EXAMPLES AND PROBLEMS IN MECHANICS OF MATERIALS STRESS-STRAIN STATE AT A POINT OF ELASTIC DEFORMABLE SOLID EDITOR-IN-CHIEF YAKIV KARPOV PDF

Preview

Summary

MINISTRY OF EDUCATION AND SCIENCE OF UKRAINE INSTITUTE OF INNOVATIONS AND CONTENTS OF EDUCATION NATIONAL AEROSPACE UNIVERSITY “KHARKIV AVIATION INSTITUTE” SERIES: ENGINEERING EDUCATION VLADISLAV DEMENKO EXAMPLES AND PROBLEMS IN MECHANICS OF MATERIALS STRESS-STRAIN STATE AT A POINT OF ELASTIC DEFORMA...

Description

MINISTRY OF EDUCATION AND SCIENCE OF UKRAINE INSTITUTE OF INNOVATIONS AND CONTENTS OF EDUCATION NATIONAL AEROSPACE UNIVERSITY “KHARKIV AVIATION INSTITUTE”

SERIES:

ENGINEERING EDUCATION

VLADISLAV DEMENKO

EXAMPLES AND PROBLEMS IN MECHANICS OF MATERIALS

STRESS-STRAIN STATE AT A POINT OF ELASTIC DEFORMABLE SOLID EDITOR-IN-CHIEF YAKIV KARPOV

Recommended by the Ministry of Education and Science of Ukraine as teaching aid for students of higher technical educational institutions

KHARKIV 2010

УДК: 539.319(075.8) UDK: 539.319(075.8) Деменко В.Ф. Задачі та приклади з механіки матеріалів. Напруженодеформований стан в околі точки пружно-деформованого твердого тіла/ В.Ф. Деменко. – Х.: Нац. аерокосм. ун-т “Харк. авіац. ін-т”, 2009. – 292 с. Examples and Problems in Mechanics of Materials. Stress-Strain State at a Point of Elastic Deformable Solid/ V. Demenko. – Kharkiv: National Aerospace University “Kharkiv Aviation Institute”, 2009. – 292 p. ISBN 978-966-662-208-5 Посібник містить важливі розділи дисципліни “Механіка матеріалів” для бакалаврів напрямів “Авіа-та ракетобудування” та “Інженерна механіка”. Починаючи з викладення основних понять цієї фундаментальної загальноінженерної дисципліни, таких, як механічне напруження, деформація, пружне деформування та інші, він охоплює всі основні практичні проблеми теорії напружено-деформованого стану на рівні, достатньому для бакалаврів авіа- і ракетобудування. Вони стосуються в першу чергу визначення діючих напружень, деформацій, а також потенціальної енергії пружної деформації в околі точки пружно-деформованого твердого тіла за умов зовнішнього навантаження різноманітних конструктивних елементів, починаючи від простих деформацій: розтягу - стиску, кручення, плоского гнуття. При розв’язанні прикладів і задач використано Міжнародну систему одиниць (СІ). Для практичної інженерної підготовки студентів, які навчаються за напрямами “Авіа- і ракетобудування” та “Інженерна механіка”, а також студентів спеціальності “Прикладна лінгвістика” при вивченні англійської мови (технічний переклад). Може бути корисним тим студентам, що готуються до стажування в технічних університетах Європи та США, а також іноземним громадянам, які навчаються в Україні. Іл. 402. Табл. 27. Бібліогр.: 24 назви The course-book includes important parts of Mechanics of Materials as an undergraduate level course of this fundamental discipline for Aerospace Engineering and Mechanical Engineering students. It defines and explains the main theoretical concepts (mechanical stress, strain, elastic behavior, etc.) and covers all the major topics of stress-strain state theory. These include the calculation of acting stresses, strains and elastic strain energy at a point of elastic deformable solid under different types of loading on structural members, starting from such simple deformations as tension-compression, torsion, and plane bending. All calculations in the examples and also the data to the problems are done in the International System of Units (SI). Intended primarily for engineering students, the book may be used by Applied Linguistics majors studying technical translation. It will be helpful for Ukrainian students preparing to continue training at European and U.S. universities, as well as for international students being educated in Ukraine. Illustrations 402. Tables 27. Bibliographical references: 24 titles Рецензенти:

д-р фіз-мат. наук, проф. П.П. Лепіхін, д-р техн. наук, проф. О.Я. Мовшович, д-р техн. наук, проф. В.В. Сухов

Reviewed by:

Doctor of Physics and Mathematics, Professor P. Lepikhin, Doctor of Technical Sciences, Professor O. Movshovich, Doctor of Technical Sciences, Professor V. Sukhov Гриф надано Міністерством освіти і науки України (лист № 1.4/18-Г-665 від 07.08.06 р.) Sealed by the Ministry of Education and Science of Ukraine (letter № 1.4/18-G-665 dated 07.08.06)

ISBN 978-966-662-208-5

© Національний аерокосмічний університет ім. М.Є. Жуковського "Харківський авіаційний інститут", 2010 © В.Ф. Деменко, 2010 © National Aerospace University “Kharkiv Aviation Institute”, 2010 © V.F. Demenko, 2010

CONTENTS Introduction to mechanics of materials and theory of stress-strain state .... 5 CHAPTER 1 Concepts of Stress and Strain in Deformable Solid ...... 7 1.1 Definition of Stress ................................................................. 7 1.2 Components of Stress.............................................................. 8 1.3 Normal Stress .......................................................................... 10 1.4 Average Shear Stress .............................................................. 12 1.5 Deformations ........................................................................... 14 1.6 Definition of Strain ................................................................. 15 1.7 Components of Strain.............................................................. 16 1.8 Measurement of Strain (Beginning)........................................ 18 1.9 Engineering Materials ............................................................. 18 1.10 Allowable Stress and Factor of Safety .................................... 20 Examples ................................................................................. 21 Problems .................................................................................. 31 CHAPTER 2 Uniaxial Stress State .................................................. 46 2.1 Linear Elasticity in Tension-Compression. Hooke’s Law and Poisson’s Ratio. Deformability and Volume Change .................. 46 2.1.1 Hooke’s Law ........................................................................... 47 2.1.2 Poisson's Ratio ........................................................................ 50 2.1.3 Deformability and Volume Change ........................................ 52 Examples ................................................................................. 53 Problems .................................................................................. 56 2.2 Stresses on Inclined Planes in Uniaxial Stress State .............. 60 Examples ................................................................................. 67 Problems .................................................................................. 71 2.3 Strain Energy Density and Strain Energy in Uniaxial Stress State .. 78 2.3.1 Strain Energy Density ............................................................. 78 Examples ................................................................................. 86 Problems .................................................................................. 92 CHAPTER 3 Two-Dimensional (Plane) Stress State ........................ 94 3.1 Stresses on Inclined Planes ..................................................... 97 3.2 Special Cases of Plane Stress .................................................. 101 3.2.1 Uniaxial Stress State as a Simplified Case of Plane Stress ........... 101 3.2.2 Pure Shear as a Special Case of Plane Stress.......................... 101 3.2.3 Biaxial Stress........................................................................... 101 Examples ................................................................................. 102 Problems .................................................................................. 107 3.3 Principal Stresses and Maximum Shear Stresses.................... 116 3.3.1 Principal Stresses .................................................................... 116 3.3.2 Maximum Shear Stresses ........................................................ 121 Examples ................................................................................. 123 Problems .................................................................................. 128 3.4 Mohr’s Circle for Plane Stress ................................................ 139 Examples ................................................................................. 145 Problems .................................................................................. 153 3.5 Hooke’s Law for Plane Stress and its Special Cases. Change of Volume. Relations between E, G, and ν ........................................ 156

Contents

4

3.5.1 3.5.2 3.5.3 3.5.4

Hooke’s Law For Plane Stress ................................................ 156 Special Cases of Hooke's Law ................................................ 158 Change of Volume .................................................................. 159 Relations between E, G, and v ................................................ 160 Examples ................................................................................. 161 Problems .................................................................................. 164 3.6 Strain Energy and Strain Energy Density in Plane Stress State .... 170 Problems .................................................................................. 173 3.7 Variation of Stress Throughout Deformable Solid. Differential Equations of Equilibrium .............................................................. 174 CHAPTER 4 Triaxial Stress ............................................................. 177 4.1 Maximum Shear Stresses ........................................................ 177 4.2 Hooke’s Law for Triaxial Stress. Generalized Hooke’s Law ....... 178 4.3 Unit Volume Change .............................................................. 181 4.4 Strain-Energy Density in Triaxial and Three – Dimensional Stress............ 182 4.5 Spherical Stress ....................................................................... 183 Examples ................................................................................. 184 Problems .................................................................................. 187 CHAPTER 5 Plane Strain ................................................................. 192 5.1 Plane Strain versus Plane Stress Relations ............................. 192 5.2 Transformation Equations for Plane Strain State ................... 195 5.2.1 Normal strain εx1 .................................................................... 195 5.2.2 Shear strain γ x1y1 ..................................................................... 197 5.2.3 Transformation Equations For Plane Strain............................ 199 5.3 Principal Strains ...................................................................... 199 5.4 Maximum Shear Strains .......................................................... 200 5.5 Mohr’s Circle for Plane Strain ................................................ 200 5.6 Measurement of Strain (Continued)........................................ 202 5.7 Calculation of Stresses ............................................................ 203 Examples ................................................................................. 203 Problems .................................................................................. 214 CHAPTER 6 Limiting Stress State. Uniaxial Limiting Stress State and Yield and Fracture Criteria for Combined Stress ....................... 224 6.1 Maximum Principal Stress Theory (Rankine, Lame) ............. 224 6.2 Maximum Principal Strain Theory (Saint-Venant) ................ 225 6.3 Maximum Shear Stress Theory (Tresca, Guest, Coulomb) .......... 226 6.4 Total Strain Energy Theory (Beltrami-Haigh)........................ 227 6.5 Maximum Distortion Energy Theory (Huber-Henky-von Mises) ..... 227 Examples ................................................................................. 228 Problems .................................................................................. 249 CHAPTER 7 Appendixes .................................................................. 254 Appendix A Properties of Selected Engineering Materials .......... 254 Appendix B Properties of Structural-Steel Shapes ................. 274 Appendix C Properties of Structural Lumber ......................... 290 References ........................................................................................................ 291

Introduction to Mechanics of Materials and Theory of Stress-Strain State Mechanics of materials is a branch of applied mechanics that deals with the behavior of deformable solid bodies subjected to various types of loading. Another name for this field of study is strength of materials. Rods with axial loads, shafts in torsion, beams in bending, and columns in compression belong to the class of deformable solid bodies. In mechanics of materials, the general aim is to calculate the stresses, strains, and displacements in structures and their components under external loading. If we can find these quantities for all the values of applied loads up to the limiting loads that cause failure, we will have a complete picture of the mechanical behavior of these structures or their components. Understanding the mechanical behavior of all types of structures is essential to design airplanes, buildings, bridges, machines, engines able to withstand an applied loads without failure. In mechanics of materials we will examine the stresses and strains inside real bodies, that is, bodies of finite dimensions being deformed under loads. To determine stresses and strains, we use the physical properties of materials as well as numerous theoretical laws and concepts, beginning from the fundamental laws of theoretical mechanics whose subject deals primarily with the forces and motions associated with particles and rigid bodies. In mechanics of materials, the most fundamental concepts are stress and strain. These concepts can be illustrated in their most elementary form by considering a prismatic bar subjected to axial forces. A prismatic bar is a straight structural member having a constant cross section throughout its length; an axial force is a load directed along the axis of the member, resulting either in its tension or compression. Other examples are the members of a bridge truss, connecting rods in automobile engines, columns in buildings. Normal and shear stresses in beams, shafts, and rods can be calculated from the basic formulas of mechanics of materials. For instance, the stresses in a beam are given by the flexure and shear formulas ( σ ( z ) = M y z I y and

τ ( z ) = Qz S *y b ( z ) I y ), and the stresses in a shaft are given by the torsion formula

(τ (ρ ) = M x ρ Iρ ). However, the stresses calculated from these formulas act on cross sections of the members, while sometimes larger stresses occur on inclined sections. The principal topics of this course-book will deal with the states of stress and strain at points located on inclined, or oblique, sections. The components of stressed and strained states also depend upon the position of the point in a loaded body.

6

Introduction

The discussions will be limited mainly to two-dimensional, or plane, stress and plane strain. The formulas derived and graphic techniques are helpful in analyzing the transformation of stress and strain at a point under various types of loading. The graphical technique will help us to gain a stronger understanding of the stress variation around a point. Also, the transformation laws will be established to obtain an important relationship between E, G, and v for linearly elastic materials. We will derive expressions for the normal and shear stresses acting on inclined sections in both uniaxial stress and pure shear. In the case of uniaxial stress, we will show that the maximum shear stresses occur on planes inclined at 45° to the axis, whereas the maximum normal stresses occur on the cross sections. In the case of pure shear, we will find that the maximum tensile and compressive stresses occur on 45° planes. Similarly, the stresses on inclined sections cut through a beam may be larger than the stresses acting on a cross section. To calculate these stresses, we need to determine the stresses acting on inclined planes under a more general stress state known as plane stress. In our discussions of plane stress, we will use infinitesimally small stress elements to represent the state of stress at a point in a deformable solid. We will begin our analysis by considering an element whose stresses are known, and then we will derive the transformation relationships to calculate the stresses acting on the sides of an element oriented in a different direction. In stress analysis, we must always keep in mind that only one intrinsic state of stress exists at a point in a stressed body, regardless of the orientation of the element being used to portray that state of stress. When we have two elements with different orientations at the same point in the body, the stresses acting on the faces of the two elements are different, but they still represent the same state of stress, namely, the stress at the point under consideration. The concept of stress is much more complex than vectors are, and in mathematics stresses are called tensors. Other tensor quantities in mechanics are strains and moments of inertia. When studying stress-strain theory, our efforts will be divided naturally into two parts: first, understanding the logical development of the concepts, and second, applying those concepts to practical situations. The former will be accomplished by studying the derivations and examples that appear in each chapter, and the latter will be accomplished by solving the problems at the ends of the chapters. In keeping with current engineering practice, this book utilizes only International System of Units (SI).

Chapter 1 Concepts of Stress and Strain in Deformable Solid 1.1 Definition of Stress A body subjected to external forces develops an associated system of internal forces. To analyze the strength of any structural element it is necessary to describe the intensity of those internal forces, which represents a particularly significant quantity. Consider one of the isolated segments of a body in equilibrium under the action of a system of forces, as shown in Figs. 1.1 and 1.2. An element of area ΔA , positioned on an interior surface passing through a point O, is acted upon by force ΔF . Let the origin of the coordinate axes be located at O, with x normal and y, z tangent to ΔA . Generally ΔF does not lie along x, y, or z. Components of ΔF parallel to x, y, and z are also indicated in the figure. The normal stress σ (sigma) and the shear, or shearing stress, τ (tau) are then defined as ΔFx dFx σ xx = σ x = lim = , dA ΔA→0 ΔA (1.1) ΔFy dFy ΔFz dFz = , τ xy = lim τ xz = lim = . dA dA ΔA→0 ΔA ΔA→0 ΔA

Fig. 1.1 Aplication of the method of sections to a body under external loading

These relations represent the stress components at the point O to which area ΔA is reduced in the limit. The primary distinction between normal and shearing stress is one of direction. From the foregoing we observe that two indices are needed to denote the components of stress. For the normal stress component the indices are identical, while for the shear stress component they are mixed. The two indices are given in double subscript notation: the

ΔFy F O

ΔFz

ΔFx A

Fig. 1.2 Components of an internal force ΔF acting on a small area centered at point O

8

Chapter 1 CONCEPTS OF STRESS AND STRAIN IN DEFORMABLE SOLID

first subscript indicates the direction of a normal to the plane, or face, on which the stress component acts; the second subscript relates to the direction of the stress itself. Repetitive subscripts will be avoided, so that the normal stress will be designated σ x , as seen in Eqs. (1.1). Note that a plane is defined by the axis normal to it; for example, the x face is perpendicular to the x axis. The limit ΔA → 0 in Eqs. (1.1) is, of course, an idealization, since the area itself is not continuous on an atomic scale. Our consideration is with the average stress on areas where size, while small as compared with the size of the body, are large as compared with the distance between atoms in the solid body. Therefore stress is an adequate ...