**Engineering Mechanics – StaticsFifteenth Edition in SI UnitsR. C. HibbelerSi Conversion byJun Hwa LeeContents1General Principles 25Chapter Objectives 251.1 Mechanics 251.2 Fundamental Concepts 261.3 The International System of Units 291.4 Numerical Calculations 321.5 General Procedure for Analysis 342Force Vectors 39Chapter Objectives 392.1 Scalars and Vectors 392.2 Vector Operations 402.3 Vector Addition of Forces 422.4 Addition of a System of CoplanarForces 542.5 Cartesian Vectors 652.6 Addition of Cartesian Vectors 682.7 Position Vectors 762.8 Force Vector Directed Along a Line 782.9 Dot Product 8618 contents3Equilibrium of aParticle 103Chapter Objectives 1033.1 Condition for the Equilibriumof a Particle 1033.2 The Free-Body Diagram 1043.3 Coplanar Force Systems 1073.4 Three-Dimensional Force Systems 1204Force SystemResultants 135Chapter Objectives 1354.1 Moment of a Force—ScalarFormulation 1354.2 Principle of Moments 1374.3 Cross Product 1454.4 Moment of a Force—VectorFormulation 1484.5 Moment of a Force about aSpecified Axis 1584.6 Moment of a Couple 1674.7 Simplification of a Force and CoupleSystem 1794.8 Further Simplification of a Force andCouple System 1904.9 Reduction of a Simple DistributedLoading 202contents 195Equilibrium of aRigid Body 217Chapter Objectives 2175.1 Conditions for Rigid-BodyEquilibrium 2175.2 Free-Body Diagrams 2195.3 Equations of Equilibrium 2305.4 Two- and Three-Force Members 2405.5 Free-Body Diagrams 2535.6 Equations of Equilibrium 2585.7 Constraints and Statical Determinacy 2596Structural Analysis 279Chapter Objectives 2796.1 Simple Trusses 2796.2 The Method of Joints 2826.3 Zero-Force Members 2886.4 The Method of Sections 2966.5 Space Trusses 3066.6 Frames and Machines 31020 contents7Internal Forces 347Chapter Objectives 3477.1 Internal Loadings 3477.2 Shear and Moment Equations andDiagrams 3637.3 Relations among Distributed Load, Shear,and Moment 3727.4 Cables 3838Friction 403Chapter Objectives 4038.1 Characteristics of Dry Friction 4038.2 Problems Involving Dry Friction 4088.3 Wedges 4308.4 Frictional Forces on Screws 4328.5 Frictional Forces on Flat Belts 4398.6 Frictional Forces on Collar Bearings, PivotBearings, and Disks 4478.7 Frictional Forces on Journal Bearings 4508.8 Rolling Resistance 452contents 219Center of Gravity andCentroid 465Chapter Objectives 4659.1 Center of Gravity, Center of Mass, and theCentroid of a Body 4659.2 Composite Bodies 4889.3 Theorems of Pappus and Guldinus 5029.4 Resultant of a General DistributedLoading 5119.5 Fluid Pressure 51210Moments of Inertia 529Chapter Objectives 52910.1 Definition of Moments of Inertia forAreas 52910.2 Parallel-Axis Theorem for an Area 53010.3 Radius of Gyration of an Area 53110.4 Moments of Inertia for CompositeAreas 54010.5 Product of Inertia for an Area 54810.6 Moments of Inertia for an Area aboutInclined Axes 55210.7 Mohr’s Circle for Moments of Inertia 55510.8 Mass Moment of Inertia 56322 contents11Virtual Work 581Chapter Objectives 58111.1 Definition of Work 58111.2 Principle of Virtual Work 58311.3 Principle of Virtual Work for a System ofConnected Rigid Bodies 58511.4 Conservative Forces 59711.5 Potential Energy 59811.6 Potential-Energy Criterion forEquilibrium 60011.7 Stability of Equilibrium Configuration 601AppendixA. Mathematical Review andFormulations 616Fundamental ProblemSolutions andAnswers 620Review ProblemAnswers 637Selected Answers 640IndexIndexAAcceleration, dynamics and, 25Active force, 105Angles, 56, 66–67, 86–88, 90, 98–99, 405–407, 432, 616–617Cartesian force vectors, 66–67coordinate direction, 66, 98–99dot product used for, 86–88, 90, 99dry friction and, 405–407, 432formed between intersecting lines, 90horizontal (u), 67impending motion and, 405, 407kinetic friction (uk), 406–407lead, 432mathematical review of, 616–617projection, parallel and perpendicular, 87Pythagorean’s theorem and, 56, 87, 617resultant forces from, 56screws, 432static friction (us), 405, 407vectors and, 56, 66–67, 86–88, 90, 98–99vertical (f), 67Applied force (P), 404–407, 430–431, 459–460Area (A), 468, 470, 474–476, 502–505, 523–524, 530–535,540–542, 548–557, 576–577axial symmetry and rotation, 502–505, 524, 548–551,577centroid (C) of an, 468, 470, 474–476, 502–505, 523–524,576centroidal axis of, 530–531composite bodies (shapes), 503, 524, 540–542, 576–577inclined axis, about, 552–554integration for, 468, 474–476, 523, 529–530Mohr’s circle for, 555–557moments of inertia (I) for, 530–535, 540–542, 548–557,576–577Pappus and Guldinus, theorems of, 502–505, 524parallel-axis theorem for, 530–532, 540, 549, 576plane, volume generated revolution of, 503polar moment of inertia, 530–531principal moments of inertia, 553–554, 577procedures for analysis of, 470, 532, 540product of inertia for, 548–551, 553, 576radius of gyration of, 531surface of revolution, 502, 504–505, 524transformation equations for, 552, 577volume of revolution, 503–505, 524Associative law, 146Axes, 158–162, 202, 212, 529–535, 540–542, 548–557, 563–570,576–577area moments of inertia for, 529–535, 548–554, 576centroidal axis of, 530–531, 576composite bodies, 540–542, 568–570, 576–577distributed load reduction, 202inclined, area about, 552–554line of action for, 158–160, 212mass moments of inertia for, 563–570, 577Mohr’s circle for, 555–557moment of a force about specified, 158–162, 212moments of inertia (I), 529–535, 540–542, 552–557,563–570, 576–577parallel-axis theorem for, 530–532, 540, 549, 567, 576principal, 553–557, 577procedures for analysis of, 532, 556, 564product of inertia and, 548–551, 553, 576radius of gyration for, 531, 568resultant forces and, 158–162, 202, 212right-hand rule for, 158–160scalar analysis, 158, 212transformation equations for, 552vector analysis, 159–162, 212Axial loads, friction analysis of, 447–449Axial revolution, 502–505, 524Axial symmetry, 502–505, 524–525axial revolution and, 502–505, 525centroid (C) and, 502–505, 524composite bodies, 503Pappus and Guldinus, theorems of, 502–505, 525rotation and, 502–505, 525surface area and, 502, 504–505, 525volume and, 503–505, 525Axis of symmetry, 469, 488–489, 524, 548–551, 553, 576area (A) of, 548–551centroid (C) and, 469, 488–489, 524, 576parallel-axis theorem for, 549, 576principal axes, 553product of inertia, 548–551, 553, 576BBall and socket connections, 253–254, 256Base units, 29, 31Beams, 347–382, 398–400bending moments (M) and, 348–349, 373–377, 398cantilevered, 347–348, 363centroid (C), 348couple moment (M) and, 374distributed loads and, 372–377, 400force equilibrium, 372–373free-body diagrams, 347–349, 398internal forces, 347–382, 398–400internal loads of, 347–354, 372–377method of sections for, 347–354, 364moments, 348–349, 372–377, 398normal force (N) and, 348–349, 398procedures for analysis of, 349, 364654 IndexBeams (Continued)resultant loadings, 348, 398shear and moment diagrams, 363–366, 372–377, 399–400shear force (V) and, 348–349, 372–377, 398sign convention for, 349, 399simply supported, 363torsional (twisting) moment, 348, 398Bearings, 253–257, 447–451, 461axial loads, 447–449collar, 447–449, 461free-body diagrams, 253–257frictional analysis of, 447–451, 461journal, 254–256, 450–451, 461lateral loads, 450–451pivot, 447–449, 461rigid-body support reactions, 253–257thrust, 255, 256Belts (flat), frictional analysis of, 439–441, 460Bending moment diagrams, 363–366. See also Shear andmoment diagramsBending moments (M), 348–349, 373–377, 398–400distributed loads and, 373–377, 400internal forces and, 348–349, 372–377, 398, 400method of sections for, 348–349shear (V) and, 373shear and moment diagrams, 372–377, 399–400Body at rest (zero), 218By inspection, determination of forces, 282, 288–289CCables, 104, 109, 131, 220, 221, 254, 383–397, 400concentrated loads, 383–385, 400connections, 220, 221continuous, 131distributed loads, 386–389, 400equilibrium of, 104, 109, 131flexibility of, 383free-body diagram for, 104, 109, 131, 220, 254inextensible, 383internal forces of, 109, 383–397, 400sagging, 383support reactions, 220, 254weight of as force, 390–393, 400Calculations, engineering importance of, 32–33Cantilevered beam, 347–348, 363Cartesian coordinate system, 55–58, 65–70, 76–81, 86, 98–99,145–152, 211addition of vectors, 55–58, 68concurrent force resultants, 55, 65–70, 99coordinate direction angles, 66–67, 98–99coplanar force resultants, 55–58cross product for, 145–147direction and, 66–68, 98–99, 145, 148dot product in, 86, 99force vector directed across a line, 78–81horizontal angles (u), 67magnitude of, 55, 66–68, 98, 145, 148moment of a force, calculations by, 148–152, 211position vectors (r), 76–77, 79–80, 99rectangular components, 55–58, 65–70, 98right-hand rule, 65, 145–146, 148sign convention for, 146three-dimensional systems, 65–70two-dimensional systems, 55–58unit vectors, 55, 65–66, 78, 98vector formulation, 146–152, 211vector representation, 65–66, 98–99vertical angles (f), 67Cartesian vector notation, 55Center of gravity (G), 29, 222, 464–527center of mass (Cm) and, 467, 524centroid (C) as, 222, 464–527composite bodies, 488–492, 525constant density and, 488coplanar forces, 222free-body diagrams of, 222location of, 465–466, 469–478, 524Newton’s law of gravitational attraction and, 29procedure for analysis of, 470, 489rigid-body equilibrium and, 222specific weight and, 488weight (W) and, 29, 222, 465–466, 488, 524Center of mass (Cm), 467, 470, 478, 523Center of pressure (P), 513, 525Centroid (C), 203, 222, 348, 464–527area in x–y plane, 468, 470, 474–476, 523axis of symmetry, 469, 488–489, 523axial symmetry, 502–505, 523–524beam cross-section location, 348center of gravity (G) as, 222, 464–527center of mass (Cm) of a body, 467, 470, 478, 523composite bodies, 488–492, 524composite shapes, 503coplanar forces, 222distributed loads and, 511–518, 525distributed loads, 203flat surfaces, 511fluid pressure and, 512–518, 525free-body diagrams for, 222integration for determination of, 467–478, 523line in x-y plane, 468–473, 523line of action for, 203, 513, 520, 525location of, 203, 467–478, 523Index 655mass of a body (Cm), 467, 469, 478method of sections and, 348Pappus and Guldinus, theorems of, 502–505, 524plates, 511–518procedure for analysis of, 470, 489Pythagorean’s theorem for, 469resultant forces and, 203, 348, 511–518, 525rigid-body equilibrium and, 222rotation of an axis, 502–505, 524surface area and, 502, 504–505, 524volume, of a, 467, 470, 503–505, 524Centroidal axis, 530–531, 576Coefficient of kinetic friction (μk), 406–407Coefficient of rolling resistance, 452–453Coefficient of static friction (μs), 405, 407, 447–448Collar bearings, frictional analysis of, 447–449, 461Collinear couple moment, 192Collinear vectors, 41, 97Commutative law, 41, 86, 146Component vectors of a force, 40, 42–48, 97Composite bodies, 488–492, 503, 524, 540–542, 568–570,576–577area (A) of, 503, 540–542, 576axial symmetry and, 488–489, 503center of gravity (G), 488–492, 524centroid (C) of, 488–492, 503, 524constant density and, 488mass moments of inertia, 568–570, 577moments of inertia (I), 540–542, 568–570, 576procedure for analysis of, 489, 540theorem of Pappus and Guldinus for parts of, 503specific weight and, 488weight (W) and, 488, 524Compressive forces (C), 280–283, 296–297method of joints and, 282–283method of sections and, 296–297truss members, 280–281Concentrated force, 27Concentrated loads, 372–373, 383–385, 399–400cables subjected to, 383–385, 400distributed loads, 372–373sagging from, 383shear and moment discontinuities from, 373, 399Concurrent forces, 40, 55, 65–70, 105, 120–124, 131, 190–192,240–241, 260addition of vectors, 40, 65–70Cartesian coordinate system for, 55, 65–70constraints and, 260equilibrium of, 108, 120–124, 131, 260equivalent systems of, 190–192free-body diagrams, 108, 120–124, 131force and couple systems, simplification of, 190–192lines of action for, 190procedure for analysis of, 192resultant couple moment, 190statical determinacy and, 260three-dimensional systems, 65–70, 120–124, 190–192three-force members, 240–241two-dimensional coplanar resultants, 55Connections, free-body diagrams of, 219–221. See also Joints;Support reactionsConservative forces, 597–598friction as nonconservative, 598spring force, 597virtual work (U) and, 597–598weight, 597Constant density, center of gravity (G) and, 488Constraints, 259–267, 276improper, 260–261, 276procedure of analysis of, 262redundant, 259statical determinacy and, 259–267, 276support reactions and, 259–267rigid-body equilibrium and, 259–267, 276Continuous cables, 131Coordinate direction angles, 66–67, 98–99Coordinates, 65–70, 76–80, 98–99, 585–590, 600, 612. See alsoCartesian coordinate systemCartesian, 65–70, 76–77, 79–80, 98–99frictionless systems, 600position, 76–77, 79–80, 585–590, 600, 612potential energy and, 600right-hand rule for, 65vector representation, 65–70, 76–78virtual work for rigid-body connections, 585–590,600, 612x, y, z positions, 65–66, 76, 98–99Coplanar distributed loads, 202–206Coplanar forces, 54–59, 98, 107–111, 131, 179–184, 190–196,202–206, 218–252, 275–276addition of systems of, 54–59, 107Cartesian vector notation, 55center of gravity, 222centroid (geometric center), 222couple moments of, 179–184, 190–196direct solution for unknowns, 230–239, 276direction of, 54–55, 107distributed load reduction, 202–206equations of equilibrium, 107, 131, 230–239, 275equilibrium of, 107–111, 131, 218–252, 275–276equivalent systems of, 179–184, 190–196free-body diagrams, 107–111, 219–228, 275idealized models of, 222–223internal forces and, 222656 IndexCoplanar forces (Continued)lines of action, 179, 190–196magnitude of, 55, 56, 107particles subjected to, 107–111, 131procedure for analysis of, 108, 181, 192, 224, 231rectangular components, 54–59, 98resultant couple moment, 190resultants, 55–59, 179–184, 190–196, 202–206rigid bodies, 218–252, 275–276scalar notation, 54, 55support reactions, 219–221, 275system components, 54–59systems, simplification of, 179–184, 190–196two-and three-force members, 240–241vectors for, 54–59, 98weight and, 222Cosine functions, 617Cosine law, 42, 44, 97Cosines, direction of, 66–67Coulomb friction, 403. See also Dry frictionCouple, 167Couple moments (M0), 167–172, 179–184, 190–196, 212–213,217–219, 374, 582–583collinear, 192concurrent force system simplification, 190–192, 213coplanar force system simplification, 179–184, 190–196,212–213distributed loading, 179–184, 190–196, 212–213, 374equivalent couples, 168equivalent systems, 179–184, 190–196, 212–213force systems and, 167–172, 217–218free vectors, 167internal forces and, 374parallel force system simplification, 191–192, 213procedure for analysis of, 181, 192resultants, 168–169, 190–196right-hand rule for, 167rigid bodies, equilibrium of, 217–218rotation of, 219, 582–583scalar formulation of, 167shear and moment diagrams, 374shear load (V) relationships, 374support reactions and, 219systems, simplification of, 179–184, 190–196, 212–213three-dimensional systems, 179–184, 190–196, 213translation of, 219, 582vector formulation of, 167–172virtual work of, 583work of, 582wrench, reduction of forces to, 192, 213Cross product, 145–147Cartesian vector formulation, 146–147direction for, 145laws of operation, 146magnitude for, 145right-hand rule for, 145–146vector multiplication using, 145–147Curved plates, fluid pressure and, 514Cylinders, rolling resistance of, 452–453DDeformation, rolling resistance and, 452, 461Derivatives, 618Derived units, 29–31Dimensional homogeneity, 32Direct solution for unknowns, 230–239, 276Direction, 39, 55–56, 66–68, 76–78, 87–90, 97–99, 105, 107,136, 145, 148, 158–160, 167, 211, 219, 407, 409, 430,432–434, 460axis, moment of a force about, 158–160Cartesian coordinate vectors, 66–68, 76–78, 98Cartesian vector notation, 55coordinate direction angles, 66–67, 98coplanar force systems, 55–56, 107cross product and, 145dot product applications, 87–90equilibrium and, 105, 107, 219force vector along a line, 78free-body diagrams, 105, 107, 219frictional forces, 407, 409, 430, 432–434, 460horizontal angle u, 56impending motion and, 432–434, 460line of action, 39–40, 78, 99, 148, 158–160moment of a couple, 167moment of a force (MO), 136, 148, 158–160, 211position vectors, 76–77, 99right-hand rule for, 65, 145, 148, 158–160, 167, 211screws, impending motion of, 432–434, 460three-dimensional systems, 66–68translation, 219vector sense of, 39, 55–56, 97vertical angle f, 56Direction cosines, 66–67Disks, 447–449, 461, 564, 566, 577frictional analysis of, 447–449, 461mass moments of inertia, 564, 566, 577Displacement (d), 583–590, 600, 612frictionless systems, 600potential energy and, 600principle of virtual work and, 583–590, 612Index 657procedure for analysis of, 586rigid bodies, connected systems of, 585–590virtual work (U) and, 583–590, 600, 612virtual work equations for, 583–584, 586Distributed loads, 202–206, 213, 372–377, 386–389, 399–400,511–518, 525axis (single) loading, 202beams subjected to, 372–377, 399–400bending moment (M) relationships, 372–377, 400cables subjected to, 386–389, 400center of pressure (P), 512, 525centroid (C) of, 203, 213, 511–518, 525concentrated loads and, 372–373, 399–400coplanar, 202, 213couple moment (M0) relationships, 374fluid pressure from, 512–518, 525force equilibrium, 372–374force system resultants, 202–206, 213incompressible fluids, 512internal forces, 372–377, 386–389, 399–400linearly, 513, 525line of action of, 203, 213loading curve for, 203, 213magnitude and, 202, 511, 525reduction of forces, 202–206, 213resultant forces of, 202–206, 213, 511, 525shear and moment diagrams, 372–377, 399–400shear force (V) relationships, 372–377, 400uniform, 372, 525Distributive law, 86, 146, 150Dot notation, 31Dot product, 86–90, 99, 159angles between intersecting lines, 87, 99applications of, 87–90Cartesian vector formulation, 86, 99laws of operation, 86moment about a specified axis, 159projections, parallel and perpendicular, 87–88, 99unit vectors and, 86–87, 99vector angles and direction from, 86–90, 99Dry friction, 403–463angles (u) of, 405–406applied force (P) and, 404–407, 459–461bearings, analysis of, 447–451, 461belts (flat), analysis of, 439–441, 460collar and pivot bearings, analysis of, 447–449, 461characteristics of, 403–407, 459coefficients of (μ), 405–407, 459direction of force, 407, 409disks, analysis of, 447–449, 461equations for friction versus equilibrium, 409–416equilibrium and, 404–405, 409frictional force, 404–407, 450impending motion, 405, 408–416, 432–434, 459–460journal bearings, analysis of, 450–451, 461kinetic force (Fk), 406–407, 459motion and, 405–416, 432–434, 447–453problems involving, 408–416procedure for analysis of, 411rolling resistance and, 452–453, 461screws, forces on, 432–434, 460sliding and, 406–416, 459slipping and, 405, 407–409, 459static force (Fs), 405, 407, 459theory of, 404tipping effect, balance of, 404, 459wedges and, 430–431, 460Dynamics, study of, 24–26EElastic potential energy (Ve), 598Engineering notation, 32Equations of equilibrium, 103, 107–108, 120–124, 218,230–239, 258, 275–276, 409–416alternative sets for, 230–231body at rest (zero), 218coplanar force systems, 107–108, 230–239, 275–276direct solution, 230–239, 276direction and, 107frictional equations and, 409–416magnitude and, 107particles, 103, 107–108, 120–124procedure for analysis using, 108, 120, 231, 411rigid bodies, 218, 230–239, 275–276scalar form, 258, 275–276three-dimensional force systems, 120–124, 258, 276two-and three-force members, 240–241vector form, 258, 276Equilibrium, 25, 102–133, 216–277, 372–373, 404–405, 409,600–606, 613concurrent forces, 105, 120–124, 131, 260conditions for, 103, 217–218, 230constraints, 259–267coplanar force systems, 107–111, 131, 218–252, 275–276direction and, 105, 107, 219distributed load relationships, 372–373free-body diagrams, 104–111, 120–124, 219–228, 253–257,275–276friction and, 404–405, 409frictionless systems, 600658 IndexEquilibrium (Continued)idealized models for, 222–223impending motion and, 409improper constraints and, 260–261, 276neutral, 601–602, 613one (single) degree-of-freedom system, 602–603particles, 102–133potential-energy (V) criterion for, 600, 613procedures for analysis of, 108, 120, 224, 231, 262, 603redundant constraints and, 259rigid bodies, 216–277shear and moment diagrams, 372–373stability of systems, 260–261, 276, 601–606, 613stable, 601–603, 613statical determinacy and, 259–267, 276statics and, 25support reactions, 219–221, 253–257, 259–267, 275–276three-dimensional force systems, 120–124, 131, 253–267,276tipping effect, balance of, 404, 459two-and three-force members, 240–241two-dimensional force systems, 107–111, 131unstable, 602–603, 613virtual work (U) and, 600–606, 613zero condition, 103, 107, 131, 218Equivalent couples, 168Equivalent systems, 179–184, 190–196concurrent force systems, 190–192coplanar force systems, 179–184, 190–196external effects of, 179force and couple moment simplification, 179–184,190–196lines of action of, 179, 190–196parallel force systems, 191–192principle of transmissibility for, 179procedures for analysis, 181, 192system of force and couple moments, 180three-dimensional systems, 179–184, 190–196wrench, reduction to, 192Exponential notation, 31External effects for equivalent systems, 179External forces, 217, 283, 296FFixed supports, 219, 221, 255Flat belts, frictional analysis of, 439–441, 460Flat plates, 511, 513, 515–516, 525constant width, 513distributed loads on, 511, 525fluid pressure and, 513, 515–516, 525variable width, 515Floor beams, truss analysis and, 280Fluid pressure, 512–518, 525acceleration due to gravity (g), 513center of pressure (P), 513centroid (C), 512–518, 525curved plate of constant width, 514flat plate of constant width, 513flat plate of variable width, 515incompressible fluids, 512line of action, 513Pascal’s law, 512plates, 512–518, 525resultant forces and, 512–518, 525Force, 25–29, 38–215, 217–218, 222, 240–241, 279–305,310–341, 346–463, 511–518, 525, 581–583, 585–590,597–598active, 105addition of vectors, 40–48, 54–59, 68–70applied (P), 404–407, 430–431, 459–460axis, about a specified, 158–162, 202, 212basic quantity of mechanics, 26beams, 347–382, 398–399bending moments (M) and, 348–349, 372–377, 398, 400by inspection, 282, 288–289cables, 104, 109, 383–397, 400Cartesian vector notation for, 55, 76–81, 149components of, 42–48, 54–59, 97compressive (C), 280–283, 296–297concentrated, 27, 372–373, 383–385, 399–400concurrent, 40, 55, 65–70, 99, 190–192, 260conservative, 597–598coplanar, 54–59, 98, 107–111, 131, 179–184, 190–196,202–206, 213couple moments and, 167–172, 179–184, 190–196, 212–213,217–218cross product, 145–147directed along a line, 78–81displacements from, 585–590distributed loads, 202–206, 213, 372–377, 386–389, 400,511–518dot product, 86–90, 99equilibrium and, 102–133, 217–218, 240–241, 372–373equivalent systems, reduction to, 179–184, 190–196external, 217, 283, 296fluid pressure, 512–518, 525frames, 310–325free-body diagrams, 104–111, 120–124, 131, 222, 310–341,347–349friction as, 402–463, 598frictional, 404–407, 459gravitational, 29idealized models for, 222–223internal, 222, 296–298, 346–401Index 659kinetic frictional (Fk), 406–407, 459line of action, 39–40, 78, 99, 148–149, 158–160, 179,190–196, 203, 212machines, 310–325mechanics of, 25method of joints and, 282–290method of sections for, 296–301, 347–354, 364moment (M) of, 135–139, 148–152, 158–162, 167–172,179–184, 190–199, 211–212, 348–349, 398, 400motion and, 405–407multiforce members, 310Newton’s laws, 28–29nonconservative, 598normal (N), 104, 348–349, 398, 404–406parallel systems, 191–192parallelogram law for, 40, 42–44, 97particles subjected to, 102–133position vectors and, 76–77, 79–80, 99principle of moments, 137–139, 150principle of transmissibility, 148, 179procedures for analysis of, 44, 105, 108, 181, 192, 231, 316,349pulleys, 104, 110reactive, 105rectangular components, 54–70, 98–99resultant, 40, 42–48, 55–59, 97, 99, 134–215, 348–349,511–518, 525rigid bodies, equilibrium of, 217–218, 222–223scalar notation for, 54, 55scalar formulation, 39, 40, 86, 97, 135–136, 158, 167, 211shear (V), 348–349, 372–377, 398, 400simplification of systems, 179–184, 190–196, 213smooth surface contact, 104spring (Fs), 104, 597springs, 104, 111static frictional (Fs), 405, 407, 459structural analysis and, 279–305, 310–341, 347–354structural members, 240–241, 279–290, 296–301, 347–382systems of, 54–59, 134–215tensile (T), 280–283, 296–297, 439–441three-dimensional systems, 65–70, 76–80, 120–124, 131,179–184, 190–196, 212trusses, 279–305, 342–343two-and three-force members, 240–241unbalanced, 28units of, 30–31unknown, 282–287, 296–301vector formulation, 38–133, 145–152, 159–162, 167–172, 211virtual work (U) and, 581–583, 585–590, 597–598weight, 29, 222, 390–393, 400, 597work (W) of, 581–583wrench, reduction to, 192Frames, 310–325, 343free-body diagrams for, 310–316, 343multiforce members of, 310, 343procedure for analysis of, 316structural analysis of, 310–325, 343Free-body diagrams, 104–116, 120–124, 131, 219–228,230–239, 253–257, 259–267, 275–276, 296–301, 310–316,342–343, 347–354, 398, 585–590beams, 347–354, 398cables, 104, 109center of gravity, 222centroid (geometric center), 222, 348, 398concurrent forces, 120–124coplanar force systems, 107–111, 131, 219–228, 230–239,275–276constraints, 259–267, 276direction and, 105, 107, 219equilibrium and, 104–116, 120–124, 131, 219–228, 230–239,253–257, 259–267, 275–276external forces and, 296–297frames, 310–316, 343idealized models of, 222–223internal forces and, 222, 296–297, 347–354, 398machines, 310–316, 343method of sections using, 296–301, 347–354particle equilibrium, 104–106procedures for analysis using, 105, 108, 120, 224, 231, 262,298, 316pulleys, 104, 110rigid bodies, 219–228, 230–239, 253–257, 259–267,275–276smooth surface contact, 104springs, 104, 111statical determinacy and, 259–267, 276structural analysis using, 296–301, 310–316, 342–343support reactions, 219–221, 253–257, 259–267,275–276three-dimensional systems, 120–124, 131, 253–257, 276trusses, 296–301virtual work, 585–590weight and, 222Free vector, 167, 224Friction (F), 402–463, 598angles (u) of, 405–406applied force (P), 404–407, 430–431, 459–460axial loads and, 447–449bearings, analysis of, 447–451, 461belts (flat), forces on, 439–441, 460characteristics of, 403–407, 459coefficients of (μ), 405–407, 447–448, 459collar bearings, analysis of, 447–449, 461Coulomb, 403660 IndexFriction (F) (Continued)disks, analysis of, 447–449, 461dry, 403–463equations for friction and equilibrium, 409–416equilibrium and, 404–405, 409impending motion, 405, 408–416, 432–434, 459–460journal bearings, analysis of, 450–451, 461kinetic force (Fk), 406–407, 459lateral loads and, 450–451nonconservative force, as a, 598point of contact, 403–409, 452–453, 459pivot bearings, analysis of, 447–449, 461procedure for analysis of, 411rolling resistance and, 452–453, 461screws, forces of, 432–434, 460shaft rotation and, 447–451, 461sliding and, 406–416, 459slipping and, 405, 407–409, 459static force (Fs), 405, 407, 459virtual work (U) and, 598wedges and, 430–431, 460Frictional circle, 450Frictional force, 404–407, 459Frictionless systems, 600GGeometric center, 203, 222, 348. See also Centroid (C)Gravitational attraction, Newton’s law of, 29Gravitational potential energy (Vg), 598Gravity, see Center of gravity (G)Gusset plate, 280–281HHinge connections, 221, 253, 255–256Hyperbolic functions, 618IIdealizations (models) for mechanics, 27, 222–223Impending motion, 405, 408–416, 432–434, 459–460all points of contact, 408angle of static friction for, 405coefficient of static friction (μs) for, 405, 459downward, 433, 460dry friction problems due to, 408–416equilibrium and frictional equations for, 409–416friction and, 405, 408–416, 432–434, 459–460no apparent, 408points of contact, 405, 408–409procedure for analysis of, 411screws and, 432–434, 460slipping, verge of, 405tipping and, 409upward, 432–433, 460Inclined axes, moment of inertia for area about, 552–554Incompressible fluids, 512Inertia, see Moments of inertiaIntegrals, 529, 619Integration, 467–478, 511, 515, 525, 529–535, 563, 576–577area (A), centroid of, 468, 474–476, 529–535center of mass (Cm), determination of using, 467, 478centroid (C), determination of using, 467–478, 511, 515,525, 576distributed loads, 511, 515, 525fluid pressure distribution from, 515, 525line, centroid of, 468–469, 471–473mass moments of inertia, determination of using, 563, 577moments of inertia, determination of using, 529–535, 576parallel-axis theorem, 530–531, 576procedure for analysis using, 470, 532resultant forces determined by, 511, 515volume (V), centroid of, 467, 477volume elements for, 563Internal forces, 222, 296–297, 346–401beams subjected to, 347–382, 398–399bending moments (M) and, 348–349, 372–377, 398, 400cables subjected to, 383–397, 400compressive (C), 296concentrated loads, 372–373, 383–385, 399–400couple moment (M0) and, 374distributed loads, 372–377, 386–390, 399–400force equilibrium, 372–373free-body diagrams, 222, 296–297, 347–354, 398method of sections and, 296–297, 347–354, 364moments (M) and, 348–349, 372–377, 398–400normal force (N) and, 348–349, 398procedures for analysis of, 349, 364resultant loadings, 348–349, 398rigid-body equilibrium and, 222shear and moment diagrams, 363–366, 372–377, 399–400shear force (V) and, 348–349, 372–377, 398, 400sign convention for, 349, 399structural members with, 296–297, 347–354, 398tensile (T), 296torsional (twisting) moment, 348, 398weight, 390–393, 400International System (SI) of units, 29–31JJoints, 279–282, 290. See also Method of jointsequilibrium of, 282–283loadings at, 280–281pin connections, 280–281procedure for analysis of, 283truss analysis and, 279–290unknown forces, 282–287Index 661zero-force members, 288–290Joules (J), unit of, 582Journal bearings, 254–256, 450–451, 461free-body diagrams, 254–256frictional analysis of, 450–451, 461support reactions, 254–256KKinetic frictional force (Fk), 406–407, 459LLateral loads, friction analysis of, 450–451Lead of a screw, 432Lead angle, 432Length, 26, 30–31, 468–473, 523basic quantity of mechanics, 26centroid (C) of lines, 468–473, 523integration for, 468–469, 471–473, 523procedure for analysis of, 470Pythagorean theorem for, 469units of, 30–31Line of action, 39–40, 78, 99, 148–149, 158–160, 179, 190–196,212, 511, 513, 525centroid (C) location from, 203, 511collinear vectors, 40distributed loads, 511, 513, 525fluid pressure and, 513force and couple system simplification, 179, 190–196force vector directed along, 78, 99moment force-vector formulation, 148–149moment of a force about an axis, 158–160, 212perpendicular to force resultants, 190–196principle of transmissibility, 148, 179resultant force, 203, 511vector representation of, 39–40, 78, 99, 159–160Linear elastic behavior, 104Linear load distribution, 513, 525Lines, centroid (C) of, 468–473. See also Lengthintegration for, 468–469, 471–473procedure for analysis of, 470Loading curve, 203, 213Loads, 202–206, 279–281, 347–354, 372–377, 383–400,447–451, 511–518, 525. See also Distributed loadsaxial, 447–449beams, 347–354, 372–377, 398–399cables, 383–397, 400concentrated, 372–373, 383–385, 399–400distributed, 202–206, 372–377, 386–389, 399–400,511–518fluid pressure, 512–518friction (F) and, 447–451internal, 347–354lateral, 450–451linear distribution of, 513, 525moment (M) relations with, 373–377, 400plates, 511–518, 525resultant forces, 202–206, 511–518reduction of distributed, 202–206shaft rotation and, 447–451shear (V), 372–377, 398, 400single axis representation, 202structural analysis and, 279–281three-dimensional, 348, 398truss joints, 279–281uniform, 525units of, 202weight, 390–393, 400MMachines, 310–325, 343free-body diagrams for, 310–316, 343multiforce members of, 310, 343vector formulation, 38–133, 145–152, 159–162, 167–172, 211procedure for analysis of, 316structural analysis of, 310–325, 343Magnitude, 39, 42–48, 54–56, 66–67, 105, 107, 131, 136, 145,148, 167, 202–203, 211, 511, 525Cartesian vectors, 55, 66–68coplanar force systems, 54–56, 107constant, 131couple moments, 167cross product and, 145distributed load reduction and, 202–203, 511, 525equilibrium and, 105, 107force components, 42, 44, 54–56, 105free-body diagrams, 105, 107integration for, 511, 525moments and, 136, 145, 148, 211Pythagorean theorem for, 56resultant forces, 42, 44, 202–203, 511, 525right-hand rule for, 148sine and cosine laws for, 42, 44vector force addition and, 42, 44–48vector representation of, 39, 42, 44, 54–56, 66units of, 136Mass, 26, 30, 467, 470, 478, 523basic quantity of mechanics, 26center of (Cm), 467, 470, 478, 523integration of, 467, 478, 523units of, 30Mass moments of inertia, 563–570, 577axis systems, 563–570, 577composite bodies, 568–570, 577disk elements, 564, 566, 577662 IndexMass moments of inertia (Continued)integration for, 563, 577parallel-axis theorem for, 567procedure for analysis of, 564Pythagorean theorem for, 567radius of gyration for, 568shell elements, 564–565, 577volume elements for integration, 563Mathematical expressions, 616–619Mechanics, study of, 25Members, 240–241, 279–290, 296–301, 310–325, 343, 347–354,**

**See also Beams**

compressive force (C), 280–282, 296–297

equilibrium of forces, 240–241

frame analysis, 310–325, 343

internal loads(forces) in, 296, 347–354, 398

joint connections, 279–290

machine analysis, 310–325, 343

method of sections for, 296–301

multiforce, 310, 343

pin connections, 280–281

procedure for analysis of, 349

tensile force (T), 280–281

three-force, 240–241

truss analysis, 279–290, 296–301

two-force, 240–241

unknown forces, 282–287, 296–301

zero-force, 288–290

Method of joints, 282–290, 306–307, 342

compressive forces, 282–283

procedures for analysis using, 283, 306

space truss analysis, 306–307

structural analysis using, 282–290, 306–307, 342

tensile forces, 282–283

truss analysis, 282–290, 306–307, 342

unknown forces, 282–287

zero-force members for, 288–290

Method of sections, 296–301, 306, 342, 347–354, 364

beam analysis using, 347–354, 364

compressive forces (C), 296–297

external forces and, 296–297

internal forces and, 296–297, 347–354

free-body diagrams for, 296–301, 347–354

procedures for analysis using, 298, 306, 349

shear and moment diagrams from, 364

space truss analysis, 306

structural analysis using, 296–301, 306, 342, 347–354

tensile forces (T), 296–297

truss analysis, 296–301, 306, 342

unknown member forces, 296–301

Models (idealizations), 27, 222–223

Mohr’s circle, 555–557

Moment arm (perpendicular distance), 135–137, 158, 212

Moment axis, 136, 148, 158–162, 212

direction and, 136, 148

force about a, 158–162, 212

force-vector formulation and, 148

right-hand rule for, 136, 148, 158–160

scalar analysis of, 158

vector analysis of, 159–162

Moments (M), 134–215, 348–349, 372–377, 398, 400. See also

Couple moments

bending (M), 348–349, 372–377, 398, 400

concentrated load discontinuities, 373

couple (M0), 167–172, 179–184, 190–196, 212–213, 374

cross product for, 145–147

direction and, 136, 145, 148, 211

distributed loads and, 202–206, 213, 372–377, 400

equivalent systems, reduction to, 179–184, 190–196

force, of, 134–215

force-vector formulation, 148–152

free vector, 167

internal forces and, 348–349, 372–377, 398, 400

magnitude and, 136, 145, 148, 211

normal force (N) and, 348–349

parallel force systems, 191–192

perpendicular to force resultants, 190–196

principle of moments, 137–139, 150, 211

principle of transmissibility, 148, 179

procedures for analysis of, 181, 192

right-hand rule for, 136–137

resultant forces and, 136, 149, 168–169, 202–206

scalar formulation of, 135–136, 158, 167, 211

shear loads (V) and, 348–349, 372–377, 400

sign convention for, 136, 146

system simplification of, 179–184, 190–196, 212–213

torque, 136

torsional (twisting), 348, 398

Varignon’s theorem, 137–139

vector formulation of, 148–152, 159–162, 167–172, 211

wrench, reduction of force and couple to, 192

Moments of inertia (I), 529–579

algebraic sum of, 540

area (A), 530–535, 540–542, 548–554, 576

axis of symmetry, 548–551, 563–570, 596

axis systems, 529–535, 540–542, 563–570

composite bodies, 540–542, 568–570, 576–577

disk elements, 564, 566

inclined axis, area about, 552–554

integrals, 529

integration and, 529–535

mass, 563–570, 577Index 663

Mohr’s circle for, 555–557

parallel-axis theorem for, 530–532, 540, 549, 567, 576

polar, 530–531

principle, 553–554, 556, 577

procedures for analysis of, 532, 540, 556, 564

product of inertia and, 548–551, 555–556, 576

radius of gyration for, 531, 568

shell elements, 564–565, 577

transformation equations for, 552–553, 577

Motion, 28, 405–416, 430–434, 439–441, 447–453, 459–461.

See also Revolution; Shaft rotation

bearings, 447–451, 461

belt drives, 439–441, 460

coefficients of friction (μ) and, 405–407, 452–453, 459

downward, 433, 460

equilibrium and frictional equations for, 409–416

friction and, 405–416, 430–434, 439–441, 447–453,

459–461

impending, 405, 408–416, 432–434, 459–460

kinetic frictional force (Fk), 406–407, 459

Newton’s laws of, 28

points of contact, 405–409

procedure for analysis of, 411

rolling resistance and, 452–453, 461

screws and, 432–434, 460

self-locking mechanisms, 430, 433

shaft rotation, 447–451, 461

sliding, 406–416, 459

slipping (impending), 405, 408–409, 459

static frictional force (Fs), 405, 407, 459

upward, 432–433, 460

verge of sliding, 405

wedges, 430–431, 460

Movement, virtual, 583

Multiforce members, 310. See also Frames; Machines

N

Neutral equilibrium, 601–602

Newton, unit of, 30

Newton’s laws, 26, 28–29

dynamics and, 26

gravitational attraction, 29

motion, 28

Nonconservative force, friction as a, 598

Normal force (N), 348–349, 398, 404–406

dry friction and, 404–406

equilibrium and, 404

impending motion and, 405–406

internal forces as, 348–349, 398

method of sections for, 348–349, 398

Numerical calculations, importance of, 32–33

P

Pappus and Guldinus, theorems of, 502–505, 524

axial revolution and symmetry, 502–505, 524

centroid (C) and, 502–505, 524

composite shapes, 503

surface area and, 502, 504–505, 524

volume and, 503–505, 524

Parallel-axis theorem, 530–532, 540, 549, 567, 576

area (A) and, 530–532, 540, 549, 576

centroidal axis for, 530–532, 576

composite parts, 540

mass moments of inertia determined by, 567

moments of inertia determined by, 530–532, 540, 567, 576

product of inertia determined by, 549, 576

Parallel force systems, 191–192, 240

equilibrium of, 240, 260–261

improper constraints, 260–261

reactive, 260–261

three-force members, 240

simplification of, 191–192

Parallelogram law, 40, 42–44, 97

Particles, 27–29, 102–133

coplanar force systems, 107–111, 131

defined, 27

equations of equilibrium, 103, 107–108, 120–124

equilibrium of, 102–133

free-body diagrams, 104–106

gravitational attraction, 29

idealized model of, 27

Newton’s laws applied to, 28–29

nonaccelerating reference of motion, 28

procedures for analysis of, 105, 120

three-dimensional force systems, 120–124, 131

two-dimensional force systems, 107–111, 131

zero condition, 103, 131

Pascal’s law, 512

Perpendicular distance (moment arm), 135–137,

158, 212

Pin connections, 219–221, 223, 253, 255–256, 280–281

concurrent forces of, 280–281

coplanar systems, 219–221, 223

free-body diagrams of, 219–221, 253, 255–256

three-dimensional systems, 253, 255–256

truss member analysis, 280–281

Pivot bearings, frictional analysis of, 447–449

Planar truss, 279

Plates, 511–518, 525

flat of constant width, 513

distributed loads on, 511, 525

centroid (C), 511–518, 525

curved of constant width, 514664 Index

Plates (Continued)

flat of constant width, 513

flat of variable width, 515

fluid pressure and, 512–518, 525

linear distribution on, 513, 525

resultant forces acting on, 511–518, 525

Point of contact, 403–409, 452–453, 459

friction and, 403–409, 452–453, 459

impending motion (slipping), 405, 408–409

kinetic friction and, 406–407

motion (sliding), 406–407

rolling resistance and, 452–453

static friction and, 405, 407

Polar moments of inertia, 530–531

Position coordinates, 585–590, 600, 612

Position vectors (r), 76–77, 79–80, 99

Cartesian vector form, 76–77, 79–80, 99

head-to-tail addition, 76–77

x, y, z coordinates, 76, 99

Potential energy (V), 598–606, 613

elastic (Ve), 598

equilibrium, criterion for, 600, 613

equilibrium configurations, 601–606

frictionless systems, 600

gravitational (Vg), 598

position coordinates for, 600

potential function equations, 599

procedure for analysis of, 603

single (one) degree-of-freedom systems, 599, 602–603

stability of systems and, 601–606, 613

virtual work (V) and, 598–606, 613

Power-series expansions, 618

Pressure, see Fluid pressure

Principal axes, 553–557, 576

Mohr’s circle for, 555–557

principal moment of inertia and, 553–557

procedure for analysis of, 556

product of inertia for, 553, 576

Principle moments of inertia, 553–554, 556, 577

Mohr’s circle for, 555–557

principal axes, 553–554, 556, 577

procedure for analysis of, 556

transformation equations for, 553, 577

Principle of moments, 137–139, 150, 211

Principle of transmissibility, 148, 179

Principle of virtual work, 581, 583–590, 612

Product of inertia, 548–551, 553, 555–556, 576

axis of symmetry for, 548–551

centroid for, 549, 576

Mohr’s circle and, 555–556

parallel-axis theorem for, 549, 576

procedure for analysis of, 556

principal axes and, 553

Procedure for analysis, 34–36

Projections, parallel and perpendicular, 87–88, 99, 159

Pulleys, free-body diagram of, 104, 110, 131

Purlins, 279

Pythagorean theorem, 56, 87, 469, 567, 617

Q

Quadratic formula, 618

R

Radius of gyration, 531, 568

Reactive force, 105, 260–261

Rectangular components, 54–70, 98–99

coplanar force systems, 54–59, 98–99

force vectors of, 54–70, 98–99

resultant force, 55–56, 98

Resultant forces, 40, 42–48, 55–59, 65–71, 97–98, 134–215,

348–349, 398, 511–518, 525

axis, moment of force about, 158–162, 202, 212

beams, 348–349

Cartesian vector components, 65–70

Cartesian vector notation for, 55

centroid (C) and, 203, 348, 511–518, 525

concurrent forces, 55, 65–70, 99, 190–192

coplanar forces, 55–56, 98, 179–184, 190–196

couple moments, 167–172, 179–184, 190–196, 213

cross product for, 146–147

direction of, 146, 211

distributed loads, 511–518, 525

fluid pressure and, 512–518, 525

force components and, 40, 42–44, 97

force system, 55–59, 99, 134–215

integration for, 511, 525

internal forces, 348–289, 398

lines of action, 158–160, 179, 190–196, 203, 511, 513

magnitude of, 146, 202, 211, 511, 525

method of sections for, 348–349

moments of a force, 135–139, 149, 158–162, 211, 348–349

parallel force systems, 191–192

parallelogram law for, 40, 42–44, 97

perpendicular to moments, 190–196

plates, 511–518, 525

principle of moments, 137–139, 211

procedure for analysis of, 44, 192

reduction of distributed loads, 202–206, 213

scalar formulation of, 135–136, 158, 167, 211

scalar notation for, 55

system reduction for, 179–184, 190–196, 213

vector addition for, 40–44, 55–59

vector formulation of, 42–48, 97, 148–152, 167–172, 211–212

vector subtraction for, 41

wrench, reduction to, 192, 213Index 665

Revolution, 502–505, 524

axial symmetry and, 502–505

centroid (C) and, 502–505, 524

composite shapes, 503

Pappus and Guldinus, theorems of, 502–505, 524

plane area, 503

surface area, 502, 504–505, 524

volume from, 503–505, 524

Right-hand rule, 65, 136–137, 145–146, 148, 158–160, 167

axis, moment of a force about, 158–160

cross product direction, 145–146

force-vector formulation, 146, 148

moment of a couple, 167

moment of a force, 136–137, 158–160

three-dimensional coordinate systems, 65

Rigid bodies, 25, 27, 216–277, 585–590, 600–606, 612

center of gravity, 222

centroid (geometric center), 222

conditions for, 217–218

connected systems of, 585–590, 612

constraints of, 259–267

coplanar force systems, 218–252, 275–276

defined, 27

displacement (d) and, 585–590, 600, 612

equations of equilibrium for, 218, 230–239, 275–276

equilibrium of, 216–277, 600–606

external forces and, 217

force and couple systems acting on, 217–218

free-body diagrams, 219–228, 253–257, 259–267,

275–276

frictionless systems, 600

idealized models of, 27, 222–223

internal forces and, 222

improper constraints for, 260–261, 276

mechanics, study of, 25

position coordinates for, 585–590, 600, 612

potential energy and, 600–606

procedures for analysis of, 224, 231, 262, 586, 603

redundant constraints for, 259

statical determinacy and, 259–267, 276

support reactions, 219–221, 253–257, 259–267, 275–276

three-dimensional systems, 253–267, 276

two-and three-force members, 240–241

uniform, 222

virtual work (V) for, 585–590, 600–606, 612

weight and, 222

Rocker connections, 220, 221

Roller connections, 219–220, 223, 254, 256

Rolling resistance, frictional forces and, 452–453, 461

Roof truss, 279–280, 342

Rotation, 219, 447–448, 450, 461, 582–583. See also

Revolution; Shaft rotation

Rounding off numbers, 33

S

Scalar notation, 54, 55

Scalar product, 86

Scalar triple product, 159

Scalars, 39, 40, 54, 86, 107, 135–136, 158, 167, 211, 258,

275–276, 582

axis, moment of force about, 158

couple moments, formulation by, 167

division of vectors by, 40

dot product and, 86

equations of equilibrium, 258, 275–276

magnitude of, 39

moment of a force, formulation by, 135–136, 158, 211

multiplication of vectors by, 40, 86

negative, 54, 107

torque, 135

vectors and, 39, 40, 86

work of a couple moment, 582

Screws, frictional forces on, 432–434, 460

Self-locking mechanisms, 430, 433

Sense of direction, 39

Shaft rotation, 447–451, 461

axial loads, 447–449

collar and pivot bearings, 447–449, 460

disks, 447–449, 460

frictional analysis of, 447–451, 460

frictional circle, 450

journal bearings, 450–451, 460

lateral loads, 450–451

Shear and moment diagrams, 363–366, 372–377, 399–400

beam analysis using, 363–366, 372–377, 399

couple moment (M0) and, 374

discontinuities in, 373

distributed load relations and, 372–377, 400

internal forces and, 363–366, 372–377, 399–400

method of sections for, 364

moment (M) relations in, 373–377, 399–400

procedure for analysis of, 364

shear force (V) relations in, 372–377, 399–400

Shear force (V), 348–349, 372–377, 398–400

beams, 348–349, 372–377, 398

bending moments (M) and, 348–349, 372–377, 398, 400

concentrated load discontinuities, 373

couple moment (M0) and, 374

distributed load relations, 372–377, 400

internal forces, 348–349, 372–377, 398–400

method of sections for, 348–349

shear and moment diagrams, 372–377, 399–400

Shell elements, mass moments of inertia, 564–565, 577

Significant figures, 32–33

Simple trusses, 279–281, 342

Simply supported beam, 363

Sine functions, 617666 Index

Sine law, 42, 44, 97

Single degree-of-freedom systems, 599, 602–603

Sliding, 405–416, 459–460

friction and, 405–416

kinetic frictional force (Fk), 406–407, 459

motion of, 406–416

problems involving, 408–416

verge of, 405

Sliding vector, 148, 179, 224

Slipping, 405, 407–416, 459

friction and, 405, 407–416, 459

impending motion of, 405, 408–409, 459

points of contact, 405, 408–409

problems involving, 408–416

static frictional force (Fs), 405, 407, 459

Smooth surface contact (support), 104, 254

Solving problems, procedure for, 34–36

Space trusses, structural analysis of, 306–307, 343

Specific weight, center of gravity (G) and, 488

Spring constant (k), 104

Spring force (Fs), virtual work and, 597

Springs, free-body diagram of, 104, 111, 131

Stable equilibrium, 601–603, 613

Stability of a system, 260–261, 276, 601–606, 613. See also

Equilibrium

equilibrium configurations for, 601–606, 613

free-body diagrams for, 260–261, 276

improper constraints and, 260–261

neutral equilibrium, 601–602, 613

one (single) degree-of-freedom system,

602–603

potential energy and, 601–606

procedure for analysis of, 603

reactive parallel forces, 260–261

rigid-body equilibrium and, 260–261, 276

stable equilibrium, 601–603, 613

statical determinacy and, 260–261, 276

unstable equilibrium, 601–603, 613

virtual work and, 601–606, 613

Static frictional force (Fs), 405, 407, 459

Statical determinacy, 259–267, 276

equilibrium and, 259–267

procedure for analysis of, 262

improper constraints and, 260–261

indeterminacy, 259, 276

redundant constraints and, 259

rigid-body equilibrium and, 259–267, 276

stability and, 260–261, 276

Statically indeterminate bodies, 259, 276

Statics, 24–37

basic quantities, 26

concentrated force, 27

equilibrium and, 25

force, 26–30

gravitational attraction, 29

historical development of, 26

idealizations, 27

length, 26, 29–31

mass, 26, 29–31

mechanics study of, 24–25

motion, 28

Newton’s laws, 28–29

numerical calculations for, 32–33

particles, 27

procedure for analysis of, 34–36

rigid bodies, 27

study of, 24–37

time, 26, 30

units of measurement, 29–31

weight, 29

Stiffness factor (k), 104

Stringers, 280

Structural analysis, 278–345, 347–382

beams, 347–382

compressive forces (C), 280–283, 296–297

frames, 310–325, 343

free-body diagrams, 296–301, 310–316, 342–343

internal forces and, 296–297, 347–382

machines, 310–325, 343

method of joints, 282–290, 306–307, 342

method of sections, 296–301, 306, 342, 347–354, 364

multiforce members, 310, 343

procedures for analysis of, 283, 298, 306, 316, 349, 364

shear and moment diagrams for, 363–366

space trusses, 306–307, 343

tensile forces (T), 280–283, 296–297

trusses, 279–309, 342–343

unknown forces, 282–287, 296–301

zero-force members, 288–290

Structural members, see Members

Support reactions, 219–221, 223, 253–257, 259–267,

275–276

coplanar force systems, 219–221, 223, 275

free-body diagrams, 219–221, 223, 253–257, 259–267,

275–276

improper constraints, 260–261

procedure for analysis of, 262

redundant constraints, 259

rigid-body equilibrium and, 219–221, 223, 253–257,

259–267

statical determinacy and, 259–267, 276

three-dimensional force systems, 253–257, 259–267, 276

Surface area, centroid (C) and, 502, 504–505, 524

Symmetry, see Axial symmetry; Axis of symmetryIndex 667

System simplification, 179–184, 190–196

concurrent force system, 190–192

coplanar force systems, 179–184, 190–196

equivalent system, reduction to, 179–184, 190–196

lines of action and, 179, 190–196

parallel force systems, 191–192

procedures for analysis, 181, 192

reduction to a wrench, 192

system of force and couple moments, 180

three-dimensional systems, 179–184, 190–196

T

Tangent functions, 617

Tensile forces (T), 280–283, 296–297, 439–441

flat belts, 439–441

method of joints and, 282–283

method of sections and, 296–297

truss members, 280–281, 296–297

Tetrahedron form, 306

Thread of a screw, 432

Three-dimensional systems, 65–70, 76–81, 86–90, 98–99,

120–124, 131, 179–184, 190–196, 253–267, 276. See also

Concurrent forces

addition of vectors, 68

Cartesian coordinate system for, 65–70, 98–99

Cartesian unit vectors, 65–66, 78, 98–99

Cartesian vector representation, 65–66

concurrent forces, 65–70, 99, 120–124, 131, 190–192, 276

constraints for, 259–267, 276

coordinate direction angles, 66–67, 98–99

direction angles for, 66–67

dot product for, 86–90, 99

equations of equilibrium, 120, 258, 276

equilibrium of, 120–124, 131, 253–267, 276

equivalent systems, 179–184, 190–196

force and couple moment system simplification, 179–184,

190–196

force vectors, 65–70, 78–81, 99

free-body diagrams, 120–124, 253–257, 276

magnitude of, 66

parallel system simplification, 191–192

particles, 120–124, 131

position vectors, 76–77, 79–80, 99

procedure for analysis of, 120

reactive parallel forces, 261

rectangular components, 65–70, 98–99

resultants, 65–70

right-hand rule, 65

rigid bodies, 253–267, 276

statical determinacy and, 259–267, 276

support reactions for, 253–257, 259–267, 276

x, y, z position coordinates, 65–66, 76, 98–99

Three-force member equilibrium, 240–241

Thrust bearing connections, 255, 256

Time, 26, 30

basic quantity of mechanics, 26

units of, 30

Tipping effect, balance of, 404, 459

Torque, 135. See also Moments (M)

Torsional (twisting) moment, 348, 398

Transformation equations, moments of inertia (I) and,

552–553, 577

Translation, 219, 582

Trapezoid, distributed loading of, 206

Triangle rule, 41, 97

Triangular truss, 281

Trigonometric identities, 618

Trusses, 279–309, 342–343

assumptions for design, 280–281, 306

bridges, 279–280

compressive force (C) and, 280–283,

296–297

floor beams, 280

gusset plate for, 280–281

joints, 279–290

method of joints, 282–290, 306–307, 342

method of sections, 296–301, 306, 342

planar, 279

procedures for analysis of, 283, 298, 306

purlins, 279

roof, 279–280, 342

simple, 279–281, 342

space trusses, 306–307, 343

stringers, 280

structural analysis for, 279–309, 342–343

tensile force (T) and, 280–283, 296–297

triangular, 281

zero-force members, 288–290

Two-dimensional systems, 54–59, 98, 107–111, 218–252.

See also Coplanar forces

Cartesian unit vectors, 55, 98

coplanar force vectors, 54–59, 98

free-body diagrams for, 107–111

particle equilibrium, 107–111

procedure for analysis of, 108, 224, 231

rigid-body equilibrium, 218–252

scalar notation for, 54

Two-force member equilibrium,

240–241

U

Unbalanced force, 28

Uniform distributed load, 372, 525

Uniform rigid bodies, 222668 Index

Unit vector (u), 55–56, 65–66, 78, 86–87, 98–99. See also

Cartesian coordinates

Cartesian vectors, 55–56, 65–66, 78, 98

dot product and, 86–87, 99

three-dimensional, 65–66, 78, 98–99

force components, 55–56

force vectors, 78, 99

Units of measurement, 29–31

base, 29, 31

derived, 29–31

International System (SI) of, 30–31

prefixes, 30

rules for use, 31

Unknown member forces, 282–287,

296–301

Unstable equilibrium, 601–603, 613

V

Varignon’s theorem, 137–139

Vectors, 38–101, 145–152, 159–162, 167–172, 211, 258, 276

addition of, 40–48, 54–59, 68–70

addition of forces, 42–48, 54–59

axis, moment of a force about, 159–162

Cartesian coordinate system, 55–58, 65–70, 76–81, 98,

145–152, 211

Cartesian notation for, 55

components of a force, 40, 42–48, 97

concurrent forces, 40, 55, 65–70, 99

coplanar force systems, 54–59

cross product method of multiplication, 145–147

collinear, 41, 97

couple moments, formulation by, 167–172

direction and, 39, 55–56, 66–68, 145, 148

division by scalars, 40, 97

dot product, 86–90, 99, 169

equations of equilibrium, 258, 276

force directed along a line, 78–81

forces and, 38–101

free, 167, 224

line of action, 39–40, 78, 99, 148–149, 159–160

magnitude and, 39, 42–48, 54–56, 66–68, 145, 148

moments of a force, formulation by, 148–152,

159–162, 211

multiplication by scalars, 40, 86, 97

operations, 40–41

parallelogram law for, 40, 42–44, 97

physical quantity requirements, 39

position (r), 76–77, 79–80, 99

principle of transmissibility, 148

procedure for analysis of, 44

projections, parallel and perpendicular, 87–88, 169

rectangular components, 54–70, 98–99

resultant couple moment, 168–169

resultant of a force, 40, 42–48, 97, 149

rigid-body equilibrium and, 258, 276

scalar notation for, 54

scalar triple product, 159

scalars and, 39, 40, 86, 97

sliding, 148, 224

subtraction of forces, 41

systems of coplanar forces, 54–59

three-dimensional systems, 65–70, 76–81, 86–90, 98–99, 276

triangle rule for, 41, 97

two-dimensional systems, 54–59, 98

unit, 55–56, 65–66, 78, 86–87, 98–99

Virtual movement, 583

Virtual work (U), 580–615

conservative forces and, 597–598

couple moment, work of, 582–583

displacement (d) and, 583–590, 600, 612

equations for, 583–584, 586

equilibrium and, 600–606, 613

force (F) and, 581–582, 585–590, 597–598, 612

friction and, 598

frictionless systems, 600

movement as, 583

position coordinates for, 585–590, 600, 612

potential energy (V) and, 598–606, 613

principle of, 581, 583–590, 612

procedures for analysis using, 586, 603

rigid-bodies, connected systems of, 585–590

single (one) degree-of-freedom systems, 599, 602–603

spring force (Fs) and, 597

stability of a system, 601–606, 613

weight (W) and, 597

work (W) of a force, 581–583

Volume (V), 467, 470, 477, 503–505, 523–524

axial rotation and symmetry, 503–505, 524

centroid of (C), 467, 470, 477, 503–505, 523–524

integration of, 467, 477, 523

Pappus and Guldinus, theorems of, 503–505, 524

plane area revolution and, 503

procedure for analysis of, 470

W

Wedges, 430–431, 460

Weight (W), 29, 222, 390–393, 400, 465–466, 488, 523–524, 597

cables subjected to own, 390–393, 400

center of gravity (G) and, 222, 465–466, 488, 523–524

composite body parts, 488, 524Index 669

conservative force of, 597

gravitational attraction and, 29

internal force of, 390–393, 400

rigid-body equilibrium and, 222

virtual work (U) and, 597

Weightless link, support reactions of, 220

Work (W) of a force, 581–583. See also Virtual work

couple moment, of a, 582

force, of a, 581–582

virtual movement and, 583

Wrench, reduction of force and moment to,

192, 213

X

x, y, z position coordinates, 65–66, 76, 98–99

Z

Zero condition of equilibrium, 103, 131, 218

Zero-force members, method of joints and, 288–290

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