THIRD SI J\l\ETRIC EDITION Vector Mechanics for Engineers FERDINAND P BEER Lehigh University E RUSSELL JOHNSTON, JR University of Connecticut With the collaboration of Elliot R Eisenberg Pennsylvania State University SI Metric adaptation by Theodore Wildl Sperika Enterprises McGraw-Hili Ryerson Toronto New York Burr Ridge Bangkok Bogota Caracas Lisbon London Madrid Mexico City Milan New Delhi Seoul Singapore Sydney Taipei McGraw-Hill Ryerson Limited A Subsidiary of The McGraw-Hill Companies Vector Mechanics for Engineers: Statics Third SI Metric Edition Copyright © 1998, 1988, 1984, 1977, 1972, 1962 McGraw-Hili Ryerson Limited, a Subsidiary of The McGraw-Hili Companies Copyright © 1996, 1988, 1984, 1977, 1972, 1962 McGraw-Hili, Inc All rights reserved No part of this pubiication may be reproduced or transmitted in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of McGraw-Hili Ryerson Limited, or in the case of photocopying or other reprographic copying, a licence from CANCOPY (the Canadian Copyright Licensing Agency), Adelaide Street East, Suite 900, Toronto, Ontario, Canada M5C 1H6 Any request for photocopying, recording, directed in writing to CANCOPY or taping of any part of this publication shall be ISBN: 0-07-560076-5 10 VH Care has been taken to trace ownership of copyright material contained in this text; however, the pubiishers wili welcome any information that enables them to rectify any reference or credit for subsequent editions Sponsoring Editor: Dave Ward Supervising Editor: Margaret Henderson Developmental Editor: Laurie Graham Proofreader: Matthew Kudelka Production Coordinator: Nicla Dattolico Designer: Merrill Haber Illustrations: FineLine Illustrations, Inc Cover Photo: Derek Croucher/First Light Typesetter: York Graphic Services, Inc Typeface: New Caledonia Printer: Van Hoffman Press, Inc The cover photograph is of the pyramid designed by the American architect I M Pei to serve as the principle entrance to the Grand Louvre museum in Paris, France It is 21 metres high with a 33-metre square base and consists of four sides made of glass that are supported by a truss system composed of thin stainless-steel tubes and cables located inside the pyramid, close to its surface This design technique and the materials used combine to give to the pyramid its remarkably graceful and translucent appearance PHOTO CREDITS: Cover: Derek Croucher/First Light Authors' photograph: B J Clark, 1995 Chapter 1: Bill Sanderson/Science Photo Library/Photo Researchers; Chapter 2: d'Arazien/lmage Bank; Chapter 3: John Coletti/Stock, Boston; Chapter 4: T Zimmermann/FPG; Chapter 5: Bruce Hands/Stock, Boston; Chapter 6: Jeff Gnass/Stock Market; Chapter 7: Brian Yarvin/Photo Researchers; Chapter 8: Wayne Hoy/Picture Cube; Chapter 9: Paul Steel/Stock Market; Chapter 10: Wolf Von Dem Bussche/lmage Block Canadian Cataloguing in Publication Beer, Ferdinand P., (date)Vector mechanics for engineers: statics 3rd SI metric ed Includes index ISBN 0-07-560076-5 Mechanics, Applied Statics Vector analysis Mechanics, Applied-Problems, exercises, etc I Johnston, E Russell (Eiwood Russell), (date)- II Eisenberg, Eliiot R III Wildi, Theodore, (date)- IV Title TA351.B441998 620.1'053'0151563 C98-930601-1 About the Authors "How did you happen to write your books together, with one of you at Lehigh and the other at UConn, and how you manage to keep collaborating on their successive revisions?" These are the two questions most often asked of our two authors The answer to the first question is simple Russ Johnston's first teaching appointment was in the Department of Civil Engineering and Mechanics at Lehigh University There he met Ferd Beer, who had joined that department two years earlier and was in charge of the courses in mechanics Born in France and educated in France and Switzerland (he holds an M.S degree from the Sorbonne and an Sc.D degree in the field of theoretical mechanics from the University of Geneva), Ferd had come to the United States after serving in the French army during the early part of World War II and had taught for four years at Williams College in The Williams-MIT joint arts and engineering program Born in Philadelphia, Russ had obtained a B.S degree in civil engineering from the University of Delaware and an Sc.D degree in the field of structural engineering from MIT Ferd was delighted to discover that the young man who had been hired chiefly to teach graduate structural engineering courses was not only willing but eager to help him reorganize the mechanics courses Both believed that these courses should be taught from a few basic principles and that the various concepts involved would be best understood and remembered by the students if they were presented to them in a graphic way Together they wrote lecture notes in statics and dynamics, to which they later added problems they felt would appeal to future engineers, and soon they produced the manuscript of the first edition of Mechanics for Engineers The second edition of Mechanics for Engineers and the first edition of Vector Mechanics for Engineers found Russ Johnston at Worcester Polytechnic Institute and the next editions at the University of Connecticut In the meantime, both Ferd and Russ had assumed administrative responsibilities in their departments, and both were involved in research, consulting, and supervising graduate students-Ferd in the area of stochastic processes and random vibrations, and Russ in the area of elastic v vi About the Authors stability and structural analysis and design Howe\"er their interest in improving the teaching of the basic mechanics courses had not subsided, and they both taught sections of these courses as they kept re\ising their texts and began writing the manuscript of the first edition of Mechanics of Materials This brings us to the second question: How did the authors manage to work together so effectively after Russ Johnston had left Lehigh? Part of the answer is provided by their phone bills and the money they have spent on postage As the publication date of a new edition approaches, they call each other daily and rush to the post office with express-mail packages There are also visits between the two families At one time there were even joint camping trips, with both families pitching their tents next to each other Now, with the advent of the fax machine, they not need to meet so frequently Their collaboration has spanned the years of the revolution in computing The first editions of Mechanics for Engineers and of Vector Mechanics for Engineers included notes on the proper use of the slide rule To guarantee the accuracy of the answers given in the back of the book, the authors themselves used oversize 20-inch slide rules, then mechanical desk calculators complemented by tables of trigonometric functions, and later four-function electronic calculators With the advent of the pocket multifunction calculators, all these were relegated to their respective attics, and the notes in the text on the use of the slide rule were replaced by notes on the use of calculators Now problems requiring the use of a computer are included in each chapter of their texts, and Ferd and Russ program on their own computers the solutions of most of the problems they create Ferd and Russ's contributions to engineering education have earned them a number of honors and awards They were presented with the Western Electric Fund Award for excellence in the instruction of engineering students by their respective regional sections of the American Society for Engineering Education, and they both received the Distinguished Educator Award from the Mechanics Division of the same society In 1991 Russ received the Outstanding Civil Engineer Award from the Connecticut Section of the American Society of Civil Engineers, and in 1995 Ferd was awarded an honorary Doctor of Engineering degree by Lehigh University A new collaborator, Elliot Eisenberg, Professor of Engineering at the Pennsylvania State University, has joined the Beer and Johnston team for this new edition Elliot holds a B.S degree in engineering and an M.E degree, both from Cornell University He has focused his scholarly activities on professional service and teaching, and he was recognized for this work in 1992 when the American Society of Mechanical Engineers awarded him the Ben C Sparks Medal for his contributions to mechanical engineering and mechanical engineering technology education and for service to that society and to the American Society for Engineering Education And finally, there are the contributions of Theodore Wildi to the integrated conversion of this Third SI Metric Edition He is Chair of the CSA Technical Committee on the International System of Units and author of Metric Units and Conversion Charts, a widely used handbook for professional engineers Contents Preface xiii List of Symbols xvii INTRODUCTION 1.1 1.2 1.3 1.4 1.5 What Is Mechanics? Fundamental Concepts and Principles Systems of Units Method of Problem Solution Numerical Accuracy 2 STATICS OF PARTICLES 11 2.1 Introduction 12 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 Forces in a Plane 12 Force on a Particle Resultant of Two Forces 12 Vectors 13 Addition of Vectors 14 Resultant of Several Concurrent Forces 16 Resolution of a Force into Components 17 Rectangular Components of a Force Unit Vectors 23 Addition of Forces by Summing x and y Components 26 Equilibrium of a Particle 31 Newton's First Law of Motion 32 Problems Involving the Equilibrium of a Particle Free-Body Diagrams 32 Forces in Space 41 2.12 Rectangular Components of a Force in Space 2.13 Force Defined by Its Magnitude and Two Points on Its Line of Action 44 2.14 Addition of Concurrent Forces in Space 45 41 vii viii Contents 2.15 Equilibrium of a Particle in Space Review and Summary for Chapter Review Problems 63 53 60 RIGID BODIES: EQUIVALENT SYSTEMS OF FORCES 67 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 *3.21 Introduction 68 External and Internal Forces 68 Principle of Transmissibility Equivalent Forces 69 Vector Product of Two Vectors 71 Vector Products Expressed in Terms of Rectangular Components 73 Moment of a Force about a Point 75 Varignon's Theorem 77 Rectangular Components of the Moment of a Force 77 Scalar Product of Two Vectors 87 Mixed Triple Product of Three Vectors 89 Moment of a Force about a Given Axis 91 Moment of a Couple 101 Equivalent Couples 102 Addition of Couples 104 Couples Can Be Represented by Vectors 104 Resolution of a Given Force Into a Force at and a Couple 105 Reduction of a System of Forces to One Force and One Couple 116 Equivalent Systems of Forces 118 Equipollent Systems of Vectors 118 Further Reduction of a System of Forces 119 Reduction of a System of Forces to a Wrench 121 Review and Summary for Chapter Review Problems 145 140 EQUILIBRIUM 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Introduction 150 Free-Body Diagram OF RIGID BODIES 149 151 Equilibrium in Two Dimensions 152 Reactions at Supports and Connections for a Two-Dimensional Structure 152 Equilibrium of a Rigid Body in Two Dimensions 154 Statically Indeterminate Reactions Partial Constraints Equilibrium of a Two-Force Body 173 Equilibrium of a Three-Force Body 174 Equilibrium in Three Dimensions 181 Equilibrium of a Rigid Body in Three Dimensions Reactions at Supports and Connections for a Three-Dimensional Structure 181 Review and Summary for Chapter Review Problems 200 198 185 156 Contents DISTRIBUTED FORCES: CENTROIDS AND CENTERS OF GRAVITY 204 5.1 5.2 5.3 5.4 5.5 5.6 5.7 *5.8 *5.9 Introduction 206 Areas and Lines 206 Center of Gravity of a Two-Dimensional Body 206 Centroids of Areas and Lines 208 First Moments of Areas and Lines 209 Composite Plates and Wires 212 Determination of Centroids by Integration 223 Theorems of Pappus-Guldinus 225 Distributed Loads on Beams 236 Forces on Submerged Surfaces 237 Volumes 247 5.10 Center of Gravity of a Three-Dimensional Body Centroid of a Volume 247 5.11 Composite Bodies 250 5.12 Determination of Centroids of Volumes by Integration Review and Summary for Chapter Review Problems 266 262 ANALYSIS OF STRUCTURES 270 6.1 6.2 6.3 6.4 *6.5 *6.6 6.7 *6.8 Introduction 271 Trusses 272 Definition of a Truss 272 Simple Trusses 274 Analysis of Trusses by the Method of Joints 275 Joints under Special Loading Conditions 277 Space Trusses 279 Analysis of Trusses by the Method of Sections 289 Trusses Made of Several Simple Trusses 290 Frames and Machines 301 6.9 Structures Containing Multiforce Members 301 6.10 Analysis of a Frame 301 6.11 Frames Which Cease to Be Rigid When Detached from Their Supports 302 6.12 Machines 317 Review and Summary for Chapter Review Problems 332 329 FORCES IN BEAMS AND CABLES 337 *7.1 *7.2 Introduction 338 Internal Forces in Members *7.3 Beams 345 Various Types of Loading and Support 338 345 250 ix X Contents *7.4 *7.5 *7.6 Shear and Bending Moment in a Beam 346 Shear and Bending-Moment Diagrams 348 Relations among Load, Shear, and Bending Moment *7.7 *7.8 *7.9 *7.10 Cables 367 Cables with Concentrated Loads 367 Cables with Distributed Loads 368 Parabolic Cable 369 Catenary 378 Review and Summary for Chapter Review Problems 389 356 386 FRICTION 392 8.1 8.2 8.3 8.4 8.5 8.6 *8.7 *8.8 *8.9 *8.10 Introduction 393 The Laws of Dry Friction Coefficients of Friction Angles of Friction 396 Problems Involving Dry Friction 397 Wedges 413 Square-Threaded Screws 413 Journal Bearings Axle Friction 422 Thrust Bearings Disk Friction 424 Wheel Friction Rolling Resistance 425 Belt Friction 432 Review and Summary for Chapter Review Problems 446 323 443 DISTRIBUTED 9.1 Introduction FORCES: MOMENTS OF INERTIA 451 452 Moments of Inertia of Areas 453 Second Moment, or Moment of Inertia, of an Area 453 Determination of the Moment of Inertia of an Area by Integration 454 9.4 Polar Moment of Inertia 455 9.5 Radius of Gyration of an Area 456 9.6 Parallel-Axis Theorem 463 9.7 Moments of Inertia of Composite Areas 464 *9.8 Product of Inertia 476 *9.9 Principal Axes and Principal Moments of Inertia 477 *9.10 Mohr's Circle for Moments and Products of Inertia 485 9.2 9.3 Moments of Inertia of Masses 491 Moment of Inertia of a Mass 491 Parallel-Axis Theorem 493 Moments of Inertia of Thin Plates 494 Determination of the Moment of Inertia of a Three-Dimensional Body by Integration 495 9.15 Moments of Inertia of Composite Bodies 495 *9.16 Moment of Inertia of a Body with Respect to an Arbitrary Axis through O Mass Products of Inertia 510 9.11 9.12 9.13 9.14 *9.17 *9.18 Ellipsoid of Inertia Principal Axes of Inertia 511 Determination of the Principal Axes and Principal Moments of Inertia of a Body of Arbitrary Shape 513 Review and Summary for Chapter Review Problems 530 524 10 METHOD OF VIRTUAL WORK 535 *10.1 *10.2 *10.3 *10.4 *10.5 *10.6 *10.7 *10.8 *10.9 Introduction 536 Work of a Force 536 Principle of Virtual Work 539 Applications of the Principle of Virtual Work 540 Real Machines Mechanical Efficiency 542 Work of a Force during a Finite Displacement 556 Potential Energy 558 Potential Energy and Equilibrium 559 Stability of Equilibrium 560 Review and Summary for Chapter 10 Review Problems 573 u.s 570 Appendix CUSTOMARY UNITS AND CONVERSIONS TO SI 577 A.1 A.2 Index U.S Customary Units 577 Conversion from One System of Units to Another 583 Answers to Problems 589 578 Contents xi Preface The main objective of a first course in mechanics should be to develop in the engineering student the ability to analyze any problem in a simple and logical manner and apply to its solution a few, well-understood basic principles It is hoped that this text, designed for the first course in statics offered in the sophomore year, and the volume that follows, Vector Mechanics for Engineers: Dynamics, will help the instructor achieve this goal t Vector algebra is introduced early in the text and is used in the presentation and the discussion of the fundamental principles of mechanics Vector methods are also used to solve many problems, particularly three-dimensional problems where these techniques result in a simpler and more concise solution The emphasis in this text, however, remains on the correct understanding of the principles of mechanics and on their application to the solution of engineering problems, and vector algebra is presented chiefly as a convenient tool.! One of the characteristics of the approach used in these volumes is that the mechanics of particles has been clearly separated from the mechanics of rigid bodies This approach makes it possible to consider simple practical applications at an early stage and to postpone the introduction of more difficult concepts In this volume, for example, the statics of particles is treated first (Chap 2); after the rules of addition and subtraction of vectors have been introduced, the principle of equilibrium of a particle is immediately applied to practical situations involving only concurrent forces The statics of rigid bodies is considered in Chaps and In Chap 3, the vector and scalar products of two vectors are introduced and used to define the moment of a force about a point and about an axis The presentation of these new concepts is followed by a thorough and rigorous discussion of equivalent systems of forces leading, in Chap 4, to many practical applications involving the equilibrium of rigid bodies tBoth texts are also available in a single volume, Vector Mechanics for Engineers: Statics and Dynamics, sixth edition JIn a parallel text, Mechanics for Engineers: Statics, fourth edition, the use of vector algebra is limited to the addition and subtraction of vectors xiii Index Resultant of forces, 12-13, 45, 117 (See also Addition, of forces; Addition, of vectors) Revolution body of, 225, 396 surface of, 225 Right-hand rule, 71, 75 Rigid body, 3, 68 equilibrium of: in a plane, 150-183 in space, 182 free-body diagram of, 151 Rigid truss, 274, 291 Rollers, 152, 182, 183 Rolling resistance, 425-426 coefficient of, 426 Rough surfaces, 152, 182, 183 Sag, 370, 380 Scalar components, 24 Scalar product, 87-89 Scalars, 13 Screws, 413-414 Second, 5, Second moment, 453-454 Sections, method of, 289-290 Self-locking screws, 414 Sense of a force, 14 Shear, 338, 346-358 Shear diagram, 348 SI units, 5-8 Significant figures, Simple trusses, 274, 279 Software, computer, 455 Space, Space truss, 279 Specific weight, 208, 248 Spring: force exerted by, 557 potential energy, 558 Spring constant, 557 Square-threaded screws, 413-414 Stable equilibrium, 560-561 Static friction, 393 angle of, 396 coefficient of, 394 Statically determinate reactions, 156 Statically determinate structures, 304 Statically determinate trusses, 291 Statically indeterminate reactions, 156, 182 Statically indeterminate structures, 304 Statically indeterminate trusses, 291 Statics, definition of, Structural shapes, properties of, 466, 581 Structures: analysis of, 271-316 determinate, 304 indeterminate, 304 internal forces in, 271, 338, 339 two-dimensional, 76, 152 Submerged surfaces, forces on, 237, 453 Subtraction of vectors, 15 Supports: ball, 182, 183 ball-and-socket, 182, 183 of beams, 345-346 reactions at, 152, 153, 182 Surface: frictionless, 152, 182, 183 of revolution, 225 rough, 152, 182, 183 submerged, forces on, 237, 453 Suspension bridges, 369 Symmetry: axis of 210 center' of 210 plane of,'248 Systems: of forces, 116-122 of units 5-8 ' Tension, 71, 273, 338 Three-force body, 174 Thrust bearings, 424-425 Time, Toggle vise, analysis of, 540-542 Ton: metric, 6n Transfer formula (see Parallel-axis theorem) Transmissibility, principle of, 3, 69- 70 Triangle rule, 15 Trusses, 272-291 compound, 290 determinate, 291 indeterminate, 291 overrigid, 291 rigid, 274, 291 simple, 274, 279 space, 279 typical, 273 587 )88 Index :\vo-dimensional structures, :\vo-force body, 173-174 76, 152 Jnit vectors, 23-24, 43 Jnits, 5-8 Jniversal joints, 182, 183, 328 Jnstable equilibrium, 560-561 Jnstable rigid bodies, 156n , belts, 434 'arignon's theorem, 77 'ector addition, 14-16 'ector components, 26 'ector product, 71- 73 determinant form for, 74 rectangular components of, 73-74 ector subtraction, 15 ectors, 13 bound, fixed, 13 coplanar, 16 couple, 104 free, 13 sliding, 14, 70 Virtual displacement, 539-543 Virtual work, 539 principle of, 536, 539-542 Wedges, 413 Weight, Wheel friction, 425-426 Wheels, 182, 183, 425 Work: of a couple, 538, 539, 556 of a force, 536-537 of force exerted by spring, 557 of forces on a rigid body, 538, 540 input and output, 542-543 virtual, 539 of a weight, 556-557 Wrench, 121-122 Zero-force member, 278 ... future engineers, and soon they produced the manuscript of the first edition of Mechanics for Engineers The second edition of Mechanics for Engineers and the first edition of Vector Mechanics for Engineers. .. Pressure Force; vector Force; vector xvii xviii List of Symbols Position vector Radius; distance; polar coordinate Resultant force; resultant vector; reaction Radius of earth Position vector Length... spanned the years of the revolution in computing The first editions of Mechanics for Engineers and of Vector Mechanics for Engineers included notes on the proper use of the slide rule To guarantee