Statics and Mechanics of Structures

Statics and Mechanics of Structures

Steen Krenk Jan Høgsberg

Statics and

Mechanics of

Structures

Prof. Steen Krenk

Department of Mechanical Engineering

Technical University of Denmark

Kongens Lyngby, Denmark

[email protected]

Prof. Jan Høgsberg

Department of Mechanical Engineering

Technical University of Denmark

Kongens Lyngby, Denmark

[email protected]

ISBN 978-94-007-6112-4 ISBN 978-94-007-6113-1 (eBook)

DOI 10.1007/978-94-007-6113-1

Springer Dordrecht Heidelberg New York London

Library of Congress Control Number: 2013933869

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Preface

The theory of statics of structures has developed from intuition via grad￾ual refinement to its current state, where the basic principles are put into a

systematic framework that enables precise analysis. Although the basic laws

governing statics of structures have been known for several centuries, the

methods of analysis have developed considerably over the last decades. At

the current state of this development an introductory book on statics should

aim at the dual goal of providing sufficient background for developing an

intuitive understanding of structures, and at the same time lay a solid foun￾dation for modern analysis, typically made by computational techniques. In

this vein the present book makes extensive use of simple but realistic exam￾ples to develop familiarity and understanding of how structures carry and

distribute the loads through the structural members to the supports. This is

then supplemented by a few simple computer programs that illustrate, how

the theories for trusses and frames are implemented, and open up to a more

general approach to computational mechanics as a natural extension of the

present book.

The book is organized as follows. The first five chapters build up a basic

understanding of the statics of structures. It starts with force systems and

reactions in Chapter 1, then proceeding to the intuitively very accessible

theory of trusses, first analyzed by hand calculation procedures and then

reformulated as a small systematic finite element program MiniTruss in

Chapter 2. Chapter 3 develops the statics of beams and introduces the con￾cept of internal forces. The internal forces are then related to deformation

mechanisms of curvature, shear and extension in Chapter 4, and the princi￾ple of virtual work is developed in a concise form and used for calculation

of specific displacements. The introductory part is rounded off in Chapter 5

on the analysis of columns, describing instability as a bifurcation problem,

solved by eigenvalue analysis, and design principles based on the existence of

a characteristic imperfection. This part of the book covers material suitable

for an introductory one-semester course on basic statics of structures.

The remaining six chapters treat various extensions, that are typically in￾cluded in one form or another in a second semester course. The Chapters 6

and 7 deal with analysis of statically indeterminate frame structures. The

vi Preface

first of these chapters gives a systematic development of the force method

and describes how simple structures can conveniently be analyzed by hand.

The following chapter then develops the deformation method in which the

displacements of individual nodes play the key role. This then serves to in￾troduce the idea of the finite element formulation of frame structures. This

development is supported by the small program MiniFrame for internal

forces and displacements, and an extension MiniFrameS for linearized sta￾bility analysis. The Chapters 8 and 9 introduce three-dimensional states of

stress and strain, and present the theory of linear elasticity and some common

failure conditions. This material provides the background for the Chapters 10

and 11, in which the simple two-dimensional beam theory used in the previ￾ous chapters is extended to flexure and torsion of non-symmetric beams, and

the associated shear stress distributions.

The three small computer programs are coded in Matlab. The syntax and

input structure are described in connection with the corresponding theory in

the text, and the code is available from the authors via e-mail.

The authors are grateful for the permission to include photographs provided

by the following companies: Chapter 7, Rafsanjan Bridge, Waagner-Biro AG,

Vienna, Austria; Chapter 8, Test of wind turbine blade, LM Wind Power,

Kolding, Denmark; Chapter 10, Wind turbine, Siemens Wind Power, Brande,

Denmark.

Kgs. Lyngby Steen Krenk

September 2012 Jan Høgsberg

Contents

1 Equilibrium and Reactions ................................ 1

1.1 Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 The parallelogram rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Parallel forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Moment from forces in a plane . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Moment from forces in space . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 Force couples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Virtual work of rigid bodies. . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.2 Equilibrium in a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.3 Distributed load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Support conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5 Reactions by equilibrium equations . . . . . . . . . . . . . . . . . . . . . . . 19

1.5.1 Plane beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5.2 Simple frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5.3 Three-hinge frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.5.4 Space structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.6 Reactions by virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2 Truss Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1 Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.1.1 Building with triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.1.2 Counting joints and bars . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.1.3 Qualitative tension-compression considerations . . . . . . . 45

2.2 Method of joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.2.1 Planar truss structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.2.2 Space trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.3 Method of sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3.1 Bar forces via the method of sections . . . . . . . . . . . . . . . 55

2.3.2 Special types of planar trusses . . . . . . . . . . . . . . . . . . . . . 57

2.4 Stiffness and deformation of truss structures . . . . . . . . . . . . . . . 64

2.4.1 Axial stress and strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

viii Contents

2.4.2 Linear elastic bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.4.3 Virtual work for truss structures . . . . . . . . . . . . . . . . . . . 67

2.4.4 Displacements of elastic truss structures . . . . . . . . . . . . . 71

2.5 Finite element analysis of trusses . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.5.1 Elastic bar element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.5.2 Finite Element Method for trusses . . . . . . . . . . . . . . . . . . 77

2.5.3 The MiniTruss program . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3 Statics of Beams and Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.1 Internal forces and moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.2 Beams with concentrated loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.2.1 Variation of internal forces for concentrated loads . . . . 99

3.3 Beams with distributed load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.3.1 Differential equations for internal forces . . . . . . . . . . . . . 107

3.3.2 Maximum moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.4 Combined loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

3.4.1 Superposition of load cases . . . . . . . . . . . . . . . . . . . . . . . . 116

3.4.2 Superimposing the distributed load . . . . . . . . . . . . . . . . . 117

3.5 Internal forces in frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.5.1 Influence of load distribution. . . . . . . . . . . . . . . . . . . . . . . 124

3.5.2 Influence of support conditions . . . . . . . . . . . . . . . . . . . . . 128

3.5.3 Three-hinge frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3.5.4 Principle of the arch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

4 Deformation of Beams and Frames . . . . . . . . . . . . . . . . . . . . . . . 143

4.1 Bending of elastic beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.1.1 Homogeneous bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.1.2 Linear kinematic relations . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.2 Bernoulli beam theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

4.2.1 Statically determinate beams . . . . . . . . . . . . . . . . . . . . . . 154

4.2.2 Statically indeterminate beams . . . . . . . . . . . . . . . . . . . . . 159

4.3 Shear flexible beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

4.4 Virtual work and displacements of beams . . . . . . . . . . . . . . . . . . 168

4.4.1 Principle of virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . 168

4.4.2 Displacements in elastic beams . . . . . . . . . . . . . . . . . . . . . 171

4.4.3 Virtual work and displacements in frames . . . . . . . . . . . 179

4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

5 Column Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5.1 Beam with normal force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5.1.1 Stiffness reduction from normal force . . . . . . . . . . . . . . . 193

5.2 Stability of the ideal column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Contents ix

5.2.1 Equivalent column length . . . . . . . . . . . . . . . . . . . . . . . . . 203

5.2.2 Buckling direction and intermediate supports . . . . . . . . 205

5.3 Design of columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

5.3.1 Column length and slenderness . . . . . . . . . . . . . . . . . . . . . 208

5.3.2 Geometric imperfections. . . . . . . . . . . . . . . . . . . . . . . . . . . 212

5.3.3 Stresses in column cross-sections . . . . . . . . . . . . . . . . . . . 215

5.3.4 Perry-Robertson’s column design criterion . . . . . . . . . . . 218

5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

6 The Force Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

6.1 Principle of the force method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

6.2 The general force method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

6.2.1 Released structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

6.2.2 The basic steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

6.2.3 Summary of the force method . . . . . . . . . . . . . . . . . . . . . . 241

6.3 Application of the Force Method . . . . . . . . . . . . . . . . . . . . . . . . . 242

6.4 The force method for frame structures . . . . . . . . . . . . . . . . . . . . 250

6.4.1 Simply supported frames . . . . . . . . . . . . . . . . . . . . . . . . . . 251

6.4.2 Frames with fixed supports . . . . . . . . . . . . . . . . . . . . . . . . 257

6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

7 Deformation and Element Methods for Frames . . . . . . . . . . . 267

7.1 Stiffness of beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

7.1.1 Symmetric and anti-symmetric bending . . . . . . . . . . . . . 269

7.1.2 Basic cases of imposed deformation . . . . . . . . . . . . . . . . . 271

7.1.3 Loads on constrained beams . . . . . . . . . . . . . . . . . . . . . . . 277

7.2 Deformation method for frames . . . . . . . . . . . . . . . . . . . . . . . . . . 278

7.3 Beam elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

7.3.1 Beam bending element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

7.3.2 Beam-column element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

7.3.3 Transformation to global form . . . . . . . . . . . . . . . . . . . . . 305

7.4 Finite element method for frames . . . . . . . . . . . . . . . . . . . . . . . . . 307

7.4.1 The MiniFrame program . . . . . . . . . . . . . . . . . . . . . . . . . . 308

7.4.2 Stability analysis of frames . . . . . . . . . . . . . . . . . . . . . . . . 312

7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

8 Stresses and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

8.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

8.1.1 The stress vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

8.1.2 General stress components . . . . . . . . . . . . . . . . . . . . . . . . . 324

8.1.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

8.2 Deformation and strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

8.2.1 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

8.2.2 Rotation at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

x Contents

8.2.3 Displacement decomposition . . . . . . . . . . . . . . . . . . . . . . . 337

8.3 Virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

8.3.1 Equation of virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . 338

8.3.2 Matrix and tensor notation . . . . . . . . . . . . . . . . . . . . . . . . 341

8.4 Special states of stress and strain . . . . . . . . . . . . . . . . . . . . . . . . . 342

8.4.1 Plane stress and plane strain . . . . . . . . . . . . . . . . . . . . . . . 342

8.4.2 Stress and strain transformations . . . . . . . . . . . . . . . . . . . 343

8.4.3 Principal stresses and strains in a plane . . . . . . . . . . . . . 349

8.4.4 Principal stresses in three dimensions . . . . . . . . . . . . . . . 354

8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

9 Material Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

9.1 Elastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

9.1.1 Internal elastic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

9.1.2 Linear isotropic elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 367

9.2 Mean and deviator components . . . . . . . . . . . . . . . . . . . . . . . . . . 376

9.3 Yield conditions for metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

9.3.1 Von Mises’ yield condition . . . . . . . . . . . . . . . . . . . . . . . . . 380

9.3.2 Tresca’s yield condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

9.4 Coulomb’s theory of friction materials . . . . . . . . . . . . . . . . . . . . . 385

9.4.1 Critical section and stress state . . . . . . . . . . . . . . . . . . . . 386

9.4.2 Coulomb failure surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

10 General Bending of Beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

10.1 Bending of non-symmetric beams . . . . . . . . . . . . . . . . . . . . . . . . . 397

10.1.1 Kinematic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

10.1.2 Stresses and section forces . . . . . . . . . . . . . . . . . . . . . . . . . 400

10.2 Cross-section analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

10.2.1 Elastic center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

10.2.2 Moments of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

10.2.3 Principal coordinate system. . . . . . . . . . . . . . . . . . . . . . . . 417

10.3 Axial stresses and strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

10.3.1 Neutral axis and line of curvature . . . . . . . . . . . . . . . . . . 434

10.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

11 Flexure and Torsion of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

11.1 Shear stresses in beam flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

11.1.1 Shear flow – Grashof’s formula . . . . . . . . . . . . . . . . . . . . . 445

11.1.2 Shear stress on cross-section . . . . . . . . . . . . . . . . . . . . . . . 449

11.2 Thin-walled cross-sections in shear . . . . . . . . . . . . . . . . . . . . . . . . 455

11.2.1 Shear center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

11.2.2 Shear flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

11.3 Torsion of circular cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

Contents xi

11.4 General homogeneous torsion of beams . . . . . . . . . . . . . . . . . . . . 472

11.4.1 The Prandtl stress function . . . . . . . . . . . . . . . . . . . . . . . . 475

11.5 Torsion of thin-walled beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

11.5.1 Open sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

11.5.2 Single-cell sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

11.5.3 Multi-cell sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

Equilibrium and Reactions 1

Statics of structures deals with structures that are exposed to loads and

develop reactions and internal forces that leave the structure stationary. The

present book deals with buildings and civil engineering structures that are

supported to prevent motion, as opposed to space structures, trains etc. where

motion is an integral part of the behavior. A fundamental tool of statics is the

concept of equilibrium. In order to remain stationary the total effect of the

loads and the reactions provided by the supports must be in equilibrium. This

applies to the full structure and also to its different parts. In this chapter the

equilibrium conditions for the full structure are used to identify requirements

for the supports and to determine the reactions provided by the supports.

The concept of equilibrium is developed further in the following chapters to

deal with hypothetical parts of the structure, and thereby obtain knowledge

of the distribution of the forces inside the structure.

First the notion of a force is introduced in Section 1.1. A force is specified by

its magnitude and its line of action, and is closely related to the mathemat￾ical concept of a vector. If two forces have intersecting lines of action they

combine as vectors, and act at the point of intersection. However, this con￾cept of intersecting forces is too limited, and it is necessary to introduce the

notion of a moment, as described in Section 1.2. When considering forces and

moments together, the concept of equilibrium takes a precise mathematical

form, discussed in Section 1.3. The direct form of the equilibrium conditions

constitutes two vector equations for the total force and the total moment,

respectively. It is explained in detail, how these equations can be combined

S. Krenk, J. Høgsberg, Statics and Mechanics of Structures,

DOI 10.1007/978-94-007-6113-1 1,

© Springer Science+Business Media Dordrecht 2013

2 Equilibrium and Reactions

into a single scalar equation of virtual work – that is the work that the forces

and moments would perform, if subjected to an arbitrary small virtual dis￾placement. The virtual work is here introduced in its basic form, but appears

in a more advanced form later in connection with deformation of beams and

frames. The concept of virtual work plays a central role in the modern formu￾lation of theories for structures and solid bodies, e.g. in connection with the

formulation of numerical methods. The two last sections of the chapter deal

with the support conditions and the reactions developed in the supports.

1.1 Forces

The notion of a force is fundamental to the theory of structures. A force

is associated with a magnitude, a direction, and a point of action. In the

analysis of forces it is convenient first to focus on the direction and magnitude,

combined in the boldface vector symbol P. The magnitude is represented by

the length of the vector and is denoted by P = |P|. In practice a force often

has a specific point of action, but it is often convenient to consider the force

as acting in a line of action, defined as the line obtained by extending the

force vector in space. This notion permits the force to be translated along

its line of action, and leads to a fairly intuitive formulation of the theory of

equilibrium of a set of one or more forces.

1.1.1 The parallelogram rule

It is an important property of a force P that it can be resolved into com￾ponents according to the parallelogram rule, known from elementary vec￾tor analysis, see e.g. Strang (2001). The parallelogram rule is illustrated in

Fig. 1.1 showing the force P and two directions intersecting the line of action

of P. For convenience and clarity the point of intersection is shown as the

point of action of the force P in the figure. If this is not the case for the initial

location of P, it is translated to the point of intersection along the line of

action.

Fig. 1.1: Decomposition of force P in given direction.

The parallelogram rule for resolving a force P into components P1 and P2

with given directions consists in forming a parallelogram with P along a

Forces 3

diagonal and the components P1 and P2 along adjoining sides as shown in

Fig. 1.1b.

The parallelogram rule can also be used to form the resulting force from two

given forces P1 and P2, when these forces have intersecting lines of action.

The construction of the parallelogram follows from sliding the forces P1 and

P2 to the point of intersection. They then form the sides of a parallelogram

with the resultant P along the diagonal as shown in Fig. 1.1b.

Example 1.1. Force on a string. Figure 1.2 shows a simple example of the force paral￾lelogram rule. A vertical force is acting on a string, which is stretched, forming two linear

parts. These parts carry constant forces T1 and T2, in the direction of the respective

strings.

Fig. 1.2: Cable carries P via the forces T1 and T2.

In a static analysis of simple structures the forces may be referred directly to

directions in the structure, e.g. as along or transverse to a beam. However, in

larger analyzes, and when using a computer for the numerical computations,

it is often convenient to represent forces by their components in a Cartesian

coordinate system. In this case two forces P1 and P2 are represented by their

xyz-components as

P1 =

⎡

⎢

⎣

P1

x

P1

y

P1

z

⎤

⎥

⎦ , P2 =

⎡

⎢

⎣

P2

x

P2

y

P2

z

⎤

⎥

⎦ , (1.1)

and it then follows from the parallelogram composition rule that the resultant

force P has the components

⎡

⎢

⎣

Px

Py

Pz

⎤

⎥

⎦ =

⎡

⎢

⎣

P1

x

P1

y

P1

z

⎤

⎥

⎦ +

⎡

⎢

⎣

P2

x

P2

y

P2

z

⎤

⎥

⎦ . (1.2)

It should be noted that this standard addition rule of vector components

must be accompanied by an account of the resultant’s line of action.

4 Equilibrium and Reactions

1.1.2 Parallel forces

In the case of two parallel forces their lines of action do not intersect, and

thus the parallelogram rule needs an extension. The problem is illustrated in

Fig. 1.3 showing two parallel forces with distance a and magnitude P1 and

P2, respectively. In principle the magnitude and location of the resulting force

P can be obtained as a limit of the two forces, if inclined slightly with their

original common direction. However, it is more direct to obtain the result by

introducing two auxiliary forces as demonstrated here.

Fig. 1.3: Parallel forces with distance a.

In order to increase the clarity of the geometric construction the two forces

P1 and P2 are first translated along their respective lines of action, until

their points of application A1 and A2 lie on a line orthogonal to the lines of

action as shown in Fig. 1.4a. Two forces of equal magnitude Q but opposite

direction along the connecting line are now added as shown in the figure. As

these forces are opposite with the same line of action they have the sum zero,

and therefore do not change the resulting force of the system. There is now

a force P1 + Q acting at A1 and a force P2 − Q acting at A2. These forces

are not parallel, and they can therefore be combined by the parallelogram

rule, whereby the resulting force P = P1 + P2 passes through the point of

intersection C of the lines of action.

Fig. 1.4: Composition of parallel forces.

The line of action of the resulting force P is characterized by the distance a1

from P1 and the distance a2 from P2. The figure contains similar triangles