Mechanics of Materials
Mechanics of Materials
This telecommunications tower is an assemblage of many members that act primarily in tension or compression
NPR. Used with permission. This telecommunications tower is an assemblage of many members that act primarily in tension or compression
Tension,Compression, and Shear CHAPTER OVERVIEW In Chapter 1,we are introduced to mechanics of materials,which exam- ines the stresses,strains,and displacements in bars of various materials acted on by axial loads applied at the centroids of their cross sections. We will learn about normal stress (o)and normal strain (e)in materials used for structural applications,then identify key properties of various materials,such as the modulus of elasticity (E)and yield (o)and ulti- mate(o)stresses,from plots of stress()versus strain(e).We will also plot shear stress(T)versus shear strain (y)and identify the shearing modulus of elasticity(G).If these materials perform only in the linear range,stress and strain are related by Hooke's Law for normal stress and strain (o =E.e)and also for shear stress and strain (T=G.y).We will see that changes in lateral dimensions and volume depend upon Poisson's ratio (v).Material properties E,G,and v,in fact,are directly related to one another and are not independent properties of the material. Assemblage of bars to form structures (such as trusses)leads to consideration of average shear (T)and bearing (p)stresses in their connections as well as normal stresses acting on the net area of the cross section (if in tension)or on the full cross-sectional area (if in compression).If we restrict maximum stresses at any point to allow- able values by use of factors of safety,we can identify allowable levels of axial loads for simple systems,such as cables and bars.Factors of safery relate actual to required strength of structural members and account for a variety of uncertainties,such as variations in material properties and probability of accidental overload.Lastly,we will con- sider design:the iterative process by which the appropriate size of structural members is determined to meet a variety of both strength and stiffness requirements for a particular structure subjected to a variety of different loadings. 3
CHAPTER OVERVIEW In Chapter 1, we are introduced to mechanics of materials, which examines the stresses, strains, and displacements in bars of various materials acted on by axial loads applied at the centroids of their cross sections. We will learn about normal stress (#) and normal strain ($) in materials used for structural applications, then identify key properties of various materials, such as the modulus of elasticity (E) and yield (#y) and ultimate (#u) stresses, from plots of stress (#) versus strain ($). We will also plot shear stress (%) versus shear strain (&) and identify the shearing modulus of elasticity (G). If these materials perform only in the linear range, stress and strain are related by Hooke’s Law for normal stress and strain (# E . $) and also for shear stress and strain (% G . &). We will see that changes in lateral dimensions and volume depend upon Poisson’s ratio (v). Material properties E, G, and v, in fact, are directly related to one another and are not independent properties of the material. Assemblage of bars to form structures (such as trusses) leads to consideration of average shear (%) and bearing (#b) stresses in their connections as well as normal stresses acting on the net area of the cross section (if in tension) or on the full cross-sectional area (if in compression). If we restrict maximum stresses at any point to allowable values by use of factors of safety, we can identify allowable levels of axial loads for simple systems, such as cables and bars. Factors of safety relate actual to required strength of structural members and account for a variety of uncertainties, such as variations in material properties and probability of accidental overload. Lastly, we will consider design: the iterative process by which the appropriate size of structural members is determined to meet a variety of both strength and stiffness requirements for a particular structure subjected to a variety of different loadings. 3 1 Tension, Compression, and Shear
4 CHAPTER 1 Tension,Compression,and Shear Chapter 1 is organized as follows: 1.1 Introduction to Mechanics of Materials 5 1.2 Normal Stress and Strain 7 1.3 Mechanical Properties of Materials 15 1.4 Elasticity,Plasticity,and Creep 24 1.5 Linear Elasticity,Hooke's Law,and Poisson's Ratio 27 1.6 Shear Stress and Strain 32 1.7 Allowable Stresses and Allowable Loads 43 1.8 Design for Axial Loads and Direct Shear 49 Chapter Summary Review 55 Problems 57
Chapter 1 is organized as follows: 1.1 Introduction to Mechanics of Materials 5 1.2 Normal Stress and Strain 7 1.3 Mechanical Properties of Materials 15 1.4 Elasticity, Plasticity, and Creep 24 1.5 Linear Elasticity, Hooke’s Law, and Poisson’s Ratio 27 1.6 Shear Stress and Strain 32 1.7 Allowable Stresses and Allowable Loads 43 1.8 Design for Axial Loads and Direct Shear 49 Chapter Summary & Review 55 Problems 57 4 CHAPTER 1 Tension, Compression, and Shear
SECTION 1.1 Introduction to Mechanics of Materials 5 1.1 INTRODUCTION TO MECHANICS OF MATERIALS Mechanics of materials is a branch of applied mechanics that deals with the behavior of solid bodies subjected to various types of loading. Other names for this field of study are strength of materials and mechanics of deformable bodies.The solid bodies considered in this book include bars with axial loads,shafts in torsion,beams in bending, and columns in compression. The principal objective of mechanics of materials is to determine the stresses,strains,and displacements in structures and their compo- nents due to the loads acting on them.If we can find these quantities for all values of the loads up to the loads that cause failure,we will have a complete picture of the mechanical behavior of these structures. An understanding of mechanical behavior is essential for the safe design of all types of structures,whether airplanes and antennas,buildings and bridges,machines and motors,or ships and spacecraft.That is why mechanics of materials is a basic subject in so many engineering fields.Stat- ics and dynamics are also essential,but those subjects deal primarily with the forces and motions associated with particles and rigid bodies.In mechanics of materials we go one step further by examining the stresses and strains inside real bodies,that is,bodies of finite dimensions that deform under loads.To determine the stresses and strains,we use the physical prop- erties of the materials as well as numerous theoretical laws and concepts. Theoretical analyses and experimental results have equally important roles in mechanics of materials.We use theories to derive formulas and equations for predicting mechanical behavior,but these expressions cannot be used in practical design unless the physical properties of the materials are known.Such properties are available only after careful experiments have been carried out in the laboratory.Furthermore,not all practical problems are amenable to theoretical analysis alone,and in such cases physical testing is a necessity. The historical development of mechanics of materials is a fascinating blend of both theory and experiment-theory has pointed the way to useful results in some instances,and experiment has done so in others. Such famous persons as Leonardo da Vinci (1452-1519)and Galileo Galilei (1564-1642)performed experiments to determine the strength of wires,bars,and beams,although they did not develop adequate theories (by today's standards)to explain their test results.By contrast,the famous mathematician Leonhard Euler(1707-1783)developed the math- ematical theory of columns and calculated the critical load of a column in 1744,long before any experimental evidence existed to show the signifi- cance of his results.Without appropriate tests to back up his theories, Euler's results remained unused for over a hundred years,although today they are the basis for the design and analysis of most columns. "The history of mechanics of materials,beginning with Leonardo and Galileo,is given in Refs.1-1.1-2.and 1-3
SECTION 1.1 Introduction to Mechanics of Materials 5 1.1 INTRODUCTION TO MECHANICS OF MATERIALS Mechanics of materials is a branch of applied mechanics that deals with the behavior of solid bodies subjected to various types of loading. Other names for this field of study are strength of materials and mechanics of deformable bodies. The solid bodies considered in this book include bars with axial loads, shafts in torsion, beams in bending, and columns in compression. The principal objective of mechanics of materials is to determine the stresses, strains, and displacements in structures and their components due to the loads acting on them. If we can find these quantities for all values of the loads up to the loads that cause failure, we will have a complete picture of the mechanical behavior of these structures. An understanding of mechanical behavior is essential for the safe design of all types of structures, whether airplanes and antennas, buildings and bridges, machines and motors, or ships and spacecraft. That is why mechanics of materials is a basic subject in so many engineering fields. Statics and dynamics are also essential, but those subjects deal primarily with the forces and motions associated with particles and rigid bodies. In mechanics of materials we go one step further by examining the stresses and strains inside real bodies, that is, bodies of finite dimensions that deform under loads. To determine the stresses and strains, we use the physical properties of the materials as well as numerous theoretical laws and concepts. Theoretical analyses and experimental results have equally important roles in mechanics of materials. We use theories to derive formulas and equations for predicting mechanical behavior, but these expressions cannot be used in practical design unless the physical properties of the materials are known. Such properties are available only after careful experiments have been carried out in the laboratory. Furthermore, not all practical problems are amenable to theoretical analysis alone, and in such cases physical testing is a necessity. The historical development of mechanics of materials is a fascinating blend of both theory and experiment—theory has pointed the way to useful results in some instances, and experiment has done so in others. Such famous persons as Leonardo da Vinci (1452–1519) and Galileo Galilei (1564–1642) performed experiments to determine the strength of wires, bars, and beams, although they did not develop adequate theories (by today’s standards) to explain their test results. By contrast, the famous mathematician Leonhard Euler (1707–1783) developed the mathematical theory of columns and calculated the critical load of a column in 1744, long before any experimental evidence existed to show the significance of his results. Without appropriate tests to back up his theories, Euler’s results remained unused for over a hundred years, although today they are the basis for the design and analysis of most columns.* * The history of mechanics of materials, beginning with Leonardo and Galileo, is given in Refs. 1-1, 1-2, and 1-3