Download - Staticsand MoM ISM FM1

Instructor's and Solutions Manual to accompany

Statics and Mechanics of Materials

Ferdinand P. Beer Late of Lehigh University

E. Russell Johnston, Jr. University of Connecticut

John T. DeWolf University of Connecticut

David F. Mazurek United States Coast Guard Academy

Prepared by Dean P. Updike

Lehigh University

PROPRIETARY AND CONFIDENTIAL This Manual is the proprietary property of The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and protected by copyright and other state and federal laws. By opening and using this Manual the user agrees to the following restrictions, and if the recipient does not agree to these restrictions, the Manual should be promptly returned unopened to McGraw-Hill: This Manual is being provided only to authorized professors and instructors for use in preparing for the classes using the affiliated textbook. No other use or distribution of this Manual is permitted. This Manual may not be sold and may not be distributed to or used by any student or other third party. No part of this Manual may be reproduced, displayed or distributed in any form or by any means, electronic or otherwise, without the prior written permission of the McGraw-Hill.

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TABLE OF CONTENTS

TO THE INSTRUCTOR .................................................................................................................v DESCRIPTION OF THE MATERIAL CONTAINED IN THE TEXT ...................................... vii TABLE I: LIST OF THE TOPICS COVERED IN STATICS.................................................... xix TABLE II: LIST OF THE TOPICS COVERED IN MECHANICS OF MATERIALS ..............xx TABLE III: CLASSIFICATION AND DESCRIPTION OF PROBLEMS FOR STATICS ...... xxi TABLE IV: CLASSIFICATION AND DESCRIPTION OF PROBLEMS FOR MECHANICS OF MATERIALS............................................................................... xxvii TABLE V: SAMPLE ASSIGNMENT SCHEDULE ........................................................................ PROBLEM SOLUTIONS ...............................................................................................................1

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TO THE INSTRUCTOR As indicated in its preface, this text is designed for a combined course in Statics and Mechanics of Materials that can be offered to engineering students in the sophomore or junior year. The text has been divided into units, each corresponding to a well-defined topic and consisting of one or several theory sections followed by sample problems and a large number of problems to be assigned. In order to accommodate courses of varying emphases, considerably more material has been included than can be covered effectively in a single three-credit-hour course. To assist the instructors in making up a schedule of assignments that best fits their classes, the various topics presented in the text have been listed in Table I for Statics and Table II for Mechanics of Materials. Both a minimum and maximum number of periods to be spent on each topic have been suggested. Topics have been divided into two categories: core topics and additional topics that can be selected to complement this core to form courses of various emphases. Since the approach used in this text differs in a number of respects from the approach used in other books, instructors will be well advised to read the preface to Statics and Mechanics of Materials in which the authors have outlined their general philosophy. In addition, instructors will find in the following pages a description, chapter by chapter, of the more significant features of this text. It is hoped that this material will help instructors in organizing their courses to best fit the needs of their students.

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DESCRIPTION OF THE MATERIAL CONTAINED IN STATICS AND MECHANICS OF MATERIALS, First Edition Chapter 1 Introduction The material in this chapter can be used as a first assignment or for later reference. The six fundamental principles needed for the study of statics are listed in Sec. 1.2. They are discussed at greater length in the following six chapters. Section 1.3 notes that the concepts needed for mechanics of deformable bodies, referred to as mechanics of materials, involve determination of the stresses and deformations. These concepts are developed in Chapters 9 through 16. Sec. 1.4 deals with the two systems of units used in the text. The SI metric units are discussed first. The base units are defined and the use of multiples and submultiples is explained. The various SI prefixes are pre-sented in Table 1.1, while the principal SI units used in statics and dynamics are listed in Table 1.2. In the second part of Sec. 1.4, the base U.S. customary units used in mechanics are defined, and in Sec. l.5, it is shown how numerical data stated in U.S. customary units can be converted into SI units, and vice versa. The SI equivalents of the principal U.S. customary units used in statics and dynamics are listed in Table 1.3. The instructor’s attention is called to the fact that the various rules relating to the use of SI units have been observed throughout the text. For instance, multiples and submultiples (such as kN and mm) are used whenever possible to avoid writing more than four digits to the left of the decimal point or zeros to the right of the decimal point. When 5-digit or larger numbers involving SI units are used, spaces rather than commas are utilized to separate digits into groups of three (for example, 20 000 km). Also, prefixes are never used in the denominator of derived units; for example, the constant of a spring which stretches 20 mm under a load of 100 N is expressed as 5 kN/m, not as 5 N/mm. In order to achieve as much uniformity as possible between results expressed respectively in SI and U.S. customary units, a center point, rather than a hyphen, has been used to combine the symbols representing U.S. customary units (for example, 10 lb · ft); furthermore, the unit of time has been represented by the symbol s, rather than sec, whether SI or U.S. customary units are involved (for example, 5 s, 50 ft/s, 15 m/s). However, the traditional use of commas to separate digits into groups of three has been maintained for 5-digit and larger numbers involving U.S. customary units. Section 1.6 describes how students should approach the solution of a mechanics problem, and Sec. 1.7 discusses the numerical accuracy to be expected in such a solution. Chapter 2 Statics of Particles This is the first of two chapters dealing with the fundamental properties of force systems. A simple, intuitive classification of forces has been used: forces acting on a particle (Chap. 2) and forces acting on a rigid body (Chap. 3).

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Chapter 2 begins with the parallelogram law of addition of forces and with the introduction of the fundamental properties of vectors. In the text, forces and other vector quantities are always shown in bold-face type. Thus, a force F (boldface), which is a vector quantity, is clearly distinguished from the magnitude F (italic) of the force, which is a scalar quantity. On the blackboard and in handwritten work, where bold-face lettering is not practical, vector quantities can be indicated by underlining. Both the magnitude and the direction of a vector quantity must be given to completely define that quantity. Thus, a force F of magnitude F = 280 lb, directed upward to the right at an angle of 25with the horizontal, is indicated as F = 280 lb 25when printed or as F = 280 lb 25when handwritten. Unit vectors i and j are introduced in Sec. 2.7, where the rectangular components of forces are considered. In the early sections of Chap. 2 the following basic topics are presented: the equilibrium of a particle, Newton’s first law, and the concept of the free-body diagram. These first sections provide a review of the methods of plane trigonometry and familiarize the students with the proper use of a calculator. A general procedure for the solution of problems involving concurrent forces is given: when a problem involves only three forces, the use of a force triangle and a trigonometric solution is preferred; when a problem involves more than three forces, the forces should be resolved into rectangular components and the equations ΣFx = 0, ΣFy = 0 should be used. The second part of Chap. 2 deals with forces in space and with the equilibrium of particles in space. Unit vectors are used and forces are expressed in the form F = Fxi + Fyj + Fzk = Fλ, where i, j, and k are the unit vectors directed respectively along the x, y, and z axes, and λ is the unit vector directed along the line of action of F. Note that since this chapter deals only with particles or bodies that can be considered as particles, problems involving compression members have been postponed with only a few exceptions until Chap. 4, where students will learn to handle rigid-body problems in a uniform fashion and will not be tempted to erroneously assume that forces are concurrent or that reactions are directed along members. It should be observed that when SI units are used a body is generally specified by its mass expressed in kilograms. The weight of the body, however, should be expressed in newtons. Therefore, in many equilibrium problems involving SI units, an additional calculation is required before a free-body diagram can be drawn (compare the example in Sec. 2.11 and Sample Probs. 2.5 and 2.9). This apparent disadvantage of the SI system of units, when compared to the U.S. customary units, will be offset in dynamics, where the mass of a body expressed in kilograms can be entered directly into the equation F = ma, whereas with U.S. customary units the mass of the body must first be determined in lb · s2/ft (or slugs) from its weight in pounds. Chapter 3

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Rigid Bodies: Equivalent Systems of Forces The principle of transmissibility is presented as the basic assumption of the statics of rigid bodies. However, it is pointed out that this principle can be derived from Newton’s three laws of motion. The vector product is then introduced and used to define the moment of a force about a point. The convenience of using the determinant form (Eqs. 3.19 and 3.21) to express the moment of a force about a point should be noted. The scalar product and the mixed triple product are introduced and used to define the moment of a force about an axis. Again, the convenience of using the determinant form (Eqs. 3.43 and 3.46) should be noted. The amount of time that should be assigned to this part of the chapter will depend on the extent to which vector algebra has been considered and used in prerequisite mathematics and physics courses. It is felt that, even with no previous knowledge of vector algebra, a maximum of four periods is adequate (see Table I). In Secs. 3.12 through 3.15 couples are introduced, and it is proved that couples are equivalent if they have the same moment. While this fundamental property of couples is often taken for granted, the authors believe that its rigorous and logical proof is necessary if rigor and logic are to be demanded of the students in the solution of their mechanics problems. In Secs. 3.16 through 3.20, the concept of equivalent systems of forces is carefully presented. This concept is made more intuitive through the extensive use of free-body-diagram equations (see Figs. 3.39 through 3.46). Note that the moment of a force is either not shown or is represented by a green vector (Figs. 3.12 and 3.27). A red vector with the symbol is used only to represent a couple, that is, an actual system consisting of two forces (Figs. 3.38 through 3.46). Since one of the purposes of Chap. 3 is to familiarize students with the fundamental operations of vector algebra, students should be encouraged to solve all problems in this chapter (two-dimensional as well as three-dimensional) using the methods of vector algebra. However, many students may be expected to develop solutions of their own, particularly in the case of two-dimensional problems, based on the direct computation of the moment of a force about a given point as the product of the magnitude of the force and the perpendicular distance to the point considered. Such alternative solutions may occasionally be indicated by the instructor (as in Sample Prob. 3.9). It should be pointed out that in later chapters the use of vector products will generally be reserved for the solution of three-dimensional problems. Chapter 4 Equilibrium of Rigid Bodies In the first part of this chapter, problems involving the equilibrium of rigid bodies in two dimensions are considered and solved using ordinary algebra, while problems involving three dimensions and requiring the full use of vector algebra are discussed in the second part of the chapter. Particular emphasis is placed on the correct drawing and use of free-body diagrams and on the types of reactions produced by various supports and connections (see Figs. 4.1 and 4.10). Note that a distinction is made between hinges used

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in pairs and hinges used alone; in the first case the reactions consist only of force components, while in the second case the reactions may, if necessary, include couples. For a rigid body in two dimensions, it is shown (Sec. 4.4) that no more than three independent equations can be written for a given free body, so that a problem involving the equilibrium of a single rigid body can be solved for no more than three unknowns. It is also shown that it is possible to choose equilibrium equations containing only one unknown to avoid the necessity of solving simultaneous equations. Sec. 4.5 introduces the concepts of statical indeterminacy and partial constraints. Sections 4.6 and 4.7 are devoted to the equilibrium of two- and three-force bodies; it is shown how these concepts can be used to simplify the solution of certain problems. This topic is presented only after the general case of equilibrium of a rigid body to lessen the possibility of students misusing this particular method of solution. The equilibrium of a rigid body in three dimensions is considered with full emphasis placed on the free-body diagram. While the tool of vector algebra is freely used to simplify the computations involved, vector algebra does not, and indeed cannot, replace the free-body diagram as the focal point of an equilibrium problem. Therefore, the solution of every sample problem in this section begins with a reference to the drawing of a free-body diagram. Emphasis is also placed on the fact that the number of unknowns and the number of equations must be equal if a structure is to be statically determinate and completely constrained. Sections. 4.10 through 4.13 are devoted to the presentation of the laws of dry friction and to their application to various problems. The different cases that can be encoun-tered are illustrated by diagrams in Figs. 4.12, 4.13, and 4.14. Particular emphasis is placed on the fact that no relation exists between the friction force and the normal force except when motion is impending or when motion is actually taking place. Following the general procedure outlined in Chap. 2, problems involving only three forces are solved by a force triangle, while problems involving more than three forces are solved by summing x and y components. In the first case the reaction of the surface of contact should be represented by the resultant R of the friction force and normal force, while in the second case it should be resolved into its components F and N. Chapter 5 Distributed Forces: Centroids and Centers of Gravity Chapter 5 starts by defining the center of gravity of a body as the point of application of the resultant of the weights of the various particles forming the body. This definition is then used to establish the concept of the centroid of an area or line. Section 5.4 introduces the concept of the first moment of an area or line, a concept fundamental to the analysis of shearing stresses in beams in the later study of mechanics of materials. All problems assigned for the first period involve only areas and lines made of simple geometric shapes; thus, they can be solved without using calculus.

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Section 5.6 explains the use of differential elements in the determination of centroids by integration. The theorems of Pappus-Guldinus are given in Sec. 5.7. Section 5.8 is optional; it shows how the resultant of a distributed load can be determined by evaluating an area and by locating its centroid. Sections 5.9 and 5.10 deal with centers of gravity and centroids of volumes of composite shapes, and is limited to problems that can be solved without using calculus. It should be noted that when SI units are used, a given material is generally characterized by its density (mass per unit volume, expressed in kg/m3), rather than by its specific weight (weight per unit volume, expressed in N/m3). The specific weight of the material can then be obtained by multiplying its density by g = 9.81 m/s2

(see the footnote in Sec. 5.3 of the text). Chapter 6 Analysis of Structures In this chapter students learn to determine the internal forces exerted on the members of pin-connected structures. The chapter starts with the statement of Newton’s third law (action and reaction) and is divided into two parts: (a) trusses, that is, structures consisting of two-force members only, (b) frames and machines, that is, structures involving multiforce members. After trusses and simple trusses have been defined in Secs. 6.2 and 6.3, the method of joints and the method of sections are explained in detail in Sec. 6.4 and Sec. 6.6, respectively. Since a discussion of Maxwell’s diagram is not included in this text, the use of Bow’s notation has been avoided, and a uniform notation has been used in presenting the method of joints and the method of sections. In the method of joints, a free-body diagram should be drawn for each pin. Since all forces are of known direction, their magnitudes, rather than their components, should be used as unknowns. Following the general procedure outlined in Chap. 2, joints involving only three forces are solved using a force triangle, while joints involving more than three forces are solved by summing x and y components. Section 6.5 shows how the analysis of certain trusses can be expedited by recognizing joints under special loading conditions. It is pointed out in Sec. 6.4 that forces in a simple truss can be determined by analyzing the truss joint by joint and that joints can always be found that involve only two unknown forces. The method of sections (Sec. 6.6) should be used (a) if only the forces in a few members are desired, or (b) if the truss is not a simple truss and if the solution of simultaneous equations is to be avoided (for example, Fink truss). Students should be urged to draw a separate free-body diagram for each section used. The free body obtained should be emphasized by shading and the intersected members should be removed and replaced by the forces they exert on the free body. It is shown that, through a judicious choice of equilibrium equations, the force in any given member can be obtained in most cases by solving a single equation. Section 6.7 is optional; it deals with the trusses obtained by combining several simple trusses and discusses the statical determinacy of such structures as well as the completeness of their constraints.

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Structures involving multiforce members are separated into frames and machines. Frames are designed to support loads, while machines are designed to transmit and modify forces. It is shown that while some frames remain rigid after they have been detached from their supports, others will collapse (Sec. 6.10). In the latter case, the equations obtained by considering the entire frame as a free body provide necessary but not sufficient conditions for the equilibrium of the frame. It is then necessary to dismember the frame and to consider the equilibrium of its component parts in order to determine the reactions at the external supports. The same procedure is necessary with most machines in order to determine the output force Q from the input force P or inversely (Sec. 6.11). Students should be urged to resolve a force of unknown magnitude and direction into two components but to represent a force of known direction by a single unknown, namely its magnitude. While this rule may sometimes result in slightly more complicated arithmetic, it has the advantage of matching the numbers of equations and unknowns and thus makes it possible for students to know at any time during the computations what is known and what is yet to be determined. Chapter 7 Distributed Forces: Moments of Inertia The purpose of Sec. 7.2 is to give motivation to the study of moments of inertia of areas. An example is considered that deals with the pure bending of a beam. It is shown that the solution of the problem reduces to the computation of the moment of inertia of an area. The other sections in the first assignment are devoted to the definition and the computation of rectangular moments of inertia, polar moments of inertia, and the corresponding radii of gyration. It is shown how the same differential element can be used to determine the moment of inertia of an area about each of the two coordinate axes. Sections 7.6 and 7.7 introduce the parallel-axis theorem and its application to the determination of moments of inertia of composite areas. Particular emphasis is placed on the proper use of the parallel-axis theorem (see Sample Prob. 7.5). Chapter 8 Concept of Stress The concept of a normal stress is introduced in Sec. 8.2 for an axially loaded element. Sec. 8.3 emphasizes the fact that stresses are inherently statically indeterminate and that, at this point, normal stresses under an axial loading can only be assumed to be uniformly distributed. Moreover, such an assumption requires that the axial load be centric. Section 8.4 discusses shearing stresses — with applications to pins and bolts in single and double shear, and Sec. 8.5 discusses bearing stresses. Sec. 8.6 is devoted to the application of these concepts to the analysis of a simple structure. Statics, emphasizing the use of a free-body diagram, is first used to find the forces in the elements. The stresses are then determined for the different elements. Design based on evaluation of stresses is introduced in Sec. 8.7. Problems included in the first lesson also serve as a

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review of the methods of analysis of trusses, frames, and mechanisms learned in the previous chapters. Section 8.8 discusses the determination of normal and shearing stresses on oblique planes under an axial loading, while Sec. 8.9 introduces the components of stress under general loading conditions. This section emphasizes the fact that the components of the shearing stresses exerted on perpendicular planes, such as τxy and τyx, must be equal. It also introduces the students to the concept of transformation of stress. However, the study of the computational techniques associated with the transformation of stress at a point is delayed until Chap. 14, after students have discovered for themselves the need for such techniques.

Section 8.10 is devoted to design considerations. It introduces the concepts of ultimate load, ultimate stress, and factor of safety. It also discusses the reasons for the use of factors of safety in engineering practice. The section ends with an optional presentation of an alternative method of design, Load and Resistance Factor Design.

Chapter 9 Stress and Strain: Axial Loading This chapter is devoted to the analysis and design of members under a centric axial loading. Sections 9.1 and 9.2 introduce the concept of normal strain, while Sec. 9.3 describes the general properties of the stress-strain diagrams of ductile and brittle materials and defines the yield strength, ultimate strength, and breaking strength of a material. Section 9.4 introduces Hooke's law, the modulus of elasticity, and the proportional limit of a material. It defines as isotropic those materials whose mechanical properties are independent of the direction considered and as anisotropic those whose mechanical properties depend upon that direction. Among the latter are fiber-reinforced composite materials, which are described in this section.

Section 9.5 discusses the elastic and the plastic behavior of a material and defines its elastic limit, while Sec. 9.6 is devoted to fatigue and the behavior of materials under repeated loadings. These two sections are optional. The first lesson of Chap. 9 ends with Sec. 9.7, which shows how Hooke's law can be used to determine the deformation of a rod of uniform or variable cross section under one or several loads, and introduces the concept of relative displacement. Sec. 9.8 discusses statically indeterminate problems involving members under an axial load. The authors believe it is important to introduce the students at an early stage to the concept of statical indeterminacy and to show them how the analysis of deformations can be used in the solution of problems that cannot be solved by the methods of statics alone. It will also help them realize that stresses, being statically indeterminate, can be computed only by considering the corresponding distribution of strains. Sec. 9.9 discusses the thermal expansion of rods and shows how to determine stresses in statically indeterminate members subjected to temperature changes.

Section 9.10 introduces the concept of lateral strain for an isotropic material and defines

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Poisson's ratio. Sec. 9.11 discusses the multiaxial loading of a structural element and derives the generalized Hooke's law for such a loading. Since this derivation is based on the application of the principle of superposition, this principle is also introduced in Sec. 9.11, and the conditions under which it can be used are clearly stated.

Section 9.12 introduces the concept of shearing strain. It should be noted that the authors define the shearing strain as the change in the angle formed by the faces of the element of material considered, and not as the angle through which one of these faces rotates. Hooke's law for shearing stress and strain and the modulus of rigidity are also introduced in this section, as well as the generalized Hooke's law for a homogeneous, isotropic material under the most general stress conditions. The optional Sec. 9.13 points out that strains, just as stresses, depend upon the orientation of the planes considered. It also establishes the fact that the constants E, v, and G are not independent from each other and derives Eq. (9.35), which expresses the relation among these three constants.

Section 9.14 discusses the distribution of the normal stresses under a centric axial loading and shows that this distribution depends upon the manner in which the loads are applied. However, except in the immediate vicinity of the points of application of the loads, the distribution of stresses can be assumed uniform. This result verifies Saint-Venant's principle. Section 9.15 discusses stress concentrations near circular holes and fillets in flat bars under axial loading.

Chapter 10 Torsion Section 10.1 introduces this type of loading, while Sec. 10.2 establishes the relation that must be satisfied, on the basis of statics, by the shearing stresses in a given section of a shaft subjected to a torque. This condition, however, does not suffice to determine the stresses, and one must analyze the deformations that occur in the shaft. This is done in Sec. 10.3, where it is proved that the distribution of shearing strains in a circular shaft is linear. It should be noted that the discussion presented in Sec. 10.3 is based solely on the assumption of rigid end plates, rather than on arbitrary and gratuitous assumptions regarding the deformations of a shaft. The results obtained in this and the following sections clearly depend upon the validity of this assumption, but can be extended to other loading conditions through the application of Saint-Venant's principle.

Section 10.4 is devoted to the analysis of the shearing stresses in the elastic range and presents the derivation of the elastic torsion formulas for circular shafts. The section ends with remarks on the transformation of stresses in torsion and the comparison between the failures of ductile and brittle materials in torsion.

The formula for the angle of twist of a shaft in the elastic range is derived in Sec. 10.5. This section also contains various applications involving the twisting of single and gear-connected shafts. Section 10.6 deals with the solution of problems involving statically indeterminate shafts.

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Chapter 11 Pure Bending Section 11.1 defines this type of loading and shows how the results obtained in the following sections can be applied to the analysis of other types of loadings as well, namely, eccentric axial loadings and transverse loadings. Sec. 11.2 establishes the relation that must be satisfied, on the basis of statics, by the normal stresses in a given section of a member subjected to pure bending. This condition, however, does not suffice to determine the stresses, and one must analyze the deformations that occur in the member. This is done in Sec. 11.3, where it is proved that the distribution of normal stresses in a symmetric member in pure bending is linear. It should be noted that no assumption is made in this discussion regarding the deformations of the member, except that the couples should be applied in such a way that the ends of the member remain plane.

Section 11.4 is devoted to the analysis of the normal stresses in the elastic range and presents the derivation of the elastic flexure formulas. It also defines the elastic section modulus and ends with the derivation of the formula for the curvature of an elastic beam.

Section 11.5 discusses the determination of stresses in members made of several materials and defines the transformed section of such members. It also shows how the transformed section can be used to determine the radius of curvature of the member. The section ends with a discussion of the stresses in reinforced-concrete beams.

Section 11.6 shows how the stresses due to a two-dimensional eccentric axial loading can be obtained by replacing the given eccentric load by a centric load and a couple, and superposing the corresponding stresses. Attention is called to the fact that the neutral axis does not pass through the centroid of the section.

Section 11.7 deals with the unsymmetric bending of elastic members. It is first shown that the neutral axis of a cross section will coincide with the axis of the bending couple if, and only if, the axis of the couple is directed along one of the principal centroidal axes of the cross section. It is then shown that stresses due to unsymmetric bending can always be determined by resolving the given bending couple into two component couples directed along the principal axes of the section and superposing the corresponding stresses.

This method of analysis is extended in Sec. 11.8 to the determination of the stresses due to an eccentric axial loading in three-dimensional space. The eccentric load is replaced by an equivalent system consisting of a centric load and two bending couples, and the corresponding stresses are superposed.

Chapter 12 Analysis and Design of Beams for Bending In Section 12.1 beams are defined as slender prismatic members subjected to transverse loads and are classified according to the way in which they are supported. It is shown that the internal forces in any given cross section are equivalent to a shear force V and a bending couple M. The bending couple M creates normal stresses in the section, while the shear force V creates shearing stresses. The former is

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determined in this chapter, using the flexure formula (12.2), while the latter will be discussed in Chap.13.

Since the dominant criterion in the design of beams for strength is usually the bending stresses in the beam, the determination of the maximum value of the bending moment in the beam is the most important factor to be considered. To facilitate the determination of the bending moment in any given section of the beam, the concept of shear and bending-moment diagrams is introduced in Sec. 12.2, using free-body diagrams of various portions of the beam.

An alternative method for the determination of shear and bending-moment diagrams, based on relations among load, shear, and bending moment, is presented in Sec. 12.3. To maintain the interest of the students, most of the problems to be assigned are focused on the engineering applications of these methods and call for the determination, not only of the shear and bending moment, but also of the normal stresses in the beam.

Section 12.4 is devoted to the design of prismatic beams based on the allowable normal stress for the material used. Sample Problems and problems to be assigned include wooden beams of rectangular cross section, as well as rolled-steel W and S beams.

Chapter 13 Shearing Stresses in Beams and Thin-Walled Members It is shown in Sec. 13.1 that a transverse load creates shearing stresses as well as normal stresses in a beam. Considering first the horizontal face of a beam element, it is shown in Sec. 13.2 that the horizontal shear per unit length q, or shear flow, is equal to VQ/I. This result is applied in Example 13.1 to the determination of the shear force in the nails connecting three planks forming a wooden beam, as well as in Probs. 13.1 through 13.4.

In Sec. 13.3 the average shearing stress τave exerted on the horizontal face of the beam element is obtained by dividing the shear flow q by the width t of the beam:

ave =VQ

It (13.6)

Note that since the shearing stresses τxy and τyx exerted at a given point are equal, the expression obtained also represents the average shearing stress exerted at a given height on a vertical section of the beam. This formula is used to determine shearing stresses in a beam made of glued planks in Sample Prob. 13.1 and to design a timber beam in Sample Prob. 13.2.

In Examples 13.2 and 13.3 the designs obtained on the basis of normal stresses, respectively, for a timber beam in Sample Prob. 12.7 and for a rolled-steel beam in Sample Prob. 12.8 are checked and found to be acceptable from the point of view of shearing stresses.

In Sec. 13.5 the expression q = VQ/I obtained in Sec. 13.2 for the shear flow on the

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horizontal face of a beam element is shown to remain valid for the curved surface of a beam element of arbitrary shape. It is then applied in Example 13.4 and in Probs. 13.25 through 13. 27 for the determination of the shearing forces and shearing stresses in nailed and glued vertical surfaces.

Section 13.6 deals with the determination of shearing stresses in thin-walled members and shows that Eq. (13.6) can be applied to the determination of the average shearing stress in a section of arbitrary orientation.

Chapter 14 Transformations of Stress and Strain After a short introduction (Sec. 14.1), formulas for the transformation of plane stress under a rotation of axes are derived in Sec. 14.2, while the principal planes of stress, principal stresses, and maximum shearing stress are determined in Sec. 14.3.

Section 14.4 is devoted to the use of Mohr's circle. It should be noted that the convention used in the text provides for a rotation on Mohr's circle in the same sense as the corresponding rotation of the element. Attention is called to Figure 14.18 and the accompanying text.

Section 14.5 deals with stresses in thin-walled pressure vessels; it is limited to the analysis of cylindrical and spherical pressure vessels.

Chapter 15 Deflection of Beams The relation derived in Chap. 11 between the curvature of a beam and the bending moment is recalled in Sec. 15.2 and used to predict the variation of the curvature along the beam. In Sec. 15.3, the equation of the elastic curve for a beam is obtained through two successive integrations, after the bending moment has been expressed as a function of the coordinate x. Examples 15.1 and 15.2 show how the boundary conditions can be used to determine the two constants of integration in the cases of a cantilever beam and of a simply supported beam. Example 15.3 indicates how to proceed when the bending moment must be represented by two different functions of x.

In the case of a beam supporting a distributed load, Sec. 15.4 shows how the equation of the elastic curve can be obtained directly from the function representing the load distribution through the use of four successive integrations.

Section 15.5 is devoted to the analysis of statically indeterminate beams and to the determination of the reactions at their supports.

Section 15.6 discusses the method of superposition for the determination of beam deflections and slopes. It shows how the expressions given in Appendix C for various simple loadings can be used to obtain the deflection and slope of a beam supporting a more complex loading. In Sec. 15.7, the method of superposition is applied to the determination of the reactions at the supports of statically indeterminate beams.

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Chapter 16 Columns Section 16.2 introduces the concept of stability of a structure. The example considered in this section consists of a block supported by two spring-connected rigid rods. It is shown that the position of equilibrium in which both rods are aligned is stable if this position is the only possible position of equilibrium of the system. The same criterion is applied to an elastic pin-ended column in Sec. 16.3 in order to derive Euler's formula. Section 16.4 shows how Euler's formula for pin-ended columns can be used to determine the critical load of columns with other end conditions.

Section 16.6 is optional. It discusses the design of columns under a centric load and presents the empirical formulas developed by various engineering associations for the design of steel columns, aluminum columns, and wood columns. As noted at the end of this section, the design formulas presented in this section are intended to provide examples of different design approaches. These formulas do not provide all the requirements that are needed for many designs.

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TABLE I: LIST OF THE TOPICS COVERED FOR STATICS

Suggested Number of Periods Additional Sections Topics Basic Course Topics 1. INTRODUCTION 1.1–7 This material may be used for the first assignment or for later reference 2. STATICS OF PARTICLES 2.1–6 Addition and Resolution of Forces 0.5–1 2.7–8 Rectangular Components 0.5–1 2.9–11 Equilibrium of a Particle 1 2.12–14 Forces in Space 1 2.15 Equilibrium in Space 1 3. RIGID BODIES: EQUIVALENT SYSTEMS OF FORCES 3.1–8 Vector Product; Moment of a Force about a Point 1–2 3.9–11 Scalar Product; Moment of a Force about an Axis 1–2 3.12–16 Couples 1 3.17–20 Equivalent Systems of Forces 1–1.5 4. EQUILIBRIUM OF RIGID BODIES 4.1–5 Equilibrium in Two Dimensions 1–1.5 4.6–7 Two- and Three-Force Bodies 1 4.8–9 Equilibrium in Three Dimensions 1–1.5 4.10–13 Laws of Friction and Applications 1–1.5 5. CENTROIDS AND CENTERS OF GRAVITY 5.1–5 Centroids and First Moments of Areas and Lines 1–1.5 5.6–7 Centroids by Integration 1–1.5 * 5.8 Beams 1 5.9–10 Centroids of Volumes 1–1.5 6. ANALYSIS OF STRUCTURES 6.1–4 Trusses by Method of Joints 1–1.5 * 6.5 Joints under Special Loading Conditions 0.25–0.5 6.6 Trusses by Method of Sections 1–1.5 * 6.7 Combined Trusses 0.25–0.5 6.8–10 Frames 1–2 6.11 Machines 1–1.5 7. MOMENT OF INERTIA OF AREAS 7.1–5 Moments of Inertia of Areas 1 7.6–7 Composite Areas 1–2 Total Number of Periods 17–26 5.5–7

xix

TABLE II: LIST OF TOPICS COVERED IN MECHANICS OF MATERIALS

Suggested Number of Periods

Core Additional

Sections Topics Topics Topics 8. INTRODUCTION – CONCEPT OF STRESS 8.1–7 Stress Under Axial Loading 1–1.5 8.8-10 Components of Stress; Factor of Safety 1 9. STRESS AND STRAIN – AXIAL LOADING 9.1–7 Stress-Strain Diagrams; Deformations Under 1–1.5

Axial Loading 9.8–9 Statically Indeterminate Problems 1 9.10–11 Poisson’s Ratio; Generalized Hooke’s Law 1 9.12–13 Shearing Strain 0.5 9.14-15 Stress Concentrations 0.5 10. TORSION 10.1–4 Stresses in Elastic Range 1 10.5–6 Angle of Twist; Statically Indeterminate Shafts 1–1.5 11. PURE BENDING 11.1–4 Stresses in Elastic Range 1–1.5 11.5 Members Made of Several Materials 1–1.5 11.6 Eccentric Axial Loading 1 11.7–8 Unsymmetric Bending; General Eccentric 1 Axial Loading 12. ANALYSIS AND DESIGN OF BEAMS FOR BENDING 12.1–2 Shear and Bending-Moment Diagrams 1–1.5 12.3 Using Relations among w, V, and M 1–1.5 12.4 Design of Prismatic Beams in Bending 1–1.5 13. SHEARING STRESSES IN BEAMS AND THIN-WALLED MEMBERS 13.1–4 Shearing Stresses in Beams 1–1.5 13.5–6 Shearing Stresses in Thin-Walled Members 1–1.5 14. TRANSFORMATION OF STRESS AND STRAIN 14.1–3 Transformation of Plane Stress 1–1.5 14.4 Mohr’s Circle for Plane Stress 1–1.5 14.5 Thin-Walled Pressure Vessels 0.5–1 15. DEFLECTION OF BEAMS 15.1–5 Equation of Elastic Curve 1–1.5 15.6–7 Method of Superposition 1–1.5 16. COLUMNS 16.1–4 Euler’s Column Formula 1–1.5 16.5 Design of Columns under a Centric Load 1–1.5

Total Number of Periods 21 – 28.5 2.5-3.5

xx

TABLE III: CLASSIFICATION AND DESCRIPTION OF PROBLEMS FOR STATICS

Problem Number* SI Units U.S. Units Problem Description

* Problems that do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxi

CHAPTER 2: STATICS OF PARTICLES

FORCES IN A PLANE

Resultant of several concurrent forces 2.1, 3 2.2, 4 graphical method 2.6, 7 2.5, 8 law of sines 2.9 2.11, 13 2.10, 12 laws of cosines and sines 2.14 2.15 special problems

2.16, 19 2.17, 18 Rectangular components of force 2.21, 23 2.20, 22 2.24, 27 2.25, 26 Resultant by ΣFx = 0, ΣFy = 0 2.28, 31 2.29, 30 Select force so that resultant has a given direction

Equilibrium, Free-Body Diagram 2.32, 34 2.33, 35 equilibrium of 3 forces 2.36 2.39 2.37, 38 equilibrium of 4 forces 2.40 2.41, 42 2.43, 44 find parameter to satisfy specified conditions 2.45, 47 2.46, 48 2.49, 51 2.50, 53 special problems 2.52, 54 2.55

FORCES IN SPACE

Rectangular components of a force in space: 2.56, 57 2.58, 59 given F, , and , find components and direction angles 2.60, 61 2.64, 67 2.62, 63 relations between components and direction angles 2.65, 66 2.68, 69 2.70, 71 direction of force defined by two points on its line of action 2.72, 73 resultant of two forces 2.74 2.76, 77 2.75 special problems 2.78, 79

Equilibrium of a particle in space 2.80,81 2.83, 84 load applied to three cables, introductory problems 2.82 2.85 2.88, 89 2.86, 87 load applied to three sables, more involved problems 2.92, 93 2.90, 91 cable problems involving full simultaneous problems 2.94, 95 2.96, 97 cable passing through ring (five forces acting on particle) 2.98, 102 2.99, 100 special problems 2.103 2.101

2.104, 107 2.105, 106 Review problems 2.110, 111 2.108, 109 2.112, 113 2.114, 115

TABLE III: CLASSIFICATION AND DESCRIPTION OF PROBLEMS FOR STATICS (CONTINUED) Problem Number* SI Units U.S. Units Problem Description

* Problems which do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxii

CHAPTER 3: RIGID BODIES; EQUIVALENT SYSTEMS OF FORCES

Moment of a force about a point: two dimensions 3.3, 4 3.1, 2 introductory problems 3.5, 6 3.9, 10 3.7, 8 direction of a force defined by two points on its line of action 3.11, 12 Moment of a force about a point: Three dimensions 3.13, 14 3.15, 17 compute M = r F 3.16, 19 3.18, 20 3.22, 24 3.21, 23 using M to find the perpendicular distance from a point to a line 3.25 3.26 Scalar Product 3.27, 28 3.29, 30 Find angle between two lines 3.31, 32 3.33 3.34 Mixed triple product 3.35, 36 3.37, 38 Moment of a force about the coordinate axes 3.39, 40 3.41, 42 3.45, 46 3.43, 44 Moment of a force about an oblique axis 3.47, 48 3.49, 51 3.50,53 Couples in two dimensions 3.52 3.54 3.55, 57 3.56, 59 Couples in three dimensions 3.58 3.60 3.62, 63 3.61,64 Replace force or force-couple system in two dimensions 3.65 3.66 3.67, 70 3.68, 69 Replacing force or force-couple system in three dimensions 3.72 3.71 3.75, 77 3.73, 74 Find resultant of parallel forces in two dimensions 3.78 3.76 3.80, 83 3.79, 81 Find the resultant and its line of action in two dimensions 3.84 3.82 3.86, 87 3.85, 88 Reduce three-dimensional system of forces to a single force-couple system 3.89, 90 3.91, 94 3.92, 93 Finding the resultant of parallel forces in three dimensions 3.95, 96

3.99, 100 3.97, 98 Review problems 3.101, 103 3.102, 3.105 3.104, 108 3.106, 107


* Problems which do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxiii

CHAPTER 4: EQUILIBRIUM OF RIGID BODIES

EQUILIBRIUM IN TWO DIMENSIONS 4.2, 4 4.1, 3 Parallel forces 4.5, 6 4.7, 8 Parallel forces, find range of values of loads to satisfy multiple criteria 4.12, 13 4.9, 10 Rigid bodies with one reaction of unknown direction and one of known direction 4.16 4.11, 14 4.15 4.17, 18 4.19, 20 Rigid bodies with three reactions of known direction 4.21, 24 4.22, 23 Rigid bodies with a couple included in the reactions 4.26 4.25 Partial constraints, statical indeterminacy

Three-force bodies 4.27, 28 4.29, 34 simple geometry 4.30, 31 4.35, 38 4.32, 33 4.39 4.36, 37 4.40, 44 4.41, 42 more involved geometry 4.45, 48 4.43, 46 4.47, 49 4.50

EQUILIBRIUM IN THREE DIMENSIONS

4.51, 52 4.53, 54 Rigid bodies with two hinges along a coordinate axis and 4.55, 56 an additional reaction parallel to another coordinate axis 4.59, 60 4.57, 58 Plate supported by three vertical wires 4.64 4.61, 62 Derrick and boom problems involving unknown tension in two cables 4.63 4.66, 69 4.65, 67 Rigid bodies with two hinges along a coordinate axis and an additional 4.70 4.68 reaction not parallel to a coordinate axis 4.72 4.71 Problems involving couples as part of the reaction at a hinge 4.74 4.73 Advanced problems

FRICTION 4.75, 76 4.77, 78 For given loading, determine whether block is in equilibrium and find friction force 4.81, 4.83 4.79, 80 Find minimum force required to start, maintain, or prevent motion 4.84 4.82 4.85, 86 4.89, 90 Sliding or tipping of a rigid body 4.87, 88 4.93, 94 4.91, 92 Problems involving rods 4.95, 96 4.97 4.98 Special problems

4.99, 100 4.101, 103 Review problems 4.102, 104 4.105, 107 4.106, 109 4.108, 110


* Problems which do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxiv

CHAPTER 5: DISTRIBUTED FORCES: CENTROIDS AND CENTERS OF GRAVITY

Centroid of an area formed by combining 5.1, 4 5.2, 3 rectangles and triangles 5.6, 7 5.5, 8 rectangles and portions of circular areas 5.9, 12 5.10, 11 rectangles, triangles, circular and parabolic areas 5.13, 16 5.14, 15 First moment of an area 5.17, 19 5.18, 20 Center of gravity of a wire figure 5.21, 22 5.23, 24 Equilibrium of wire figures

Use integration to find centroid of 5.25, 26 5. 27 simple areas 5.28 5.31 5.29, 30 areas shown in Fig. 5.8A 5.32 5.33, 34 parabolic area 5.35, 36 areas defined by hyperbola Find areas or volumes by Pappus - Guldinus 5.39, 38 5.37, 40 rotate simple geometric figures 5.41, 42 5.43, 44 rotate arc or area of circle 5.47, 48 5.45, 46 rotate trapezoid or two areas of circles Distributed load on beams 5.49, 50 resultant of loading 5.53, 56 5.51, 52 reactions at supports 5.54, 55 Centroids and centers of gravity of three-dimensional bodies 5.58, 60, 5.57, 59 composite bodies formed from two common shapes 5.63, 64 5.61, 62 composite bodies formed from four or more elements 5.65, 68 5.66, 67 sheet-metal forms 5.69 5.70 wire figures 5.72 5.71 composite bodies made of two different materials 5.73, 75 5.74, 76 Review problems 5.78, 79 5.77, 81 5.80, 84 5.82, 83


* Problems which do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxv

CHAPTER 6: ANALYSIS OF STRUCTURES

TRUSSES Method of joints 6.2, 4 6.1, 3 6.6, 7 6.5, 9 6.8, 10 6.11, 12 6.14, 17 6.13,15 6.18 6.16 6.19 6.20 designate simple trusses 6.21, 22 6.23, 24 find zero-force members Method of sections 6.27, 28 6.25, 26 6.31, 32 6.29, 30 6.33, 34 6.35, 36 6.37, 38 6.39, 40 6.41, 42 K-type trusses 6.43 6.44, 45 trusses with counters 6.46 6.47 6.48 Classify trusses according to constraints

FRAMES AND MACHINES Analysis of Frames 6.49, 51 6.50, 52 6.55, 56 6.53, 54 6.58 6.57 6.59 6.60, 61 problems where internal forces are changed by repositioning a couple or by 6.62 moving a force along its line of action 6.63, 64 analysis of hydraulic control systems 6.67, 68 6.65, 66 analysis of frames supporting pulleys or pipes 6.71, 72 6.69, 70 analysis of highway vehicles Analysis of Machines 6.73, 75 6.74, 76 toggle-type machines 6.77, 80 6.78, 79 tangs and gear-pulling machines 6.81, 82 6.83, 84 machines involving cranks 6.85, 87 6.86, 88 wrenches, pliers, and shears 6.91, 92 6.89, 90 find force to release brace or to hold a toggle 6.93, 95 6.94, 96 large mechanical equipment 6.97, 98 6.99, 100 Review problems 6.101, 103 6.102, 104 6.105, 108 6.106, 107


* Problems which do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxvi

CHAPTER 7: DISTRIBUTED FORCES: MOMENTS OF INERTIA OF AREAS Find by direct integration 7.1, 4 7.2, 4 moments of inertia of an area 7.5, 8 7.6, 7 7.9, 11 7.10, 12 moments of inertia and radii of gyration of an area 7.13, 15 7.14, 16 7.17, 18 7.19, 20 polar moments of inertia and polar radii of gyration of an area 7.21, 22 *7.23, *24 Special problems Parallel-axis theorem applied to composite areas to find 7.25, 27 7.26, 28 moment of inertia and radius of gyration 7.29, 31 7.30, 32 7.33, 34 centroidal moment of inertia, given I or J 7.36 7.35, 37 centroidal moments of inertia 7.39 7.38, 40 centroidal polar moment of inertia centroidal moments of inertia of composite areas consisting of rolled-steel shapes: 7.43, 44 7.41, 42 symmetrical composite areas 7.47, 48 7.45, 46 singly-symmetrical composite areas (first locate centroid of area) Review problems 7.49, 50 7.51, 52 7.53, 54 7.55, 58 7.56, 57 7.59, 60

TABLE IV: CLASSIFICATION AND DESCRIPTION OF PROBLEMS FOR Mechanics of Materials .

Problem Number* SI Units U.S. Units Problem description

* Problems that do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxvii

CHAPTER 8: INTRODUCTION - CONCEPT OF STRESS

Normal stress under axial loading: 8.1, 2 8.3,4 in bars 8.5, 6 8.7, 8 in pin-connected structures 8.11, 12 8.9, 10 in trusses and mechanisms 8.13, 14 8.15, 16 Shearing stress 8.17, 18 8.19, 20 Bearing stress between flat surfaces 8.23, 24 8.21, 22 Shearing and bearing stresses at pin-connected joints 8.27, 28 8.25, 26 Stresses on an oblique plane 8.31, 32 8.29, 30

Factor of safety: 8.33, 36 8.34, 35 in tension 8.38, 40 8.37, 39 in shear 8.43, 44 8.41, 42 in structures involving links and pins 8.47, 48 8.45, 46 8.49, 52 8.50, 51 Review problems 8.54, 57 8.53, 55 8.58, 59 8.56, 60



* Problems that do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxviii

CHAPTER 9: STRESS AND STRAIN - AXIAL LOADING

Stresses and deformations in statically determinate structures:

9.2, 4 9.1, 3 simple rods and wires 9.5, 8 9.6, 7 9.10, 11 9.9, 12 multiple-criteria problems 9.14, 15 9.13, 16 composite rods and members 9.18 9.17 9.19, 22 9.20, 21 members of trusses and simple frames 9.23 9.24

Statically indeterminate structures (constant temperature): 9.25, 26 9.27, 28 with members undergoing equal deformations 9.30, 31 9.29, 32 composite rods with both ends restrained 9.35, 36 9.33, 34 with members undergoing unequal deformations

Statically indeterminate structures (with temperature changes): 9.37, 38 9.39, 40 with members undergoing equal deformations 9.41 9.42, 43 composite rods with both ends restrained 9.44, 45 9.46 rods with gaps 9.48 9.47 with unequal deformations

Poisson’s ratio and generalized Hooke’s law: 9.50, 51 9.49, 54 uniaxial loading 9.52, 53 9.57, 58 9.55,56 biaxial loading 9.59, 60 derivation of formulas 9.61, 64 9.62, 63 Hooke’s law for shearing stress and strain 9.67, 68 9.65, 66 Stress concentrations in flat bars 9.69, 70 9.71, 72

9.75, 77 9.73, 74 Review problems 9.78, 79 9.76, 80 9.83, 84 9.81, 82



* Problems that do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxix

CHAPTER 10: TORSION

Shearing stresses: 10.1, 2 10.3, 4 in simple shafts 10.5, 6 10.7, 8 10.9, 10 in shafts subjected to several torques 10.11, 12 10.15, 16 10.13, 14 in composite shafts 10.17, 18 10.21, 22 10.19, 20 in gear-connected shafts 10.23, 24

Angle of twist: 10.27 10.25, 26 in simple shafts 10.28 10.29, 30 10.31 in shafts subjected to several torques, composite shafts 10.32 10.33, 34 10.35, 36 in gear-connected shafts 10.37, 38 10.39, 40 Design of shafts based on allowable stress and allowable angle of twist 10.45, 46 10.41, 42 Statically indeterminate shafts: 10.47, 48 10.43, 44 10.53, 54 10.49, 50 Review problems 10.55, 58 10.51, 52 10.59, 60 10.56, 57



* Problems that do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxx

CHAPTER 11: PURE BENDING

Normal stresses: 11.1, 3 11.2, 6 in beams with horizontal plane of symmetry 11.4, 5 11.7, 8 11.10, 12 11.9, 11 in unsymmetrical beams (first locate centroid) 11.15, 16 11.13, 14 Resultant force on portion of cross section 11.17, 20 11.18, 19 Beams with different allowable stresses in tension and compression 11.21 11.22 11.24 11.23 Maximum stress and radius of curvature

Stresses in composite beams:

11.25, 26 11.29, 30 symmetric beams of two materials 11.27, 28 11.31, 32 11.33, 34 11.35, 36 unsymmetric beams of two materials 11.37, 38 11.39, 40 Radius of curvature in composite beams 11.41, 42 11.45, 46 Stresses in reinforced concrete beams 11.43, 44 11.47, 48 Beams of three materials

Eccentric loading in plane of symmetry of member: 11.49, 52 11.50, 51 find stress in symmetric section 11.55, 56 11.53, 54 11.57, 58 11.59, 60 design of symmetric section 11.63, 64 11.61, 62 11.65, 66 11.67, 68 find stress in unsymmetric section 11.69, 72 11.70, 71 computation of loads from strain measurements

Unsymmetric bending with one or two planes of symmetry: 11.74, 78 11.73, 75 bending moment at an angle with horizontal 11.76, 77 11.79, 80 11.81, 83 section at an angle with horizontal 11.82, 84

General eccentric bending: 11.85, 86 symmetric beam; find stresses 11.87, 88 11.89, 90 symmetric beam; find allowable load or dimension 11.91, 92 1 1.94, 95 11.93, 96 Review problems 11.97, 101 11.98, 99 11.102, 103 11.100, 104



* Problems that do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxxi

CHAPTER 12: ANALYSIS AND DESIGN OF BEAMS FOR BENDING

Using the free-body diagram of a portion of a beam: 12.1, 2 12.3, 4 draw V and M diagrams (easy problems) 12.5, 6 12.7, 8 draw V and M diagrams and determine maximum values of |V | and |M | 12.9, 11 12.10, 12 12.14, 15 12.13, 16 find maximum normal stress in given beam section 12.17, 18 12.20, 22 12.19, 21 draw V and M diagrams and find maximum normal stress in beam 12.23, 24 12.25, 26 12.27, 28 determine given parameter to minimize normal stress in beam

Using relations among w, V and M whenever appropriate: 12.29, 30 12.31, 32 draw V and M diagrams (easy problems) 12.33, 34 12.35, 36 draw V and M diagrams and determine maximum values of |V | and |M | 12.39, 40 12.37, 38 12.42,43 12.41,44 find maximum normal stress in a given beam section 12.45, 46 12.47, 48 write equations for V and M and find maximum value of |M | 12.49, 50 12.51, 52 draw V and M diagrams and find maximum normal stress in beam 12.55, 56 12.53, 54 12.57, 58 12.59, 60 Design of timber beams 12.61 12.62 12.65, 66 12.63, 64 Design of steel beams, W shapes 12.67, 68 12.69, 70 Design of steel beams, S shapes 12.71, 72 12.73, 74 Design of steel beams, miscellaneous shapes 12.75 12.76 Design of beams resting on ground 12.78, 79 12.77, 80 Review Problems 12.81, 83 12.82, 84 12.86, 87 12.85, 88



* Problems that do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxxii

CHAPTER 13: SHEARING STRESSES IN BEAMS AND THIN-WALLED MEMBERS 13.1, 2 13.3, 4 Shearing forces in nails and bolts, using horizontal cuts 13.5, 6 13.7, 8 13.11, 12 13.9, 10 Shearing stresses in beams 13.14, 16 13.13, 15 Checking earlier designs of beams for shearing stresses 13.17, 19 13.18, 20 Beams with singly-symmetric sections 13.21, 22 13.23, 24 Beams with various geometric sections 13.25, 28 13.26, 27 Shearing forces and shearing stresses on arbitrary cuts due to vertical shear 13.29, 30 13.31, 32 Shearing stresses in extruded beams 13.33, 34 13.35, 36 13.37, 38 13.39, 40 Shearing stresses in bolts 13.41, 42 13.43, 44 Shearing stresses and shear flow in thin-walled members 13.45, 46 13.47, 49 Shearing stresses in composite beams 13.48 13.50, 52 13.51, 54 Review problems 13.53, 56 13.55, 57 13.60, 61 13.58, 59



* Problems that do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxxiii

CHAPTER 14: TRANSFORMATION OF STRESS 14.1, 2 14.3, 4 Find stresses on oblique plane from equilibrium of wedge 14.5, 6 14.7, 8 Find principal planes and stresses 14.9, 10 14.11, 12 Find planes of maximum shearing stress and corresponding stresses 14.15, 16 14.13, 14 Find stresses on a given plane 14.17, 19 14.18, 20 Stresses on oblique planes - simple applications 14.23, 24 14.21, 22 Find principal stresses and/or maximum shearing stress in loaded shaft

Using Mohr’s circle, determine: 14.25, 26 14.27, 28 principal planes and stresses, and maximum shearing stress 14.31, 32 14.29, 30 14.33, 35 14.34, 36 stresses on oblique plane - simple applications 14.39, 40 14.37, 38 principal stresses and/or maximum shearing stress in 14.42, 43 14.41 solve special problems involving determination of a stress to satisfy a 14.44 given requirement 14.46 14.45, 47 find principal planes and stresses resulting from superposition of two

14.48 states of stress

14.51, 53 14.49, 50 Spherical pressure vessels (easy problems) 14.54 14.52 14.55, 56 14.57, 59 Cylindrical pressure vessels (easy problems) 14.58 14.60 14.61, 62 14.63, 64 Stresses in welds in cylindrical pressure vessels 14.65, 66 14.67, 68 14.69, 70 Stresses on oblique planes - simple applications

14.71, 72 14.73, 75 14.74, 76 Review problems 14.78, 80 14.77, 79 14.81, 84 14.82, 83



* Problems that do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxxiv

CHAPTER 15: DEFLECTION OF BEAMS

Using the integration method, determine the equation of the elastic curve and the deflection and/or slope at specific points for:

15.1, 2 15.3, 4 cantilever beams 15.5, 6 15.7, 8 overhanging beams simply supported beams: 15.9 15.10 symmetrical loading 15.11 15.12 unsymmetrical loading 15.13, 14 beams and loadings requiring the use of 2 equations and 4 constants of integration 15.15, 16 direct determination of the elastic curve from an analytic function of w(x)

For a statically indeterminate beam (first degree), determine:

15.17, 19 15.18, 20 reaction at the roller support 15.21 15.22 reaction at the roller support and draw the M diagram (use of 2 equations

and 4 constants of integration required) 15.23 15.24 reaction at the roller support and the deflection at a given point (use of 2

equations and 4 constants of integration required) 15.25 15.26 for a statically indeterminate beam (second degree), determine the reaction

at one end and draw the M diagram Using method of superposition, determine the deflection and slope at

specified points in: 15.29, 30 15.27, 28 simply supported beams 15.32, 34 15.31, 33 cantilever beams 15.35, 36 cantilever beams (with numerical data) 15.37, 38 simply supported beams (with numerical data) 15.39, 40 15.41, 42 statically indeterminate beams (first degree) 15.44 15.43 statically indeterminate beams (second degree) 15.46 15.45 combined beams, determinate (with numerical data) 15.48 15.47 special problems, statically indeterminate beams (with numerical data) 15.50 15.49 combined bending and torsion of rods

15.51, 52 15.54, 55 Review problems 15.53, 57 15.56, 58 15.59, 61 15.60, 62

CHAPTER 16: COLUMNS



* Problems that do not involve any specific system of units have been indicated by underlining their number. Answers are not given to problems with a number set in italic type. xxxv

Stability of rigid-rod-and-spring systems:

16.1, 2 16.3, 4 easy problems, single spring 16.6 16.5 systems with two or more springs

Application of Euler’s formula to the critical loading or pin-ended columns: 16.8 16.7 short struts 16.10, 11 16.9, 12 comparison of critical loads for various cross sections

Allowable loading for pin-ended columns: 16.13, 16 16.14, 15 rolled-steel shapes 16.17, 18 multiple-member structures 16.19, 20 16.21, 22 columns with various end conditions 1.23, 24

Analysis of columns under centric load: columns with simple cross section:

16.25, 27 16.26, 28 steel columns 16.29 16.30 wood columns 16.31 16.32 aluminum columns

columns with built-up cross sections: 16.33, 34 16.36 steel columns 16.35 16.37 wood columns 16.38 aluminum columns

Design of columns under a centric load:

16.40 16.39 wood columns 16.42 16.41 aluminum columns 16.44, 46 16.43, 45 steel columns 16.47, 48 16.49, 50 16.51, 52 Review problems 16.54, 56 16.53, 55 16.57, 59 16.58, 60

xxxvi

TABLE V: SAMPLE ASSIGNMENT SCHEDULE

ANSWERS TO ALL OF THESE PROBLEMS ARE GIVEN IN THE BACK OF THE BOOK NO ANSWERS ARE GIVEN IN THE BOOK FOR ANY OF THESE PROBLEMS

Group Sections Topics List 1 List 2 List 3 List 4 List 5 List 6 List 7 List 8

1 1.1–6 Introduction 2 2.1–8 Addition and Resolution of Forces 2.3, 12, 27 2.1, 10, 24 2.4, 13, 26 2.2, 19, 25 2.7, 18, 31 2.5, 20, 29 2.6, 22, 28 2.8, 21, 30 3 2.9–11 Equilibrium of a Particle 2.34, 43, 51 2.32, 40, 49 2.35, 42, 48 2.33, 41, 50 2.36, 46, 54 2.38, 45, 55 2.39, 44, 52 2.37, 47, 53 4 2.12–14 Forces in Space 2.60, 66, 76 2.56, 65, 77 2.59, 67, 73 2.58, 64, 72 2.57, 71, 79 2.63, 69, 75 2.61, 70, 78 2.62, 68, 74 5 2.15 Equilibrium in Space 2.82, 87, 98 2.80, 91, 95 2.84, 89, 99 2.83, 93, 97 2.81, 90, 103 2.86, 92, 101 2.88, 96, 102 2.85, 94, 100 6 3.1–8 Vector Product, Moment of a Force about a Point 3.5, 16, 18 3.6, 17, 19 3.1, 10, 15 3.2, 9, 20 3.3, 12, 24 3.7, 14, 23 3.4, 11, 22 3.8, 13, 21 7 3.9–11 Scalar Product, Moment of a Force about an Axis 3.28, 34, 42 3.27, 33, 41 3.25, 40, 44 3.26, 39, 43 3.31, 37, 46 3.29, 35, 48 3.32, 38, 45 3.30, 36, 47 8 3.12–16 Couples 3.52, 59, 65 3.49, 56, 67 3.53, 58, 66 3.50, 57, 68 3.51, 61, 72 3.54, 63, 71 3.55, 64, 70 3.60, 62, 69 9 3.17–20 Equivalent Systems of Forces 3.75, 81, 95 3.77, 82, 91 3.73, 83, 90 3.74, 84, 89 3.80, 85, 96 3.76, 87, 92 3.78, 88, 94 3.79, 86, 93

10 4.1–5 Equilibrium in Two Dimensions 4.5, 9, 18 4.2, 10, 23 4.7, 15, 24 4.1, 12, 20 4.6, 14, 17 4.3, 16, 19 4.4, 11, 21 4.8, 13, 22 11 4.6–7 Two- and Three-Force Bodies 4.31, 43, 48 4.28, 42, 45 4.34, 37, 47 4.29, 33, 46 4.27, 40, 44 4.32, 38, 50 4.30, 39, 41 4.35, 36, 49 12 4.8–9 Equilibrium in Three Dimensions 4.56, 61, 72 4.51, 62, 70 4.57, 64, 71 4.54, 59, 67 4.52, 65, 74 4.53, 66, 73 4.58, 63, 69 4.55, 60, 68 13 4.10–13 Friction Forces 4.83, 90, 94 4.75, 81, 93 4.80, 85, 91 4.78, 88, 97 4.76, 82, 95 4.77, 87, 98 4.84, 89, 96 4.79, 86, 92 14 5.1–5 Centroids and First Moments 5.6, 11, 17 5.1, 10, 19 5.3, 9, 18 5.2, 12, 20 5.7, 14, 21 5.5, 13, 23 5.4, 15, 22 5.8, 16, 24 15 5.6–7 Centroids by Integration 5.25, 36, 41 5.26, 35, 42 5.29, 34, 43 5.30, 33, 44 5.31, 37, 48 5.32, 39, 45 5.28, 40, 47 5.27, 38, 46 16 5.8–10 Distributed Loads on Beams; Centroids of Volumes 5.49, 59, 69 5.53, 57, 72 5.51, 60, 70 5.54, 58, 71 5.56, 61, 65 5.55, 63, 66 5.50, 62, 68 5.52, 64, 67 17 6.1–5 Trusses: Method of Joints 6.4, 11, 19 6.2, 12, 18 6.3, 8, 15 6.1, 10, 20 6.6, 13, 21 6.9, 17, 24 6.7, 16, 23 6.5, 14, 22 18 6.6–7 Trusses: Method of Sections 6.28, 39, 41 6.27, 40, 42 6.26, 33, 44 6.25, 34, 45 6.31, 35, 47 6.29, 37, 48 6.32, 36, 43 6.30, 38, 46 19 6.8–10 Analysis of Frames 6.49, 57, 71 6.51, 61, 67 6.50, 58, 66 6.52, 59, 69 6.55, 62, 72 6.53, 64, 65 6.56, 60, 68 6.54, 63, 70 20 6.11 Analysis of Machines 6.73, 83, 95 6.75, 86, 91 6.76, 82, 96 6.74, 85, 89 6.80, 88, 92 6.78, 81, 94 6.77, 84, 93 6.79, 87, 90 21 7.1–5 Moments of Inertia of Areas 7.1, 14, 17 7.5, 10, 18 7.2, 13, 23 7.6, 9, 24 7.4, 12, 21 7.3, 11, 20 7.8, 16, 22 7.7, 15, 19 22 7.6–7 Composite Areas 7.25, 40, 43 7.27, 35, 44 7.26, 36, 41 7.30, 33, 45 7.31, 38, 47 7.32, 39, 46 7.29, 37, 48 7.28, 34, 42 23 8.1-7 Stresses Under Axial Loading 8.1, 10, 18 8.2, 9, 24 8.3, 12, 20 8.4, 11, 22 8.6, 16, 17 8.7, 14, 21 8.5, 15, 23 8.8, 13, 19 24 8.8-13 Components of Stress; Factor of Safety 8.27, 34, 48 8.28, 35, 47 8.25, 33, 46 8.26, 36, 42 8.31, 39, 44 8.30, 38, 41 8.32, 37, 43 8.29, 40, 45 25 9.1-7 Stress-Strain Diagram 9.2, 12, 18 9.4, 9, 19 9.1, 15, 20 9.3, 14, 21 9.8, 13, 23 9.6, 11, 17 9.5, 16, 22 9.7, 10, 24 26 9.8-9 Statically Indeterminate Problems 9.25, 39, 41 9.26, 33, 44 9.27, 36, 42 9.28, 35, 46 9.30, 40, 48 9.32, 37, 47 9.31, 34, 45 9.29, 38, 43 27 9.10-15 General Hooke’s Law; Poisson’s Ratio; Stress Conc. 9.50, 60, 67 9.52, 59, 68 9.55, 61, 65 9.54, 58, 71 9.53, 63, 70 9.56, 64, 72 9.51, 62, 69 9.49, 57, 66 28 10.1-4 Stresses in Torsion 10.2, 13, 21 10.1, 9, 17 10.4, 15, 20 10.3, 16, 24 10.7, 10, 18 10.6, 12, 23 10.8, 14, 22 10.5, 11, 19 29 10.5-6 Angle of Twist, Indeterminate Shafts 10.30, 36, 45 10.27, 40, 47 10.28, 33, 44 10.25, 37, 41 10.32, 39, 48 10.26, 34, 42 10. 29, 35, 46 10.31, 38, 43 30 11.1-4 Stresses and Deformations 11.3, 13, 20 11.1, 9, 17 11.6, 10, 19 11.2, 16, 18 11.4, 14, 24 11.7, 12, 22 11.5, 11, 21 11.8, 15, 23 31 11.5 Members Made of Several Materials 11.27, 35, 42 11.26, 36, 41 11.31, 33, 46 11.30, 34, 45 11.25, 40, 44 11.29, 38, 48 11.28, 39, 43 11.32, 37, 47 32 11.6 Eccentric Axial Loading in a Plane of Symmetry 11.56, 60, 66 11.52, 59, 65 11.51, 64, 70 11.50, 58, 68 11.49, 62, 72 11.54, 57, 71 11.55, 61, 69 11.53, 63, 67 33 11.7-8 Unsymmetric Bending 11.78, 81, 91 11.74, 83, 89 11.75, 82, 87 11.73, 84, 85 11.80, 87, 89 11.76, 83, 91 11.79, 86, 92 11.77, 88, 90 34 12.1-2 Shear and Bending-Moment Diagrams 12.5, 16, 20 12.9, 13, 25 12.7, 15, 23 12.10, 14, 24 12.11, 19, 26 12.12, 18, 27 12.6, 17, 21 12.8, 22, 28 35 12.3 Using Relations Among w, V, and M 12.33, 44, 49 12.39, 41, 50 12.35, 43, 51 12.37, 42, 52 12.40, 47, 55 12.38, 45, 54 12.34, 48, 56 12.36, 46, 53 36 12.4 Design of Prismatic Beams in Bending 12.57, 64, 71 12.58, 63, 72 12.60, 66, 73 12.60, 65, 74 12.58, 69, 72 12.62, 67, 74 12.61, 70, 75 12.59, 68, 76 37 13.1-4 Shearing Stresses in a Beam 13.1, 10, 17 13.2, 9, 19 13.3, 12, 18 13.4, 11, 20 13.5, 13, 22 13.7, 14, 24 13.6, 15, 21 13.8, 16, 23 38 13.5-6 Shearing Stresses in Thin-Walled Members 13.25, 36, 41 13.28, 35, 42 13.26, 34, 43 13.27, 33, 47 13.29, 39, 46 13.31, 37, 49 13.30, 40, 48 13.32, 38, 44 39 14.1-3 Transformation of Plane Stress 14.1, 13, 17 14.2, 14, 19 14.4, 9, 18 14.3, 16, 20 14.5, 12, 24 14.7, 10, 22 14.6, 11, 23 14.8, 15, 21 40 14.4 Mohr’s Circle for Plane Stress 14.26, 34, 44 14.32, 36, 43 14.30, 35, 48 14.28, 33, 41 14.25, 38, 46 14.29, 40, 47 14.31, 37, 42 14.27, 39, 45 41 14.5 Stresses in Thin-Walled Pressure Vessels 14.51, 59, 66 14.53, 60, 65 14.49, 56, 72 14. 50, 58, 71 14.54, 64, 68 14.52, 62, 69 14.55, 63, 67 14.57, 61, 70 42 15.1-5 Deflection of Beams by Integration 15.1, 12, 17 15.2, 10, 21 15.3, 11, 18 15.4, 9, 20 15.6, 15, 23 15.7, 13, 22 15.5, 16, 19 15.8, 14, 24 43 15.6-7 Method of Superposition; Statically Indet. Beams 15.29, 36, 46 15.30, 35, 48 15.27, 39, 43 15.28, 37, 45 15.32, 42, 50 15.31, 40, 49 15.34, 41, 44 15.33, 38, 47 44 16.1-4 Columns: Euler’s Formula 16.1, 12, 17 16.2, 14, 20 16.5, 13, 23 16.3, 16, 22 16.4, 9, 18 16.7, 10, 24 16.6, 15, 19 16.8, 11, 21 45 16.5 Design of Columns Under a Centric Load 16.27, 39, 42 16.25, 41, 44 16.28, 37, 43 16.26, 34, 48 16.31, 36, 40 16.32, 33, 47 16.29, 38, 46 16.30, 35, 45