Appendix

Vector A can be represented through its three orthogonal components

A ¼ Axiþ Ay jþ Azk

where the unit vectors i, j, and k correspond to the coordinate axes in x, y, andz directions.

Its magnitude can be represented as

��A

�� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A2x þ A2

y þ A2z

q

And its unit vector

λ ¼ A

Aj jThus, vector A may be represented also as

A ¼ Aj jλIts direction cosines (cosine of the angle between the vector and corresponding

axis) are

λx ¼ Ax

Aj j λy ¼ Ay

Aj j λz ¼ Az

Aj jTwo vectors are equal if there corresponding components are equal, i.e.,

A¼B, when Ax¼Bx, Ay¼By, and Az¼Bz.

Vector addition

A +B¼(Ax+Bx) i+(Ay +By) j+(Az+Bz) k

# Springer Science+Business Media New York 2016

I. Emri, A. Voloshin, Statics, DOI 10.1007/978-1-4939-2101-0563

Vector may be multiplied by a scalar

nA¼ nAxi+ nAyj+ nAzk

Cross product (vector product) is defined as

C ¼ A� B ¼ λ Aj j Bj j sinαwhere α is the angle between the vectors, λ is the unit vector perpendicular to the

plane formed by the vectors A and B. Its direction is defined by the right hand rule.

From the definition of cross product follows

i� i ¼ 0 j� j ¼ 0 k� k ¼ 0

i� j ¼ k j� i ¼ �k k� i ¼ j

The rectangular components of the cross product are

Cx ¼ AyBz � AzBy

Cy ¼ AzBx � AxBz

Cz ¼ AxBy � AyBx

Using a determinant

C ¼i j jAx Ay Az

Bx By Bz

������

������

Scalar or dot product of vectors A and B is defined as

A � B ¼ Aj j Bj j cosθwhere θ is the angle between the vectors.

A � B ¼ AxBx þ AyBy þ AzBz

From the definition of dot product follows

i � i ¼ 1 j � j ¼ 1 k � k ¼ 1

i � j ¼ 0 j � k ¼ 0 k � i ¼ 0

It should be noted that the main purpose of this Appendix is to provide a

refreshment of the rules on vectors and matrix algebra; it cannot serve as a tutorial.

564 Appendix

Index

AAcceleration, 9

Angle of kinetic friction, 504

Angle of static friction, 504

BBall joint, 44

Beams

classification, 322

coordinate system and sign convention,

323–324

curved beams

free body diagram, 356

region AB, 356

region BC, 358

roof of shelter, 356

in engineering applications, 322

internal forces and moments

arbitrary distributed load, 337

axial force N(x), 329

intuitive approach, 341–346

mathematical corner, 338, 339

moment M(x), 332

physical model, 333

positive and negative sign, 340

region AC, 335

region CD, 335–336

region DE, 336

region EB, 336–337

shear force T(x), 331, 337, 339

in shelf supporting a radio and five

books, 333

within region, 340

piece-wise straight and curved beams

axial (normal) direction, 364

external loads, 361

free body diagram, 361

geometry, 359

intuitive rules, 365

MATLAB/software package, 361

part AB, 363–365

part BC, 365

physical model, 361

structure, 359

transverse (shear) direction, 364

regions, 322

straight, 324–329

superposition principle

beam loading, 370, 376

bending moment, 377

distributed load, 370, 376, 378

external reaction force, 377

guidelines, 381

internal axial force, 377

internal moment and forces, 370

intuitive approach, 370, 377

length and geometry, 370

normal forces, 377

preprepared solutions, 370

transverse forces, 377

CCables

concentrated forces

approach of joints, 393

appropriate safety factor, 390

axial force, 397

cable structure, 391

equilibrium equation, 393, 395, 396, 401

external forces, 392, 401

free body diagram, 394

geometry, 393

guidelines, 402

horizontal component, 395, 397

inserting numerical values, 401

internal forces, 392

# Springer Science+Business Media New York 2016

I. Emri, A. Voloshin, Statics, DOI 10.1007/978-1-4939-2101-0565

Cables (cont.)length of cable, 401

method of joints, 394

negative sign, 400

previously developed equations, 399

reaction force, 399

tensile force, calculation, 396

two forces, 390, 397

unknown reactions, 391

vertical deflection, 396

definition, 390

distributed loads, 432

Brooklyn Suspension Bridge, 406

free body diagram, 403

geometry, 404, 405

horizontal component, 404

hyperbolic solution, 423–428

infinitesimal segment, 404

internal tensile force, 409–412

internal tensile force,S(x), 428–430length of the cable,L, 430–432origin of, 403

parabolic solution, 4, 7–409

sag of the cable, 416–423

sag of the catenary, f (see Sag of the

catenary, f)

vertical component, 404

Cardan joint, 44

Composite bodies, 196–201

Compound structures

beam elements, 460

frames

definition, 473

equilibrium equations, 461, 462, 474

free body diagram, 472–474

geometry of, 475

guidelines, 476

physical model, 473

free body diagram, 458

interaction forces, 459

mechanisms, 479–488

reaction forces, 461

schematically representation, 458

types of, 459

Compressive force, 269

Conical axial bearings, 527–528

DDensity, 6

Distributed forces

centre of gravity

concentrated force equivalent, 241

definition, 241

of flat plate, 185–189

mathematical procedure, 231

moment of body, 231

centroids

area, 191

composite bodies, 196–201

definition, 241

first moments, 190

flat wire, 190

geometrical characteristics, 190

guidelines, 200

integration, 191–196

location of, 232–234

segments, 235

Theorems of Pappus, 236–241

distributed loads

vs. concentrated loads, 241

fluid pressure, effect of, 217–220

guidelines, 220

oddly shaped cargo, 216

two-dimensional problems, 216

load and rigid body, 184

Door hinge, 44

EEquilibrium forces

in-plane forces

equilibrium equations, 63

force diagram, 65

graphical solution, 66, 69

horizontal component and direction, 64

initial conditions, 63

magnitude F and direction α, 63, 64MATLAB functions, 67, 70

numerical solution, 67, 69

orthogonal components Fx and Fy, 63, 64

physical model, 64, 65

rigid body equilibrium, 63

solving 2D problems, 71–87

vertical component and direction, 64

in space, solving 3-D problems, 91–108

Equilibrium pair of forces, 12

equilibriumPoint 2D, 70

FFree body diagram (FBD), 71, 257

basketball stand, 58

the brake, 57

bridge, 54, 272

bus stop cover, 52

calendar stand, design, 46

canopy support, 46, 50

car jack, 53

caster wheel, 35

566 Index

C clamp, 52

closet, 50

coat hanger, 47

door handle, 53

earth mover, 57

fixed bar, 56

front loader, 56

guidelines, 45

highway signs, 49

house entry, 51

joint A, 273

joint B, 275, 281

joint C, 274, 282

joint D, 278

joint E, 276

joint F, 277

joint G, 277

joint H, 276

lifting system, 280

mechanical systems

definition, 30

external forces and moments, 31

idealization process, 31

loads, 33–34

physical model, 30, 31

rigid body, 30

seat design, 32

selected element, 31

simplification process, 31

swing, 31

upper bar design, 32

reactions, 35

street lamp, 58

swing set, 55

terrace cover, 55

three dimensional systems, 40–45

traffic light, 49

two-dimensional systems, 36–40

wine barrels, 51

Friction

angles of friction

angle of plane’s inclination, 502

coefficients, 499

components, 500

coordinate system, 500

definition, 501

free body diagram, 500

kinetic, 502

line of action, 499

point of action of force, 499

resultant external moment, 499

resultant reaction force, 501

rigid body, 502

static, 501

belts and ropes friction

acting clockwise, 540

acting counterclockwise, 541

angle of contact, 535

flat belts passing over cylinders/

cylindrical drums, 536

forces and moments, 538

free body diagram, 534

guidelines, 538

ideally flexible and non-deformable,

533, 534

infinitesimal rope segment, 534

physical model, 539

V-shaped belts, 536–537

components, 508

dry friction, 491

dynamic component, 508

experimental facts, 491

external forces, 492, 493

free body diagram, 492, 505

guidelines, 503

with inclination, 504

intended movement, 492

kinetic-friction force, 494, 495

limits, determination of, 508, 509

lubricated friction, 491

physical model, 492, 505

plain bearings

axial/trust bearings, 527

bearing surface and shaft, 523

collar bearings, 530

end bearing, 529

external load Q and weight

of hoist, 532

guidelines, 531

oldest wooden wheel dates, 523

physical model, 532

radial/journal bearing, 524–527

replacing kinetic friction

coefficient, 533

solid polymer, 523

reactions, 492

rolling and rolling resistance

changing geometry, 516

coefficient of, 517, 518

equilibrium equations, 520,

521, 523

free body diagram, 519, 522

guidelines, 519

kinetic friction coefficients, 517

rolling conditions, 517

rolling moment, 516, 517

static friction coefficient,

517, 521

Index 567

Friction (cont.)rollover moment, 506

sliding friction, 490

without slipping/rolling over, 508

slope, 505

stabilizing moment, 506

static friction coefficient, 507, 509

static friction force, 493, 495

stick-slip effect

definition, 497, 498

engineering applications, 498

high precision manufacturing systems,

development, 496

instrumental mechanism, 497

physical model and free body

diagram, 496

rubber band, deformation of, 496

stringed instruments, 498

threshold magnitude, 493

wedges

free body diagrams, 511

guidelines, 511

moving object A upwards,

512, 513

object B, moving upwards and

downwards, 513–515

pair of forces, 511

wedge inclination angle, 511

Fundamantal law of nature

couple of forces

experiment demonstration, 16

moment, definition of, 16–20

moment of a couple, 21, 22

moment of a force, 22–24

Varignon’s theorem, 21

equilibrium pair of forces, 12

First Fundamental Law, 10

First Newton’s Law, 9

internal and external forces, 11

matter, 6

numerical calculations, 26–27

parallelogram of forces, 13–15

space, 6

statics, 7

systems of units, 24–25

time, 6

transmissibility, principle of, 11

GGraphical approach, 71

HHardy-Spicer joint, 44

Hook’s joint, 44

IInternal forces, 254–259

KKinetic-friction force, 494

LLearning approach, 1

MMass, 10

MATLAB, 3, 70

Matter, 6

Mechanical systems

definition, 30

external forces and moments, 31

idealization process, 31

physical model, 30, 31

rigid body, 30

seat design, 32

selected element, 31

simplification process, 31

swing, 31

upper bar design, 32

Mechanics laws, 1

Meter, 6

Moment arm, 17

NNewton’s First Law, 63

Numerical approach, 71

PPhysical model, 2

Plain bearings

axial/trust bearings, 527

bearing surface and shaft, 523

collar bearings, 530

conical axial bearings, 527–528

end bearing, 529

guidelines, 531

568 Index

oldest wooden wheel dates, 523

radial/journal bearing, 524–527

solid polymer, 523

RResultant forces

in-plane forces, solving 2D problems,

60–63, 71–87

in space

Cartesian coordinate system, 88

orthogonal components, 90

parallel vectors, 88

resulting force, 89

solving 3-D problems, 93–108

unit vector, 88, 89

Resultant moment, 114

Right-hand coordinate system, 10

Rigid bodies equilibrium

force-moment systems

Force-couple system, 117–120

moment of force, 122–129

moving force, 110–114

system of forces, reduction of, 114–116

three dimension, 170–173

two dimension

brake assembly, 135

brake pedal, 133

components and moment, 132

equilibrium equations, 134

free body diagram, 134

guidelines, 137

gymnast on the beam, 129

locations, magnitudes,

and directions, 137

MATLAB routine equilibriumBody2D.m., 135

physical model, 135

plane of symmetry, 129, 130

real-life problems, 131

single resultant force and resultant

moment, 131

three-force body, 138–143

two-force members, 137

SSag of the Catenary, f

appropriate safety factor, 440

guidelines, 436

hyperbolic solution, 437, 439, 443

location of, 432, 433

mathematical corner, 432, 433, 439, 440

parabolic and hyperbolic solutions, 433,

440, 446

parabolic solution, 438, 441

quadratic equation, 447

transcendental equation, 440

vertical reaction forces, 441

Self-locking, 503, 509

Sliding friction, 490

Static friction coefficient, 498

Static friction force, 493, 498

Statics, 3, 7

Structural elements

guidelines, 259

internal forces, 254–259

truss elements

beams, 252–253

bending, 251, 252

cables, 253–254

segment model, 251

types of, 250

TTensile force, 269

Theorems of Pappus, 236–241

Truss structures

compound trusses, 306–308

compressive force, 269

external loads and reactions, 267

FBD (see Free body diagram (FBD))

mechanism, 268

method of joints

bridge and truck, 271

concurrent system of forces, 270

external load, 283

forces acting on joint B, 270

guidelines, 271

MATLAB, 279

outside lifting system, 279

two-dimensional truss, 270, 271

unknown internal forces, 273

zero-force members, 271

method of sections

guidelines, 295

imaginary cut, 293

right hand side and coordinate

system, 295

step-by-step procedure, 293

racing car, 266

rigid, 267

simple, 268, 269

space truss

definition, 308

Index 569

Truss structures (cont.)free body diagram, 309–311

glass roof, 309

guidelines, 313

physical model, 310, 311

tetrahedrons, 308

tensile force, 269

two-dimensional view, 266

UUniversal coupling, 43

Universal joint, 44

VVarignon’s theorem, 21

570 Index