## MECHANICS OF MATERIALS

## BY FERDINAND P. BEER,E. RUSSELL JOHNSTON JR,JOHN T. DEWOLF, DAVID F. MAZUREK

## FREE DOWNLOAD PDF

**CONTENTS**

**INTRODUCTION - CONCEPT OF STRESS****STRESS AND STRAIN - AXIAL LOADING****TORSION****PURE BENDING****ANALYSIS AND DESIGN OF BEAMS FOR BENDING****SHEARING STRESSES IN BEAMS AND THIN WALLED MEMBERS****TRANSFORMATIONS OF STRESS AND STRAIN****PRINCIPAL STRESSES UNDER A GIVEN LOADING****DEFLECTION OF BEAMS****COLUMNS****ENERGY METHODS**

**INTRODUCTION TO MECHANICS OF MATERIALS**

**OBJECTIVES AND GENERAL APPROACH OF THIS BOOK :**

**OBJECTIVES**

The main objective of a basic mechanics course should be to develop in the engineering student the ability to analyze a given problem in a simple and logical manner and to apply to its solution a few fundamental and well-understood principles. This text is designed for the first course in mechanics of materials—or strength of materials—offered to engineering students in the sophomore or junior year. The authors hope dial it will help instructors achieve this goal in that particular course in the same way that meir other texts may have helped them in statics and dynamics.

**GENERAL APPROACH**

In this text the study of the mechanics of materials is based on the understanding of a few basic concepts and on the use of simplified models. This approach makes it possible to develop all the necessary formulas in a rational and logical manner, and to clearly indicate the conditions under which diey can be safely applied to the analysis and design of actual engineering structures and machine components. Free-body Diagrams Are Used Extensively. Throughout the text free-body diagrams are used to determine external or internal forces. The use of "picture equations" will also help the students understand (he superposition of loadings and the resulting stresses and deformations. Design Concepts Are Discussed Throughout the Text Whenever Appropriate. A discussion of the application of the factor of safety to design can be found in Chap. 1, where the concepts of both allowable stress design and load and resistance factor design are presented. A Careful Balance Between SI and U.S. Customary Units Is Consistently Maintained. Because it is essential that students be able to handle effectively both SI metric units and U.S. customary units, half the examples, sample problems, and problems to be assigned have been stated in SI units and half in U.S. customary units. Since a large number of problems are available, instructors can assign problems using each system of units in whatever proportion they find most desirable for their class. Optional Sections Offer Advanced or Specialty Topics. Topics such as residual stresses, torsion of noncircular and thin-walled members, bending of curved beams, shearing stresses in non-symmetrical

members, and failure criteria, have been included in optional sections for use in courses of varying emphases. To preserve the integrity of the subject, these topics are presented in the proper sequence, wherever they logically belong. Thus, even when not covered in die course, diey are highly visible and can be easily referred to by the students if needed in a later course or in engineering practice. For convenience all optional sections have been indicated by asterisks.

CHAPTER ORGANIZATION

It is expected that students using this text will have completed a course in statics. However. Chap. 1 is designed to provide them with an opportunity to review the concepts learned in that course, while shear and bending-momenl diagrams are covered in detail in Sees. 5.2 and 5.3. The properties of moments and centroids of areas are described in Appendix A; this material can be used to reinforce the discussion of the determination of normal and shearing stresses in beams (Chaps. 4. 5. and 6).

The first four chapters of the text are devoted to the analysis of the stresses and of the corresponding deformations in various structural members, considering successively axial loading, torsion, and pure bending. Each analysis is based on a few basic concepts, namely, the conditions of equilibrium of the forces exerted on the member, the relations existing between stress and strain in the material. and the conditions imposed by the supports and loading of the member. The study of each type of loading is complemented by a large number of examples, sample problems, and problems to be assigned, all designed to strengthen the students' understanding of

the subject.

The concept of stress at a point is introduced in Chap. 1. where it is shown that an axial load can produce shearing stresses as well as normal stresses, depending upon the section considered. The fact that stresses depend upon the orientation of the surface on which they are computed is emphasized again in Chaps. 3 and 4 in the

cases of torsion and pure bending. However, the discussion of computational techniques—such as Mohr's circle—used for the transformation of stress at a point is delayed until Chap. 7, after students have had the opportunity to solve problems involving a combination of the basic loadings and have discovered for themselves the need for such techniques. The discussion in Chap. 2 of the relation between stress and strain in various materials includes fiber-reinforced composite materials.

Also, the study of beams under transverse loads is covered in two separate chapters. Chapter 5 is devoted to the determination of the normal stresses in a beam and to the design of beams based on the allowable normal stress in the material used (Sec. 5.4). The chapter begins with a discussion of the shear and bending-moment diagrams

(Sees. 5.2 and 5.3) and includes an optional section on the use of singularity functions for the determination of the shear and bending moment in a beam (Sec. 5.5). The chapter ends with an optional section on nonprismatic beams (Sec. 5.6). Chapter 6 is devoted to the determination of shearing stresses in beams and thin-walled members under transverse loadings. The formula for the shear How, q = VQ/I, is derived in the traditional way. More advanced aspects of the design of beams, such as the determination of the principal stresses at the junction of the flange and web of a W-heam, are in Chap. 8, an optional chapter that may be covered after the transformations of stresses have been discussed in Chap. 7. The design of transmission shafts is in that chapter for the same reason, as well as the determination of stresses under combined loadings that can now include the determination of the principal stresses, principal planes, and maximum shearing stress at a given point. Statically indeterminate problems are first discussed in Chap. 2 and considered throughout the text for the various loading conditions encountered. Thus, students are presented at an early stage with a method of solution that combines the analysis of deformations with the conventional analysis of forces used in statics. In this way. they will have become thoroughly familial" with this fundamental method by the end of the course. In addition, this approach helps the students realize that stresses themselves are statically indeterminate and can be computed only by considering the corresponding distribution of strains. The concept of plastic deformation is introduced in Chap. 2. where it is applied to the analysis of members under axial loading. Problems involving the plastic deformation of circular shafts and of prismatic beams are also considered in optional sections of Chaps. 3, 4. and 6. While some of this material can be omitted at the choice of the instructor, its inclusion in the body of the text will help students realize the limitations of the assumption of a linear stress-strain relation and serve to caution them against the inappropriate use of the elastic torsion and flexure formulas. The determination of the deflection of beams is discussed in Chap. 9. The first part of the chapter is devoted to the integration method and to the method of superposition, with an optional section (Sec. 9.6) based on the use of singularity functions. (This section should be used only if Sec. 5.5 was covered earlier.) The second part of Chap. 9 is optional. It presents the moment-area method in two lessons. Chapter 10 is devoted to columns and contains material on the design of steel, aluminum, and wood columns. Chapter 11 covers energy methods, including Castigliano's theorem.

**STRESS - INTRODUCTION**

**INTRODUCTION**

The main objective of the study of the mechanics of materials is to provide the future engineer with the means of analyzing and designing various machines and load-bearing structures. Both the analysis and the design of a given structure involve the determination of stresses and deformations. This first chapter is devoted to the concept of stress. Section 1.2 is devoted to a short review of the basic methods of statics and to their application to the determination of the forces in the members

of a simple structure consisting of pin-connected members. Section 1.3 will introduce you to die concept of stress in a member of a structure, and you will be shown how that stress can be determined from the farce in the member. After a short discussion of engineering analysis and design , you will consider successively the normal stresses in a member under axial loading , the shearing stresses caused by the application of equal and opposite transverse forces , and the bearing stresses created by bolts and pins in the members they connect . These various concepts will be applied in Sec. 1.8 to the determination of the stresses in the members of the simple structure considered earlier in Sec. 1.2. The first part of the chapter ends with a description of die method you should use in the solution of an assigned problem and with a discussion of the numerical accuracy appropriate in engineering calculations.

In Sec. 1.11, where a two-force member under axial loading is considered again, it will be observed that the stresses on an oblique plane include both normal and shearing stresses, while in Sec. 1.12 you will note that six components are required to describe the state of stress at a point in a body under the most general loading conditions. Finally. Sec. 1.13 will be devoted to the determination from test specimens of the ultimate strengih of a given material and to the use of a factor

of safety in the computation of the allowable load for a structural component made of that material.

**STRAIN - INTRODUCTION**

In Chap. 1 - Concepts of Stress we analyzed die stresses created in various members and connections by the loads applied lo a structure or machine. We also learned to design simple members and connections so that they would not fail under specified loading conditions. Another important aspect of the analysis and design of structures relates to the deformations caused by the loads applied to a structure. Clearly, it is important to avoid deformations so large that they may prevent the structure from fulfilling the purpose for which it was intended. But the analysis ol deformations may also help us in the determination of stresses. Indeed, it is not always

possible lo determine die forces in die members of a structure by applying only the principles of statics. This is because statics is based on the assumption of undeformable, rigid structures. By considering engineering structures as deformable and analyzing the deformations in their various members, ii will be possible for us to compute forces thai are statically indeterminate, i.e., indeterminate within die framework of statics. Also, as we indicated in Sec. 1.5, the distribution of stresses in a given member is statically indeterminate, even when the force in that member is known. To determine die actual distribution of stresses within

a member, it is thus necessary to analyze die deformations that lake place in thai member. In this chapter, you will consider the deformations of a structural member such as a rod, bar, or plate under axial loading.

**TORSION - INTRODUCTION**

In the two preceding chapters you studied how to calculate the stresses and strains in structural members subjected to axial loads, that is, to forces directed along the axis of the member. In this chapter structural members and machine parts that are in torsion will be considered. More specifically, you will analyze the stresses and strains in members of circular cross section subjected to twisting couples, or torques, T and T'. These couples have a common magnitude 7, and opposite senses. They are vector quantities and can be represented either by curved arrows. or by couple vectors.

**PURE BENDING - INTRODUCTION**

In the preceding chapters you studied how to determine the stresses in prismatic members subjected to axial loads or to twisting couples. In this chapter and in the following two you will analyze the stresses and strains in prismatic members subjected to bending. Bending is a major concept used in the design of many machine and structural components, such as beams and girders. This chapter will be devoted to the analysis of prismatic members subjected to equal and opposite couples M and M' acting in the same longitudinal plane. Such members are said to be in pure bending. In most of the chapter, the members will be assumed to possess a plane of symmetry

and the couples M and M' to be acting in that plane

**DEFLECTION IN BEAMS - INTRODUCTION**

In the preceding chapter we learned to design beams lor strength. In this chapter we will be concerned with another aspect in the design of beams, namely, the determination of the deflection. Of particular interest is the determination of the maximum deflection of a beam under a given loading, since the design specifications of a beam will generally include a maximum allowable value for its deflection. Also of interest is that a knowledge of the deflections is required to analyze indeterminate beams. These are beams in which the number of reactions at the supports exceeds the number of equilibrium equations available to determine these unknowns.

**COLUMNS - INTRODUCTION**

In the preceding chapters, we had two primary concerns: (I) the strength of the structure, i.e., its ability to support a specified load without experiencing excessive stress: (2) the ability of the structure to support a specified load without undergoing unacceptable deformations. In this chapter, our concern will be with Ihe stability of the structure, i.e., with its ability to support a given load without experiencing a sudden change in its configuration. Our discussion will relate chiefly to columns, i.e., to the analysis and design of vertical prismatic members supporting axial loads.

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