forces 47 Force Systems 49 Distributed Forces 49 External and Internal Forces 54 Resultant of Concurrent Forces 55 Resolution of Forces 65 Finding Resultant of Forces Using their Rectangular Components 68

Chapter 3 Equilibrium of Building Structures 71

Introduction 71 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

Introduction 97 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

Introduction 127 Wind Loads 128 Calculations of Wind Forces 138 Approximate Method 142 Seismic Loads 143 Fundamental Period 145 Building Configuration Effects 147 Summary 151 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

Introduction 175 Fundamentals 176 Anchorage and load path 176 Stability 180 Analysis 182 Design Notes 194

Chapter 7 Structural Elements: Arches 197

Introduction 197 Fundamentals 200 Modern Arches 202 Stability 209 Analysis 209 Design Notes 219

Chapter 8 Structural Elements: Trusses 221

Introduction 221 Fundamentals 223 Load Path 226 Lateral Stability 228 Analysis 228 Joints under Special Loading Conditions 239 The Method of Sections 242 Design Notes 247

Chapter 9 Shape Factors: Properties of Sections 251

Introduction 251 Centroids of Areas 251 Moment of Inertia 255 Radius of Gyration 257 Section Modulus 258

Chapter 10 Structural Materials: Strength and Behavior 259

Introduction 259 Concept of Stress 259 Shear Stress 260 Bearing Stress 261 Bending Stress 262 Torsion Stress 263 Concept of Deformation 264 Strain 265 Stress-Strain Relationship 265 Thermal Strain and Stresses 271 Design Notes 277

Chapter 11 Structural Elements: Beams 279

Introduction 279 Analysis 285 Relationships among Distributed Load (w), Shear Force (V), and Bending Moment (M) 298 Determinant and Indeterminate Beams 308 Conclusions 310 Stresses in Beams 315 Section Modulus 318 Shear Stresses 320 Shear Stresses in Other Sections 322 Vertical and Horizontal Shear Stresses 324 Summary 326 Beam deformation 327 Deflection Formula 329 Analysis Conclusions 334 Stability 335 Beam Grid Systems 337 Beam Design 340 Design Notes 349

Chapter 12 Structural Elements: Columns 355

Introduction 355 Analysis 357 Slenderness Ratio 360 Stability 362 Design Notes 372

Chapter 13 Vaults and Domes 370

Introduction 370 Fundamentals 381 Domes 386 Stability 388 References 391

Appendix A 393

Units of Measurement

Appendix B 395

Wood Section Properties

Appendix C 397

Wide-flange steel sections properties

Index 407 Th is book strives to elucidate the principles that bridge between building structures and architectural design. Th is is what we callcrossover design in architectural structures. Th e book represents our view of how to critically en- gage the subject and how to understand building structures as an integral component of architectural space-making strategies and building design concepts. Th is 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 Th e 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 PReFAce forms, using simple hand calculation techniques with a minimum of mathematics while determining the preliminary design of a structural member with reasonable accuracy. Th e mathematics is deliberately kept to a basic level so that the main emphasis on behavioral and conceptual concerns is not con- cealed 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 diff erent related fi elds such as building construction and engineering to explore the relationships between structural behavior and architectural design ideas at diff erent scales. Th is 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 fi elds. ∙ 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 as- sociated 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 suc- cess 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 dBuil ing 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 omplementary material to this book is provided in the Analyzing Building Structures: An Exercise and Solutions Manual, which will be published Cin the fall of 2012. Th e book contains exercises and solved problems that will facilitate comprehension of essential information in this book. Th e book will also help improve profi ciency in many aspects of building design. coMPAnIon MAteRIAL coMPAnIon

CoMPAnion MAteRiAL xiii e would like to express our gratitude to many students who have as- sisted in making this publication. Th ey are too numerous for all to be mentionedW 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 eff ort in reviewing illustrations. Greatly appreciated is the help of Luis P. Delfi n, who was a second-year civil engineering student when the fi rst 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

ACKnoWLeDGMents xv this work is dedicated to our parents. DeDIcAtIon

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

n general, building structures are deceptively complex. At their best, they connect us with the past and represent the greatest legacy for the future. ThI ey provide shelter, encourage productivity, embody our culture, and cer- tainly 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. Th ey are incredibly expensive to build and maintain and must constantly be adjusted to function eff ectively over their life cycle. Th e eco- nomics of building depend largely on proper design and understanding of its structural components. A structure in general can be defi ned in many diff erent manners, all of which, however, center on the single theme “the arrangement of individual components (or particles) in a defi nite (or random) pattern to form the whole.” Structure is an integral part of nature; it establishes order. It relates various en- tities 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: IntRoDUctIon legislative, executive, judicial branches); language has structure; and the inter- relationship 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 specifi c to them. Form in the realm of the corporeal is the distinctive distribution of the structural substance in three dimensions. It is geometric. Th e material forms in nature and technique perform each in a distinct manner; they fulfi ll 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 eff ects. Th e specifi c function is tied to the specifi c form. Th us, if form is encroached upon or annihilated, the functions will likewise be affl icted. Th e 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 dBuil ing Structures Natural Structures Manmade Structures

growth, mutation, planning, design, evolution construction

fusion, ssures, cracks, distresses, and decay 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 -oc curring 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 spec- tacular 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 (c) Diamond

(d) iceberg (e) naturally occurring arch

(f) solar system (g) Free-standing trees

Figure 1.2: examples of natural structures.

4 BUiLDinG stRUCtURes 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: Animal- Made 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.)

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 dBuil ing 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 space- enclosing 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, geol- ogy, orientation, sustainability, existing urban fabric) ∙∙ Aesthetic experience, and massing (architectural massing is the act of com- posing 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 dBuil ing Structures Slab: Concrete on metal deck

Foundation Figure 1.5: Typical Building Structure.

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 dBuil ing 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 crystal- line 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-on- grade, 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 Tr us s e s

12 dBuil ing Structures Load-bearing walls Beam element

One-way slab element Two-way slab element Hyperbolic paraboloid

Cable suspension element 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

14 dBuil ing 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 dBuil ing 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 sky- scraper, 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

Introduction 17 (d) Diagrid framing systems

Figure 1.15: Types of Building Frames.

18 dBuil ing 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 combination- type joints. The advantage of trusses lies in thelightness 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 Tr us s e s.

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 dBuil ing Structures Figure 1.20: Surface Structures—Arches.

Figure 1.21: Surface Structures—Vaults.

Pantheon (124 AD) and the Colosseum’s amphi- theater (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

22 dBuil ing 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 shell- like 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 dBuil ing 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 crea- tures), 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 Tr us s e s. Figure 1.29a: Arched Tr us s e s.

Figure 1.29b: Arched Tr us s e s. Figure 1.29c: Arched Tr us s e s.

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 dBuil ing 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 dBuil ing 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 dBuil ing Structures Order: It functions as a spatial and dimensional organizer besides iden- tifying 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 dBuil ing 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 associ- ated 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

Thestructural 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 dBuil ing 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 dBuil ing 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 Pompidou- Metz, France.

38 dBuil ing 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

iNtroductioN

he external loads that act on buildings will result in internal forces within buildings. Th e forces fl ow through the structural members to the subsurface,T requiring foundations to act as discharge media to the supporting soils. Th e structural elements must be strong and stiff enough to resist the internal forces. In other words, building structures must provide the necessary strength and stiff ness 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 stiff ness, stability is a necessary requirement for building structures to maintain their shapes and forms. Th erefore, one of the fi rst 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. Th is 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 eff ects of forces on building structures.

loadS oN BuildiNgS FoRceS on BUILDInGS FoRceS on Structures in nature and technique serve the purpose of not only controlling their own object weight but also of receiving additional loads (forces). Th is mechanical action is what is termed “Resistance and Stability.” Th e 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 occu- pants 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 dBuil ing 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 Table 2.2 Average dead loads of structural components

Material lb/ft3 (pcf) Component lb/ft2 (psf)

Metals Roofs Asphalt shingles 2.0

Steel (hot rolled) 490 Cement asbestos shingles 4.0

Aluminum (cast) 165 3-ply and gravel 5.5

Concrete 5-ply and gravel 6.5

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 Floors Concrete slab (per inch of thickness) 12.5

Soil Hollow-core concrete planks 45–50

Dry clayey soil 63 Plywood (1 inch) 3.0

Moist clayey soil 110 Steel decking 2-10

Sand and gravel 120 Walls 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 dBuil ing Structures 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 = 12 ft -- Live load per typical floor is = 60 psf -- Steel framing and walls load per floor = 20 psf -- Lightweight concrete on metal deck = 45 psf -- MEP (mechanical, electrical, and plumbing) = 10 psf -- Ceiling, finish = 5 psf -- Partitions = 15 psf

Total dead load per floor = 95 psf Total live load per floor = 60 psf

The oofr 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) xarea = (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 dBuil ing 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 dBuil ing 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 forcesF 1, 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 dBuil ing 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 dBuil ing 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 dBuil ing 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 dBuil ing 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 ad- jacent 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 onA 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 particleA 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 dBuil ing 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 dBuil ing 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

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 dBuil ing 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 dBuil ing 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

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 theRectangular 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 y- components 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 22 FF=+xyF (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 Fy = F sin θ

66 dBuil ing Structures 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

Theresultant 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:

Rx = ∑Fx and Ry = ∑Fy (2.4)

68 dBuil ing Structures 22 And the resultant is determined from: RR=+xyR (2.5)

-1 Ry Then, corresponding direction of the resultant is i = tan (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 forceF 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 forceR is given by: 22 22 RR=+xyRk=+(4.33) (6.5)7= .81 ips

The direction of the resultantR is determined from:

--11Ry 6.5 i ==tantan = 56.33° Rx 4.33

70 dBuil ing Structures Learning objectives: to understand the meaning of static equilibrium of forces and its application in analyzing building forces and stability. 3

iNtroductioN

quilibrium refers essentially to a state of rest or balance. Th e fundamental requirement of equilibrium is concerned with the guarantee that a build- ingE 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 en- sure the equilibrium of a simple system can be easily visualized in the popular games displayed in Figures 3.1 and 3.2. StRUctUReS

State of Equilibrium eQUILIBRIUM oF BUILDInG BUILDInG eQUILIBRIUM oF

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 dBuil ing 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 dBuil ing 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 =F 1 x d1 (counterclockwise rotation)

The moment produced by the right-side force =F 2 x d2 (clockwise rotation)

At equilibrium, we have F1 x d1 = F2 x d2

Figure 3.6: Balanced S e e s aw.

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 dBuil ing 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 = F x d

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 dBuil ing Structures Th e equivalent resultantR 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 eff ect. Since forces by defi nition 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:

Fa12Fa()bF3 ()abc x = ××+++++ R (3.3)

Th e 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 dBuil ing 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 forceR 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 resultantR 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 dBuil ing 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 =-3.4kft 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 dBuil ing 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 y-

components), which reduces this case into two unknown forces (Rx and Ry).

86 dBuil ing 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 dBuil ing 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 dBuil ing 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)300 lb 5 Fy ==4 (500lb)400 lb 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 400lb Using symmetry, the reactions Ay = By = 2 = 200 lb

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 dBuil ing 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 TTx ==() 10 22 10 T TTy ==() 10 2 2

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 2 or

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 T 2(1000lb ) Ax == = 1000lb 22 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 T A – (500 lb) – = 0 or y 2

2(1000lb ) → Ay – (500 lb) – 2 = 0 Ay = 1,500 lb

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 dBuil ing 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

iNtroductioN

building is a cellular aggregate of spaces that must be dimensionally co- ordinated so that it can be constructed. Th is dimensional network forms patternsA of a certain order. Th e 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 (fl oor and roof structures), the supporting vertical planes (columns, walls, frames, etc.), and the foundations. Th e horizontal planes tie the vertical planes together to achieve some type of a rigid 3D eff ect, 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. Th e designer may treat the structure not just in the mini- mal sense as support, but superimpose other layers of meaning to enrich its expression. Th e 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. Th e horizontal and vertical structural building members must transmit the external and internal loads to the ground. Th e load path is the course that loads FoRceS VeRtIcAL PAtH: LoAD

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.

(a) (b)

98 dBuil ing 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 dBuil ing 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 dBuil ing 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 dBuil ing 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 dBuil ing 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 dBuil ing Structures (b) Steel framing plan showing load distributions.

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 dBuil ing 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 dBuil ing 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 dBuil ing 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 dBuil ing 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.

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 dBuil ing 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 dBuil ing 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

TheFBD 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

TheFBD 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 dBuil ing 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 dBuil ing 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 dBuil ing Structures