Engineering mechnics : Volume 2 : Stresses, strains, displacements

ENGINEERING MECHANICS

Engineering Mechanics

Volume 2: Stresses, Strains, Displacements

by

C. HARTSUIJKER

Delft University of Technology, Delft, The Netherlands

and

J.W. WELLEMAN

Delft University of Technology, Delft, The Netherlands

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-1-4020-4123-5 (HB)

ISBN 978-1-4020-5763-2 (e-book)

Published by Springer,

P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

www.springer.com

This is a translation of the original Dutch work “Toegepaste Mechanica, Deel 2: Spanningen, Vervormingen, Verplaatsingen”, 2001,

Academic Service, The Hague, The Netherlands.

Printed on acid-free paper

All Rights Reserved

© 2007 Springer

No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming,

recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and

executed on a computer system, for exclusive use by the purchaser of the work.

Table of Contents

Preface ix

Foreword xiii

1 Material Behaviour 1

1.1 Tensile test 1

1.2 Stress-strain diagrams 5

1.3 Hooke’s Law 11

2 Bar Subject to Extension 15

2.1 The fibre model 16

2.2 The three basic relationships 18

2.3 Strain diagram and normal stress diagram 24

2.4 Normal centre and bar axis 26

2.5 Mathematical description of the extension problem 30

2.6 Examples relating to changes in length and displacements 34

2.7 Examples relating to the differential equation for extension 45

2.8 Formal approach and engineering practice 52

2.9 Problems 54

3 Cross-Sectional Properties 71

3.1 First moments of area; centroid and normal centre 74

3.2 Second moments of area 91

3.3 Thin-walled cross-sections 121

3.4 Formal approach and engineering practice 132

3.5 Problems 135

4 Members Subject to Bending and Extension 151

4.1 The fibre model 153

4.2 Strain diagram and neutral axis 155

4.3 The three basic relationships 157

4.4 Stress formula and stress diagram 168

4.5 Examples relating to the stress formula for bending with

extension 171

4.6 Section modulus 184

4.7 Examples of the stress formula related to bending without

extension 186

4.8 General stress formula related to the principal directions 198

4.9 Core of the cross-section 203

vi ENGINEERING MECHANICS. VOLUME 2: STRESSES, DEFORMATIONS, DISPLACEMENTS

4.10 Applications related to the core of the cross-section 208

4.11 Mathematical description of the problem of bending with

extension 219

4.12 Thermal effects 223

4.13 Notes for the fibre model and summary of the formulas 228

4.14 Problems 234

5 Shear Forces and Shear Stresses Due to Bending 271

5.1 Shear forces and shear stresses in longitudinal direction 272

5.2 Examples relating to shear forces and shear stresses in

the longitudinal direction 282

5.3 Cross-sectional shear stresses 300

5.4 Examples relating to the shear stress distribution in a

cross-section 310

5.5 Shear centre 367

5.6 Other cases of shear 377

5.7 Summary of the formulas and rules 382

5.8 Problems 385

6 Bar Subject to Torsion 411

6.1 Material behaviour in shear 412

6.2 Torsion of bars with circular cross-section 415

6.3 Torsion of thin-walled cross-sections 426

6.4 Numerical examples 445

6.5 Summary of the formulas 468

6.6 Problems 471

7 Deformation of Trusses 483

7.1 The behaviour of a single truss member 484

7.2 Williot diagram 487

7.3 Williot diagram with rigid-body rotation 504

7.4 Williot–Mohr diagram 514

7.5 Problems 521

8 Deformation Due to Bending 541

8.1 Direct determination from the moment distribution 543

8.2 Differential equation for bending 557

8.3 Forget-me-nots 576

8.4 Moment-area theorems 598

8.5 Simply supported beams and the M/EI diagram 633

8.6 Problems 648

9 Unsymmetrical and Inhomogeneous Cross-Sections 679

9.1 Sketch of the problems and required assumptions 679

9.2 Kinematic relationships 682

9.3 Curvature and neutral axis 686

9.4 Normal force and bending moments – centre of force 690

9.5 Constitutive relationships for unsymmetrical and/or

inhomogeneous cross-sections 695

9.6 Plane of loading and plane of curvature – neutral axis 701

9.7 The normal centre NC for inhomogeneous cross-sections 706

9.8 Stresses due to extension and bending – a straightforward

method 714

9.9 Applications of the straightforward method 715

9.10

method 734

9.11 Transformation formulae for the bending stiffness tensor 736

9.12 Application of the alternative method based on the

principal directions 752

9.13 Displacements due to bending 761

Stresses in the principal coordinate system – alternative

Table of Contents vii

9.14 Maxwell’s reciprocal theorem 773

9.15 Core of a cross-section 777

9.16 Thermal effects 791

9.17 Shear flow and shear stresses in arbitrary cross-sections –

shear centre 809

9.18 Problems 845

Index 865

Preface

This Volume is the second of a series of two:

• Volume 1: Equilibrium

• Volume 2: Stresses, deformations and displacements

These volumes introduce the fundamentals of structural and continuum

mechanics in a comprehensive and consistent way. All theoretical devel￾opments are presented in text and by means of an extensive set of figures.

Numerous examples support the theory and make the link to engineering

practice. Combined with the problems in each chapter, students are given

ample opportunities to exercise.

The book consists of distinct modules, each divided into sections which are

conveniently sized to be used as lectures. Both formal and intuitive (engi￾neering) arguments are used in parallel to derive the important principles.

The necessary mathematics is kept to a minimum however in some parts

basic knowledge of solving differential equations is required.

The modular content of the book shows a clear order of topics concerning

stresses and deformations in structures subject to bending and extension.

Chapter 1 deals with the fundamentals of material behaviour and the intro￾duction of basic material and deformation quantities. In Chapter 2 the fibre model is introduced to describe the behaviour of line elements subject to ex- tension (tensile or compressive axial forces). A formal approach is followed in which the three basic relationships (the kinematic, constitutive and static relationships) are used to describe the displacement field with a second order differential equation. Numerous examples show the influence of the boundary conditions and loading conditions on the solution of the displace- ment field. In Chapter 3 the cross-sectional quantities such as centre of mass or centre of gravity, centroid, normal (force) centre, first moments of area or static moments, and second moments of area or moments of inertia are introduced as well as the polar moment of inertia. The influence of the translation of the coordinate system on these quantities is also investigated, resulting in the parallel axis theorem or Steiner’s rule for the static moments and moments of inertia. With the definitions of Chapters 1 to 3 the complete theory for bending and extension is combined in Chapter 4 which describes the fibre model subject to extension and bending (Euler–Bernoulli theory). The same framework is used as in Chapter 2 by defining the kinematic, constitutive and static relationships, in order to obtain the set of differential equations to describe the combined behaviour of extension and bending. By

x ENGINEERING MECHANICS. VOLUME 2: STRESSES, DEFORMATIONS, DISPLACEMENTS

choosing a specific location of the coordinate system through the normal

(force) centre, we introduce the uncoupled description of extension and

bending. The strain and stress distribution in a cross-section are introduced

and engineering expressions are resolved for cross-sections with at least

one axis of symmetry. In this chapter also some special topics are covered

like the core of a cross-section, and the influence of temperature effects.

For non-constant bending moment distributions, beams have to transfer

shear forces which will lead to shear stresses in longitudinal and transversal

section planes. Based on the equilibrium conditions only, expressions for

the shear flow and the shear stresses will be derived. Field of applications

are (glued or dowelled) interfaces between different materials in a compos￾ite cross-section and the stresses in welds. Special attention is also given to

thin-walled sections and the definition of the shear (force) centre for thin￾walled sections. This chapter focuses on homogeneous cross-sections with

at least one axis of symmetry. Shear deformation is not considered.

Chapter 6 deals with torsion, which is treated according to the same concept

as in the previous chapters; linear elasticity is assumed. The elementary

theory is used on thin-walled tubular sections. Apart from the deformations

also shear stress distributions are obtained. Special cases like solid circular

sections and open thin-walled sections are also treated.

Structural behaviour due to extension and or bending is treated in Chap￾ters 7 and 8. Based on the elementary behaviour described in Chapters 2 and

4 the structural behaviour of trusses is treated in Chapter 7 and of beams in

Chapter 8. The deformation of trusses is treated both in a formal (analytical)

way and in a practical (graphical) way with aid of a relative displacement

graph or so-called Williot diagram. The deflection theory for beams is

elaborated in Chapter 8 by solving the differential equations and the in￾troduction of (practical) engineering methods to obtain the displacements

and deformations based on the moment distribution. With these engineering

formulae, forget-me-nots and moment-area theorems, numerous examples

are treated. Some special cases like temperature effects are also treated in

this chapter.

Chapter 9 shows a comprehensive description of the fibre model on un￾symmetrical and or inhomogeneous cross-sections. Much of the earlier

presented derivations are now covered by a complete description using

a two letter symbol approach. This formal approach is quite unique and

offers a fast and clear method to obtain the strain and stress distribution

in arbitrary cross-sections by using an initially given coordinate system

with its origin located at the normal centre of the cross-section. Although a

complete description in the principal coordinate system is also presented, it

will become clear that a description in the initial coordinate system is to be

preferred. Centres of force and core are also treated in this comprehensive

theory, as well as the full description of the shear flow in an arbitrary cross￾section. The last part of this chapter shows the application of this theory

on numerous examples of both inhomogeneous and unsymmetrical cross￾sections. Special attention is also given to thin-walled sections as well as

the shear (force) centre of unsymmetrical thin-walled sections which is of

particular interest in steel structures design.

This latter chapter is not necessarily regarded as part of a first introduction

into stresses and deformations but would be more suitable for a second or

third course in Engineering Mechanics. However, since this chapter offers

the complete and comprehensive description of the theory, it is an essential

part of this volume.

We do realise, however, that finding the right balance between abstract

fundamentals and practical applications is the prerogative of the lecturer.

He or she should therefore decide on the focus and selection of the topics

treated in this volume to suit the goals of the course in question.

Preface xi

The authors want to thank especially the reviewer Professor Graham M.L.

Gladwell from the University of Waterloo (Canada) for his tedious job to

improve the Dutch-English styled manuscript into readable English. We

also thank Jolanda Karada for her excellent job in putting it all together

and our publisher Nathalie Jacobs who showed enormous enthusiasm and

patience to see this series of books completed and to have them published

by Springer.

Coenraad Hartsuijker

Hans Welleman

Delft, The Netherlands

July 2007

Foreword

Structural or Engineering Mechanics is one of the core courses for new

students in engineering studies. At Delft University of Technology a joint

educational program for Statics and Strength of Materials has been devel￾oped by the Koiter Institute, and has subsequently been incorporated in

the curricula of faculties like Civil Engineering, Aeronautical Engineering,

Architectural Engineering, Mechanical Engineering, Maritime Engineering

and Industrial Design.

In order for foreign students also to be able to benefit from this pro￾gram an English version of the Dutch textbook series written by Coenraad

Hartsuijker, which were already used in most faculties, appeared to be nec￾essary. It is fortunate that in good cooperation between the writers, Springer

and the Koiter Institute Delft, an English version of two text books could

be realized, and it is believed that this series of books will greatly help the

student to find his or her way into Engineering or Structural Mechanics.

Indeed, the volumes of this series offer some advantages not found

elsewhere, at least not to this extent. Both formal and intuitive approaches

are used, which is more important than ever. The books are modular and can

also be used for self-study. Therefore, they can be used in a flexible manner

and will fit almost any educational system. And finally, the SI system is

used consistently. For these reasons it is believed that the books form a

very valuable addition to the literature.

René de Borst

Scientific Director, Koiter Institute Delft

Material Behaviour

1

To calculate the stresses and deformations in structures, we have to know

the material behaviour, which can be obtained only by experiments.

Through standardised tests, the material properties are laid down in a num￾ber of specific quantities. One of these tests is the tensile test, described in

Section 1.1, resulting in a so-called stress-strain diagram.

Section 1.2 looks at stress-strain diagrams for a number of materials.

This book addresses mainly materials with a linear-elastic behaviour, which

obey Hooke’s Law. Section 1.3 devotes attention to the linear behaviour of

materials, such as steel, aluminium, concrete and wood.

1.1 Tensile test

Strength, stiffness and ductility are important material properties and can

be described as follows:

• strength – the resistance that has to be overcome to break the cohesion

of the material;

• stiffness – the resistance against deformation;

• ductility – the capacity to undergo large strains before fracture occurs.

2 ENGINEERING MECHANICS. VOLUME 2: STRESSES, DEFORMATIONS, DISPLACEMENTS

Figure 1.1 Prismatic bar subject to tension.

Figure 1.2 Test bar.

Figure 1.3 Force-elongation diagram (F-
diagram).

An important and often-used test for determining the strength, stiffness and

ductility of a material is the tensile test. In tensile tests, a specimen of the

material in the form of a bar is placed in a so-called tensile testing machine.

The test bar is slowly stretched until fracture occurs. For each applied elon￾gation
the required strength F1 is measured, and both values are plotted

in a so-called F-
diagram or force-elongation diagram.

Figure 1.1 shows a prismatic bar; prismatic means that the bar has a uniform

cross-section. To prevent fracture of the bar near the ends a test bar is

shaped at the ends as in Figure 1.2. In that case,
is the elongation of

the distance between two measuring points on the prismatic part of the

bar.

Figure 1.3 shows the force-elongation diagram or F-
diagram (not to

scale) for hot rolled steel (mild steel) in tension.

There are four stages in this F-
diagram.

• Linear-elastic stage – path OA

This part of the diagram is practically straight. Up to point A there

seems to be a proportionality (linear relationship) between the force

F and elongation
. If the load in A is removed, the same path is

followed in opposite direction until point O is again reached. In other

words, if the force is removed, the bar springs back to its original

length. This type of behaviour is known as elastic.

• Yield stage or plastic stage – path AB

Path AB of the diagram generally includes a number of “bumps” but

is otherwise virtually horizontal. This means that the elongation of the

1 If a change in length is applied and the required force is measured, the test is

referred to as being deformation-driven. If, however, a load is applied and the

associated change in length is measured, the test is said to be load-driven. In

general, deformation-driven tests and load-driven tests give different results.

1 Material Behaviour 3

Figure 1.4 Local necking of a test bar.

Figure 1.5 F-
diagrams for bars with various dimensions.

Figure 1.6 Due to the same force F, a member that is twice as

long has an elongation that is twice as large.

bar increases with a nearly constant load. This phenomenon is known

as yielding or plastic flow of the material.

• Strain hardening stage – path BC

When the deformation becomes larger, the material may offer addi￾tional resistance. The required force to obtain the elongation increases.

This is called strain hardening.

• Necking stage – path CD

Beyond point C, the load decreases with increasing elongation. Locally

the bar produces a pronounced necking (see Figure 1.4) that increases

until fracture occurs at D. At fracture the load falls away and both parts

of the bar spring back a little elastically.

If somewhere between point A (the limit of proportionality) and point D

at which fracture occurs) the load is removed, the test bar reverts a little

elastically. The return path (unloading path) is a nearly straight line parallel

to OA. In Figure 1.3 this is shown by means of the dashed line. Once the

load has been released to zero the bar demonstrates a permanent set or

plastic elongation
p; the elastic elongation was
e.

The F-
diagram depends not only on the material, but also on the dimen￾sions of the test bar, namely the length between the measuring points on

the prismatic part of the bar, and the area A of the cross-section. Figure 1.5

shows the F-
diagrams for three bars made of the same material but with

different dimensions.

If the (prismatic) bar is chosen twice as long without changing the load

F, then the elongation is twice as large. We can see this by looking at

the behaviour of the two bars in Figure 1.6, attached one behind the other.

The total elongation is the sum of the elongations of each of the bars. The

elongation
is therefore proportional to the length of the bar.

4 ENGINEERING MECHANICS. VOLUME 2: STRESSES, DEFORMATIONS, DISPLACEMENTS

Figure 1.7 For a cross-section that is twice as large, the required

force to get the same elongation
is twice as large.

Figure 1.8 The normal stress σ = N/A due to extension is con￾stant in a homogeneous cross-section.

To eliminate the influence of the length of the test bar, we plot ε =
/ on

the horizontal axis instead of
. The (dimensionless) deformation quantity

ε =

=

elongation

original length

is referred to as the strain of the bar.

If the cross-section A of the bar is chosen twice as large, a doubled load

F is required to get the same elongation
. Refer to the behaviour of the

two parallel bars in Figure 1.7. To get an elongation
each bar has to be

subjected to a normal force F, and the total load on the system of two bars is

2F. Therefore the force F is proportional to the area A of the cross-section

of the bar.

To eliminate the influence of the area of the cross-section, we plot the

quantity

σ =

F

A

along the vertical axis instead of F; σ is the normal stress in the cross￾section.

In general, the normal stress varies across the cross-section and σ = F/A

should be seen as the “average” normal stress in the cross-section. If the

cross-section is homogeneous (the cross-section consists of the same mate￾rial everywhere) and the cross-section in question is far enough away from

the ends of the bar where the loads are applied (these are disruption zones),

then the normal stress due to the tensile force is roughly constant over the

cross-section (see Figure 1.8).

By converting the force-elongation diagram (F-
diagram) into a stress￾strain diagram (σ-ε diagram) we eliminate the influence of the bar dimen￾Figure 1.6 Due to the same force F, a member that is twice as long has an elongation that is twice as large.