Statics and Strength of Materials for Architecture and Building Construction PDF

Statics and Strength of Materials for Architecture and Building Construction  Definition of Terms Measurement U.S. Units Metric (S.I.) a measure of length in...

Statics and Strength of Materials for Architecture and Building Construction  Definition of Terms Measurement U.S. Units Metric (S.I.) a measure of length inch (in. or ′′) millimeter (mm) feet (ft. or ′) meter (m) a measure of area square inches (in. ) 2 square millimeters (mm2) square feet (ft.2) square meters (m2) a measure of mass pound mass (lbm) kilogram (kg) a measure of force pound (lb. or #) newton (N) kilopound = 1,000 lb. (k) kilonewton = 1,000 N (kN) a measure of stress (force/area) 2 psi (lb./in. or #/in. ) 2 pascal (N/m2) ksi (k/in.2) a measure of pressure psf (lb./ft.2 or #/ft.2) kilopascal = 1,000 Pa moment (force × distance) pound-feet (lb.-ft. or #-ft.) newton-meter (N-m) kip-feet (k-ft.) kilonewton-meter (kN-m) a load distributed over length ω (lb./ft., #/ft., or plf) ω (kN/m) density (weight/volume) γ (lb./ft. or #/ft. ) 3 3 γ (kN/m3) force = (mass) × (acceleration); acceleration due to gravity: 32.17 ft./sec.2 = 9.807 m/sec.2 Conversions 1 m = 39.37 in. 1 ft. = 0.3048 m 1 m2 = 10.76 ft.2 1 ft.2 = 92.9 × 10-3 m2 1 kg = 2.205 lb.-mass 1 lbm = 0.4536 kg 1 kN = 224.8 lb.-force 1 lb. = 4.448 N 1 kPa = 20.89 lb./ft.2 1 lb./ft.2 = 47.88 Pa 1 MPa = 145 lb./in.2 1 lb./in.2 = 6.895 kPa 1 kg/m = 0.672 lbm/ft. 1 lbm/ft. = 1.488 kg/m 1 kN/m = 68.52 lb./ft. 1 lb./ft = 14.59 N/m Prefix Symbol Factor giga- G 109 or 1,000,000,000 mega- M 106 or 1,000,000 kilo- k 103 or 1,000 milli- m 10-3 or 0.001 Also, refer to Appendix Table A-7. xi Contents CHAPTER 1 INTRODUCTION 1 1.1 Definition of Structure 1 1.2 Structural Design 2 1.3 Parallels in Nature 3 1.4 Loads on Structures 5 1.5 Basic Functional Requirements 9 1.6 Architectural Issues 11 CHAPTER 2 STATICS 15 2.1 Characteristics of a Force 15 2.2 Vector Addition 23 2.3 Force Systems 29 2.4 Equilibrium Equations: Two-Dimensional 61 2.5 Free-Body Diagrams of Rigid Bodies 74 2.6 Statical Indeterminacy and Improper Constraints 86 CHAPTER 3 ANALYSIS OF SELECTED DETERMINATE STRUCTURAL SYSTEMS 96 3.1 Equilibrium of a Particle 96 3.2 Equilibrium of Rigid Bodies 111 3.3 Plane Trusses 119 3.4 Pinned Frames (Multiforce Members) 153 3.5 Three-Hinged Arches 164 3.6 Retaining Walls 175 CHAPTER 4 LOAD TRACING 195 4.1 Load Tracing 195 4.2 Lateral Stability Load Tracing 231 CHAPTER 5 STRENGTH OF MATERIALS 251 5.1 Stress and Strain 251 5.2 Elasticity, Strength, and Deformation 267 5.3 Other Material Properties 274 5.4 Thermal Effects 289 5.5 Statically Indeterminate Members (Axially Loaded) 294 xiii xiv Contents CHAPTER 6 CROSS-SECTIONAL PROPERTIES OF STRUCTURAL MEMBERS 300 6.1 Center of Gravity—Centroids 300 6.2 Moment of Inertia of an Area 311 6.3 Moment of Inertia of Composite Areas 318 6.4 Radius of Gyration 329 CHAPTER 7 BENDING AND SHEAR IN SIMPLE BEAMS 332 7.1 Classification of Beams and Loads 332 7.2 Shear and Bending Moment 337 7.3 Equilibrium Method for Shear and Moment Diagrams 340 7.4 Relationship Between Load, Transverse Shear, and Bending Moment 346 7.5 Semigraphical Method for Load, Shear, and Moment Diagrams 348 CHAPTER 8 BENDING AND SHEAR STRESSES IN BEAMS 365 8.1 Flexural Strain 366 8.2 Flexural (Bending) Stress Equation 368 8.3 Shearing Stress—Longitudinal and Transverse 382 8.4 Development of the General Shear Stress Equation 384 8.5 Deflection in Beams 402 8.6 Lateral Buckling in Beams 419 8.7 Introduction to Load Resistance Factor Design (LRFD) 422 CHAPTER 9 COLUMN ANALYSIS AND DESIGN 438 9.1 Short and Long Columns—Modes of Failure 439 9.2 End Support Conditions and Lateral Bracing 446 9.3 Axially Loaded Steel Columns 456 9.4 Axially Loaded Wood Columns 474 9.5 Columns Subjected to Combined Loading or Eccentricity 487 CHAPTER 10 STRUCTURAL CONNECTIONS 494 10.1 Steel Bolted Connections 495 10.2 Welded Connections 519 10.3 Common Framing Details in Steel 531 CHAPTER 11 STRUCTURE, CONSTRUCTION, AND ARCHITECTURE 538 11.1 Initiation of Project—Predesign 539 11.2 Design Process 540 11.3 Schematic Design 542 11.4 Design Development and Construction Documents 544 11.5 Integration of Building Systems 555 11.6 Construction Sequence 561 11.7 Conclusion 563 Contents xv APPENDIX TABLES FOR STRUCTURAL DESIGN 565 Lumber Section Properties 567 (a) Dimensioned Sizes—Rafters, Joists, and Studs 567 (b) Beams and Columns 567 Allowable Stress Design for Shapes Used as Beams 568 Structural Steel—Wide-Flange Shapes 570 Structural Steel—American Standard Shapes and Channels 573 Structural Steel—Tubing (Square) and Pipe 574 Structural Steel—Angles 575 Definition of Metric (S.I.) Terms and Conversion Tables 576 Wide Flange Shapes (Abridged Listing)—S.I. Metric 577 Elastic Section Modulus—U.S. and S.I. Metric 578 Western Glue-Laminated Sections—U.S. and S.I. Metric 579 Plastic Section Modulus—Selected Beam Shapes 581 ANSWERS TO SELECTED PROBLEMS 583 INDEX 589 1 Introduction 1.1 DEFINITION OF STRUCTURE Structure is defined as something made up of interdepen- dent parts in a definite pattern of organization (Figures 1.1 and 1.2)—an interrelation of parts as determined by the general character of the whole. Structure, particularly in the natural world, is a way of achieving the most strength from the least material through the most appropriate arrangement of elements within a form suitable for its intended use. The primary function of a building structure is to support Figure 1.1 Radial, spiral pattern of the and redirect loads and forces safely to the ground. spider web. Building structures are constantly withstanding the forces of wind, the effects of gravity, vibrations, and sometimes even earthquakes. The subject of structure is all-encompassing; everything has its own unique form. A cloud, a seashell, a tree, a grain of sand, the human body—each is a miracle of structural design. Buildings, like any other physical entity, require structural frameworks to maintain their existence in a recognizable physical form. To structure also means to build—to make use of solid materials (timber, masonry, steel, concrete) in such a way as to assemble an interconnected whole that creates space suitable to a particular function or functions and to protect the internal space from undesirable external elements. Figure 1.2 Bow and lattice structure of A structure, whether large or small, must be stable and the currach, an Irish workboat. Stresses on the durable, must satisfy the intended function(s) for which it hull are evenly distributed through the was built, and must achieve an economy or efficiency— longitudinal stringers, which are held that is, maximum results with minimum means (Figure 1.3). together by steam-bent oak ribs. As stated in Sir Isaac Newton’s Principia: Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes. Figure 1.3 Metacarpal bone from a vulture wing and an open-web steel truss with web members in the configuration of a Warren Truss. 1 2 Chapter 1 1.2 STRUCTURAL DESIGN Structural design is essentially a process that involves bal- ancing between applied forces and the materials that resist these forces. Structurally, a building must never collapse under the action of assumed loads, whatever they may be. Furthermore, tolerable deformation of the structure or its elements should not cause material distress or psychologi- cal harm. Good structural design is more related to correct intuitive sense than to sets of complex mathematical equa- tions. Mathematics should be merely a convenient and validating tool by which the designer determines the physical sizes and proportions of the elements to be used in the intended structure. The general procedure of designing a structural system (called structural planning) consists of the following phases: Conceiving of the basic structural form. Devising the gravity and lateral force resisting Figure 1.4 Eiffel Tower. strategy. Roughly proportioning the component parts. Developing a foundation scheme. Determining the structural materials to be used. Detailed proportioning of the component parts. Devising a construction methodology. After all of the separate phases have been examined and modified in an iterative manner, the structural elements within the system are then checked mathematically by the structural consultant to ensure the safety and economy of the structure. The process of conceiving and visualizing a structure is truly an art. There are no sets of rules one can follow in a linear man- ner to achieve a so-called “good design.” The iterative approach is most often employed to arrive at a design solution. Nowadays, with the design of any large struc- ture involving a team of designers working jointly with specialists and consultants, the architect is required to function as a coordinator and still maintain a leadership role even in the initial structural scheme. The architect needs to have a broad general understanding of the structure with its various problems and a sufficient under- standing of the fundamental principles of structural behavior to provide useful approximations of member sizes. The structural principles influence the form of the Figure 1.5 Nave of Reims Cathedral building, and a logical solution (often an economical one (construction begun in 1211). as well) is always based on a correct interpretation of these principles. A responsibility of the builder (constructor) is to have the knowledge, experience, and inventiveness to resolve complex structural and constructional issues with- out losing sight of the spirit of the design. A structure need not actually collapse to be lacking in integrity. For example, a structure indiscriminately employ- ing inappropriate materials or an unsuitable size and pro- portion of elements would reflect disorganization and a Introduction 3 sense of chaos. Similarly, a structure carelessly overdesigned would lack truthfulness and reflect a wastefulness that seems highly questionable in our current world situation of rapidly diminishing resources. It can be said that in these works (Gothic Cathedrals, Eiffel Tower, Firth of Forth Bridge), forerunners of the great architecture of tomorrow, the relationship between technology and aesthetics that we found in the great buildings of the past has remained intact. It seems to me that this relationship can be defined in the following manner: the objective data of the problem, technology and statics (empirical or scientific), suggest the solutions and forms; the aesthetic sensitivity of the designer, who understands the intrinsic beauty and validity, welcomes the suggestion and models it, emphasizes it, proportions it, in a personal manner which constitutes the artistic element in architecture. Quote from Pier Luigi Nervi, Aesthetics and Technology in Figure 1.6 Tree—a system of cantilevers. Architecture, Harvard University Press; Cambridge, Massachusetts, 1966. (See Figures 1.4 and 1.5.) 1.3 PARALLELS IN NATURE There is a fundamental “rightness” in the structurally cor- rect concept, leading to an economy of means. Two kinds of “economy” are present in buildings. One such economy is based on expediency, availability of materials, cost, and constructability. The other “inherent” economy is dictated by the laws of nature (Figure 1.6). Figure 1.7 Beehive—cellular structure. In his wonderful book On Growth and Form, D’Arcy Wentworth Thompson describes how Nature, as a response to the action of forces, creates a great diversity of forms from an inventory of basic principles. Thompson says that in short, the form of an object is a diagram of forces; in this sense, at least, that from it we can judge of or deduce the forces that are acting or have acted upon it; in this strict and particular sense, it is a diagram. The form as a diagram is an important governing idea in the application of the principle of optimization (maximum output for minimum energy). Nature is a wonderful venue to observe this principle, because survival of a species depends on it. An example of optimization is the honeycomb of the bee (Figure 1.7). This system, an arrangement of hexagonal cells, contains the greatest amount of honey with the least amount of beeswax and is the structure that requires the least energy for the bees to construct. Galileo Galilei (16th century), in his observation of animals and trees, postulated that growth was maintained within a relatively tight range—that problems with the organism would occur if it were too small or too large. In his Dialogues Concerning Two New Sciences, Galileo hypothe- sizes that Figure 1.8 Human body and skeleton. 4 Chapter 1 it would be impossible to build up the bony structures of men, horses, or other animals so as to hold together and perform their normal functions if these animals were to be increased enormously in height; for this increase in height can be accomplished only by employing a material which is harder and stronger than usual, or by enlarging the size of the bones, thus changing their shape until the form and appearance of the animals suggest monstrosity.... If the size of a body be diminished, the strength of that body is not diminished in the same proportion; indeed, the smaller the body the greater its relative strength. Thus a small dog could probably carry on its back two or three dogs of his own size; but I believe that a horse could not carry even one of his own size. Economy in structure does not just mean frugality. Without the economy of structure, neither a bird nor an airplane Figure 1.9 Flying structures—a bat and could fly, for their sheer weight would crash them to earth. Otto Lilienthal’s hang glider (1896). Without economy of materials, the dead weight of a bridge could not be supported. Reduction in dead weight of a structure in nature involves two factors. Nature uses mate- rials of fibrous cellular structure (as in most plants and ani- mals) to create incredible strength-to-weight ratios. In inert granular material such as an eggshell, it is often used with maximum economy in relation to the forces that the struc- ture must resist. Also, structural forms (like a palm leaf, a nautilus shell, or a human skeleton) are designed in cross- section so that the minimum of material is used to develop the maximum resistance to forces (Figure 1.8). Nature creates slowly through a process of trial and error. Living organisms respond to problems and a changing environment through adaptations over a long period of time. Those that do not respond appropriately to the envi- ronmental changes simply perish. Historically, human development in the area of structural forms has also been slow (Figure 1.9). For the most part, Figure 1.10 The skeletal latticework of the limited materials and knowledge restricted the develop- radiolarian (Aulasyrum triceros) consists of ment of new structural elements or systems. Even within hexagonal prisms in a spherical form. the last 150 years or so, new structural materials for build- ings have been relatively scarce—steel, reinforced con- crete, prestressed concrete, composite wood materials, and aluminum alloys. However, these materials have brought about a revolution in structural design and are currently being tested to their material limit by engineers and architects. Some engineers believe that most of the significant structural systems are known and, therefore, that the future lies in the development of new materials and the exploitation of known materials in new ways. Advances in structural analysis techniques, especially with the advent of the computer, have enabled designers to explore very complex structures (Figures 1.10 and 1.11) under an array of loading conditions much more rapidly and accurately than in the past. However, the computer is Figure 1.11 Buckminster Fuller’s Union Tank still being used as a tool to validate the intent of the Car dome, a 384-ft.-diameter geodesic dome. designer and is not yet capable of actual “design.” A Introduction 5 human designer’s knowledge, creativity, and understand- ing of how a building structure is to be configured are still essential for a successful project. 1.4 LOADS ON STRUCTURES Structural systems, aside from their form-defining func- tion, essentially exist to resist forces that result from two general classifications of loads: 1. Static. This classification refers to gravity-type forces. 2. Dynamic. This classification is due to inertia or momentum of the mass of the structure (like earth- quakes). The more sudden the starting or stopping of the structure, the greater the force will be. Note: Other dynamic forces are produced by wave action, landslides, falling objects, shocks, blasts, vibration from heavy machinery, and so on. A light, steel frame building may be very strong in resist- ing static forces, but a dynamic force may cause large dis- tortions to occur because of the frame’s flexible nature. On the other hand, a heavily reinforced concrete building may be as strong as the steel building in carrying static loads but have considerable stiffness and sheer dead weight, which may absorb the energy of dynamic forces with less distortion (deformation). All of the following forces must be considered in the de- sign of a building structure (Figure 1.12). Figure 1.12 Typical building loads. Dead Loads. Loads resulting from the self-weight of the building or structure and of any perma- nently attached components, such as partition walls, flooring, framing elements, and fixed equip- ment, are classified as dead loads. Standard weights of commonly used materials for building are known, and a complete building’s dead weight can be calculated with a high degree of certainty. However, the weight of structural elements must be estimated at the beginning of the design phase of the structure and then refined as the design process proceeds toward completion. A sampling of some standard building material weights used for the initial structural design process is: concrete = 150 pounds per cubic foot (pcf) timber = 35 pcf steel = 490 pcf built-up roofing = 6 pounds per square foot (psf) half-inch gypsum wallboard = 1.8 psf plywood, per inch of thickness = 3 psf suspended acoustical ceiling = 1 psf When activated by earthquake, static dead loads take on a dynamic nature in the form of horizontal inertial forces. Buildings with heavier dead loads 6 Chapter 1 generate higher inertial forces that are applied in a horizontal direction. Live Loads. Transient and moving loads that include occupancy loads, furnishings, and storage are classified as live loads. Live loads are extremely variable by nature and normally change during a structure’s lifetime as occupancy changes. Building codes specify minimum uniform live loads for the design of roof and floor systems based on a history of many buildings and types of occupancy condi- tions. These codes incorporate safety provisions for overload protection, allowance for construction loads, and serviceability considerations, such as vi- bration and deflection behavior. Minimum roof live loads include allowance for minor snowfall and construction loads. (See Table 1.2 for an additional listing of common live loads for buildings.) Snow Loads. Snow loads represent a special type of live load because of the variability involved. Local building officials or applicable building codes pre- scribe the design snow for a specific geographical jurisdiction. Generally, snow loads are determined from a zone map reporting 50-year recurrence inter- vals of an extreme snow depth. Snow weights can vary from approximately 8 pcf for dry powder snow to 12 pcf for wet snow (Figure 1.13). Design loads can vary from 10 psf on a horizontal surface to 400 psf in some specific mountainous regions. In many areas of the United States, design snow loads can range from 20 to 40 psf. The accumulation depth of the snow depends on the slope of the roof. Steeper slopes have smaller ac- cumulations. Special provisions must also be made for potential accumulation of snow at roof valleys, parapets, and other uneven roof configuration. Except for a building’s dead load, which is fixed, the other forces listed above can vary in duration, magnitude, and point of application. A building structure must neverthe- less be designed for these possibilities. Unfortunately, a large portion of a building structure exists for loads that will be present at much lower magnitudes—or may never Figure 1.13 Failure from snow load. occur at all. The structural efficiency of a building is often measured by its dead load weight in comparison to the live load car- ried. Building designers have always strived to reduce the ratio of dead to live load. New methods of design, new and lighter materials, and old materials used in new ways have contributed to the dead/live load reduction. The size of the structure has an influence on the ratio of dead to live load. A small bridge over a creek, for example, can carry a heavy vehicle—a live load representing a large portion of the dead/live load ratio. The Golden Gate Bridge in San Francisco, on the other hand, spans a long Introduction 7 distance, and the material of which it is composed is used chiefly in carrying its own weight. The live load of the vehicular traffic has a relatively small effect on the bridge’s internal stresses. With the use of modern materials and construction meth- ods, it is often the smaller rather than the larger buildings that show a high dead/live load ratio. In a traditional house, the live load is low, and much of the dead load not only supports itself but also serves as weather protection and space-defining systems. This represents a high dead/live load ratio. In contrast, in a large factory build- ing, the dead load is nearly all structurally effective, and the dead/live load ratio is low. The dead/live load ratio has considerable influence on the choice of structure and especially on the choice of beam types. As spans increase, so do the bending effects caused by dead and live loads; therefore, more material must be introduced into the beam to resist the increased bending effects. This added material weight itself adds further dead load and pronounced bending effects as spans in- crease. The dead/live load ratio not only increases but may eventually become extremely large. Wind Loads. Wind is essentially air in motion and creates a loading on buildings that is dynamic in nature. When buildings and structures become ob- stacles in the path of wind flow, the wind’s kinetic energy is converted into potential energy of pres- sure on various parts of the building. Wind pres- sures, directions, and duration are constantly changing. However, for calculation purposes, most wind design assumes a static force condition for more conventional, lower rise buildings. The fluc- tuating pressure caused by a constantly blowing wind is approximated by a mean pressure that acts on the windward side (the side facing the wind) and leeward side (the side opposite the windward Figure 1.14 Wind loads on a structure. side) of the structure. The “static” or nonvarying external forces are applied to the building structure and simulate the actual varying wind forces. Direct wind pressures depend on several variables: wind velocity, height of the wind above ground (wind velocities are lower near the ground), and the nature of the building’s surroundings. Wind pres- sure on a building varies as the square of the veloc- ity (in miles per hour). This pressure is also referred to as the stagnation pressure. Buildings respond to wind forces in a variety of complex and dynamic ways. The wind creates a negative pressure, or suction, on both the leeward side of the building and on the side walls parallel to the wind direction (Figure 1.14). Uplift pressure occurs on horizontal or sloping roof surfaces. In addition, the corners, edges, and eave overhangs of 8 Chapter 1 a building are subjected to complicated forces as the wind passes these obstructions, causing higher localized suction forces than generally encountered on the building as a whole. Wind is a fluid and acts like other fluids—a rough surface causes friction and slows the wind velocity near the ground. Wind speeds are measured at a standard height of 10 meters (33 feet) above the ground, and adjustments are made when calculat- ing wind pressures at higher elevations. The wind pressure increases with the height of the building. Other buildings, trees, and topography affect how the wind will strike the building. Buildings in vast open areas are subject to larger wind forces than are those in sheltered areas or where a building is sur- rounded by other buildings. The size, shape, and surface texture of the building also impact the de- sign wind forces. Resulting wind pressures are treated as lateral loading on walls and as downward pressure or uplift forces (suction) on roof planes. Earthquake Loads (seismic). Earthquakes, like wind, produce a dynamic force on a building. During an actual earthquake, there are continuous ground mo- tions that cause the building structure to vibrate. The dynamic forces on the building are a result of the violent shaking of the ground generated by seis- mic shock waves emanating from the center of the fault (the focus or hypocenter) (Figure 1.15). The point directly above the hypocenter on the earth’s surface is known as the epicenter. The rapidity, magnitude, and duration of these shakes depend on the intensity of the earthquake. During an earthquake, the ground mass suddenly Figure 1.15 Earthquake loads on a structure. moves both vertically and laterally. The lateral movements are of particular concern to building designers. Lateral forces developed in the structure are a function of the building’s mass, configuration, building type, height, and geographic location. The force from an earthquake is initially assumed to de- velop at the base of the building; the force being known as the base shear (Vbase). This base shear is then redistributed equal and opposite at each of the floor levels where the mass of the building is assumed concentrated. All objects, including buildings, have a natural or fundamental period of vibration. It represents the time it takes an object or building to vibrate through one cycle of vibration (or sway) when subjected to an applied force. When an earthquake ground motion 1-story 4-story 15-story causes a building to start vibrating, the building be- T~0.1 sec. T~0.4 sec. T~1.5 sec. gins to displace (sway) back and forth at its natural period of vibration. Shorter, lower buildings have Figure 1.16 Approximate building periods of very short periods of vibration (less than one sec- vibration. ond), while tall high rises can have periods of Introduction 9 vibration that last several seconds (Figure 1.16). Fundamental periods are a function of a building’s height. An approximate estimate of a building’s period is equal to T = 0.1N where N represents the number of stories and T represents the period of vibration in seconds. The ground also vibrates at its own natural period of vibration. Many of the soils in the United States have periods of vibration in the range of 0.4 to 1.5 seconds. Short periods are more characteristic of hard soils (rock), while soft ground (some clays) may have periods of up to two seconds. Many common buildings can have periods within the range of the supporting soils, making it possible for the ground motion to transmit at the same nat- ural frequency as that of the building. This may cre- ate a condition of resonance (where the vibrations increase dramatically), in which the inertial forces might become extremely large. Inertial forces develop in the structure due to its weight, configuration, building type, and geo- graphic location. Inertial forces are the product of mass and acceleration (Newton’s second law: F = m * a). Heavy, massive buildings will result in larger inertial forces; hence, there is a distinct advantage in using a lighter weight construction when seismic considerations are a key part of the design strategy. For some tall buildings or structures with complex configurations or unusual massing, a dynamic structural analysis is required. Computers are used to simulate earthquakes on the building to study how the forces are developed and the re- sponse of the structure to these forces. Building codes are intended to safeguard against major fail- ures and loss of life; they are not explicitly for the protection of property. 1.5 BASIC FUNCTIONAL REQUIREMENTS The principal functional requirements of a building struc- ture are: 1. Stability and equilibrium. 2. Strength and stiffness. 3. Continuity and redundancy. Figure 1.17 Stability and the strength of a 4. Economy. structure—the collapse of a portion of the 5. Functionality. University of Washington Husky stadium 6. Aesthetics. during construction (1987) due to lack of Primarily, structural design is intended to make the building adequate bracing to ensure stability. Photo by “stand up” (Figure 1.17). In making a building “stand up,” author. 10 Chapter 1 the principles governing the stability and equilibrium of buildings form the basis for all structural thinking. Strength and stiffness of materials are concerned with the stability of a building’s component parts (beams, columns, walls), whereas statics deals with the theory of general stability. Statics and strength of materials are actually intertwined, because the laws that apply to the stability of the whole structure are also valid for the individual components. The fundamental concept of stability and equilibrium is concerned with the balancing of forces to ensure that a building and its components will not move (Figure 1.18). Figure 1.18 Equilibrium and Stability?— In reality, all structures undergo some movement under sculpture by Richard Byer. Photo by author. load, but stable structures have deformations that remain relatively small. When loads are removed from the struc- ture (or its components), internal forces restore the struc- ture to its original, unloaded condition. A good structure is one that achieves a condition of equilibrium with a min- imum of effort. Strength of materials requires knowledge about building material properties, member cross-sections, and the abil- ity of the material to resist breaking. Also of concern is that the structural elements resist excessive deflection and/or deformation. (a) Continuity—loads from the roof beams are redistributed to the roof columns below. A Continuity in a structure refers to a direct, uninterrupted continuous path is provided for the column path for loads through the building structure—from the loads to travel directly to the columns below roof level down to the foundation. Redundancy is the con- and then on to the foundation. cept of providing multiple load paths in a structural framework so that one system acts as a backup to another in the event of localized structural failure. Structural re- dundancy enables loads to seek alternate paths to bypass structural distress. A lack of redundancy is very haz- ardous when designing buildings in earthquake country (Figure 1.19). On 9/11, both of the World Trade Center towers were able to withstand the impact of jetliners crashing into them and continue standing for some time, permitting many people (b) Discontinuity in the vertical elevation can to evacuate. The towers were designed with structural re- result in very large beam bending moments dundancy, which prevented an even larger loss of life. and deflection. Structural efficiency is However, the process by which the collapse of the im- enhanced by aligning columns to provide a pacted story level led to the progressive collapse of the en- direct path to the foundation. Beam sizes can tire building may have led some investigators to hint that thus be reduced significantly. In this example, an inadequate degree of structural redundancy existed. missing or damaged columns could also The requirements of economy, functionality, and aesthetics represent how structural frameworks can have are usually not covered in a structures course and will not the ability to redistribute loads to adjacent be dealt with in this book. Strength of materials is typi- members without collapse. This is referred to cally covered upon completion of a statics course. as structural redundancy. Figure 1.19 Examples of continuity and redundancy. Introduction 11 1.6 ARCHITECTURAL ISSUES A technically perfect work can be aesthetically inexpressive but there does not exist, either in the past or in the present, a work of architecture which is accepted and recognized as excellent from an aesthetic point of view which is not also excellent from the technical point of view. Good engineering seems to be a necessary though not sufficient condition for good architecture. —Pier Luigi Nervi The geometry and arrangement of the load-bearing mem- bers, the use of materials, and the crafting of joints all rep- resent opportunities for buildings to express themselves. The best buildings are not designed by architects who, after resolving the formal and spatial issues, simply ask the structural engineer to make sure it does not fall down. An Historical Overview It is possible to trace the evolution of architectural space Figure 1.20 Stonehenge. and form through parallel developments in structural en- gineering and material technology. Until the 19th century, this history was largely based on stone construction and the capability of this material to resist compressive forces. Less durable wood construction was generally reserved for small buildings or portions of buildings. Neolithic builders used drystone techniques, such as coursed masonry walling and corbelling, to construct monuments, dwellings, tombs, and fortifications. These structures demonstrate an understanding of the material properties of the various stones employed (Figure 1.20). Timber joining and dressed stonework were made possi- ble by iron and bronze tools. Narrow openings in masonry building walls were achieved through corbelling and tim- ber or stone lintels. The earliest examples of voussoir arches and vaults in both stone and unfired brick construction have been Figure 1.21 Construction of a Greek found in Egypt and Greece (Figure 1.21). These materials peristyle temple. and structural innovations were further developed and refined by the Romans. The ancient Roman architect Vitruvius, in his Ten Books, described timber trusses with horizontal tie members capable of resisting the outward thrust of sloping rafters. Roman builders managed to place the semicircular arch atop piers or columns; the larger spans reduced the num- ber of columns required to support the roof. Domes and barrel and groin vaults were improved through the use of modular fired brick, cement mortar, and hydraulic con- crete. These innovations enabled Roman architects to cre- Figure 1.22 Stone arch, barrel vault, and ate even larger unobstructed spaces (Figure 1.22). groin vault. 12 Chapter 1 Gradual refinements of this technology by Romanesque ma- son builders eventually led to the structurally daring and expressive Gothic cathedrals. The tall, slender nave walls with large stained glass openings, which characterize this architecture, are made possible by improvements in concrete foundation construction; the pointed arch, which reduces lateral forces; flying arches and buttresses, which resist the remaining lateral loads; and the ribbed vault, which rein- forces the groin and creates a framework of arches and columns, keeping opaque walls to a minimum (Figure 1.23). The medium of drawing allowed Renaissance architects to work on paper, removed from construction and the site. Existing technical developments were employed in the search for a classical ideal of beauty and proportion. Structural cast iron and larger, stronger sheets of glass became available in the late 18th century. These new materi- Figure 1.23 Construction of a Gothic als were first employed in industrial and commercial cathedral. buildings, train sheds, exhibition halls, and shopping ar- cades. Interior spaces were transformed by the delicate long-span trusses supported on tall, slender, hollow columns. The elements of structure and cladding were more clearly articulated, with daylight admitted in great quantities. Wrought iron and, later, structural steel pro- vided excellent tensile strength and replaced brittle cast iron. Art Nouveau architects exploited the sculptural po- tential of iron and glass, while commercial interests capi- talized on the long-span capabilities of rolled steel sections. The tensile properties of steel were combined with the high compressive strength of concrete, making a compos- ite section with excellent weathering and fire-resistive properties that could be formed and cast in almost any shape (Figure 1.24). Steel and reinforced concrete struc- tural frames enabled builders to make taller structures with more stories. The smaller floor area devoted to struc- Figure 1.24 Sports Palace, reinforced ture and the greater spatial flexibility led to the develop- concrete arena, by Pier Luigi Nervi. ment of the modern skyscraper. Today, pretensioned and posttensioned concrete, engi- neered wood products, tensile fabric, and pneumatic structures and other developments continue to expand the architectural and structural possibilities. The relationship between the form of architectural space and structure is not deterministic. For example, the devel- opment of Buckminster Fuller’s geodesic dome did not immediately result in a proliferation of domed churches or office buildings. As history has demonstrated, vastly dif- ferent spatial configurations have been realized with the same materials and structural systems. Conversely, simi- lar forms have been generated utilizing very different structural systems. Architects as well as builders must de- velop a sense of structure (Figure 1.25). Creative collabo- ration between architect, builder, and engineer is necessary to achieve the highest level of formal, spatial, and structural integration. Introduction 13 Criteria for the Selection of Structural Systems Most building projects begin with a client program outlin- ing the functional and spatial requirements to be accom- modated. Architects typically interpret and prioritize this information, coordinating architectural design work with the work of other consultants on the project. The architect and structural engineer must satisfy a wide range of fac- tors in determining the most appropriate structural sys- tem. Several of these factors are discussed here. Nature and magnitude of loads The weight of most building materials (Table 1.1) and the self-weight of structural elements (dead loads) can be cal- culated from reference tables listing the densities of vari- ous materials. Building codes establish design values for the weight of the occupants and furnishings—live loads (Table 1.2)—and other temporary loads, such as snow, wind, and earthquake. Building use/function Sports facilities (Figure 1.26) require long, clear span areas free of columns. Light wood framing is well suited to the Figure 1.25 Hong Kong Bank, by Norman relatively small rooms and spans found in residential con- Foster. struction. Site conditions Topography and soil conditions often determine the design of the foundation system, which in turn influences the way loads are transmitted though walls and columns. Low soil-bearing capacities or unstable slopes might sug- gest a series of piers loaded by columns instead of conven- tional spread footings. Climatic variables, such as wind speed and snowfall, affect design loads. Significant move- ment (thermal expansion and contraction) can result from extreme temperature fluctuations. Seismic forces, used to calculate building code design loads, vary in different parts of the country. Building systems integration All building systems (lighting, heating/cooling, ventilation, plumbing, fire sprinklers, electrical) have a rational basis that governs their arrangement. It is generally more elegant and cost-effective to coordinate these systems to avoid con- Figure 1.26 Sports Palace interior, by Pier flict and compromise in their performance. This is espe- Luigi Nervi (1955). cially the case where the structure is exposed and dropped ceiling spaces are not available for duct and pipe runs. Fire resistance Building codes require that building components and struc- tural systems meet minimum fire-resistance standards. The combustibility of materials and their ability to carry design loads when subjected to intense heat are tested to ensure that buildings involved in fires can be safely evacuated in a 14 Chapter 1 Table 1.1 Selected building material weights. given period of time. Wood is naturally com- bustible, but heavy timber construction Assembly lb./ft.2 kN/m2 maintains much of its strength for an ex- Roofs: tended period of time in a fire. Steel can be Three-ply and gravel 5.5 0.26 weakened to the point of failure unless pro- Five-ply and gravel 6.5 0.31 tected by fireproof coverings. Concrete and Wood shingles 2 0.10 masonry are considered to be noncom- Asphalt shingles 2 0.10 bustible and are not significantly weakened Corrugated metal 1–2.5 0.05–0.12 Plywood 3 #/in. 0.0057 kN/mm in fires. The levels of fire resistance vary from Insulation—fiberglass batt 0.5 0.0025 unrated construction to four hours and are Insulation—rigid 1.5 0.075 based on the type of occupancy and size of a Floors: building. Concrete plank 6.5 0.31 Concrete slab 12.5 #/in. 0.59 kN/mm Construction variables Steel decking w/concrete 35–45 1.68–2.16 Wood joists 2–3.5 0.10–0.17 Cost and construction time are almost always Hardwood floors 4 #/in. 0.19 kN/mm relevant issues. Several structural systems Ceramic tile w/thin set 15 0.71 will often accommodate the load, span, and Lightweight concrete 8 #/in. 0.38 kN/mm fire-resistance requirements for a building. Timber decking 2.5 #/in. 0.08 kN/mm Local availability of materials and skilled Walls: construction trades typically affect cost and Wood studs (average) 2.5 0.012 schedule. The selected system can be refined Steel studs 4 0.20 to achieve the most economical framing Gypsum drywall 3.6 #/in. 0.17 kN/mm Partitions (studs w/drywall) 6 0.29 arrangement or construction method. The use of heavy equipment, such as cranes or concrete trucks and pumps, may be restricted Table 1.2 Selected live load requirements.* by availability or site access. Occupancy/Use (Uniform load) lb./ft.2 kN/m2 Architectural form and space Apartments: Social and cultural factors that influence the Private dwellings 40 1.92 architect’s conception of form and space ex- Corridors and public rooms 100 4.79 tend to the selection and use of appropriate Assembly areas/theaters: materials. Where structure is exposed, the lo- Fixed seats 60 2.87 cation, scale, hierarchy, and direction of fram- Stage area 100 4.79 ing members contribute significantly to the Hospitals: expression of the building. Private rooms and wards 40 1.92 Laboratories/operating rooms 60 2.87 This book, Statics and Strengths of Materials for Architecture and Building Construction, covers Hotels: the analysis of statically determinate systems Private guest rooms 40 1.92 Corridors/public rooms 100 4.79 using the fundamental principles of free- body diagrams and equations of equilib- Offices: rium. Although during recent years have General floor area 50 2.40 Lobbies/first floor corridor 100 4.79 seen an incredible emphasis on the use of computers to analyze structures by matrix Residential (private): analysis, it is the author’s opinion that a clas- Basic floor area and decks 40 1.92 Uninhabited attics 20 0.96 sical approach for a beginning course is nec- Habitable attics/sleeping areas 30 1.44 essary. This book’s aim is to give the student an understanding of physical phenomena Schools: Classrooms 40 1.92 before embarking on the application of so- Corridors 80–100 3.83–4.79 phisticated mathematical analysis. Reliance on the computer (sometimes the “black or Stairs and exits: Single family/duplex dwellings 40 1.92 white box”) for answers that one does not All other 100 4.79 fully understand is a risky proposition at best. Application of the basic principles of * Loads are adapted from various Code sources and are listed here for statics and strength of materials will enable illustrative purposes only. Consult the governing Code in your local the student to gain a clearer and, it is hoped, jurisdiction for actual design values. more intuitive sense about structure. 2 Statics 2.1 CHARACTERISTICS OF A FORCE Figure 2.1 Sir Isaac Newton (1642–1727). Force Born on Christmas Day in 1642, Sir Isaac Newton is viewed by many as the greatest What is force? Force may be defined as the action of one scientific intellect who ever lived. Newton said body on another that affects the state of motion or rest of the of himself, “I do not know what I may appear body. In the late 17th century, Sir Isaac Newton (Figure 2.1) to the world, but to myself I seem to have been summarized the effects of force in three basic laws: only like a boy playing on the seashore, and diverting myself in now and then finding First Law: Any body at rest will remain at rest, a smoother pebble or a prettier shell than and any body in motion will move uniformly in a ordinary, whilst the great ocean of truth lay straight line, unless acted upon by a force. all undiscovered before me.” (Equilibrium) Second Law: The time rate of change of momen- Newton’s early schooling found him tum is equal to the force producing it, and the fascinated with designing and constructing change takes place in the direction in which the mechanical devices such as water clocks, force is acting. (F = m * a) sundials, and kites. He displayed no unusual Third Law: For every force of action, there is a re- signs of being gifted until his later teens. action that is equal in magnitude, opposite in In the 1660s, he attended Cambridge but direction, and has the same line of action. (Basic without any particular distinction. In his last concept of force.) undergraduate year at Cambridge, with no more than basic arithmetic, he began to study Newton’s first law involves the principle of equilibrium of mathematics, primarily as an autodidact, forces, which is the basis of statics. The second law formu- deriving his knowledge from reading with lates the foundation for analysis involving motion or little or no outside help. He soon assimilated dynamics. Written in equation form, Newton’s second law existing mathematical tradition and began may be stated as to move beyond it to develop calculus F = m * a (independent of Leibniz). At his mother’s farm, where he had retired to avoid the plague where F represents the resultant unbalanced force acting that had hit London in 1666, he watched an on a body of mass m with a resultant acceleration a. apple fall to the ground and wondered if there Examination of this second law implies the same meaning was a similarity between the forces pulling on as the first law, because there is no acceleration when the the apple and the pull on the moon in its force is zero and the body is at rest or moves with a orbit around the Earth. He began to lay the constant velocity. foundation of what was later to become the concept of universal gravitation. In his three laws of motion, he codified Galileo’s findings and provided a synthesis of celestial and terrestrial mechanics. 15 16 Chapter 2 The third law introduces us to the basic concept of force. It states that whenever a body A exerts a force on another body B, body B will resist with an equal magnitude but in the opposite direction. For example, if a building with a weight W is placed on the ground, we can say that the building is exerting a downward force of W on the ground. However, for the building to remain stable on the resisting ground surface without sinking completely, the ground must resist with an upward force of equal magnitude. If the ground re- sisted with a force less than W, where R < W, the building would settle. On the other hand, if the ground exerted an Figure 2.2 Ground resistance on a building. upward force greater than W (R > W ), the building would rise (levitate) (Figure 2.2). Characteristics of a Force A force is characterized by its (a) point of application, (b) magnitude, and (c) direction. The point of application defines the point where the force is applied. In statics, the point of application does not imply the exact molecule on which a force is applied but a location that, in general, describes the origin of a force (Figure 2.3). In the study of forces and force systems, the word particle will be used, and it should be considered as the location or Figure 2.3 Rope pulling on an eyebolt. point where the forces are acting. Here, the size and shape of the body under consideration will not affect the solu- tion. For example, if we consider the anchor bracket shown in Figure 2.4(a), three forces—Fl, F2, and F3—are applied. The intersection of these three forces occurs at point O; therefore, for all practical purposes, we can rep- resent the same system as three forces applied on particle O, as shown in Figure 2.4(b). Magnitude refers to the quantity of force, a numerical measure of the intensity. Basic units of force that will be used throughout this text are the pound (lb. or #) and the kilo pound (kip or k = 1,000#). In metric (SI) units, force is expressed as the newton (N) or kilonewton (kN) where Figure 2.4(a) An anchor device with three ap- 1 kN = 1,000 N. plied forces. The direction of a force is defined by its line of action and sense. The line of action represents an infinite straight line along which the force is acting. In Figure 2.5, the external effects on the box are essentially the same whether the person uses a short or long cable, provided the pull exerted is along the same line of action and of equal magnitude. If a force is applied such that the line of action is neither vertical nor horizontal, some reference system must be es- tablished. Most commonly accepted is the angular symbol of θ (theta) or φ (phi) to denote the number of degrees the line of action of the force is in relation to the horizontal or Figure 2.4(b) Force diagram of the anchor. vertical axis, respectively. Only one (θ or φ ) needs to be Statics 17 indicated. An alternative to angular designations is a slope relationship. The sense of the force is indicated by an arrowhead. For example, in Figure 2.6, the arrowhead gives the indication that a pulling force (tension) is being applied to the bracket at point O. By reversing only the arrowhead (Figure 2.7), we would have a pushing force (compression) applied on the bracket with the same magnitude (F  10 k), point of application Figure 2.5 Horizontal force applied to a box. (point O), and line of action (θ = 22.6° from the horizontal). Figure 2.6 Three ways of indicating direction for an angular Figure 2.7 Force in compression. tension force. Rigid Bodies Practically speaking, any body under the action of forces undergoes some kind of deformation (change in shape). In statics, however, we deal with a body of matter (called a continuum) that, theoretically, undergoes no deformation. This we call a rigid body. Deformable bodies under loads will be studied in depth, in Chapter 5, under the heading Strength of Materials. When a force of F = 10# is applied to a box, as shown in Figure 2.8, some degree of deformation will result. The de- formed box is referred to as a deformable body, whereas in Figure 2.8(b) we see an undeformed box called the rigid body. Again, you must remember that the rigid body is a purely theoretical phenomenon but necessary in the study of statics. (a) Original, unloaded box. (b) Rigid body (example: stone). (c) Deformable body (example: foam). Figure 2.8 Rigid body/deformable body. 18 Chapter 2 Principle of Transmissibility An important principle that applies to rigid bodies in par- ticular is the principle of transmissibility. This principle states that the external effects on a body (cart) remain un- changed when a force Fl acting at point A is replaced by a force F2 of equal magnitude at point B, provided that both forces have the same sense and line of action (Figure 2.9). In Figure 2.9(a), the reactions Rl and R2 represent the reac- tions of the ground onto the cart, opposing the weight of Figure 2.9 An example of the principle of the cart W. Although in Figure 2.9(b) the point of applica- transmissibility. tion for the force changes (magnitude, sense, and line of action remaining constant), the reactions Rl and R2 and also the weight of the cart W remain the same. The princi- ple of transmissibility is valid only in terms of the external effects on a body remaining the same (Figure 2.10); inter- nally, this may not be true. External and Internal Forces Let’s consider an example of a nail being withdrawn from a wood floor (Figure 2.11). If we remove the nail and examine the forces acting on it, Figure 2.10 Another example of the principle we discover frictional forces that develop on the embed- of transmissibility. ded surface of the nail to resist the withdrawal force F (Figure 2.12). Treating the nail as the body under consideration, we can then say that forces F and S are external forces. They are being applied outside the boundaries of the nail. External forces represent the action of other bodies on the rigid body. Let’s consider just a portion of the nail and examine the forces acting on it. In Figure 2.13, the frictional force S plus the force R (the resistance generated by the nail internally) resist the applied force F. This internal force R is responsi- Figure 2.11 Withdrawal force on a nail. ble for keeping the nail from pulling apart. Figure 2.12 External forces on the nail. Figure 2.13 Internal resisting forces on the nail. Statics 19 Examine next a column-to-footing arrangement with an applied force F, as illustrated in Figure 2.14. To appropri- ately distinguish which forces are external and which are internal, we must define the system we are considering. Several obvious possibilities exist here: Figure 2.15(a), column and footing taken together as a system; Figure 2.15(b), column by itself; and Figure 2.15(c), footing by itself. In Figure 2.15(a), taking the column and footing as the sys- tem, the external forces are F (the applied force), Wcol, Wftg, and Rsoil. Weights of bodies or members are considered as external forces, applied at the center of gravity of the mem- ber. Center of gravity (mass center) will be discussed in a later section. Figure 2.14 A column supporting The reaction or resistance the ground offers to counteract an external load. the applied forces and weights is Rsoil. This reaction occurs on the base of the footing, outside the imaginary system’s boundary; therefore, it is considered separately. When a column is considered separately, as a system by itself, the external forces become F, Wcol, and Rl. The forces F and Wcol are the same as in Figure 2.15(b), but the force Rl is a result of the resistance the footing offers to the col- umn under the applied forces (F and Wcol) shown. (b) Column. (a) Column and footing. (c) Footing. The last case, Figure 2.15(c), considers the footing as a sys- tem by itself. External forces acting on the footing are R2, Figure 2.15 Different system groupings. Wftg, and Rsoil. The R2 force represents the reaction the col- umn produces on the footing, and Wftg and Rsoil are the same as in Figure 2.15(a). 20 Chapter 2 Now let’s examine the internal forces that are present in each of the three cases examined above (Figure 2.16). Examination of Figure 2.16(a) shows forces Rl and R2 oc- curring between the column and footing. The boundary of the system is still maintained around the column and foot- ing, but by examining the interaction that takes place be- tween members within a system, we infer internal forces. Force Rl is the reaction of the footing on the column, while R2 is the action of the column on the footing. From Newton’s third law, we can then say that Rl and R2 are equal and opposite forces. Internal forces occur between bodies within a system, as in Figure 2.16(a). Also, they may occur within the mem- bers themselves, holding together the particles forming the rigid body, as in Figure 2.16(b) and 2.16(c). Force R3 represents the resistance offered by the building material (stone, concrete, or steel) to keep the column intact; this acts in a similar fashion for the footing. (a) Relationship of forces between the column and footing. (b) Column. (c) Footing. Figure 2.16 External and internal forces. Statics 21 Types of Force Systems Force systems are often identified by the type or types of systems on which they act. These forces may be collinear, coplanar, or space force systems. When forces act along a straight line, they are called collinear; when they are ran- domly distributed in space, they are called space forces. Force systems that intersect at a common point are called concurrent, while parallel forces are called parallel. If the forces are neither concurrent nor parallel, they fall under the classification of general force systems. Concurrent force systems can act on a particle (point) or a rigid body, whereas parallel and general force systems can act only on a rigid body or a system of rigid bodies. (See Figure 2.17 for a diagrammatic representation of the various force sys- tem arrangements.) One intelligent hiker observing three other hikers Collinear—All forces acting along the same dangling from a rope. straight line. Figure 2.17(a) Particle or rigid body. Coplanar—All forces acting in the same plane. Forces in a buttress system. Figure 2.17(b) Rigid bodies. Coplanar, parallel—All forces are parallel A beam supported by a series of columns. and act in the same plane. Figure 2.17(c) Rigid bodies. 22 Chapter 2 Loads applied to a roof truss. Coplanar, concurrent—All forces intersect at a common point and lie in the same plane. Figure 2.17(d) Particle or rigid body. Column loads in a concrete building. Noncoplanar, parallel—All forces are parallel to each other, but not all lie in the same plane. Figure 2.17(e) Rigid bodies. One component of a three-dimensional Noncoplanar, concurrent—All forces intersect at a space frame. common point but do not all lie in the same plane. Figure 2.17(f ) Particle or rigid bodies. Array of forces acting simultaneously Noncoplanar, nonconcurrent—All forces are skewed. on a house. Figure 2.17(g) Rigid bodies. Statics 23 2.2 VECTOR ADDITION Characteristics of Vectors An important characteristic of vectors is that they must be added according to the parallelogram law. Although the idea of the parallelogram law was known and used in some form in the early 17th century, the proof of its validity was Figure 2.18 Cross-section through a supplied many years later by Sir Isaac Newton and the gravity-retaining wall. French mathematician Pierre Varignon (1654–1722). In the case of scalar quantities, where only magnitudes are consid- ered, the process of addition involves a simple arithmetical summation. Vectors, however, have magnitude and direc- tion, thus requiring a special procedure for combining them. Using the parallelogram law, we may add vectors graphi- cally or by trigonometric relationships. For example, two forces W and F are acting on a particle (point), as shown in Figure 2.18; we are to obtain the vector sum (resultant). Because the two forces are not acting along the same line of action, a simple arithmetical solution is not possible. The graphical method of the parallelogram law simply in- volves the construction, to scale, of a parallelogram using forces (vectors) W and F as the legs. Complete the parallel- ogram, and draw in the diagonal. The diagonal represents the vector addition of W and F. A convenient scale is used in drawing W and F, whereby the magnitude of R is scaled off using the same scale. To complete the representation, the angle θ must be designated from some reference axis— in this case, the horizontal axis (Figure 2.19). Figure 2.19 Another illustration of the parallelogram. An Italian mathematician and engineer, Giovanni Poleni (1685–1761), published a report in 1748 on St. Peter’s dome using a method of illustration shown in Figure 2.20. Poleni’s thesis of the absence of friction is demonstrated by the wedge-shaped voussoirs with spheres, which are arranged exactly in accordance with the line of thrust, thus supporting one another in an unstable equilibrium. In his report, Poleni refers to Newton and his theorem of the parallelogram of forces and deduces that the line of thrust resembles an inverted catenary. Figure 2.20 Poleni’s use of the parallelogram law in describing the lines of force in an arch. From Giovanni Poleni, Memorie istoriche della Gran Cupola del Tempio Vaticano, 1748. 24 Chapter 2 Example Problems: Vector Addition 2.1 Two forces are acting on a bolt as shown. Determine graphically the resultant of the two forces using the paral- lelogram law of vector addition. 1. Draw the 500# and 1,200# forces to scale with their proper directions. 2. Complete the parallelogram. 3. Draw the diagonal, starting at the point of origin O. 4. Scale off the magnitude of R. 5. Scale off the angle θ from a reference axis. 6. The sense (arrowhead direction) in this example moves away from point O. Another vector addition approach, which preceded the parallelogram law by 100 years or so, is the triangle rule or tip-to-tail method (developed through proofs by a 16th century Dutch engineer/mathematician, Simon Stevin). To follow this method, construct only half of the parallelo- gram, with the net result being a triangle. The sum of two vectors A and B may be found by arranging them in a tip- to-tail sequence with the tip of A to the tail of B or vice versa. Figure 2.21 Tip-to-tail method. In Figure 2.21(a), two vectors A and B are to be added by the tip-to-tail method. By drawing the vectors to scale and arranging it so that the tip of A is attached to the tail of B, as shown in Figure 2.21(b), the resultant R can be obtained by drawing a line beginning at the tail of the first vector, A, and ending at the tip of the last vector, B. The sequence of which vector is drawn first is not important. As shown in Figure 2.21(c), vector B is drawn first, with the tip of B touching the tail of A. The resultant R obtained is identical in both cases for magnitude and inclination θ. Again, the sense of the resultant moves from the origin point O to the tip of the last vector. Note that the triangle shown in Figure 2.21(b) is the upper half of a parallelogram, and the triangle shown in Figure 2.21(c) forms the lower half. Because the order in which the vectors are drawn is unim- portant, where A + B = B + A, we can conclude that the vector addition is commutative. 2.2 Solve the same problem shown in Example Problem 2.1, but use the tip-to-tail method. Tip-to-tail solution. Statics 25 Graphical Addition of Three or More Vectors The sum of any number of vectors may be obtained by ap- plying repeatedly the parallelogram law (or tip-to-tail method) to successive pairs of vectors until all of the given vectors are replaced by a single resultant vector. Note: The graphical method of vector addition requires all vec- tors to be coplanar. (a) (b) (c) Figure 2.22 Parallelogram method. Assume that three coplanar forces A, B, and C are acting at point O, as shown in Figure 2.22(a), and that the resultant of all three is desired. In Figure 2.22(b) and 2.22(c), the par- allelogram law is applied successively until the final resul- tant force R¿ is obtained. The addition of vectors A and B yields the intermediate resultant R; R is then added vecto- rially to vector C, resulting in R¿. A simpler solution may be obtained by using the tip-to-tail method, as shown in Figure 2.23. Again, the vectors are drawn to scale but not necessarily in any particular sequence. (a) (b) (c) Figure 2.23 Illustration of the tip-to-tail method. 26 Chapter 2 Example Problems: Graphical Addition of Three or More Vectors 2.3 Two cables suspended from an eyebolt carry 200# and 300# loads as s