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PART TWO ANALYSIS OF STATICALLY DETERMINATE STRUCTURES 413 Equilibrium and Support Reactions 43 3.1 Equilibrium of Structures 433.2 External and Internal Forces 463.3 Types of Supports f

Temperature Conversion Formulas T(°C) 5

Moment of inertia (area)

inch to fourth power in.4 0.416231 106 0.416 106 meter to fourth power m4

Moment of inertia (mass)

Power

Pressure; stress

Section modulus

inch to third power in.3 16.3871 106 16.4 106 meter to third power m3

Velocity (linear)

Volume

*An asterisk denotes an exact conversion factor

Note: To convert from SI units to USCS units, divide by the conversion factor

Structural Analysis

Fourth Edition, SI

Aslam Kassimali

Southern Illinois University—Carbondale

SI Edition prepared by Amit Prashant,

Indian Institute of Technology, Kanpur

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SI Edition prepared by Amit Prashant

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1 2 3 4 5 6 7 13 12 11 10 09

Preface to the SI Edition xiiiPreface xiv

1 Introduction to Structural Analysis 3

1.1 Historical Background 41.2 Role of Structural Analysis in Structural Engineering Projects 61.3 Classification of Structures 7

1.4 Analytical Models 12

Summary 16

2 Loads on Structures 17

2.1 Dead Loads 182.2 Live Loads 212.3 Impact 242.4 Wind Loads 242.5 Snow Loads 322.6 Earthquake Loads 352.7 Hydrostatic and Soil Pressures 372.8 Thermal and Other E¤ects 372.9 Load Combinations 37

Summary 38Problems 39

vii

PART TWO ANALYSIS OF STATICALLY DETERMINATE STRUCTURES 41

3 Equilibrium and Support Reactions 43

3.1 Equilibrium of Structures 433.2 External and Internal Forces 463.3 Types of Supports for Plane Structures 473.4 Static Determinacy, Indeterminacy, and Instability 473.5 Computation of Reactions 60

3.6 Principle of Superposition 783.7 Reactions of Simply Supported Structures Using Proportions 79

Summary 80Problems 82

4 Plane and Space Trusses 89

4.1 Assumptions for Analysis of Trusses 914.2 Arrangement of Members of Plane Trusses—Internal

Stability 954.3 Equations of Condition for Plane Trusses 1004.4 Static Determinacy, Indeterminacy, and Instability of Plane

Trusses 1004.5 Analysis of Plane Trusses by the Method of Joints 1064.6 Analysis of Plane Trusses by the Method of Sections 1214.7 Analysis of Compound Trusses 128

4.8 Complex Trusses 1334.9 Space Trusses 134

Summary 144Problems 145

5 Beams and Frames: Shear and Bending Moment 160

5.1 Axial Force, Shear, and Bending Moment 1615.2 Shear and Bending Moment Diagrams 1675.3 Qualitative Deflected Shapes 172

5.4 Relationships between Loads, Shears, and Bending Moments 1735.5 Static Determinacy, Indeterminacy and Instability of Plane

Frames 195

5.6 Analysis of Plane Frames 201

Summary 215Problems 217

6 Deflections of Beams: Geometric Methods 226

6.1 Di¤erential Equation for Beam Deflection 2276.2 Direct Integration Method 230

6.3 Superposition Method 2336.4 Moment-Area Method 2346.5 Bending Moment Diagrams by Parts 2486.6 Conjugate-Beam Method 253

Summary 269Problems 269

7 Deflections of Trusses, Beams, and Frames: Work–Energy Methods 275

7.1 Work 2767.2 Principle of Virtual Work 2787.3 Deflections of Trusses by the Virtual Work Method 2827.4 Deflections of Beams by the Virtual Work Method 2937.5 Deflections of Frames by the Virtual Work Method 3017.6 Conservation of Energy and Strain Energy 312

7.7 Castigliano’s Second Theorem 3167.8 Betti’s Law and Maxwell’s Law of Reciprocal Deflections 325

Summary 326Problems 328

8 Influence Lines 337

8.1 Influence Lines for Beams and Frames by Equilibrium

Method 3388.2 Mu¨ller-Breslau’s Principle and Qualitative Influence Lines 3538.3 Influence Lines for Girders with Floor Systems 367

8.4 Influence Lines for Trusses 3778.5 Influence Lines for Deflections 389

Summary 392Problems 393

9 Application of Influence Lines 401

9.1 Response at a Particular Location Due to a Single Moving

Concentrated Load 4019.2 Response at a Particular Location Due to a Uniformly

Distributed Live Load 4039.3 Response at a Particular Location Due to a Series of Moving

Concentrated Loads 4089.4 Absolute Maximum Response 415

Summary 421Problems 422

10 Analysis of Symmetric Structures 425

10.1 Symmetric Structures 42610.2 Symmetric and Antisymmetric Components of Loadings 43210.3 Behavior of Symmetric Structures under Symmetric and

Antisymmetric Loadings 44310.4 Procedure for Analysis of Symmetric Structures 447

Summary 455Problems 456

11 Introduction to Statically Indeterminate Structures 461

11.1 Advantages and Disadvantages of Indeterminate Structures 46211.2 Analysis of Indeterminate Structures 465

Summary 470

12 Approximate Analysis of Rectangular Building Frames 471

12.1 Assumptions for Approximate Analysis 47212.2 Analysis for Vertical Loads 475

12.3 Analysis for Lateral Loads—Portal Method 48112.4 Analysis for Lateral Loads—Cantilever Method 497

Summary 504Problems 505

13 Method of Consistent Deformations—Force Method 508

13.1 Structures with Single Degree of Indeterminacy 50913.2 Internal Forces and Moments as Redundants 53113.3 Structures with Multiple Degrees of Indeterminacy 54413.4 Support Settlements, Temperature Changes and Fabrication

Errors 568Summary 577Problems 578

14 Three-Moment Equation and the Method of Least Work 586

14.1 Derivation of Three-Moment Equation 58714.2 Application of Three-Moment Equation 59214.3 Method of Least Work 599

Summary 606Problems 607

15 Influence Lines for Statically Indeterminate Structures 609

15.1 Influence Lines for Beams and Trusses 61015.2 Qualitative Influence Lines by Mu¨ller-Breslau’s Principle 627

Summary 631Problems 632

16 Slope-Deflection Method 635

16.1 Slope-Deflection Equations 63616.2 Basic Concept of the Slope-Deflection Method 64416.3 Analysis of Continuous Beams 651

16.4 Analysis of Frames without Sidesway 67316.5 Analysis of Frames with Sidesway 681

Summary 702Problems 702

17 Moment-Distribution Method 707

17.1 Definitions and Terminology 70817.2 Basic Concept of the Moment-Distribution Method 71717.3 Analysis of Continuous Beams 725

17.4 Analysis of Frames without Sidesway 74117.5 Analysis of Frames with Sidesway 744

Summary 761Problems 762

18 Introduction to Matrix Structural Analysis 767

18.1 Analytical Model 76818.2 Member Sti¤ness Relations in Local Coordinates 77218.3 Coordinate Transformations 780

18.4 Member Sti¤ness Relations in Global Coordinates 78618.5 Structure Sti¤ness Relations 787

18.6 Procedure for Analysis 795

Summary 813Problems 814

Appendix A Areas and Centroids of Geometric Shapes 817

Appendix B Review of Matrix Algebra 821

B.1 Definition of a Matrix 821B.2 Types of Matrices 822B.3 Matrix Operations 824B.4 Solution of Simultaneous Equations by the Gauss-Jordan

Method 831Problems 835

Appendix C Computer Software 837

Starting the Computer Software 837Inputting Data 837

Results of the Analysis 844Problems 849

Bibliography 851Answers to Selected Problems 853Index 863

This edition of Structural Analysis has been adapted to incorporate the tional System of Units (Le Syste`me International d’Unite´s or SI) throughout thebook Amit Prashant wishes to acknowledge the contributions made by his col-leagues, Arindam Dey, and Kaustubh Dasgupta to this SI Edition

Interna-Le Syste`me International d’Unite´s

The United States Customary System (USCS) of units uses FPS second) units (also called English or Imperial units) SI units are primarily theunits of the MKS (meter-kilogram-second) system However, CGS (centimeter-gram-second) units are often accepted as SI units, especially in textbooks

(foot-pound-Using SI Units in this Book

In this book, we have used both MKS and CGS units USCS units or FPS unitsused in the US Edition of the book have been converted to SI units throughoutthe text and problems However, in case of data sourced from handbooks, gov-ernment standards, and product manuals, it is not only extremely di‰cult toconvert all values to SI, it also encroaches upon the intellectual property of thesource Also, some quantities such as the ASTM grain size number and Jominydistances are generally computed in FPS units and would lose their relevance ifconverted to SI Some data in figures, tables, examples, and references, there-fore, remains in FPS units For readers unfamiliar with the relationship betweenthe FPS and the SI systems, conversion tables have been provided inside thefront and back covers of the book

To solve problems that require the use of sourced data, the sourced values can

be converted from FPS units to SI units just before they are to be used in a lation To obtain standardized quantities and manufacturers’ data in SI units, thereaders may contact the appropriate government agencies or authorities in theircountries/regions

calcu-Instructor Resources

A Printed Instructor’s Solution Manual in SI units is available on request Anelectronic version of the Instructor’s Solutions Manual, and PowerPoint slides ofthe figures from the SI text are available through www.cengage.com/engineering.The readers’ feedback on this SI Edition will be highly appreciated and willhelp us improve subsequent editions

The Publishers

The objective of this book is to develop an understanding of the basicprinciples of structural analysis Emphasizing the intuitive classical ap-proach, Structural Analysis covers the analysis of statically determinateand indeterminate beams, trusses, and rigid frames It also presents anintroduction to the matrix analysis of structures.

The book is divided into three parts Part One presents a generalintroduction to the subject of structural engineering It includes a chap-ter devoted entirely to the topic of loads because attention to this im-portant topic is generally lacking in many civil engineering curricula.Part Two, consisting of Chapters 3 through 10, covers the analysis ofstatically determinate beams, trusses, and rigid frames The chapters ondeflections (Chapters 6 and 7) are placed before those on influence lines(Chapters 8 and 9), so that influence lines for deflections can be included

in the latter chapters This part also contains a chapter on the analysis

of symmetric structures (Chapter 10) Part Three of the book, Chapters

11 through 18, covers the analysis of statically indeterminate structures.The format of the book is flexible to enable instructors to emphasizetopics that are consistent with the goals of the course

Each chapter of the book begins with an introductory section fining its objective and ends with a summary section outlining its salientfeatures An important general feature of the book is the inclusion ofstep-by-step procedures for analysis to enable students to make an easiertransition from theory to problem solving Numerous solved examplesare provided to illustrate the application of the fundamental concepts

de-A computer program for analyzing plane framed structures isavailable on the publisher’s website www.cengage.com/engineering.This interactive software can be used to simulate a variety of structuraland loading configurations and to determine cause versus e¤ect rela-tionships between loading and various structural parameters, therebyenhancing the students’ understanding of the behavior of structures.The software shows deflected shapes of structures to enhance students’xiv

understanding of structural response due to various types of loadings Itcan also include the e¤ects of support settlements, temperature changes,and fabrication errors in the analysis A solutions manual, containingcomplete solutions to over 600 text exercises, is also available for theinstructor.

A NOTE ON THE REVISED EDITION

In this fourth edition, while the major features of the third editon havebeen retained, over 15 percent of the problems from the previous editionhave been replaced with new ones The chapter on loads has been revised

to meet the provisions of the ASCE 7-05 Standard, and the treatment ofthe structures with internal hinges has been expanded in Chapter 3 Thecomputer software has been upgraded to make it compatible with thelatest versions of Microsoft Windows Finally, most of the photographshave been replaced with new ones, and the page layout of the book hasbeen redesigned to enhance clarity

ACKNOWLEDGMENTS

I wish to express my thanks to Christopher Carson and Hilda Gowans

of Cengage Learning for their constant support and encouragementthroughout this project, and to Rose Kernan for all her help during theproduction phase The comments and suggestions for improvementfrom colleagues and students who have used previous editions aregratefully acknowledged All of their suggestions were carefully consid-ered, and implemented whenever possible Thanks are due to thefollowing reviewers for their careful reviews of the manuscripts of thevarious editions, and for their constructive suggestions:

Ayo AbatanVirginia Polytechnic Institute andState University

Riyad S AboutahaGeorgia Institute of TechnologyOsama Abudayyeh

Western Michigan University

Thomas T BaberUniversity of VirginiaGordon B BatsonClarkson UniversityGeorge E BlandfordUniversity of Kentucky

Ramon F BorgesPenn State/Altoona CollegeKenneth E Buttry

University of WisconsinSteve C S Cai

Louisiana State UniversityWilliam F CarrollUniversity of Central FloridaMalcolm A CutchinsAuburn UniversityJack H EmanuelUniversity of Missouri—RollaFouad Fanous

Iowa State UniversityLeon Feign

Fairfield UniversityRobert FleischmanUniversity of Notre DameGeorge Kostyrko

California State University

E W LarsonCalifornia State University/

Northridge

Eugene B LoverichNorthern Arizona University

L D LutesTexas A&M UniversityDavid Mazurek

US Coast Guard AcademyAhmad Namini

University of MiamiArturo E SchultzNorth Carolina State UniversityKassim Tarhini

Valparaiso UniversityRobert TaylorNortheastern University

C C TungNorth Carolina State UniversityNicholas Willems

University of KansasJohn ZacharMilwaukee School of EngineeringMannocherh Zoghi

University of Dayton

Finally, I would like to express my loving appreciation to my wife.Maureen, for her constant encouragement and help in preparing thismanuscript, and to my sons, Jamil and Nadim, for their enormous un-derstanding and patience

Aslam Kassimali

Part One

Introduction to

Structural Analysis and Loads

Introduction to Structural Analysis

1.1 Historical Background1.2 Role of Structural Analysis in Structural Engineering Projects1.3 Classification of Structures

1.4 Analytical ModelsSummary

3

Structural analysis is the prediction of the performance of a given ture under prescribed loads and/or other external e¤ects, such as supportmovements and temperature changes The performance characteristicscommonly of interest in the design of structures are (1) stresses or stressresultants, such as axial forces, shear forces, and bending moments; (2)deflections; and (3) support reactions Thus, the analysis of a structureusually involves determination of these quantities as caused by a givenloading condition The objective of this text is to present the methodsfor the analysis of structures in static equilibrium

This chapter provides a general introduction to the subject of tural analysis We first give a brief historical background, includingnames of people whose work is important in the field Then we discussthe role of structural analysis in structural engineering projects We de-scribe the five common types of structures: tension and compressionstructures, trusses, and shear and bending structures Finally, we con-sider the development of the simplified models of real structures for thepurpose of analysis

struc-Marina City District, Chicago

Hisham Ibrahim / Photographer’s Choice RF /Getty Images

1.1 HISTORICAL BACKGROUND

Since the dawn of history, structural engineering has been an essentialpart of human endeavor However, it was not until about the middle ofthe seventeenth century that engineers began applying the knowledge

of mechanics (mathematics and science) in designing structures Earlierengineering structures were designed by trial and error and by using rules

of thumb based on past experience The fact that some of the nificent structures from earlier eras, such as Egyptian pyramids (about

mag-3000 b.c.), Greek temples (500–200 b.c.), Roman coliseums and ducts (200 b.c.–a.d 200), and Gothic cathedrals (a.d 1000–1500), stillstand today is a testimonial to the ingenuity of their builders (Fig 1.1).Galileo Galilei (1564–1642) is generally considered to be the origi-nator of the theory of structures In his book entitled Two New Sciences,which was published in 1638, Galileo analyzed the failure of some sim-ple structures, including cantilever beams Although Galileo’s predic-tions of strengths of beams were only approximate, his work laid thefoundation for future developments in the theory of structures and

aque-FIG.1.1 The Cathedral of Notre Dame

in Paris Was Completed in the

Thirteenth Century

ushered in a new era of structural engineering, in which the analyticalprinciples of mechanics and strength of materials would have a majorinfluence on the design of structures.

Following Galileo’s pioneering work, the knowledge of structuralmechanics advanced at a rapid pace in the second half of the seven-teenth century and into the eighteenth century Among the notable in-vestigators of that period were Robert Hooke (1635–1703), who devel-oped the law of linear relationships between the force and deformation

of materials (Hooke’s law); Sir Isaac Newton (1642–1727), who lated the laws of motion and developed calculus; John Bernoulli (1667–1748), who formulated the principle of virtual work; Leonhard Euler(1707–1783), who developed the theory of buckling of columns; and C

formu-A de Coulomb (1736–1806), who presented the analysis of bending ofelastic beams

In 1826 L M Navier (1785–1836) published a treatise on elasticbehavior of structures, which is considered to be the first textbook onthe modern theory of strength of materials The development of struc-tural mechanics continued at a tremendous pace throughout the rest ofthe nineteenth century and into the first half of the twentieth, when most

of the classical methods for the analysis of structures described in thistext were developed The important contributors of this period included

B P Clapeyron (1799–1864), who formulated the three-moment tion for the analysis of continuous beams; J C Maxwell (1831–1879),who presented the method of consistent deformations and the law ofreciprocal deflections; Otto Mohr (1835–1918), who developed the con-jugate-beam method for calculation of deflections and Mohr’s circles

equa-of stress and strain; Alberto Castigliano (1847–1884), who formulatedthe theorem of least work; C E Greene (1842–1903), who developedthe moment-area method; H Mu¨ller-Breslau (1851–1925), who pre-sented a principle for constructing influence lines; G A Maney (1888–1947), who developed the slope-deflection method, which is considered

to be the precursor of the matrix sti¤ness method; and Hardy Cross(1885–1959), who developed the moment-distribution method in 1924.The moment-distribution method provided engineers with a simple iter-ative procedure for analyzing highly statically indeterminate structures.This method, which was the most widely used by structural engineersduring the period from about 1930 to 1970, contributed significantly totheir understanding of the behavior of statically indeterminate frames.Many structures designed during that period, such as high-rise buildings,would not have been possible without the availability of the moment-distribution method

The availability of computers in the 1950s revolutionized structuralanalysis Because the computer could solve large systems of simulta-neous equations, analyses that took days and sometimes weeks in theprecomputer era could now be performed in seconds The development

of the current computer-oriented methods of structural analysis can beattributed to, among others, J H Argyris, R W Clough, S Kelsey,

R K Livesley, H C Martin, M T Turner, E L Wilson, and O C.Zienkiewicz.

1.2 ROLE OF STRUCTURAL ANALYSIS IN STRUCTURAL ENGINEERING PROJECTS

Structural engineering is the science and art of planning, designing, andconstructing safe and economical structures that will serve their intendedpurposes Structural analysis is an integral part of any structural engi-neering project, its function being the prediction of the performance ofthe proposed structure A flowchart showing the various phases of atypical structural engineering project is presented in Fig 1.2 As this di-agram indicates, the process is an iterative one, and it generally consists

of the following steps:

1 Planning Phase The planning phase usually involves the ment of the functional requirements of the proposed structure, the

establish-FIG.1.2 Phases of a Typical Structural

Engineering Project

general layout and dimensions of the structure, consideration of thepossible types of structures (e.g., rigid frame or truss) that may befeasible and the types of materials to be used (e.g., structural steel

or reinforced concrete) This phase may also involve consideration

of nonstructural factors, such as aesthetics, environmental impact

of the structure, and so on The outcome of this phase is usually astructural system that meets the functional requirements and is ex-pected to be the most economical This phase is perhaps the mostcrucial one of the entire project and requires experience and knowl-edge of construction practices in addition to a thorough under-standing of the behavior of structures

2 Preliminary Structural Design In the preliminary structural designphase, the sizes of the various members of the structural system se-lected in the planning phase are estimated based on approximateanalysis, past experience, and code requirements The member sizesthus selected are used in the next phase to estimate the weight of thestructure

3 Estimation of Loads Estimation of loads involves determination ofall the loads that can be expected to act on the structure

4 Structural Analysis In structural analysis, the values of the loadsare used to carry out an analysis of the structure in order to de-termine the stresses or stress resultants in the members and thedeflections at various points of the structure

5 Safety and Serviceability Checks The results of the analysis are used

to determine whether or not the structure satisfies the safety andserviceability requirements of the design codes If these requirementsare satisfied, then the design drawings and the construction specifi-cations are prepared, and the construction phase begins

6 Revised Structural Design If the code requirements are not isfied, then the member sizes are revised, and phases 3 through 5are repeated until all the safety and serviceability requirements aresatisfied

sat-Except for a discussion of the types of loads that can be expected toact on structures (Chapter 2), our primary focus in this text will be onthe analysis of structures

1.3 CLASSIFICATION OF STRUCTURES

As discussed in the preceding section, perhaps the most important sion made by a structural engineer in implementing an engineering pro-ject is the selection of the type of structure to be used for supporting ortransmitting loads Commonly used structures can be classified into fivebasic categories, depending on the type of primary stresses that maydevelop in their members under major design loads However, it should

deci-be realized that any two or more of the basic structural types descrideci-bed

in the following may be combined in a single structure, such as a ing or a bridge, to meet the structure’s functional requirements

build-Tension Structures

The members of tension structures are subjected to pure tension underthe action of external loads Because the tensile stress is distributed uni-formly over the cross-sectional areas of members, the material of such

a structure is utilized in the most e‰cient manner Tension structurescomposed of flexible steel cables are frequently employed to supportbridges and long-span roofs Because of their flexibility, cables havenegligible bending sti¤ness and can develop only tension Thus, underexternal loads, a cable adopts a shape that enables it to support the load

by tensile forces alone In other words, the shape of a cable changes

as the loads acting on it change As an example, the shapes that a singlecable may assume under two di¤erent loading conditions are shown inFig 1.3

Figure 1.4 shows a familiar type of cable structure—the suspensionbridge In a suspension bridge, the roadway is suspended from two maincables by means of vertical hangers The main cables pass over a pair

of towers and are anchored into solid rock or a concrete foundation attheir ends Because suspension bridges and other cable structures lacksti¤ness in lateral directions, they are susceptible to wind-induced oscil-lations (see Fig 1.5) Bracing or sti¤ening systems are therefore provided

to reduce such oscillations

Besides cable structures, other examples of tension structures includevertical rods used as hangers (for example, to support balconies or tanks)and membrane structures such as tents

FIG.1.3

Compression Structures

Compression structures develop mainly compressive stresses under theaction of external loads Two common examples of such structures arecolumns and arches Columns are straight members subjected to axiallycompressive loads, as shown in Fig 1.6 When a straight member issubjected to lateral loads and/or moments in addition to axial loads, it iscalled a beam-column

An arch is a curved structure, with a shape similar to that of an verted cable, as shown in Fig 1.7 Such structures are frequently used tosupport bridges and long-span roofs Arches develop mainly compres-

in-FIG.1.4 Suspension Bridge

FIG.1.5 Tacoma Narrows Bridge

Oscillating before Its Collapse in 1940

Smithsonian Institution Photo No 72-787

sive stresses when subjected to loads and are usually designed so thatthey will develop only compression under a major design loading How-ever, because arches are rigid and cannot change their shapes as cancables, other loading conditions usually produce secondary bending andshear stresses in these structures, which, if significant, should be con-sidered in their designs.

Because compression structures are susceptible to buckling or stability, the possibility of such a failure should be considered in theirdesigns; if necessary, adequate bracing must be provided to avoid suchfailures

in-Trusses

Trusses are composed of straight members connected at their ends byhinged connections to form a stable configuration (Fig 1.8) When theloads are applied to a truss only at the joints, its members either elon-gate or shorten Thus, the members of an ideal truss are always either

in uniform tension or in uniform compression Real trusses are usuallyconstructed by connecting members to gusset plates by bolted or weldedconnections Although the rigid joints thus formed cause some bending

in the members of a truss when it is loaded, in most cases such dary bending stresses are small, and the assumption of hingedjoints yields satisfactory designs

secon-Trusses, because of their light weight and high strength, are amongthe most commonly used types of structures Such structures are used in

a variety of applications, ranging from supporting roofs of buildings toserving as support structures in space stations

Shear Structures

Shear structures, such as reinforced concrete shear walls (Fig 1.9), areused in multistory buildings to reduce lateral movements due to windloads and earthquake excitations Shear structures develop mainly in-plane shear, with relatively small bending stresses under the action ofexternal loads

FIG.1.6 Column

FIG.1.7 Arch

FIG.1.8 Plane Truss

FIG.1.9 Shear Wall

Bending Structures

Bending structures develop mainly bending stresses under the action ofexternal loads In some structures, the shear stresses associated with thechanges in bending moments may also be significant and should be con-sidered in their designs

Some of the most commonly used structures, such as beams, rigidframes, slabs, and plates, can be classified as bending structures A beam

is a straight member that is loaded perpendicular to its longitudinal axis(Fig 1.10) Recall from previous courses on statics and mechanics ofmaterials that the bending (normal) stress varies linearly over the depth

of a beam from the maximum compressive stress at the fiber farthestfrom the neutral axis on the concave side of the bent beam to the max-imum tensile stress at the outermost fiber on the convex side For ex-ample, in the case of a horizontal beam subjected to a vertically down-ward load, as shown in Fig 1.10, the bending stress varies from themaximum compressive stress at the top edge to the maximum tensilestress at the bottom edge of the beam To utilize the material of a beamcross section most e‰ciently under this varying stress distribution, thecross sections of beams are often I-shaped (see Fig 1.10), with most ofthe material in the top and bottom flanges The I-shaped cross sectionsare most e¤ective in resisting bending moments

Rigid frames are composed of straight members connected togethereither by rigid (moment-resisting) connections or by hinged connections

to form stable configurations Unlike trusses, which are subjected only

to joint loads, the external loads on frames may be applied on themembers as well as on the joints (see Fig 1.11) The members of a rigidframe are, in general, subjected to bending moment, shear, and axialcompression or tension under the action of external loads However, thedesign of horizontal members or beams of rectangular frames is oftengoverned by bending and shear stresses only, since the axial forces insuch members are usually small

Frames, like trusses, are among the most commonly used types ofstructures Structural steel and reinforced concrete frames are commonlyused in multistory buildings (Fig 1.12), bridges, and industrial plants.Frames are also used as supporting structures in airplanes, ships, aero-space vehicles, and other aerospace and mechanical applications

It may be of interest to note that the generic term framed structure isfrequently used to refer to any structure composed of straight members,including a truss In that context, this textbook is devoted primarily tothe analysis of plane framed structures

FIG.1.10 Beam

FIG.1.11 Rigid Frame

1.4 ANALYTICAL MODELS

An analytical model is a simplified representation, or an ideal, of a realstructure for the purpose of analysis The objective of the model is tosimplify the analysis of a complicated structure The analytical modelrepresents, as accurately as practically possible, the behavioral char-acteristics of the structure of interest to the analyst, while discardingmuch of the detail about the members, connections, and so on, that

is expected to have little e¤ect on the desired characteristics lishment of the analytical model is one of the most important steps of theanalysis process; it requires experience and knowledge of design practices

Estab-in addition to a thorough understandEstab-ing of the behavior of structures.Remember that the structural response predicted from the analysis of themodel is valid only to the extent that the model represents the actualstructure

Development of the analytical model generally involves eration of the following factors

consid-Plane Versus Space Structure

If all the members of a structure as well as the applied loads lie in asingle plane, the structure is called a plane structure The analysis ofplane, or two-dimensional, structures is considerably simpler than theanalysis of space, or three-dimensional, structures Fortunately, manyFIG.1.12 Skeletons of Frame Buildings

Racheal Grazias / Shutterstock

actual three-dimensional structures can be subdivided into plane tures for analysis.

struc-As an example, consider the framing system of a bridge shown inFig 1.13(a) The main members of the system, designed to supportvertical loads, are shown by solid lines, whereas the secondary bracingmembers, necessary to resist lateral wind loads and to provide stability,are represented by dashed lines The deck of the bridge rests on beamscalled stringers; these beams are supported by floor beams, which, inturn, are connected at their ends to the joints on the bottom panels ofthe two longitudinal trusses Thus, the weight of the tra‰c, deck, string-ers, and floor beams is transmitted by the floor beams to the supportingtrusses at their joints; the trusses, in turn, transmit the load to the foun-dation Because this applied loading acts on each truss in its own plane,the trusses can be treated as plane structures

As another example, the framing system of a multistory building isshown in Fig 1.14(a) At each story, the floor slab rests on floor beams,which transfer any load applied to the floor, the weight of the slab, andtheir own weight to the girders of the supporting rigid frames This ap-plied loading acts on each frame in its own plane, so each frame can,therefore, be analyzed as a plane structure The loads thus transferred toeach frame are further transmitted from the girders to the columns andthen finally to the foundation

Although a great majority of actual three-dimensional structuralsystems can be subdivided into plane structures for the purpose ofanalysis, some structures, such as latticed domes, aerospace structures,and transmission towers, cannot, due to their shape, arrangement ofmembers, or applied loading, be subdivided into planar components.Such structures, called space structures, are analyzed as three-dimen-sional bodies subjected to three-dimensional force systems

Line Diagram

The analytical model of the two- or three-dimensional body selectedfor analysis is represented by a line diagram On this diagram, eachmember of the structure is represented by a line coinciding with itscentroidal axis The dimensions of the members and the size of theconnections are not shown on the diagram The line diagrams of thebridge truss of Fig 1.13(a), and the rigid frame of Fig 1.14(a) areshown in Figs 1.13(b) and 1.14(b), respectively Note that two lines( * *) are sometimes used in this text to represent members on the line

diagrams This is done, when necessary, for clarity of presentation; insuch cases, the distance between the lines does not represent the mem-ber depth

Two types of connections are commonly used to join members of tures: (1) rigid connections and (2) flexible, or hinged, connections (Athird type of connection, termed a semirigid connection, although rec-ognized by structural steel design codes, is not commonly used in prac-tice and, therefore, is not considered in this text.)

struc-FIG.1.13 Framing of a Bridge

A rigid connection or joint prevents relative translations and tions of the member ends connected to it; that is, all member ends con-nected to a rigid joint have the same translation and rotation In otherwords, the original angles between the members intersecting at a rigidjoint are maintained after the structure has deformed under the action ofloads Such joints are, therefore, capable of transmitting forces as well

rota-as moments between the connected members Rigid joints are usuallyrepresented by points at the intersections of members on the line dia-gram of the structure, as shown in Fig 1.14(b)

A hinged connection or joint prevents only relative translations ofmember ends connected to it; that is, all member ends connected to ahinged joint have the same translation but may have di¤erent rotations.Such joints are thus capable of transmitting forces but not moments be-tween the connected members Hinged joints are usually depicted bysmall circles at the intersections of members on the line diagram of thestructure, as shown in Fig 1.13(b)

FIG.1.14 Framing of a Multistory Building

The perfectly rigid connections and the perfectly flexible frictionlesshinges used in the analysis are merely idealizations of the actual con-nections, which are seldom perfectly rigid or perfectly flexible (see Fig.1.13(c)) However, actual bolted or welded connections are purposelydesigned to behave like the idealized cases For example, the connec-tions of trusses are designed with the centroidal axes of the membersconcurrent at a point, as shown in Fig 1.13(c), to avoid eccentricitiesthat may cause bending of members For such cases, the analysis based

on the idealized connections and supports (described in the followingparagraph) generally yields satisfactory results

Supports

Supports for plane structures are commonly idealized as either fixedsupports, which do not allow any movement; hinged supports, which canprevent translation but permit rotation; or roller, or link, supports, whichcan prevent translation in only one direction A more detailed descrip-tion of the characteristics of these supports is presented in Chapter 3.The symbols commonly used to represent roller and hinged supports online diagrams are shown in Fig 1.13(b), and the symbol for fixed sup-ports is depicted in Fig 1.14(b)

of computers has revolutionized structural analysis

Structural engineering is the science of planning, designing, andconstructing safe, economical structures Structural analysis is an in-tegral part of this process

Structures can be classified into five basic categories, namely, tensionstructures (e.g., cables and hangers), compression structures (e.g., col-umns and arches), trusses, shear structures (e.g., shear walls), and bend-ing structures (e.g., beams and rigid frames)

An analytical model is a simplified representation of a real structurefor the purpose of analysis Development of the model generally involves(1) determination of whether or not the structure can be treated as aplane structure, (2) construction of the line diagram of the structure, and(3) idealization of connections and supports

Loads on Structures

2.1 Dead Loads2.2 Live Loads2.3 Impact2.4 Wind Loads2.5 Snow Loads2.6 Earthquake Loads2.7 Hydrostatic and Soil Pressures2.8 Thermal and Other Effects2.9 Load CombinationsSummary

Problems

17

Earthquake-Damaged Building

Robert Yager / Stone / Getty Images

The objective of a structural engineer is to design a structure that will

be able to withstand all the loads to which it is subjected while serving itsintended purpose throughout its intended life span In designing a struc-ture, an engineer must, therefore, consider all the loads that canrealistically be expected to act on the structure during its planned life span.The loads that act on common civil engineering structures can be groupedaccording to their nature and source into three classes: (1) dead loads due

to the weight of the structural system itself and any other material manently attached to it; (2) live loads, which are movable or moving loadsdue to the use of the structure; and (3) environmental loads, which arecaused by environmental e¤ects, such as wind, snow, and earthquakes

per-In addition to estimating the magnitudes of the design loads, anengineer must also consider the possibility that some of these loadsmight act simultaneously on the structure The structure is finally de-signed so that it will be able to withstand the most unfavorable combi-nation of loads that is likely to occur in its lifetime

The minimum design loads and the load combinations for which thestructures must be designed are usually specified in building codes Buildingcodes vary from country to country and also, owing to geographical varia-tions, from region to region within a country The US national codes pro-viding guidance on loads for buildings, bridges, and other structures include

ASCE Standard Minimum Design Loads for Buildings and Other Structures(ASCE/SEI 7-05) [1],* Manual for Railway Engineering [26], StandardSpecifications for Highway Bridges [36], and International Building Code[15].

Although the load requirements of most local building codes aregenerally based on those of the national codes listed herein, local codesmay contain additional provisions warranted by such regional conditions

as earthquakes, tornadoes, hurricanes, heavy snow, and the like Localbuilding codes are usually legal documents enacted to safeguard publicwelfare and safety, and the engineer must become thoroughly familiarwith the building code for the area in which the structure is to be built.The loads described in the codes are usually based on past experi-ence and study and are the minimum for which the various types ofstructures must be designed However, the engineer must decide if thestructure is to be subjected to any loads in addition to those considered

by the code, and, if so, must design the structure to resist the additionalloads Remember that the engineer is ultimately responsible for the safedesign of the structure

The objective of this chapter is to describe the types of loads monly encountered in the design of structures and to introduce the basicconcepts of load estimation We first describe dead loads and then dis-cuss live loads for buildings and bridges We next consider the dynamice¤ect, or the impact, of live loads We describe environmental loads,including wind loads, snow loads, and earthquake loads We give a briefdiscussion of hydrostatic and soil pressures and thermal e¤ects andconclude with a discussion about the combinations of loads used fordesign purposes

com-The material presented herein is mainly based on the ASCE ard Minimum Design Loads for Buildings and Other Structures (ASCE/SEI 7-05), which is commonly referred to as the ASCE 7 Standard and

Stand-is perhaps the most widely used standard in practice Since the intenthere is to familiarize the reader with the general topic of loads on struc-tures, many of the details have not been included Needless to say, thecomplete provisions of the local building codes or the ASCE 7 Standard†

must be followed in designing structures

2.1 DEAD LOADS

Dead loads are gravity loads of constant magnitudes and fixed positionsthat act permanently on the structure Such loads consist of the weights

of the structural system itself and of all other material and equipment

* The numbers in brackets refer to items listed in the bibliography.

† Copies of this standard may be purchased from the American Society of Civil Engineers,

1801 Alexander Bell Drive, Reston, Virginia 20191-4400.

permanently attached to the structural system For example, the deadloads for a building structure include the weights of frames, framing andbracing systems, floors, roofs, ceilings, walls, stairways, heating and air-conditioning systems, plumbing, electrical systems, and so forth.The weight of the structure is not known in advance of design and isusually assumed based on past experience After the structure has beenanalyzed and the member sizes determined, the actual weight is com-puted by using the member sizes and the unit weights of materials Theactual weight is then compared to the assumed weight, and the design

is revised if necessary The unit weights of some common constructionmaterials are given in Table 2.1 The weights of permanent serviceequipment, such as heating and air-conditioning systems, are usuallyobtained from the manufacturer

Example 2.1

The floor system of a building consists of a 15-cm-thick reinforced concrete slab resting on four steel floor beams, which

in turn are supported by two steel girders, as shown in Fig 2.1(a) The cross-sectional areas of the floor beams and thegirders are 94.8 cm2and 337.4 cm2, respectively Determine the dead loads acting on the beams CG and DH and thegirder AD

TABLE 2.1 UNIT WEIGHTS OF

Beam CG As shown in Fig 2.1(a), the portion of the slab supported by beam CG has a width of 3 m (i.e., half thedistance between beams CG and BF plus half the distance between beams CG and DH) and a length of 8 m This sur-face area (8 3 ¼ 24 m2) supported by beam CG (the shaded rectangular area in Fig 2.1(a)) is referred to as the trib-utary area for beam CG

We use the unit weights of reinforced concrete and structural steel from Table 2.1 to compute the dead load permeter of length of beam CG as follows:

This load is uniformly distributed on the beam, as shown in Fig 2.1(b) This figure also shows the reactions exerted bythe supporting girders at the ends of the beam As the beam is symmetrically loaded, the magnitudes of the reactions areequal to half of the total load acting on the beam:

RC¼ RG¼1

2ð11:35 kN=mÞð8 mÞ ¼ 90:8 kNNote that the magnitudes of these end reactions represent the downward loads being transmitted to the supportinggirders AD and EH at points C and G, respectively

Beam DH The tributary area for beam DH is 1.5 m wide and 8 m long The dead load per foot of length of thisbeam is computed as follows:

Concrete slab: ð23:6 kN=m3Þð1:5 mÞ 15

100 m

¼ 5:31 kN=mSteel beam: ðsame as for beam CGÞ ¼ 0:73 kN=m

As shown in Fig 2.1(c), the end reactions are

RD¼ RH¼1

2ð6:04 kN=mÞð8 mÞ ¼ 48:32 kNGirder AD Because of the symmetry of the framing system and loading, the loads acting on beams BF and AE arethe same as those on beams CG and DH, respectively The load on girder AD consists of the uniformly distributed loaddue to its own weight, which has a magnitude of

2.2 LIVE LOADS

Live loads are loads of varying magnitudes and/or positions caused bythe use of the structure Sometimes, the term live loads is used to refer toall loads on the structure that are not dead loads, including environ-mental loads, such as snow loads or wind loads However, since theprobabilities of occurrence for environmental loads are di¤erent fromthose due to the use of structures, the current codes use the term liveloads to refer only to those variable loads caused by the use of thestructure It is in the latter context that this text uses this term

The magnitudes of design live loads are usually specified in buildingcodes The position of a live load may change, so each member of thestructure must be designed for the position of the load that causes themaximum stress in that member Di¤erent members of a structure mayreach their maximum stress levels at di¤erent positions of the givenload For example, as a truck moves across a truss bridge, the stresses inthe truss members vary as the position of the truck changes If member A

is subjected to its maximum stress when the truck is at a certain position

x, then another member B may reach its maximum stress level when thetruck is in a di¤erent position y on the bridge The procedures for de-termining the position of a live load at which a particular responsecharacteristic, such as a stress resultant or a deflection, of a structure ismaximum (or minimum) are discussed in subsequent chapters

Live Loads for Buildings

Live loads for buildings are usually specified as uniformly distributedsurface loads in kilopascals Minimum floor live loads for some com-mon types of buildings are given in Table 2.2 For a comprehensive list

of live loads for various types of buildings and for provisions regardingroof live loads, concentrated loads, and reduction in live loads, thereader is referred to the ASCE 7 Standard

TABLE 2.2 MINIMUM FLOOR LIVE LOADS FOR BUILDINGS

Live Load

Hospital patient rooms, residentialdwellings, apartments, hotel guest rooms, schoolclassrooms

1.92

Library reading rooms, hospital operating roomsand laboratories

2.87Dance halls and ballrooms, restaurants, gymnasiums 4.79Light manufacturing, light storage warehouses,

wholesale stores

6.00

Source: Adapted with permission from ASCE/SEI 7-05, Minimum Design Loads for Buildings and Other Structures.